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Removal And Separation Of Particles (In Solid Organic Chemicals And Metals) By Crystallization

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Removal And Separation Of Particles (In Solid Organic Chemicals And Metals) By Crystallization

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Content R E M O V A L A N D SEPARATIO N O F PARTICLES B Y CRYSTALLIZATION by Vincent H. S. Kuo A D issertation Presented to the FACULTY O F TH E G R A D U A T E S C H O O L UNIVERSITY O F S O U T H E R N CALIFORNIA In P a rtia l F u lfillm e n t of the Requirements fo r the Degree D O C T O R O F PHILO SO PHY (Chemical Engineering) February 1973 INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or “target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an imege on the film is obliterated with a large round bleck mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed a s received. Xerox University Microfilms 300 North ZMb Road Ann Arbor, Michigan 48106 I I 73-14,421 KUO, V incent H. S . , 1938- RBCVAL A N D SEPARATION OF PARTICLES BY CRYSTALLIZATION. U n iv e rs ity o f S outhern C a lifo rn ia , P h.D ., 1973 E n gineering, chem ical University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EX A CTLY AS RECEIVED. U N IVER SITY O F S O U TH E R N C A LIFO R N IA TH E G RADUATE SCHO O L U N IV E R S IT Y PARK LOS ANOELES. C A L IF O R N IA 9 0 0 0 7 This dissertation, •written by ..................... .V i.nc e n.t. J H - J S - ........................ under the direction of hA&.... Dissertation Com mittee, and approved by a ll its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of requirements of the degree of D O C T O R O F P H IL O S O P H Y D t» * Date F e b ru ary 1973 DISSERTATION COMMITTEE J & X Z * ..... ACKNOWLEDGMENTS I w ish to g r a t e f u lly acknowledge my a d v is o r, Dr. W . R. W ilc o x , f o r h is a d v ic e , guidance and encouragement th ro u g h o u t th is w o rk, and a ls o f o r h is warm personal concern d u rin g the course o f t h is stu d y. I a ls o w ish to acknowledge my com m ittee members, Dr. C. J . Rebert and Dr. S. M. C opley, f o r t h e ir in te r e s t and con s tr u c tiv e su g g e stio n s. Dr. R. C. B in d e r served the com m ittee u n t il h is re tire m e n t (August 1972). S pecial thanks are due P ro fe sso r N. Kharasch f o r p e rm ittin g me to use h is gas chrom atograph; to Mr. P. Weidman f o r the h e lp fu l d iscu ssio n s on r o ta tio n o f th e f l u i d ; and to Messrs. G. M u e lle r, J . Emerson and J. S c o tt f o r a s s is tin g in c o n s tru c tio n o f the exp erim enta l apparatus. T his research was f in a n c ia lly supported by the Petroleum Research Fund (1969-1971), a d m in iste re d by the American Chemical S o c ie ty ; and by the Advanced Research P ro je c ts Agency (1971-1972) o f th e Department o f Defense under G rant No. DAHC 15-72-G7. The t t M C p a r tic le s iz e d is tr ib u tio n s were k in d ly perform ed by th e Mi H i pore C o rp o ra tio n , and th e scanning e le c tro n m icrographs o f th e carbon p a r tic le s were taken by the Aerospace C o rp o ra tio n . P re p a ra tio n o f the f in a l m anuscript by M rs. Shari W ilc o x , and th e rough d r a ft by Mrs. G eorgia Lum are th a n k fu lly acknowledged. F in a lly , I am p a r t ic u la r ly g ra te fu l to my w ife , Jane, f o r her encouragement and understanding d u rin g the course o f th is w ork. She assumed part of the fin a n c ia l re s p o n s ib ility and f u ll re s p o n s ib ility fo r taking care of our two children, so th a t I could concentrate on pursuing m y research and in te re sts. iii TABLE OF CONTENTS Page ACKNOW LEDGM ENTS ........................................................................................................ 11 LIST OF FIGURES........................................................................................................v11 LIST OF T A B L E S ...................................................................................................... x ABSTRACT........................................................................................................................ x i CHAPTER I - INTRODUCTION ................................................................................. 1 CHAPTER I I - LITERATURE STUDIES ...................................................................... 3 A. Review o f Previous W o rk ................................................................ 3 1. C r y s ta lliz a tio n P r e s s u r e ...........................................................3 2. C orren's Phase-Boundary Force ............................................. 6 3. Work o f C o r t e ............................................................................... 7 4. Work o f Uhlmann, Chalmers, and J a c k s o n ............................ 8 5. Work o f Hoekstra and M i l l e r .......................................................9 6. Work o f B o llin g and C is s e .........................................................13 B. Some T h e o re tic a l C onsiderations ................................................. 27 1. S tru c tu re o f the S o lid -L iq u id In te rfa c e ....................... 27 2. In te r fa c ia l Temperature G r a d ie n t ........................................29 CHAPTER I I I - EXPERIMENTAL.................................................................................. 34 A. Bridgm an-Stockbarger Technique ................................................. 35 1. A p p a ra tu s .......................................................................................* 3 5 2. Sample P reparation .................................................................. 38 3. Experim ental P r o c e d u r e .............................................................39 4. Temperature Measurements ..................................................... 40 B. H o rizo n ta l Zone-R efining w ith R o t a t io n .................................. 43 1. A p p a ra tu s .......................................................................................... 44 2. Sample P r e p a r a t io n .....................................................................44 3. Experim ental P r o c e d u r e ............................................................ 47 C. M a t e r ia ls ...................................................................................................48 1. P a r t ic le s .......................................................................................... 48 iv Page i 2. Organic Compounds .............................................................. 53 D. S o lid ifie d Products ................................................................... 53 E. Zone-Refining of Organic Compounds ..................................... 56 1. Camphor.................................................................................. 56 2. Naphthalene......................................................................... 57 C H A P T E R IV - R ESU LTS A N D DISCUSSIONS........................................................61 A. Bridgman-Stockbarger ............................................................... 61 1. G eneral................................................................................. 61 2. S olid-Liquid Interface Morphology ............................... 62 i 3. Temperature G rad ients............................................................. 68 j 4. S t i r r i n g ................................................................................ 73 j 5. M atrix-P a rticle Dependence ............................................. 75 | 6. Gas Bubbles and Surface-Driven Flows ......................... 77 ] 7. Grain S u rfa c e s .........................................................................81 j 8. Growth O r ie n ta tio n .................................................................82 J 9. Surface o f S o lid ifie d Products ....................................... 82 I 10. Cracking and H e a lin g ............................................................. 88 j 11. Bouncing P articles ............................................................ 88 1 12. Summary........................................................................................ 90 ! B. Horizontal Zone-Refining with R o ta tio n ....................................91 j 1. Interface Shape......................................................................... 91 1 2. R o ta tio n .................... 95 3. Tube D iam eter......................................................................... 103 4. Bubbles and V o i d ................................................................. 103 5. Summary.....................................................................................104 j C H A P T E R V - POTENTIAL APPLICATIONS A N D E C O N O M IC S ............................. 105 1 A. Separation o f Mixed P articles ................................................... 105 I B. Size C la s s ific a tio n ....................................................................... 108 I C. Economics .......................................................................................109 I C H A P T E R VI - INTERPRETATION...........................................................................114 I A. Pushing Mechanisms ............................................................... 114 v Page 1. Mass T r a n s p o r t .............................................................................114 2. Heat T r a n s fe r ................................................................................. 115 3. H o rizo n ta l R o ta tio n ................................................................. 122 B. C a lc u la tio n s o f Vc ............................................................................. 125 C. Formation o f Gas B u b b le s .................................................................132 CHAPTER V II - CONCLUSIONS..................................................................................... 142 CHAPTER V III - SUGGESTED FUTURE W ORK.............................................................. 145 NOMENCLATURE ............................................................................................................ 149 REFERENCES................................... 154 APPENDIX A - INFLUENCE OF CRYSTAL DIMENSIONS O N THE INTERFACIAL TEMPERATURE GRADIENT ....................................... 159 APPENDIX B - ECONOMIC CALCULATIONS ............................................................. 164 s vi LIST O F FIGURES Figure Page 1 Schematic cross section of a spherical p a rtic le in contact with a s o lid -liq u id interface w ith a magnified contact area ............................................................... 17 2 Sketch showing variations o f dP/dr and A P with respect to r in the contact a re a ........................................... 20 3 Atomic s o lid -liq u id Interface ................................................ 28 4 C onstitutional supercooling .................................................... 3 1 5 Bench-scale u n it of improved Bridgman- Stockbarger technique ............................................................... 36 6 Schematic diagram of the heater and the cooler ................. 37 7 Temperature measurements by thermocouples during Bridgman-Stockbarger crystal growth ........................ 42 8 A naphthalene molten zone containing carbon pa rticle s tra ve lin g in the right-hand d ire ctio n w ith a concave freezing interface .................... 45 9 Preparation of the so lid charge fo r horizontal zone-refining with rota tion ................................................... 46 10 Micrographs of carbon p a rticle s ............................................ 50 1 1 Size d is trib u tio n s fo r carbon and iron oxide p a r t ic le s ...................................................................................... 5 1 12 Size d is trib u tio n fo r copper p a rticle s ................................. 52 13 Native p a rticle s collected from naphthalene ..................... 55 14 Chromatograms of camphor............................................................ 58 15 Chromatograms of naphthalene.................................................... 60 16 Shapes of the freezing interface in the Bridgman-Stockbarger experiments ........................................... 69 17 Carbon p a rticle s trapped next to the tube wall during Bridgman growth of naphthalene ................................ 70 vi i 74 76 79 80 83 84 85 86 87 89 92 94 99 Carbon p a rtic le s trapped im m ediately a fte r each a d d itio n during Bridgman growth o f s a lo l w ith s t i r r i n g .................................................... The concave in te rfa c e o f growing s a lo l con s is tin g o f many la rg e fa c e ts ................................... S u rfa ce -d rive n flow s c ir c u la tin g p a rtic le s around a gas bubble d u rin g fre e z in g o f naphthalene ..................................................................... A la rg e gas worm and a crack across the gas worm in s o lid if ie d naphthalene ............................... Surface o f s a lo l g ra in s ....................................... . Surface o f naphthalene g ra in s .............................. E ffe c t o f growth o rie n ta tio n on p a r tic le pushing .............................................................................. Surfaces o f s o lid ifie d naphthalene a t tube wal 1 ....................................................................................... Gas bubbles trapped between the s o lid if ie d naphthalene and tube w a ll ....................................... Crack and p a r t ia lly healed crack in s o lid if ie d naphthalene ..................................................................... Change o f zone shape w ith tra v e l ra te in h o riz o n ta l z o n e -re fin in g o f naphthalene w ith r o t a t i o n .............................................................................. H o riz o n ta l z o n e -re fin in g naphthalene w ith carbon p a rtic le s pushed from the l e f t to the r ig h t e n d .......................................................................... The in flu e n c e o f tube ro ta tio n on in c o rp o ra tio n o f copper p a rtic le s by naphthalene du rin g h o riz o n ta l zone m e ltin g ............................................ The In flu e n c e o f tube ro ta tio n on in c o rp o ra tio n o f iro n oxide p a rtic le s by naphthalene d u rin g h o riz o n ta l zone m e l t i n g ............................................ v i i i Figure Page 32 The influence o f tube ro ta tio n on incorporation o f carbon p a rticle s by naphthalene during horizontal zone m elting ............................................................ 101 33 Carbon and copper p a rtic le s were separated during programmed s o lid ific a tio n o f naphthalene in horizontal zone-refining w ith ro ta tio n ........................ 107 34 Size c la s s ific a tio n of spherical Ag p a rtic le s by v e rtic a l Bridgman growth o f naphthalene ........................ 110 35 The presence o f an a ir bubble in fro n t o f an advancing ice-water in te rfa ce ................................................ 116 36 E ffe ct o f p a rtic le thermal conductivity on the shape o f the equilibrium in te r fa c e .............................................118 37 Flow patterns in a horizontal ro ta tin g z o n e .......................... 124 38 V isco sitie s of supercooled organic melts ............................. 128 39 P lo t o f *(a ) versus a w ith 6 as a parameter from 4>(ct) = a ( l- a ) 2(S-Jtn a ) ........................................................ 130 40 Schematic diagram of bubble formation a t a freezing in te rface ....................................................................... 134 LIST O F TABLES Table Page 1 Summary o f C r itic a l V e lo c itie s from P revious S tudies ....................................................................................... 10 2 P ro p e rtie s o f the P a r t i c l e s ........................................................... 49 3 P ro p e rtie s o f the O rganic C o m po un ds.......................................... 54 4 Summary o f V e rtic a l Bridgman Experiments f o r C arbon-Salol System . . . . . ................................................ 63 5 Summary o f V e rtic a l Bridgman Experiments f o r Carbon-Naphthalene System ......................................................... 66 6 E ffe c t o f Temperature G ra d ie n t on C r itic a l V e lo c it y ......................................................................................................... 72 7 C r itic a l Trapping V e lo c itie s in V e rtic a l Bridgman C ry s ta l G r o w t h ...................................................................... 78 8 E ffe c t o f R o ta tio n Rate on C r itic a l Freezing Rate Vc f o r Trapping o f Carbon in Naphthalene using H o riz o n ta l Z o n e -R e fin in g ..................................................... 96 9 E ffe c t o f R o ta tio n Rate on C r itic a l Freezing Rate Vc f o r Trapping o f Copper in Naphthalene using H o riz o n ta l Z o n e -R e fin in g ..................................................... 97 10 E ffe c t o f R o ta tio n on Vc f o r Iro n O xide- Naphthalene in H o riz o n ta l Zone-R efining ................................... 98 11 Optimum R o ta tio n Rates f o r S eparation o f P a rtic le s from Naphthalene by a H o riz o n ta l Zone R e fin e r w ith R o ta tio n .............................................................. 102 12 P rice o f S elected Zone-Refined O rganic Compounds f o r Prim ary S ta n d a r d s ............................................................................113 13 In te r fa c ia l Temperature G radients f o r Rapid S o lid ific a tio n ........................................................................................ 121 14 E stim a tio n o f Vc f o r P a r t ic l e s ........................................................... 127 15 V a ria tio n o f d /a Q w ith a and 8 .................................................. 131 x ABSTR AC T Foreign p a rticle s are present in nearly a ll solid organic chemicals, including laboratory-grade chemicals. Metals may s im ila rly contain p a rticle s of slag and furnace re fra cto rie s follow ing extrac tiv e m etallurgical processes. These p a rticle s are not only sources of im purities but may induce defects and change many properties of a solid. I t has been known fo r many years that growing crystals some times re je ct and push foreign p a rticle s in addition to segregation of soluble im purities. The actual in te ra ctio n between p a rticle s and a s o lid -liq u id interface was recently studied experimentally and th e o re tic a lly , and some p a rtic le pushing and trapping phenomena c la rifie d . At s u ffic ie n t low growth rates, nearly a ll p a rticle s were pushed by a freezing in te rfa ce . As the growth rate was increased, a c r itic a l ve lo city V was reached beyond which pa rticle s started being v trapped in to the growing c ry s ta l. The measured values o f V were d iffe re n t fo r each p a rti cle-m atrix m aterial system and possessed no discernible pattern. P rio r to the present work, in s u ffic ie n t in fo r mation was known about p a rtic le pushing to permit design o f a separa tio n process fo r producing p a rtic le -fre e ultrapure m aterials. Two d iffe re n t experimental methods were employed in th is study. An improved Bridgman-Stockbarger apparatus was f i r s t constructed per m itting microscopic observation of p a rtic le pushing and trapping at the freezing in te rface. Interface shape, growth container, growth o rie n ta tio n , and presence o f bubbles a ll s ig n ific a n tly influenced V . xi For the carbon-naphthalene system V was 21 m m /hr., b u t I t was re - duced to 18 mm/hr. when the p a r tic le s were trapped n e xt to the tube w a ll w ith a convex In te rfa c e . For the copper-naphthalene system V was about 18 mm/hr. The In te r fa c ia l tem perature g ra d ie n t had no d is c e rn ib le e ffe c t on Vc> The fo rm a tio n o f bubbles was prevented by evacuating the growth c o n ta in e r. In an atte m p t to in crea se V by s t ir r in g o f the m e lt, In s e rtio n o f a p ro p e lle r o r paddle a g ita to r was found to be Im p ra c tic a l. H o riz o n ta l z o n e -re fin in g w ith r o ta tio n was then adopted to in crease Vc . This a p p a re n tly was the f i r s t tim e th is technique was a p p lie d f o r removing p a r tic le s . T his method e lim in a te d the d e t r i mental e ffe c ts re s u ltin g from s e ttlin g o f the p a r tic le s and fo rm a tio n o f bubbles which were encountered in the f i r s t method. Thus, the measured values o f Vc were increased 300% f o r copper and 50% f o r carbon p a rtic le s in naphthalene. This new se p a ra tio n process has proven to be sim p le , e ffe c tiv e and economical (p ro d u ct co st is about 22 cents per gram f o r p a r tic le - fr e e naphthalene). Furtherm ore, h o riz o n ta l zone- r e fin in g w ith r o ta tio n and v e r tic a l Bridgman technique were used to separate carbon p a r tic le s from a m ixtu re o f carbon and copper p a rtic le s in naphthalene by ta k in g advantage o f t h e ir d iffe re n c e in V . This new se p a ra tio n technique is c h ris te n e d " p a r tic le chrom atography." The measured fig u re s o f Vc were 1n good agreement w ith those c a lc u la te d by B o llin g and C isse 's e q u a tio n . T h e o re tic a l treatm ents on fo rm a tio n o f a i r bu bble s, e ffe c t o f c ry s ta l dim ension on the in t e r fa c ia l tem perature g ra d ie n t, e ffe c t o f tem perature g ra d ie n t and x ii horizontal ro ta tio n on p a rtic le pushing were made in th is study. In general, they agreed with the present experimental results and previous data. At le ast three objectives have been accomplished in th is d is serta tion . (1) Discovery of some Important phenomena leading to bette r understanding o f pushing o f p a rtic le s by a freezing in te rfa ce . (2) The development of a new separation process enabling economical removal of foreign p a rticle s from organic compounds. (3) The inven tio n o f a new technique to separate mixtures of d iffe re n t p a rtic le s , opening up a p o te n tia lly new fie ld —p a rtic le chromatography. xi i i C H A P TE R I INTRODUCTION The demand fo r highly pure s o lid m aterials is continuously growing fo r research and ap plica tions, p a rtic u la rly in the fie ld s o f semiconductors and pharmaceuticals. While separation o f soluble im purities has been extensively investigated during the past two decades, removal of insoluble im purities has been la rg e ly neglected. Foreign p a rticle s are present in nearly a ll organic compounds, in cluding laboratory-grade chemicals. Metals may s im ila rly contain p a rticle s of slag and furnace re fra cto rie s follow ing extra ctive metal lu rg ic a l processes. These p a rtic le s are not only source o f im purities but may induce defects such as dislocations [1 ], and change many properties of a s o lid . I t has been known fo r many years tha t growing crystals some times re je c t and push foreign p a rtic le s in addition to segregation o f soluble im purities [2 ]. This observation suggested the p o s s ib ility of removing p a rtic le s by s o lid ific a tio n processes [3 ]. The actual in te ra ctio n between p a rtic le s and a freezing in te rfa ce has recently been studied [4 -1 0 ], and some p a rtic le pushing phenomena have been c la rifie d . At s u ffic ie n tly low crysta l growth rate s, nearly a ll par tic le s were observed to be pushed by the s o lid -liq u id in te rfa ce . As the growth rate was increased, a c r itic a l v e lo c ity V c was reached beyond which a ll p a rticle s o f a p a rtic u la r type were trapped. The measured values o f V were d iffe re n t fo r each p a rtic le -m a trix m aterial system, and possessed no discern ible pattern. P rio r to the present 1 study In s u ffic ie n t in fo rm a tio n was known about p a r tic le re je c tio n to p e rm it engineering design o f such a se p a ra tio n process. The purpose o f th is research was to le a rn more about the funda mental nature o f p a r tic le pushing and tra p p in g phenomena, and to study the e ffe c ts o f some o f the design v a ria b le s o f a s e p a ra tio n process. Selected p a rtic le s were mixed in tra n s p a re n t o rg a n ic m e lts . In ord er to measure the c r it ic a l ra te fo r tra p p in g , the s o lid if ic a t io n ra te was g ra d u a lly increased. Two d iffe r e n t techniques were employed. An improved v e r tic a l Bridgm an-Stockbarger growth apparatus was f i r s t s e t up f o r d ir e c t o b se rva tio n o f the p a r tic le pushing and tra p p in g a t the s o lid - liq u id in te rfa c e . A h o riz o n ta l z o n e -re fin e r w ith r o ta tio n was then employed. Since some o f the o rg a n ic compounds used here fro z e non-faceted lik e most m etals [1 1 ], th e re s u lts o f th is work should be e q u a lly a p p lic a b le to removal o f p a r tic le s from m e ta ls. In the next chapter the p r io r lit e r a t u r e is c r i t i c a l l y re viewed, and some th e o re tic a l c o n s id e ra tio n s needed f o r th is work are de scrib ed . This is fo llo w e d by Chapter I I I which de scrib es the d e ta ile d experim ental methods and m a te ria ls employed f o r the research. Chapter IV shows the experim ental re s u lts . Chapter V exp lo res p o te n tia l a p p lic a tio n s w ith some experim ental evidence and a lso gives economic aspects o f th e developed process. In te rp re ta tio n o f experim ental re s u lts is given in Chapter V I. Accomplishments and im p o rta n t re s u lts drawn from th is research are summarized in Chapter V II. F in a lly , the e xte n sio n o f the p re se n t work to se ve ral fie ld s is suggested in the la s t ch a p te r. C H A P TE R II LITERATURE STUDIES This chapter consists o f two sections: a review of previous work and some theoretical considerations required fo r the present work. The p rio r lite ra tu re is reviewed in chronological order. E a rlie r work is discussed under "C ry s ta lliz a tio n Pressure", since before 1960 many investigators interpreted that a growing crystal could exert a force which pushed an external body. This is followed by a review o f more recent papers in which the in te ra ctio n between particles and a s o lid -liq u id interface was examined. Of these recent studies, the work of Bolling and Cisse [8,9,10] appears most s ig n ific a n t fo r p a rtic le pushing and trapping. Their theory is thus reviewed in d e ta il. Som e theoretical considerations needed fo r th is research are discussed in the second section o f th is chapter, such as the structure of a sol id -liq u id in terface and the in te rfa c ia l tempera ture gradient. A. Review of Previous Work 1. C ry s ta lliz a tio n Pressure I t has been known fo r many years th a t in addition to segrega tio n of soluble im p u ritie s, a growing crysta l e ith e r repels or entraps foreign solids [2 ,4 ]. Many in te restin g examples have been observed [12]. Crystals o f epsomite (MgSO^THgO) grown on the bottom of s a lt lakes repelled p a rticle s of mud and were therefore colorless and 3 tra n s p a re n t. On the o th e r hand, c ry s ta ls o f a s tra k a n lte (NagSO^-MgSO^^HgO) form ed under s im ila r c o n d itio n s were u s u a lly dark colore d due to entrapm ent o f mud p a r tic le s . Gypsum c ry s ta ls (CaSO^-ZHgO) growing in c la y re je c te d th e p a rtic le s in t h e ir p ro x im ity and were o fte n c le a r, w h ile the same c ry s ta ls formed in sand e x h ib ite d abundant in c lu s io n s o f sand p a r tic le s . These phenomena o f p a r tic le re p u ls io n o r entrapm ent may be re la te d to a fo rc e exe rted by the c ry s ta l on the p a r tic le d u rin g c r y s ta ll iz a tio n . Becker and Day [1 4 ] were th e f i r s t to perform experim ents 1n 1905. They p u t fo rw a rd the view th a t a c ry s ta l could e x e rt a pressure when grow ing. They placed a loaded g la ss p la te on th e top fa ce o f an alum c r y s ta l. The c ry s ta l continu ed growing even when the load was increased to one kilo g ra m . They a ttr ib u te d the e ffe c t to a " lin e a r c r y s ta lliz a tio n fo rc e ". The c r y s t a lliz a tio n pressure was d e fin e d as the maximum pressure e xe rte d by a c ry s ta l a t which the growth ceased a t a g ive n s u p e rs a tu ra tio n (o r s u p e rc o o lin g ). Since then many in v e s tig a to rs have pursued the measurement o f c r y s t a lliz a tio n p re ssu re , b u t have no t agreed on th e values o f the pressures th a t m ig ht occur. p Some found values o f about 20 Kg/cm [4 ,1 4 ,1 5 ,1 6 ] w h ile o th e rs found 2 o n ly a few g/cm [1 7 ,1 8 ]. The fo rm e r used Becker and Day's method, the la t t e r used S hubnikov's method [1 6 ]. Shubnikov used a grow ing c ry s ta l to encounter a g la ss b a ll which was attached to a s p rin g . The c r y s ta lliz a tio n pressure was determ ined when the b a ll began e n te rin g the c r y s ta l. Those who took the pressure to be la rg e , re la te d the pushing fo rc e to the energy o f the phase change. Those who took the pressure to be sm all e xp la in e d the e ffe c t as a su rfa ce in te r a c tio n . 5 In fa c t, the supply o f supersaturated solutio n to the contact region Is the lim itin g process 1n the la tte r case. Correns and Stelnborn [19 ] thermodynamically related the c ry s ta lliz a tio n pressure to the supersaturation. The conditions fo r reaching equilibrium were obtained by calculating in two d iffe re n t ways the amount of work needed to transform a supersaturated solution Into a saturated solutio n. The maximum work W gained to perform th is trans formation Isotherm ally and re ve rsib ly is W = N kT in (2-1) oo where C is the actual concentration, is the saturated concentra tio n , N is the number of molecules transformed, k is the Boltzmann's constant and T is the absolute temperature. The crysta l grows under a load B and the work gained by lif t in g the load fo r a distance of d is W = Bd = PAd = P N vs (2-2) where P is the c ry s ta lliz a tio n pressure, A is the contact area between the load and c ry s ta l, v is the molecular volume o f the c ry s ta l. Equating the above two equations, the c ry s ta lliz a tio n pressure is obtained as P = tn f - . (2-3) vs 0 0 The above re la tio n indicates the c ry s ta lliz a tio n pressure in creases with the supersaturation. Experimental results [4 ,1 9 ] on alum covered by a g lass p la te confirm ed the r e la tio n a t sm all and medium s u p e rs a tu ra tio n (C/C„ <. 1 .2 ). However, the th e o re tic a l p re d ic tio n was to o la rg e a t high s u p e rs a tu ra tio n [P = 55 Kg/cm f o r th e o r e tic a l, w h ile 42 Kg/cm was measured on a (111) face and 32 Kg/cm on a (110) fa c e ]. The (100) face o f alum gave zero pressure a t a ll super s a tu ra tio n s [1 9 ,2 0 ]. I t was a ls o found th a t the pressure depended on the n a tu re o f the e x te rn a l body. Alum covered by mica y ie ld e d zero pressure f o r (1 1 1 ), (110) and (100) faces a t a ll s u p e rs a tu ra tio n s . T h e re fo re , the c r y s t a lliz a tio n pressure cannot be d e scrib e d e n t ir e ly by the above r e la tio n . I t depends n o t o n ly on s u p e rs a tu ra tio n , b u t a lso on th e type o f c r y s ta l, on the s o lv e n t, on the e x te rn a l body, and on the tra n s p o rt c o n d itio n s a t the c o n ta c t area between the c ry s ta l and th e e x te rn a l body. Khaimov-Mal1kov [2 1 ] a ls o s tu d ie d c r y s ta lliz a tio n pressure therm odynam ically. A lthough h is th e o re tic a l r e la tio n in clu d e d the a n is o tro p y e ffe c ts o f e la s tic b e h a v io r, these e ffe c ts became appre c ia b le o n ly a t pressures o f g e o lo g ic a l o rd e r (g re a te r than 1000 Kg/ cm2). 2. C orrens1 Phase-Boundary Force As in d ic a te d above, a (111) face o f alum pushes a g la s s p la te w h ile a (100) fa ce s tic k s to such a p la te . F u rth e r, both faces s tic k to a mica p la te . These experim ental re s u lts g iv e the im pression th a t th e c r y s ta lliz a tio n pressure m ig h t be re la te d to su rfa ce e n e rg ie s. Correns [2 2 ] e s ta b lis h e d another c o n d itio n to show w hether o r n o t an 7 external body could be pushed. W hen a p a rtic le (p) is 1n Intim ate contact with a crystal (s) In liq u id (&), an amount o f work 1s necessary to separate the p a rtic le from the cry s ta l. Work has to be done against the surface energies In order that two new s o lid -liq u id and p a rtic le -liq u id interfaces appear and the o rig in a l p a rtic le -s o lid Interface disappears. I f the surface energy between the so lid and p a rtic le 1s c rSp, the surface energy between the so lid and liq u id 1s osA, and the surface energy between the p a rtic le and liq u id Is Op^, the work done is w ' (osp - A • (z- 4) ^ asp > as£ + apJt* t * ien W P os^ ve ant* wor^ gained so th a t the crystal w ill l i f t the p a rtic le , provided tha t s u ffic ie n t solution (o r melt) enters in to the contact region between the crystal and the p a rticle . O n the other hand, when aSp < asA + a ^ is v a lid , the p a rtic le simply adheres to the crysta l. Unfortunately, there are no experimental data available fo r a . There are few values available fo r c r „ and a „0[ 23]. The rela- sp pt Sx tionship agp > + c only indicates whether or not i t is possible fo r the growing crystal to push the external body. 3. Work of Corte Corte [5 ] investigated pushing of seven d iffe re n t p a rticle s by upward freezing of water. For each p a rtic le i t was found tha t trapping occurred only when a c r itic a l freezing rate was exceeded. Using d if f e r e n t shapes o f s il ic a (g la s s bead, broken g la s s and q u a rtz ), i t was dem onstrated th a t an Im p o rta n t fa c to r in p a r tic le re p u ls io n and ca p tu re was the shape o f the p a r t ic le . Glass beads, having th e s m a lle s t c o n ta c t a re a , showed the s m a lle s t fre e z in g ra te f o r tra p p in g . Broken g la s s and q u a rtz , having a g re a te r c o n ta c t a re a , were pushed a t la rg e r ra te s . Mica p a r tic le s , which had th e la rg e s t c o n ta c t area o f the p a r tic le s ( a ll 0 .1 5 m m 1n d ia m e te r), e x h ib ite d the g re a te s t c r i t i c a l fre e z in g ra te . The r e la tio n o f c r i t i c a l fre e z in g ra te to p a r t ic le s iz e was a ls o s tu d ie d . F ine p a r tic le s m ig rated under a wide range o f ra te s o f fre e z in g , w h ile coarse p a r tic le s m ig rated a t s lo w e r and narrow er ranges o f ra te s o f fre e z in g . He p o in te d o u t th a t a la y e r o f w a te r must be c o n tin u o u s ly p re se n t between the p a r tic le and the ic e f r o n t d u rin g p a r t ic le m ig ra tio n . 4. Work o f Uhlmann, Chalm ers, and Jackson Uhlmann, Chalm ers, and Jackson [ 6 ] s tu d ie d the In te ra c tio n between p a r tic le s and an advancing s o lid - liq u id in te rfa c e both e x p e rim e n ta lly and th e o r e tic a lly . They observed the pushing o f eleven d if f e r e n t types o f p a r tic le s in th re e o rg a n ic compounds con ta in e d as a t h in f ilm (300-500 pm) between g la s s s lid e s . They a ls o found th a t f o r each p a r tic le - m e lt com bination a c r i t i c a l v e lo c ity e x is te d , below w hich th e p a r tic le s were re je c te d by the in te r fa c e , and above w hich the y were trapped in the c r y s ta l. The dependence o f Vc on v a rio u s p ro p e rtie s o f p a r tic le s and m a trix m a te ria l was s tu d ie d . 9 A theory was developed based on the assumptions of a p a rtic le -in te rfa c e repulsion (due to the diffe re n ce o f surface energies, AoQ = agp - asi ” C T pfc) combined w ith d iffu s io n o f m elt in to the contact region. The c r itic a l trapping v e lo c ity fo r a smooth spherical p a rtic le on a planar in te rfa ce was expressed as V c = \ (n+l)(La v£D/kT R2) (2-5) where D is the d iffu s io n c o e ffic ie n t, L the la te n t heat o f fu sio n , v^ the atomic volume o f m e lt, a the molecular diameter, R the radius o f p a rtic le , k the Boltzmann's constant, and n a constant which 1s an exponent re la tin g v a ria tio n o f AoQ w ith separation distance. They also treated the problem in more d e ta il, Including the effects o f viscous drag and p a rtic le roughness. However, the re su lts were not s a tis fa c to ry . One o f the reasons is a lack o f microscopic properties fo r the supercooled th in film between the p a rtic le and the freezing s o lid . In th e ir c a lc u la tio n s , they had to assume a d iffu s io n c o e ffic ie n t 400 times sm aller than the value in the bulk liq u id in order to obtain agreement w ith the experiment re su lts. Table 1 summarizes th e ir measured V c> along w ith those observed by others [5 ,7 ,9 ,1 0 ,2 4 ]. 5. Work o f Hoekstra and M ille r Hoekstra and M ille r [7 ] postulated th a t the transpo rt o f water in a th in film between a glass p a rtic le and 1ce is by d iffu s io n . They took the fre e energy gradient as a d riv in g force and related i t to the temperature gradient, de riving T A B LE 1 S U M M A R Y O F CRITICAL VELO C ITIES F R O M P R E V IO U S S T U D IE S Investigators Uhlmann, Chalmers & Jackson Ref. 6 Particles Salol Agl not pushed Fe2 °3 9.0 M g O 10.8 N i 8.3 Si 2.9 S n 3.6 Zn 25.2 Diam ond 7.2 (0-2 yn) Diam ond 7.6 (3-5 pm ) Graphite not pushed S ilt 2.5 V , mm/hr. V Thym ol Orthoterphenyl 21.6 not pushed 7.2 9.0 28.8 1.8 28.8 7.2 36.0 2.9 14.4 3.6 21.6 9.0 4.7 32.4 5.0 43.2 1.1 57.6 2.5 Piphenylamine Water TABLE 1 (continued) S U M M A R Y O F CRITICAL VELOCITIES F R O M PR EVIO U S STUDIES V , mm/hr. Investigators Particles Salol Thymol Orthoterphenyl Diphenyl amine Water Pikunov A12°3 7 35 Ref. 24 4 ! 7 Coal 7 Lycopodi u rn 7 Corte Mica 0.4 - 3.2 Ref. 5 (1 - 0.15 m m ) Shale 0.2 - 1.1 (1 - 0.15 m m ) Quartz 0.2 - 0.7 (1 - 0.15 m m ) R utile 0.2 - 0.7 (0.6 - 0.15 m m ) C alcite 0.3 - 0.6 (0.6 - 0.15 m m ) Broken Glass 0 « 1 o • cn (0.6 - 0.15 m m ) T A B LE 1 (continued) S U M M A R Y O F CRITICAL VELO C ITIES F R O M P R E V IO U S S T U D IE S V c, mm/hr. Investigators Hoekstra and M iller Ref. 7 Cisse and Bolling Ref. 9 & 10 Particles Salol Thymol Orthoterphenyl Piphenyl amine Water Glass Beads 0.1 - 1.3 (230 - 120 urn ) Glass 0.5 - 3.0 Cylinders (270 - 160 ym ) Si0« 0.6 - 1.1 1.4 - 2.3 (90 - 35 ym) (100 - 40 ym) C u 0.6 - 1.1 0.6 - 18 (80 - 10 yin) (125 - 5 ym) W 0.6 - 1.3 0.7 - 12 (10 - 7 ym) (60 - 5 ym) w2c A £ 0.6 - 1.1 (20 - 10 ym) 0.6 - 1.1 A g (80 - 10 ym) 0.6 (80 ym) ro 13 2dXcp( vc ■ kT . P s (l?) 3 r (2‘ 6) where d is the thickness o f the th in film , X is the m o b ility o f the liq u id film , p$ is the density o f ic e , is the density o f water, and c is a constant. Their experiments, executed lik e Corte's [5 ], confirmed the lin e a r re la tio n sh ip between Vc and 1/R fo r both glass beads and glass cylinders. The glass cylind ers gave higher values fo r V than did the glass beads. However, they did not te s t experim entally the dependence o f Vc on the temperature gradient. 6. Work o f B o lling and Cisse B olling and Cisse [8 ,9 ] combined flu id -flo w viscous drag and d iffu s io n considerations to derive th e ore tical re la tio n s fo r Vc< They also considered the indentation o f the in te rfa ce shape generated by the in te ra ctio n between the p a rtic le and the in te rfa ce . Using an upward moving in te rfa ce to push fo u r d iffe re n t p a rtic le s by freezing water, th e ir experimental re su lts [9 ] supported w ell the deduced re la tio n s. They established a simple inverse power law between V and V size fo r smooth p a rtic le s . W hen the theory was made more complex by considering the e ffe cts o f p a rtic le roughness, g ra v ity and thermal co n d u ctivitie s, the f i t w ith experiment was improved. As predicted by th e ir theory, the ra tio V : V : V was measured fo r the c r it ic a l C v L v e lo c itie s when trapping o f the same type and size o f p a rtic le s occurred a t a grain surface, a grain boundary groove and a grain 14 boundary t r i p l e p o in t. T h e ir th e o re tic a l development appeared to be s ig n ific a n t f o r u n de rstand in g tra p p in g and r e je c tio n o f In s o lu b le p a r tic le s , b u t 1 t is very d i f f i c u l t to fo llo w and understand. Thus, th e study o f t h e ir th e o ry became p a rt o f th e ta sk o f th is d is s e r ta tio n . P a rt o f th e th e o ry re le v a n t to th is work is e xp la in e d here in d e ta il f o r a b e tte r unde rstand in g o f th e phenomena o f p a r t ic le pushing. The s te a d y -s ta te in te rfa c e shape d u rin g th e pushing o f a fo re ig n p a r tic le depends on th e in te r a c tio n between the p a r tic le and the fre e z in g in te r fa c e . The in te rfa c e must be resp on sive to a ll fo rc e s e xe rte d on th e p a r tic le such as g r a v ity and viscous drag. T his q u ite o fte n re s u lts in a smooth sh a llo w In d e n ta tio n in the in te rfa c e a d ja c e n t to th e p a r tic le . C ry s ta l growth can be d e scrib e d by s p e c ify ing th e s o lid - liq u id In te rfa c e tem perature as a fu n c tio n o f p o s itio n . For th e s im p le s t case o f pure growing m a te ria l and an in e r t smooth p a r tic le o f th e same therm al c o n d u c tiv ity as the s o lid and m e lt, the in t e r f a c ia l tem perature T^ can be expressed as T. = T - AT - AT - AT. (2 -7 ) i m s c 1 ' ' where Tm is th e m e ltin g tem perature o f pure m a te ria l, ATg is the s u p e rco o lin g needed to p ro v id e a d r iv in g fo rc e f o r c ry s ta l grow th and is a fu n c tio n o f growth ra te V, AT = aK/AS is th e tem perature depres- V sio n due to c u rv a tu re e ffe c ts , AS is th e e n tro p y o f fu s io n per u n it volum e, K is th e t o t a l c u rv a tu re a t any p o in t on th e in te r fa c e , and AT. is th e tem perature change due to in te r a c tio n o f the p a r tic le w ith 15 the Interface and is evaluated as follow s. The thermodynamic c rite ria fo r equilibrium requires th a t the free energies of the two phases (s o lid and liq u id ) are equal a t the melting point. The fundamental re la tio n fo r a small change in free energy of the system d G is expressed as where S is the entropy o f system, V the volume, and P the pressure. I t follow s that the condition fo r equilibrium between two phases is that dG $ * dG^, or where the subscripts s and £ re fe r to so lid and liq u id respectively. I f we assume that the liq u id pressure in the p a rtic le -in te rfa c e con tact area is sim ila r to the bulk pressure, then the V dP may be X » J O neglected. Further, T& = T^ at the equilibrium state. Thus, the above equation becomes dG = -SdT + V dP (2- 8) (2-9) I f we substitute A S = S s-S^/Vs , an<* ^ ~ F ^ltro t * ie e9ua't ‘*on» the temperature change due to a force acting on the crystal is (2- 10) s £' s AT. = ------ ~ i f r < r fiS* r 0 “ ' 0 ( 2 - 1 1 ) 0 i f r > r. 0 16 where r Q is an e ffe c tiv e c o n ta c t rad iu s beyond which no In te ra c tio n occurs between the p a r tic le and the in te rfa c e , as d e fin e d by Uhlmann e t a l. [ 6 ] . F 1s the in te ra c tin g fo rc e w ith in r Q. The search f o r a com plete s o lu tio n o f F is the n e xt step in development o f th e th e o ry . The viscous drag on th e p a r t ic le , which produces a fo rc e on the fre e z in g in te rfa c e , is considered next. A hydrodynamic approach is used to solve th e problem o f flo w around a sphere to a nearby u n ifo rm s in k (In te rfa c e ). F ig u re 1 shows a schem atic cross s e c tio n o f a s p h e ric a l p a r tic le in c o n ta c t w ith an In te rfa c e . The c o n ta c t re g io n w ith in r Q 1s expanded. The form ula fo r viscous drag F f o r th e plane sin k (0 = 0) as d e rive d in Uhlmann's d is s e rta tio n by G. C a rrie r [2 5 ] 1s F = 6irnV R 2/d ( 2- 12) which g ive s the viscous drag upon a sphere o f ra d iu s R lo c a te d a d is tance d from a plane in te rfa c e advancing a t growth ra te V. The problem considered here re q u ire s a sin k which is concave due to in t e r a c tio n w ith in the c o n ta c t area r < _ r Q. The s e p a ra tio n d ista n ce h between p a r tic le and in te rfa c e a t any p o in t in the c o n ta c t region 1s h = d + R - (R2- r 2) 1/2 - E (r) (2-13) where E (r) is th e v e r tic a l d ista n ce between a concave in te rfa c e and it s lo w e st p o in t, and is a fu n c tio n o f th e ra d ia l d is ta n c e . For the case o f r « R, (R ^ -r^ )" * ^ is ap pro xim a tely equal to R - r^/2 R by a M acLaurin's s e rie s and thus 17 t Particle Liquid Interface E(r) Solid Figure 1 Schematic cross section of a spherical particle in contact with a solid-liquid interface with a magnified contact area* 18 h « d + r 2/2R - E (r) . (2 -1 4 ) A p a ra b o lic v e lo c ity p r o f ile U is assumed f o r the flo w in the c o n ta c t re g io n , w ith zero ta n g e n tia l v e lo c ity a t the s o lid - liq u id in te rfa c e (z = E{ r )) and the p a r t ic le - liq u id in te rfa c e (z = h + E ( r ) ) . That is , U - C (r) (z - E (r)) (z - h - E (r)) . (2 -1 5 ) C onsider th e re g io n o f c o n ta c t as a s o r t o f c y lin d e r w ith a ra d iu s o f r . C onservation o f mass re q u ire s th a t the liq u id e n te rin g th e c y lin d r ic a l w a lls equals th a t le a vin g a t the bottom o f the c y lin d e r under the steady s ta te c o n d itio n s . r h+ E (r) 2 2tt rU p 0 dz = irr V p0 . (2 -1 6 ) JE (r) % s S u b s titu tin g Eq. (2-15) in to th e above eq ua tion and in te g ra tin g the le ft-h a n d s id e , th e fu n c tio n C (r) is found to be c (r )' ^ f e ) ‘ ( 2 ' 17> Hence, the com plete expression f o r U is u = - E<r ) ) ( z - h - El r >) • (2 -1 8 ) The pressure g ra d ie n t in the r d ir e c tio n is found from th e N avier-S tokes eq ua tion f o r a f lu i d in s te a d y -s ta te la m in a r m otion [2 6 ] to be 19 (2-19) The pressure d iffe re n ce AP between the inside o f the contact region and the outside o f the contact region is Since h increases w ith r in real s itu a tio n s , dP/dr in Eq. (2-19) increases from zero and then decreases, approaching zero again as r approaches R. On the other hand, AP is the la rg e s t a t r = 0 and decreases as r increases. Their va ria tio n s are shown schem atically in Figure 2. These trends, as given by Eqs. (2-19) and (2-2 0), hold whether the approximate formula Eq. (2-14) is employed o r not. In e ith e r case r/h approaches zero as r approaches R. The major con trib u tio n to the in te g ra l Eq. (2-20) comes from the region r << R. Because the h yd ro sta tic pressure P(R) can be considered con sta n t over the surface o f a small p a rtic le , AP is e s s e n tia lly the net pressure exerted on the p a rtic le in the v e rtic a l d ire c tio n . This pressure is m u ltip lie d by the projected area o f the p a rtic le , and thus gives a net force F, which is Since we have assumed r « R, we may w ell take the lower lim it as dr . ( 2- 20) 20 Figure 2 Particle Solid-liquid interface dP/dr /i 0 - r R A P Sketch showing variations of dP/dr and AP with respect to r in the contact region. 21 < » fo r R to obtain F » f° 2irr fr ----------------------- r ~ Y — ) dx dr . (2-22) J- L (d+jg- E(r)) 3 W The above equation cannot be Integrated unless we know the fun ction E (r). Because we are searching fo r an outer bound, i t 1s possible to fin d a lim it fo r th is function which may give a maximum force. At the lim it r = 0 and z = 0 , the contact region between the p a rtic le and in te rface may approximate a pa rt o f a sphere, and E (r) has a maximum value 1f the Interface would clo se ly contact w ith a spherical p a rtic le w ith a radius o f R /a where a < 1. At the lim it d « h, we obtain from Eq. (2-14) 2 E(r) « . (2-23) Equation (2-22) is then integrated by s u b s titu tin g the above re la tio n fo r E(r) to obtain , 2 /P. F(ct) * ~ n— 7 ( ~ ) (2-24) d ( l- a ) 2 \ p£ ' where F(a) > _ F. W hen a has a value o f zero, the in te rfa ce 1s a plane sink and not responsive to the presence of the p a rtic le . Equation (2-24) thus becomes Eq. (2-12) o f C a rrie r's expression fo r viscous drag. At the other extreme, when a has a value o f u n ity , the in te rfa ce and the p a rtic le are in an intim ate contact, and Eq. (2-24) y ie ld s an extremely large force. Therefore, we have to lim it the use of F(a)» p a rtic u la rly as a approaches u n ity . 22 The visco u s drag d e riv e d above appears to depend on th e s e p a ra tio n d is ta n c e d a t r = 0. The value o f d Is d i f f i c u l t to e v a lu a te so th a t we may r e la te I t to some system param eters. The In te rfa c e discussed here 1s re sp o n sive to th e e x is te n c e o f th e nearby p a r t ic le ; and th e re fo re d has an In flu e n c e on the In te rfa c e shape. I f we r e la te the In te rfa c e shape param eter a to some c h a r a c te r is tic value o f d , le t us say a t th e va lu e o f Na^ where a^ o o is the In te ra to m ic d is ta n c e and N 1s a number depending on a system param eter 0 (0 is some c h a r a c te r is tic o n ly o f th e liq u id f o r in e r t p a r t ic le b u t o f liq u id and p a r t ic le i f th e re 1s a chem ical In t e r a c tio n ) , and choose a decay fu n c tio n to re p re s e n t th e v a r ia tio n between th e two lim it s f o r a (a = 0 and 1) , we o b ta in w hich g iv e s a p la n a r in te rfa c e (a = 0 ) a t d » 0aQ and In tim a te con t a c t (a = 1) between the p a r t ic le and the in te rfa c e a t d = 0aQ. R earranging the above r e la tio n we o b ta in Thus, Eq. (2 -2 4 ) can be expressed in terms o f 0 by means o f Eq. (2 -2 6 ) W e ju s t com pleted th e hydrodynamic approach to the tra n s p o rt problem in the c o n ta c t re g io n . Uhlmann e t a l. [ 6] tre a te d th e same tra n s p o rt problem from a d iffu s io n v ie w p o in t. I f th e re Is a d iffu s io n o « exp C -(d -6a0 )/a 0] (2 -2 5 ) d = a ( 8- in a ) . (2 -2 6 ) (2 -2 7 ) 23 lim it , Eq. (2-24) may be used to compare w ith th is d iffu s io n lim it . For the purpose of comparison, F(a) is m athem atically manipulated w ith The fo rce F(a) is divided in to two p o rtio n s. One arises w ith in the contact region r £ r Q, the other one from the region r > r Q. N ow we may compare F(a) w ith a supposed fo rc e which is required fo r the d iffu s io n process, where Fp is the force needed fo r d iffu s io n in the contact area r < r Q . This force is less than th a t fo r flo w w ith in the same region, o r flow would occur. Thus, F(a) _ > F vl-rtu a -|* we work a t the lim it , i t becomes immaterial to d is tin g u is h between flow and d iffu s io n . The movement o f molecules on the surface fo llo w in g a ran dom walk w ith no preferred d ire c tio n is [27] 2 the assumption o f w = r (l-a )/2 R , 6 t t n V R F , v irtu a l (2-29) Ar^ = 4 Dt (2-30) where t is the time necessary fo r a mean displacement Ar. I f we assume th a t b is the height fo r growth o f a monoatomic la y e r, then the time required fo r molecules to d iffu s e in to the contact region (w ith an e ffe c tiv e ra d iu s r Q) is t = b /V . The le a s t advantageous d iffu s io n sequence is then r 0 2 < 4 D b/V . (2 -3 1 ) I f we r e la te f lu i d flo w and d iffu s io n by means o f th e S to k e s -E in s te in e q u a tio n , th a t i s , D = kT/3-nna^ th e above r e la tio n becomes r o ± W v • <2- 32> W ith th is r e la tio n in m ind, we may determ ine th e c o n d itio n s under which the fo rc e F (a) is dom inant d u rin g p a r tic le pushing. The fo rc e w ith in r < _ r 0 where the d iffu s io n m ig ht occur can be e va lu ated from th e second term o f Eq. (2 -2 8 ) c / * - 6tt n V R2 / ps \ f° dw 0 t0 r o ( 1- a ) ^ vpA ' J r^ (d+w) 2 o = 6rr n V R2 ( £ ) lo ------- (2-33) (1 -a ) ' p£ / 2 R d [d + (l-a )rf/2 R l (1 -a ) ' p£ / 2 R d [d + (l-a )rJ /2 R ] The fo rc e F(a) t is s e n s ib ly the fo rc e F(a) o f Eq. (2 -2 4 ) I f to r Q r 2 ( l- a ) » d . (2 -3 5 ) A t th e l i m i t o f the d iffu s io n sequence, the above r e la tio n when com bined w ith Eq. (2 -3 2 ) becomes 25 Based on Eq. (2-2 4), the viscous drag Increases w ith Increase o f the growth ra te o r the In te rfa c e curvature. F(a) reaches a maximum when the growth 1s a t the c r it ic a l trapping v e lo c ity . At th is p o in t the region o f close contact gives a maximum radius R/a. I f we equate the e ffe c t o f the maximum force to th a t o f the maximum possible curvature, o r ATC = AT^, we obtain the conditions fo r in itia tio n o f p a rtic le trapping. Thus fo r p re d ictio n o f the c r it ic a l trapping v e lo c ity . The re la tio n o f a and $ 1s found a t the extremum o f 4>(a), or a t the condition o f d<|>(a)/da = 0. They are or (2-36) I f Eq. (2-35) holds, we are allowed to use Eq. (2-32) to 2 evaluate the q u a n tity r Q w hile m aintaining the conditions o f maximum force and c r it ic a l trapping v e lo c ity . That 1s, (2-37) 2 where <f>(a) is a (l a) ( 6-£n a ). This 1s the fin a l equation to be used 3 = tn a + ( l - a ) / ( l - 3 a ) (2-38) and <Ma)max = a ( l- a ) 3/( l- 3 a ) . (2-39) 26 The above r e la tio n s r e s t r i c t th e q u a n tity a , w hich must be le s s than 1 /3 o r e ls e th e c u rv a tu re o f th e in te r fa c e would be la rg e enough to e n g u lf th e p a r t ic le . I f we a p p ly th is lim it a t io n to Eq. (2 -2 5 ), a u s e fu l r e la tio n s h ip is o b ta in e d as o r d > (1.1 + e)aQ . (2 -4 0 ) The above r e la tio n g iv e s th e th ic k n e s s o f th e th in f ilm e x is tin g between th e p a r tic le s and in te r fa c e . S ince th e t h in f ilm c o n s is ts o f se ve ra l in te ra to m ic la y e rs , i t is lo g ic a l to assume a minimum o f u n ity f o r $ to s o lv e th e problem . The r e la tio n s h ip between d and a can be o b ta in e d fro m Eqs. (2 -2 6 ) and (2 -3 8 ), d = ( l- a ) a 0/ ( l- 3 a ) . (2 -4 1 ) The in e q u a lity presented in Eq. (2 -3 5 ) to be e va lu a te d then becomes VR « 2 (l-3 a )k T /3 tr n aQ . (2 -4 2 ) I f we choose 8 = 1, th e va lu e o f a g iv e s 0.229 a t <f>(a) = O <f>(a)max* For w a te r as a m a trix m a te r ia l, aQ = 3 .8 x 10 cm, n = 1 .8 c .p . , T = 273°K, and VR < _ 3 .7 x 10“ ® cm ^/sec. T his re q u ire ment was met by s e ve ra l d if f e r e n t p a r tic le s in fre e z in g o f w a te r [ 7 , 9 ] . E q u a tio n (2 -2 7 ) was d e riv e d w ith o u t a cco u n tin g f o r e ffe c ts o f g r a v ity and p a r t ic le roughness. C isse and B o llin g [ 9 ] s tu d ie d smooth 27 spherical tungsten and copper p a rticle s in water. The data supported Eq. (2-37) qu ite well fo r the small p a rticle s with R < 10 pm. B. S om e Theoretical Considerations 1. Structure of the Solid-Liquid Interface The importance of interface morphology to th is study is obvious, since the sol id -liq u id in terface was employed to re je ct and push foreign p a rtic le s . The nature o f the interface is known to have a decisive influence on the kin etics and morphology o f crystal growth. According to one viewpoint, a ll molecules e ith e r cle a rly belong to the crystal or to the m elt, yie ld ing a sharp w ell-defined interface between the two. Such an interface may e ith e r be atom ically smooth or atom ically rough. These two possible interfaces are shown in Figure 3 where the white spheres represent the atoms of the m elt, the black spheres are the crystal atoms. Figure 3(A) is an atom ically smooth in te rfa ce , while (B) represents a rough interface. With the present knowledge o f crystal growth, two possible growth mechanisms have been discussed [28,29]. A crystal whose equilibrium interface is atom ically smooth grows by the formation and la te ra l growth o f layers. In a perfect c ry s ta l, new layers are formed discontinuously by two-dimensional nucleation. W hen one or more screw dislocations are present, new layers may be formed con tinuously. Rapid la te ra l growth o f layers leads to formation of macroscopic facets. When, on the other hand, the equilibrium interface is atom ically (A) Atomically smooth interface Melt Crystal (B) Atomically rough interface* Melt Crystal Figure 3 Atomic solid-liquid interface 29 rough, the crysta l can grow w ithout form ation o f new layers. The interface advances uniform ly and is not faceted. The theory o f the equilibrium in te rfa c e roughness was s a tis fa c to rily examined by Jackson in 1958 [3 0 ]. He considered an in it ia lly atomic smooth in te rfa c e and calculated the change in surface free energy on adding extra molecules. The re s u lts o f his elementary s ta tis tic a l thermodynamic approach predicted th a t the equilibrium in te rface should be atom ically rough and non-faceted fo r m aterials having entropies o f fusion (AS^) less than 2R o r 16.8 J/mole°K (where R is the gas constant). For A S ^. > 4R the in te rfa ce was predicted to be atom ically smooth, faceted and e s s e n tia lly c ry s ta llo g ra p h ic a lly perfect. Most metals and ce rta in organic compounds have low entropies of fusion (aS^ < 16.8 J/mole°K). Their non-faceted and planar in te r faces were confirmed by recent experiments [29,31J. A d iffe re n t approach to the problem o f the stru ctu re o f the interface was discussed by Cahn [32] who assumed a d iffu s e in te rface o f f in ite thickness. The tra n s itio n from c ry s ta l to m elt takes place over several layers o f atoms. According to the theory, la te ra l growth occurs a t low undercooling, but continuous growth takes place at high undercooling. A tra n s itio n from la te ra l growth kin e tics to continuous growth k in e tic s was predicted fo r a ll m aterials. Unfor tunately, there is no known way of determining the diffuseness of the in te rfa ce , so th a t a te s t o f the theory is impossible. 2. In te rfa c ia l Temperature Gradient The temperature gradient at the s o lid -liq u id in te rfa ce is an 3 0 Im p o rta n t param eter in the grow th o f c r y s ta ls . I t in flu e n c e s the p e rfe c tio n and p ro p e rtie s o f c ry s ta ls [3 3 ], e u te c tic s tru c tu re s [3 4 ], th e m icro sco p ic shape s t a b i l i t y o f the in te rfa c e [3 5 ], and in te r fa c e fa c e tin g [3 6 ], The r o le o f tem p eratu re g ra d ie n t in pushing and tra p p in g is n o t c le a r. H oekstra and M ille r [7 ] p re d ic te d th a t Vc should be p ro p o rtio n a l to th e tem perature g ra d ie n t. However, th e o re tic a l tre a tm e n ts by o th e r w orkers [ 6 , 8] d id n o t c o n s id e r th is term . The im portance to th is work is discussed below. In te rfa c e S t a b ilit y . Even s o -c a lle d "p u re " m a te ria ls are n o t co m p le te ly pure. Traces o f im p u ritie s are always p re se n t. I f , f o r exam ple, th e c o n c e n tra tio n o f th e s o lu te (im p u rity ) in the s o lid is le s s than th a t o f th e liq u id from w hich I t 1s fo rm in g , th e re 1s a r e je c tio n o f s o lu te in to th e liq u id a t the s o lid - liq u id in te r fa c e . The liq u id in the v ic i n it y o f th e in te rfa c e becomes en rich ed in s o lu te . Unless the grow th ra te is slo w e r than the d iffu s io n ra te o f s o lu te in th e liq u id , a c o n c e n tra tio n g ra d ie n t develops in th e liq u id ahead o f the In te rfa c e . S ince the presence o f s o lu te in th e m e lt low ers the fre e z in g p o in ts , th e fre e z in g tem perature in cre a se s from th e in te rfa c e w h ile the im p u rity c o n c e n tra tio n decreases as shown in F ig u re 4. The imposed tem p eratu re n o rm a lly Increases as one moves in to th e m e lt. When the s lo p e o f th is a c tu a l tem perature is le ss than th e slo p e o f the fre e z in g p o in t, th e m e lt ahead o f the in te r fa c e is below i t s normal fre e z in g te m p e ra tu re . T h is excess s u p e rc o o lin g , termed c o n s titu tio n a l s u p e rco o lin g [3 7 ], is a d ir e c t r e s u lt o f the c o n c e n tra tio n g ra d ie n t which e x is ts in the m e lt. Under such c o n d itio n s Concentration 31 (A) Solute distribution during normal freezing. Crystal Melt | Distance into melt (B) Corresponding equilibrium freezing temperature distribution. Actual temperature a > fa + » c d fa & e 9 E h Freezing temperature Constitutional supercooling | --- >• > Distance into melt Figure 4 Constitutional supercooling. 32 th e in te r fa c e is u n s ta b le w ith re s p e c t to sm all p e rtu rb a tio n . The s u p e rc o o lin g can be re lie v e d by th e growth o f sm all p ro je c tio n s in to th e m e lt. A fre e z in g in te rfa c e w hich is rough on an atom ic s ca le may develop pox s tru c tu re s * e lon ga te d c e lls , hexagonal c e llu la r s tru c tu re s , o r even d e n d rite s , depending on th e degree o f c o n s titu tio n a l super c o o lin g . Such in te rfa c e breakdown has a d e trim e n ta l in flu e n c e on th e process o f p a r tic le pushing and th e re fo re was avoided in th is work. The c o n d itio n s f o r th e o n se t o f c o n s titu tio n a l su p e rco o lin g may be expressed in sim p le m athem atical terms [3 8 ,3 9 ]. When the imposed tem p erature g ra d ie n t is le s s than th e g ra d ie n t o f e q u ilib riu m fre e z in g tem perature a t the in te r fa c e , the fo llo w in g r e la tio n s h ip between the Imposed grow th c o n d itio n s and the system param eters is obtained f o r th e c r i t i c a l c o n d itio n o f s u p e rc o o lin g i f th e re is no liq u id m ix in g in th e liq u id phase. where ( C L and (C0) 4 are th e in te r f a c ia l s o lu te c o n c e n tra tio n s in 5 I )C I th e s o lid and the m e lt, and m is th e slo p e o f the liq u id u s lin e on th e phase diagram . Some q u a n tita tiv e s tu d ie s [4 0 ,4 1 ] have v e r ifie d the above r e la tio n s h ip . In o rd e r to a vo id the e ffe c ts o f c o n s titu tio n a l s u p e rc o o lin g , the imposed tem perature g ra d ie n t should be increased to exceed the g ra d ie n t o f fre e z in g tem perature a t th e in te rfa c e . In te rfa c e F a c e tin g . Ja ckso n's th e o ry o f in te rfa c e roughness was d e riv e d f o r an iso th e rm a l and e q u ilib riu m environm ent. In rea l s o lid if ic a t io n s itu a tio n s , th e system is n e ith e r iso th e rm a l no r a t (2-43) 33 equilibrium . W ilcox's th e ore tical discussions [36] pointed out th a t crystals predicted to grow in a faceted manner (high entropy o f fusion m aterials) would grow non-faceted in the presence o f a s u ffic ie n tly large temperature gradient. From equilibrium considerations, faceting in a temperature gradient would e x is t only i f the edge energy was considered in addition to the surface energy and the bulk free energy. From the standpoint of k in e tic e ffe c ts , faceting is favored by anisotropic growth kin e tics and rapid growth. There is widespread q u a lita tiv e experimental evidence to support the e ffe c t o f temperature gradient on in te rfa ce faceting. CHAPTER I I I EXPERIMENTAL Two c r y s ta l grow th methods were employed f o r t h is e xp e rim e n ta l w ork. A v e r tic a l B ridgm an-S tockbarger te c h n iq u e was f i r s t used to in v e s tig a te the fundam ental n a tu re o f p a r t ic le pushing and tra p p in g phenomena. H o riz o n ta l z o n e -re fin in g w ith r o ta tio n was then em ployed to s tu d y some o f th e design v a ria b le s f o r a s e p a ra tio n process. Uhlmann, e t a l. [ 6 ] conducted p a r t ic le pushing experim ents by means o f a h o riz o n ta l th in f ilm (h e ld between g la s s s lid e s ) w ith a v e r tic a l fre e z in g in te r fa c e . T h e ir re s u lts m ig h t have been s tro n g ly i n f l u enced by th e g la ss s lid e s and no t c h a r a c te r is tic o f b u lk m a te ria ls . O thers [5 ,7 ,9 ,1 0 ] used B rid gm a n-S tockbarg er grow th w ith th e s o lid - liq u id in te r fa c e moving upward, re d u cin g th e p o s s ib ilit y o f th e p a r tic le s d ra g g in g on th e g la s s w a ll. W e in c o rp o ra te d f a c i l i t i e s f o r d ir e c t o b s e rv a tio n and te m p e ra tu re measurements. T h is was th e f i r s t tim e th a t a h o riz o n ta l z o n e -re fin e r w ith r o ta tio n was used to in v e s t i g a te removal o f p a r tic le s from th e m a trix m a te ria ls . F o llo w in g th e d e s c rip tio n o f th e two e xp e rim e n ta l te c h n iq u e s , th e m a te ria ls used f o r t h is stu d y a re p re se n te d , in c lu d in g th e p a r tic le s and th e o rg a n ic compounds. The tre a tm e n t o f s o lid if ie d p ro du cts is then d iscu sse d . The f in a l s e c tio n o f th is c h a p te r is devoted to a d e s c rip tio n o f th e z o n e -re fin in g o f o rg a n ic compounds. 34 35 A. Bridgman-Stockbarqer Technique 1. Apparatus A Bridgman-Stockbarger apparatus was constructed as shown in Figure 5. This u n it was designed to provide an upward moving so lid - liq u id in te rfa ce which could be constantly observed with a micro scope. E ssentially the u n it consisted of a-heater, a cooler and a tube lowering mechanism. The heater was made by winding nichrome wire on a piece o f flanged Pyrex glass tube, which was insulated by another glass tube. The cooler was constructed s im ila r to a laboratory condenser, as shown in Figure 6 . The cooling flu id was circulated from a refrigerated constant temperature bath (0 to -30°C), passed through the annular space o f the cooler, and returned to the c irc u la to r. A crystal-grow th tube tra v e llin g through the heater steadily displaced the heat-transfer flu id which overflowed from a side tube. This flu id was placed in the inner tube of the cooler to elim inate the a ir gap between the inner tube and the crystal-grow th tube, thus improving the heat tran sfe r. The c irc u la tin g flu id and the heat-transfer flu id were not mixed, but both consisted o f 50 volume % ethylene glycol-w ater. The position of the side tube was adjustable, so th a t the level of the flu id in the inner tube could be controlled. Two variable-speed motors provided lowering rates of 1 to 15 mm/hour and 10 to 80 mm/hour. The heater power was controlled by an auto-transformer which was connected to a Sola constant- voltage transform er (30 VA, Harmonic Neutralized Type CVS) to obtain a constant voltage A.C. input. The power (3 to 12 watts) was 36 Figure 5 Bench-scale unit or improved Bridgman- Stockbarger technique. (1) Drive mechanism (2) Heater and cooler (Details shown in Fig.6) (3) Constant temperature circulator (4) Wattmeter (5) Electronic controlled stirrer (6) Microscope (7) Movie camera equipped with time-lapse controller 37 Figure Heater j : Crystal growth tube traveling through the heater and cooler Heat-transfer fluid overflowing from the side-tube Circulating cooling fluid 6 Schematic diagram of the heater and the cooler* 38 m on itore d by a w a ttm e te r placed between th e h e a te r and th e a u to - tra n s fo rm e r. Most runs were perform ed w ith a c o o le r made o f brass (37 m m 1 .0 . x 17 cm long f o r the in n e r tu b e ) w h ile a few runs were made w ith a sm all g la s s c o o le r (17 m m I.D . x 15 cm long f o r the in n e r tu b e ). Two h e a te rs were employed— one sm all (1 3 .5 m m I.D . x 12 cm long f o r th e in n e r tube) and one la rg e (31 m m I.D . x 14 cm long f o r the in n e r tu b e ). The la rg e h e a te r was used f o r some runs w ith a g lass s t i r r e r in s e rte d in to the growth tub e. The s t ir r in g m otor was placed on a a d ju s ta b le s te p le s s p la tfo rm in o rd e r th a t the d is ta n c e between the s t ir r in g blad e and th e fre e z in g in te r fa c e could be c o n tro lle d . 2. Sample P re p a ra tio n The s o lid if ic a t io n tube was made from Pyrex g la s s tub ing w ith a w e ig h t atta ch e d to the round bottom to keep th e growth tube v e r t ic a l. Most runs were made w ith 10.5 m m I.D . (13 m m O .D .) x 10 and 15 cm long tub es. Some runs w ith s t ir r in g used la rg e tu b e s , 26.5 m m I.D . (30 m m O .D .) x 22 cm lo n g . In the e a rly exp e rim e n ts, p a r tic le s were d ir e c t ly added in to an o rg a n ic m e lt and the m ix tu re then poured in to th e growth tub e. As d e scrib e d in S e ctio n C, these p a r tic le s were agglom erates. They should have been broken in to in d iv id u a l p a r tic le s b e fo re using f o r p a r t ic le pushing stu d y. In la t e r experim ents th e p a r ti c l e -m e lt m ix tu re was f i r s t poured in to a 10 cc c y lin d r ic a l Pyrex b o t t le , sealed w ith a screw cap, and then placed in an u ltr a s o n ic c le a n e r c o n ta in in g h o t w a te r in o rd e r to keep 39 the organic molten. The m ixture was u ltra s o n ic a lly vibrated about 2 minutes, and then tran sfe rre d in to the growth tube by a dropping pi pet. Molten organic chemicals dissolve gases and release them as bubbles a t the freezing in te rfa ce during s o lid ific a tio n . These bubbles were found to seriou sly in te rfe re w ith the p a rtic le pushing process. Bubbles were avoided 1n la te r experiments by the fo llo w in g procedure: A pressure s lig h tly less than one atmosphere was applied to the organic m elt in a tube so as to gently degas the m elt. The organic was then s o lid ifie d and a vacuum (50 Torr) applied to the tube, which was sealed w ith a torch . Gas bubbles were not generated during s o lid ific a tio n in these vacuum-sealed tubes. Gas bubbles are fu rth e r discussed in Chapters IV and VI. 3. Experimental Procedure « A s o lid ific a tio n tube containing a m ixture o f p a rtic le s and organic compound was placed in the growth u n it as illu s tra te d in Figure 6. The length of the tube submerged in the cooler was in it ia lly about one tube diameter. The reason fo r th is is described in the next chapter. As soon as a steady s o lid - liq u id in te rfa ce covered w ith a la ye r o f p a rtic le s was esta blishe d, the tube was lowered at a predetermined rate . The in te rfa ce was observed through a stereo scopic microscope a t 10-50 X and the in te rfa c e p o s itio n determined by the eyepiece. Each run started w ith a low lowering rate (about 6 mm/hour) which was increased step by step (about 2 mm/hour every 30 minutes) u n til a ll p a rtic le s were trapped by the growing s o lid . 40 For some tim e a fte r each Increase in low ering ra te the grow th ra te was le ss than the lo w e ring ra te . R e la tiv e to the o b se rve r, th e in te rfa c e moved s lo w ly downward a fte r a ra te change u n t il a new s te a d y -s ta te p o s itio n was reached (s ta tio n a ry w ith re sp e ct to the h e a te r and c o o le r). Under th is s te a d y -s ta te c o n d itio n , the growth ra te and the lo w e rin g ra te were eq ua l. N o rm a lly, th is tra n s ie n t p e rio d was 20 to 30 m inutes in le n g th . The minimum growth ra te f o r p a r tic le tra p p in g , V , was u s u a lly determ ined by d ir e c t o b se rva tio n o f pushing and tra p p in g , b u t sometimes i t was estim ated by m icro scop ic exam ination o f the s o lid if ie d p ro d u ct. The e rro rs in such measurements were le ss than 10% . M otion p ic tu re s were taken o f in te re s tin g phenomena a t the fre e z in g in te rfa c e , such as bubble fo rm a tio n and s u rfa c e -d riv e n flo w s using a 16 m m movie camera on the m icroscope, as shown in F ig u re 5. T im e-lapse cinem atography was employed f o r very slow p a r tic le pushing processes. To avoid la rg e d is tu rb a n c e o f the heat balances a t the advancing in te r fa c e , a w ater c e ll was placed in fr o n t o f th e 500-w att photo lamp. Water flow ed through the c e ll c o n tin u o u s ly and absorbed heat generated from the lamp. The tim e -la p s e movie showed th a t the tr a v e llin g m otion o f th e growth tube was no t smooth, presumably because o f backlash in th e g e a rin g . A d d itio n a l w e ig h t was placed between the m otor and th e tube to reduce th is problem. 4. Temperature Measurements Thin 36-gauge copper and constantan therm ocouple w ire s were employed to measure the tem perature d is tr ib u tio n in the Bridgman 41 growth o f naphthalene. These th in wires were supported by a c a p illa ry tube and positioned a t the center o f the cross section o f the growth container as shown in Figure 7(A). The thermocouple emf was measured by a m illiv o lt recorder against an ice bath ju n c tio n . Because the thermocouple was fix e d in the growth co n ta in e r, i t tra v e lle d w ith the container as the la tte r passed through the growth u n it (Figure 6). Therefore, temperature measurements started in the m elt, reached the s o lid - liq u id in te rfa c e , and fin is h e d in the s o lid . The temperature versus time was continuously recorded. The rate o f change o f temperature divided by the growth ra te yie ld e d the temperature gradient. Thus, fo r measuring temperature gradients a t the in te rfa c e , knowledge o f the time a t which the thermocouple was at the in te rfa ce was required. For Instance, i f the thermocouple jo in t was 0.5 m m in diameter and the growth rate was 10 mm/hour, i t took 3 minutes fo r the in te rfa c e to pass by the thermocouple. This made i t d if f ic u lt to determine the in te rfa c e from the temperature recording, p a rtic u la rly fo r experiments a t very slow growth ra te s. With the help o f m icroscopic observation, the in te rfa c e p o s itio n was determined when the thermocouple was contacting the in te rfa c e . The errors in the temperature measurements were less than 0.1°C, but they might be up to 20% in the temperature gradients. Another fa c to r in flu e ncin g the accuracy o f the above measure ments was heat conduction down the thermocouple w ire s, so th a t the measured temperatures were higher than they should have been. An improved measuring technique was the refore developed as shown in < r ' i Melt I I I A i f e CCS Dewar Crystal (A) Speedomax W Recorder Copper ■Constantan (B) Figure 7 Temperature measurements by thermocouples during Bridgman-Stockbarger crystal growth. 43 Figure 7(B). The bare thermocouple wires were stretched across the cross section o f the container w ith the welded ju n c tio n positioned ★ at the center. The wires were passed through two opposing small holes in the container tube and the holes were sealed w ith epoxy cement. This improved method was not a c tu a lly used in the present study because pre lim ina ry re su lts based on the method o f Figure 7(A) indicated th a t the c r it ic a l v e lo c ity Vc does not depend on the temperature g ra d ie n t. However, the method was used in th is laboratory by Chong E. Chang. The re su lts were q u ite s a tis fa c to ry even a t very slow growth ra te s, less than 1 mm/hour. B. H orizontal Zone R efining with Rotation Some experiments were made 1n the v e rtic a l Bridgman apparatus by in s e rtin g a s t ir r e r in the growth tube (Figure 6). The purpose o f s tirrin g o f the m elt was to increase the c r it ic a l trapping v e lo c ity . However, th is method o f s tir r in g made operation q u ite d if f ic u lt . I t was im possible to keep the distance between the s t ir r e r and the freezing in te rfa c e close and constant. I f the distance was la rg e , s tir r in g would not be expected to influence V . I f the s tir r e r and the in te rfa c e contacted each oth er, e ith e r the s t ir r e r o r the growth tube was damaged. A d iffe re n t method o f s tir r in g was needed. The method o f h o rizo n ta l zone-refining w ith ro ta tio n was thus adopted to study s tir r in g e ffe c t on re je c tio n o f p a rtic le s . * 44-gauge copper-constantan thermocouple w ith 0.15 m m welded ju n c tio n diameter. 44 1. Apparatus The method o f h o riz o n ta l z o n e -re fin in g w ith r o ta tio n was developed in 1966 by P fa n n , M ille r and Hunt [4 2 ]. In th is te ch n iq u e a h o riz o n ta l tube is ro ta te d about th e tube a x is . I f p a r tic le s are p re s e n t in the m olte n zone, th e y move w ith th e m e lt. The p a r tic le s a re suspended in the zone and a re n o t always in c o n ta c t w ith th e fre e z in g in te r fa c e . A Lepel flo a t in g zone r e f in e r (model FLZ-100) was m o d ifie d to s u it th e p re s e n t in v e s tig a tio n . The m olten zone was heated w ith a lo o p o f nichrom e w ire (3 m m w id th ) which was passed around th e r o ta tin g tu b e , as shown in F igu re 8. Two h e a te rs , w hich co u ld be c o n tro lle d in d e p e n d e n tly , were needed to c o n tro l the shape o f the fre e z in g in te r fa c e . One was used to h e a t the zone; th e o th e r to c o n tro l th e h e a t flo w from th e in te r fa c e in to the grow ing c r y s t a l. The e le c t r ic a l c i r c u i t f o r th e h e a tin g system was s im ila r to th e one used in th e B ridgm an-S tockbarger a p p a ra tu s, exce p t th a t a step-dow n tra n s fo rm e r (5 .2 v o lts c .p . - 24A) was placed between th e c o n s ta n t v o lta g e tra n s fo rm e r and th e a u to -tra n s fo rm e r. 2. Sample P re p a ra tio n Two size s o f Pyrex tube (13 and 22 m m O .D .) w ith 30 cm le n g th were used to s tu d y the e f f e c t o f tube d ia m e te r on p a r t ic le pushing w ith s t ir r in g . As shown in F ig u re 9 , a m ix tu re o f u ltr a s o n ic a lly d is p e rs e d p a r tic le s and o rg a n ic m e lt was poured in to a tube w ith a s lid a b le T e flo n p lu g about 3 on above th e sealed bottom . O rg a n ic m e lt w ith o u t p a r tic le s was then poured on top o f th is m ix tu re w hich Figure 8 A naphthalene molten zone containing carbon particles traveling in the right-hand direction with a concave freezing interface (Only left-hand heater used( the white material appearing in the melting zone is part of the heater support). Figure 9 46 / , In situ zone-refined * organic compound \ Mixture of particles fand organic ^Teflon plug Preparation of the solid charge for horizontal zone-refining with rotation. 47 had s o lid ifie d . The purpose of the slid a b le plug was to prevent tube breakage [42] due to increases in volume upon m elting. Because the in it ia l so lid charge was porous, a bubble or void appeared at the top o f the molten zone during horizontal zone- refining with ro ta tio n . As the zone advanced, the porous s o lid was melted and a nonporous so lid formed behind the zone. The void thus gradually increased in size and eventually occupied more than h a lf of the zone at the end o f the run. The void adversely affected the p a rtic le separation process when i t occupied more than one-third o f the zone volume. Prelim inary in -s itu zone-melting of the p a rtic le -fre e portion of the so lid charge (see Figure 9) was therefore used p rio r to the particle-pushing experiments in order to reduce and control the size o f the void. 3. Experimental Procedure For a typica l experiment, a charge tube prepared as above was placed h o rizo n ta lly, rotated about its own a xis, and moved through a stationary heater. The p a rticle s were suspended in the molten zone and moved w ith 1t a t travel rates below Vc> The crysta l growth rate equalled the movement rate under steady-state conditions. The growth rates would have some flu ctu a tio n due to d ra fts in the laboratory. The rates were increased step by step (about 3 mm/hour every 20 minutes) with p a rtic le trapping again observed both by microscopic observations and by sectioning of the s o lid ifie d m aterial. The errors in the measured V c were less than 5% . 48 C. M a te ria ls 1 • P a rtic le s P ro p e rtie s o f the p a r tic le s employed f o r these experim ents are summarized in Table 2. Carbon p a r tic le s (bone b la c k ) were e x te n s iv e ly used as fo re ig n p a r tic le s in t h is study because they appear to be one o f the common im p u ritie s in o rg a n ic compounds. Scanning e le c tro n m icrographs o f these p a r tic le s are shown in F igu re 10. S pecial p re p a ra tio n s were needed to ta ke these p ic tu re s . The carbon sample O was placed on a piece o f brass which was coated w ith a 300-500 A g o ld f ilm . I t appears th a t the carbon p a r tic le s were a c tu a lly agglom erates o f very sm all p a r tic le s . The agglom erates were porous and o f ir r e g u la r shape. The in d iv id u a l p a r tic le s were n o t s p h e ric a l and t h e ir surfa ces were not smooth. These agglom erates were d ir e c t ly added in to o rg a n ic m e lts f o r e a rly B ridgm an-S tockbarger run s. L a te r, th e carbon was d is persed in to the m e lt using u ltr a s o n ic v ib r a tio n s , w hich broke the agglom erates in to in d iv id u a l p a r tic le s and removed most o f the gases entrapped in the carbon. Copper and red iro n o x id e were a ls o s tu d ie d e x te n s iv e ly in th e naphthalene m e lt fo r experim ents o f h o riz o n ta l z o n e -re fin in g w ith r o ta tio n . O ther types o f p a r tic le s such as s ilv e r , z in c and s ilic o n were a lso used fo r some p re lim in a ry s tu d ie s . The s iz e d is tr ib u tio n s o f u ltr a s o n ic a lly dispersed carbon, iro n o x id e and copper p a r tic le s were determ ined by a Mi H i pore t t M C p a r tic le measurement computer system [4 5 ]. F igures 11 and 12 show th e d is t r ib u tio n s based on a count. The average diam ete rs were 1 .0 , 1.7 and TABLE 2 PR O PER TIES O F T H E PARTICLES Materi al Size Range v im Shape Density^ ^ g/cc Thermal' 1 ' Conductivity cal/cm sec °C In Melt of Naphthalene c(2) <13 Irregular 1.8 - 2.1 (amorphous) 0.057 Fines suspended C u <80 Irregular & Spherical 9.0 0.94 A ll particles s e ttl ed Fe2°3 (red) < 3 Spherical 5.2 Fi nes suspended A g <74 Spherical 10.5 1.0 - Zn 7.1 0.27 - Si <150 2.3 0.2 _ Source Matheson Coleman & Bell Alcan, MD-301 P fizer, R2200 Alcan, MD-201-S M allinckrodt, A R Alcan, MD-101 (1) (2) Data from Reference 43. Density from 1.03 to 1.55 and thermal conductivity from 0.004 to 0.012 fo r carbon with a porosity o f 48 to 23% . Data from Reference 44. VO (B) Carbon particle enlarged from (A) at arrow, 6000X 10 Mtn Figure 10 Micrographs of carbon particles. ©— Carbon * — Iron Oxide 0.8 0. 6 0.4. 40 60 80 30 95 96 R Figure 11 Size distribution for carbon and iron oxide particles. PARTICLE SIZE Copper £ 4 0 1 0 — % GREATER Figure 12 Size distribution for copper particles. 26.0 urn fo r iro n oxide, carbon and copper, respectively. 53 2. Organic Compounds The organic compounds studied as m atrix m aterials were naphthalene, camphor, salol and benzophenone. Their properties are summarized in Table 3. Prelim inary re su lts indicated th a t only naphthalene could re a d ily push these p a rtic le s and thus i t was used fo r most o f the experiments. I n it ia lly , an attempt was made to zone refine naphthalene and camphor, but iro n ic a lly the very small native p a rticle s present in the reagents were d if f ic u lt to remove thereby. Therefore, these chemicals were used d ire c tly in experiments w ithout fu rth e r p u rific a tio n . Som e o f these native p a rtic le s collected from naphthalene in s o lid ific a tio n run BNA-11B are shown in Figure 13. Fortunately, such p a rtic le s d iffe re d s u ffic ie n tly from those added that no d iff ic u lt y was encountered in measuring V o f added p a rtic le s . Details o f zone-refining camphor and naphthalene are discussed in the next se ctio n , along w ith gas chromatographic analyses o f these com pounds and th e ir refined products. No im p u ritie s were detected in Baker reagent grade naphthalene by conventional gas chromatography. D. S o lid ifie d Products In addition to d ire c t observation during the s o lid ific a tio n experiment, the s o lid ifie d products in th e ir glass containers were examined m icroscopically (up to 500 X) both through the glass tube and a fte r sectioning to determine c r itic a l v e lo c itie s and p a rtic le TA B LE 3 O R G A N IC CO M PO UNDS^ Melting Poi nt Material °C Naphthalene 80.2 Cam phor Salol 179.8 42.0 Benzophenone 48.1 Density g/cc Melt 0.98 Solid 1.25 1.15 Thermal Conductivity cal/cm sec °C Melt Solid 1.03 0.00032 0.00068 0.99 Viscosity {at M.P.), ______ 0.97 5 . 0 ^ (4) 8.5 5.7 (4) Source Baker analyzed reagent Baker, Baker grade Mallinckrodt N.F. Matheson Coleman & Bell (D (2) (3) (4) Data from REference 46 and 47. Data from Reference 48. Estimated by the present author. Obtained from Figure 38. - • 'a g f c - t - ■ .VMV«r‘ JpfrJi, £ >*S ' J f f? Figure 13 Native particles collected from naphthalene (Baker analyzed reagent)* 5 0X* 56 tra p p in g c h a r a c te r is tic s . Samples were c u t e ith e r tr a v e rs e ly o r lo n g itu d in a lly by means o f a r o ta tin g blad e w hich 1s mounted on a M lc ro -M a tic P re c is io n W atering Machine (Mlcromech M fg. C o rp .). Some tim es th e naphthalene s lic e s were d is s o lv e d 1n acetone and the s o lu t io n f i l t e r e d through a 1 pm M lllip o r e f i l t e r . The f i l t e r p lu s c o lle c te d p a r tic le s was placed between two g la s s s lid e s and examined under th e m icroscope. E. Z o n e -R e fin in g o f O rganic Compounds A F is h e r zone r e f in e r was used to p u r ify o rg a n ic compounds f o r th e p a rtic le -p u s h 1 n g e xp e rim e n ts. Camphor and naphthalene were zone- re fin e d u sin g a 2 cm O.D. g la ss tube w ith two h e a te rs t r a v e llin g upward a t low speeds (2 .5 to 10 m m /hour). * 1. Camphor A ir o x id a tio n o f camphor o c c u rre d d u rin g th e i n i t i a l m e ltin g and tu b e f i l l i n g and d u rin g the f i r s t few zone passes. The brown- c o lo re d o x id a tio n p ro d u cts were r e a d ily removed as z o n e -re fin in g c o n tin u e d . F ill in g th e tube w ith argon d id n o t c o m p le te ly pre ve n t th e d is c o lo r a tio n , because some a i r was entrapped in th e camphor. When th e m e lt was u n c o lo re d , the fre e z in g in te r fa c e was p la n a r and smooth i f th e grow th ra te was le s s than 10 mm/hour. Above th is ra te th e in te r fa c e became d e n d r itic . C e lls were observed on the In te rfa c e when th e m e lt was c o lo re d and th e grow th r a te exceeded 2 .5 mm/hour. F oreign p a r tic le s o r ig in a lly p re se n t in th e rea ge nt were v e ry d i f f i c u l t to remove. Most o f these p a r tic le s were a t f i r s t 57 pushed by the planar in te rfa ce a t 9.7 mm/hour, but were then subse quently incorporated in to the crystal a fte r less than 10 m m o f pushing. The p a rtic le separation was not g re a tly Improved when the average zoning ra te was decreased to 2.5 mm/hour, which indicates th a t there was a flu c tu a tin g freezing rate due to d ra fts in the room, power flu c tu a tio n s , and the in te rm itte n t motion of the heater. Analyses by gas chromatography indicated th a t the Baker-grade DL-camphor contained about fiv e d iffe re n t trace im p u ritie s. Three of these im p u ritie s were removed by zone-refining a t 2.5 mm/hour (RC-7). W e id e n tifie d one o f the remaining im puritie s as camphene, the con tent of which was lower in the refined camphor. The concentration of one other im purity was higher in the product than in the s ta rtin g m aterial. The chromatograms o f Baker, refined and colored camphor are shown in Figure 14. The colored oxidation products did not show up in the chromatograms. However, they coagulated as brownish floes in the CSg so lu tio n used fo r chromatography, a fte r the so lu tio n had stood fo r several weeks. W hen reheated in a ir , the zone-refined camphor became yellow w hile no change occurred i f oxygen was excluded, proving th a t a ir oxida tion is the o rig in o f the colored m ate rial. 2. Naphthalene This compound is much more stable than camphor when heated in a ir. At a zone speed o f 4 mm/hour (RN-6) , the native p a rtic le s present in Baker-analyzed reagent were d if f ic u lt to remove. The freezing in te rfa ce was smooth and planar. O ccasionally, surface- 5 8 (A) DL-Camphor (Baker grade» 97*3 mg/100 cc CS2). (B) Zone-refined portion (105*^ mg/100 cc CS2)• (C) Colored portion ( 1 0 mg/100 cc CS2), Figure I** Chromatograms of camphor (Aerograph HY-F1, Model 600, column $ ft x 1/8 in, 5% SE-30 60/80 a/w DMCS Chrome W, each injection **.5 J&). driven flows caused p a rtic le s to vigorously c irc u la te around bubbles. Gas chromatographic analyses revealed no im p u ritie s in the naphthalene (Figure 15), even w ith the use o f a boosting technique and high temperature (360°C) in je c tio n . Thus, the Im purity content must be less than 0. 1% , which is the minimum detectable value fo r th is gas chromatograph. The zone-refined (bottom) and s lig h tly yellow (top) portions (RN-9) gave chromatograms s im ila r to the reagent, w ith no im purity peak. W hen 0.5 wt % o f anthracene, a lik e ly im p u rity, was added to the naphthalene, the peak showed up as shown in Figure 15(B). 60 (A) Naphthalene (Baker analysed reagent, high concentration in CS2 )« (B) Anthracene (0*5 wt %) added to naphthalene (High concentration in CS2 K Figure 15 Chromatograms of naphthalene (Aerograph HY-F1• Model 600D, same column as used in camphor). C H APTER IV RESULTS A N D DISCUSSIONS A systematic experimental study was o rig in a lly planned to determine the e ffe c t o f two important parameters— in te rfa c ia l tempera ture gradient and s tir r in g —on the c r itic a l trapping v e lo c ity . Many prelim inary experiments were conducted to fin d a su ita b le p a rtic le - organic m atrix p a ir. The ideal p a ir should have a measurable c r itic a l v e lo c ity , non-faceted s o lid -liq u id in te rfa ce and easy handling fo r sample preparations. The p a ir o f carbon and naphthalene was found to f u l f i l l most o f the requirements. Many phenomena, which help in the understanding o f p a rtic le pushing, were discovered during the experiments and la te r treatment o f the s o lid ifie d products. Experiments were designed not only to pro vide q u a n tita tiv e data on the e ffe c t o f temperature gradient and s tirrin g , but also to a ssist in the discovery o f new and im portant phenomena through d ire c t microscopic observation. The experimental re su lts obtained from the v e rtic a l Bridgman growth method are discussed f ir s t . Those obtained from horizontal zone-refining w ith ro ta tio n fo llo w . A summary o f re su lts is given a t the end o f the discussion fo r each experimental method. A. Bridqman-Stockbarger Technique 1. General Consider in e r t p a rtic le s more dense than the organic m elt added 6 1 62 In to th e upper p o rtio n o f the grow th tube (see F ig u re 6 ). The p a r tic le s are acted upon by va rio u s fo rc e s (such as g r a v ity , viscous drag and bouyancy) and s e ttle 1n th e m e lt. Most o f them a r r iv e on th e h o riz o n ta l so11d-H qu1d In te rfa c e , and a new s te a d y -s ta te I n t e r fa c e in c o n ta c t w ith th e p a rtic le s 1s form ed. Very f in e p a r tic le s a re suspended 1n the m e lt due to Brownian m otion and g e n tle co n ve ctive c u rre n ts which e x is t 1n the m e lt about 1 cm above th e fre e z in g in t e r fa c e . As s o lid if ic a t io n s ta r ts a t slow speeds, the p a r tic le s cove rin g th e In te rfa c e are swept along as th e fre e z in g In te rfa c e advances. The moving In te rfa c e a lso con tacts th e f in e p a r tic le s suspended 1n th e m e lt and pushes them. As the growth ra te 1s g ra d u a lly In cre a se d , a speed 1s reached a t which some p a r tic le s cease to be pushed by th e In te rfa c e and are trapped 1n the grow ing c r y s ta l. W ith many d iff e r e n t s iz e s and shapes o f p a r tic le s , the observed tr a n s itio n from pushing to tra p p in g takes place over a wide range o f v e lo c itie s . The s ig n ific a n t experim ents perform ed in the v e r tic a l Bridgman method are ta b u la te d 1n Table 4 ( s a lo l) and Table 5 (n a p h th a le n e ), to g e th e r w ith th e observed re s u lts . 2. S o lid -L iq u id In te rfa c e Morphology The so l i d - 11q u id in te rfa c e was m aintaine d by h e a tin g the upper p o rtio n o f th e growth tube and c o o lin g th e low er p o rtio n o f the tube. The shape o f th is In te rfa c e was s ig n ific a n tly a ffe c te d by power in p u t to th e h e a te r and by c o o la n t tem perature. Three kin d s o f in te rfa c e shapes were observed 1n the v e r tic a l Bridgman grow th method, as shown TABLE 4 S U M M A R Y O F VERTICAL B R ID G M A N EXPER IM E N TS F O R C A R B O N -S A LO L S YSTEM ^1* Run S tirrin g R P M C irculator Temp. X Heater Input, w Change of V, mm/hr. browm Length, m m Observed Results BSA-1 no -20 5 3 to 8 33 1. Vc = 7 mm/hr. fo r the observed grain surface. 2. At high temperature gradient, the in te r face was planar and smooth when V was less than 6 mm/hr., but i t became faceted when V increased to 8 mm/hr. 3. Many bubbles grew on the interface and produced tube "worms" in the crystal. BSA-2A 160 -20 5 6 1 1 1. Particles were pushed. 2. Particles were suspended BSA-3A 160 -20 3 6 20 1. The interface moved downward in to the cooler and could not be observed. 2. The shape of interface was concave. Increase o f s tirrin g did not change the shape. BSA-4A 160 -20 5 • — 1. A brass cooler was used, which improved heat transfer and moved the interface upward in to the heater. a * CO 2. Agitation made interface more faceted. TABLE 4 (continued) S U M M A R Y O F VERTICAL B R ID G M AN EXPERIM ENTS F O R CARBON-SALOL SYSTEM^ R un Sti rring R P M Circulator Tem p. °C Heater Input, w Change of V, mm/hr. uruwui Length, m m Observed Results BSA -5A BSA-6A 160 -10 5 6 1. Motion picture w as taken showing that particles were moving upward around the bulged interface at beginning of stirrin g . The interface becam e con cave when i t reached steady-state. 2, At the end of run, the melt w as de canted. The concave interface w as faceted and consisted of m any large grains. BSA -7A 160 -10 6 6 18 1. Particles were added w hen the growth unit reached steady-state. Most of the particles were trapped immediately. BSA -8A 160 -10 4 6 18 1. Small growth tube (13 m m O.D.) w as used to give a higher temperature gradient and a smoother interface. 2. Most particles were trapped immediately after addition. B SA-9A 160 -10 2 6 49 1. Three additions of particles were m ade. Each addition w as trapped immediately and produced a black band in the crystal. TABLE A (continued) S U M M A R Y O F VERTICAL B R ID G M A N EXPER IM E N TS F O R C A R B O N -S A LO L SYSTEM^ Run Sti r r i ng R P M C irculator Temp. °C Heater Input, w Change of V, mm/hr. uruwui Length, m m Observed Results BSA-1 O A 160 -10 2-3 8-15 7.5 1. The interface moved away from the s tirrin g blade when V was increased. The distance between the interface and the blade should be controlled. BSA-12A 160 -10 2.5 6-12 37 1. Carbon was dispersed in to the melt by an ultrasonic cleaner. The melt was dark and the solid was also dark. BSA-13-1 N o -10 2.5 6 29 1. D ifferent particles were used fo r the follow ing runs. 2. R ed iron oxide particles were added. They were trapped and the solid was lig h t red. BSA-13-2 N o -10 2.5 8-12 40 1. Zn dust was used, and particles were continuously trapped BSA-14 N O -10 2.5 6-12 58 1. Si powder was used and trapped. 2. The structure of s o lid ifie d salol appeared d iffe re n t from those of previous runs. ^ ^Large growth tube (26 m m I.D .) was used fo r runs BSA-1 to 7A. Small tube (10.5 m il I.D .) fo r runs BSA-8A to 14. Each tube size had its own heater. Glass cooler was used fo r runs BSA-1 to 3A, but brass cooler was used fo r the remainder o f the runs. TABLE 5 S U M M A R Y O F V E R TIC A L B R ID G M A N E X P E R IM E N T S F O R C A R B O N -N A P H T H A L E N E SYSTEM^ R u n Stirring R ? M Change of V, mm/hr. Growth Length, m m Observed Results B N A-1A 160 6-13 34 1. O ne grain pushed particles, while another grain trapped particles continuously. 2. Sometimes trapping w as due to a bubble floating aw ay causing momentary rapid freezing. B N A -3A 200 8-16 50 1. The distance (about 1 m m ) between the interface and the stirrin g blade w as controlled by a support jack. 2. V c for one grain w as 15 mm/hr,, while the other grain pushed particles until the end of the run. B N A-4 N o 8-24 80 1. M any bubbles grew on the interface w hen V exceeded 16 mm/hr. 2, V c for one grain w as 2 1 m/hr. B N A-5 N o 13-38 85 1. V c next to the wall w as 18 mm/hr. 2. Vc for one grain w as 2 1 mm/hr. 3. V c for another grain w as 26 mm/hr. TABLE 5 (continued) S U M M A R Y O F VERTICAL B R ID G M A N EXPE R IM E N TS F O R C A R B O N -N A P H T H A LE N E SYSTEM *1) Run S tirrin g R P M Change o f V, mm/hr. Growth Length, m m Observed Results BNA-6V N o 15-30 60 1. The growth tube was evacuated at 0.3 Torr. Bubbles formed at 25 mm/hr. 2. V c = 18 mm/hr. next to the w all. 3. V c = 21 mm/hr. fo r one grain. 4. V c = 25 mm/hr, fo r another grain. BNA-7V N o 13-35 60 1. S am e as BNA-6V, but had a larger vacuum space in the sealed tube. Bubbles grew at 27 mm/hr. 2. V c = 17 mm/hr. next to the w all. 3. One grain trapped more particles than the other grain. Vc varied from 19 to 35 mm/hr. BNA-8A 160 18-39 62 1. The distance between the interface and the s tirrin g blade was controlled at less than 1 m m . 2. S om e particles trapped at 30 mm/hr; most trapped at 39 mm/hr. 3. S om e bubbles formed on the interface. * Hhe circ u la to r temperature was -10°C. The heater input was 6 watts. The heater (14 m m I.D .) was placed on the top of the brass cooler. The growth tube was 13 m m O.D. and 14 to 23 cm long. Carbon particles were dispersed into the melt by an ultrasonic cleaner, except fo r run BNA-1A. 6 8 in F ig u re 16. A t a g ive n grow th r a te , the fre e z in g in te rfa c e moved upward in to th e h e a tin g zone when th e c o o la n t tem perature was lowered and became convex (F ig u re 16A). On th e o th e r hand, when th e heater power was in cre a se d , th e fre e z in g in te rfa c e moved downward in to the c o o lin g zone and became concave (F ig u re 16C). A p la n a r in te rfa c e c o u ld be o b ta in e d by a d ju s tin g c o n d itio n s so th a t th e in te rfa c e was in the v ic in it y o f the ju n c tio n o f th e h e a te r and c o o le r (F ig u re 16B). The shape o f th e in te rfa c e in flu e n c e s the c r it ic a l tra p p in g v e lo c ity . A convex in te rfa c e caused most o f the p a rtic le s to f a l l in t o the groove cre a te d between th e in te r fa c e and g la s s w a ll. The p a r tic le s a t th e g la s s - s o lid - liq u id in te r fa c e were trapped sooner tha n those in the c e n te r o f th e tub e. The form er gave a Vc o f 18 mm/ h o u r, and the la t t e r a value o f 21 mm/hour f o r u ltr a s o n ic a lly d is persed carbon in naphthalene. F ig u re 17 shows carbon trapped between th e c ry s ta l and the tube w a ll. The fre e z in g in te rfa c e o f naphthalene was always smooth under th e imposed tem perature g ra d ie n ts o f th is s tu d y , even though naphtha le n e has a la rg e e n tro p y o f fu s io n (AS^ = 53.4 J/m ole °K ). The in te r fa c e m orphology o f s a lo l was in flu e n c e d by the growth r a te in a d d itio n to th e tem perature g ra d ie n t. A t h ig h tem perature g ra d ie n ts th e in te rfa c e became smooth when th e growth ra te was less th a n 6 mm/ h o u r, b u t i t was fa ce te d when th e grow th ra te increased above 8 mm/ h o u r. 3. Tem perature G radients D ire c t a d d itio n o f carbon in to th e m e lt o f naphthalene was used \ (A) Convex l l (B) (C) Plane Concave Figure 16 Shapes of the freezing interface in the Bridgman-Stockharger experiments. \o Figure 17 Carbon particles trapped next to the tube wall during Bridgman growth of naphthalene (13 mm OD tube* BN-5, Vc* 12 mm/hr). 71 to study the e ffe c t o f temperature gradient on the c r itic a l v e lo c ity . With varied gradients, the c r itic a l v e lo c itie s were a ll about 12 mm/hour as shown in Table 6 . Higher gradients were obtained by m eans of decreasing the temperature of the cooling flu id and increasing the heater power in p u t. The measured in te rfa c ia l gradients are only approximate. At the end of th is prelim inary study, Cisse and B o lling [9 ] presented s im ila r re su lts based on p a rti cle-water systems. Their results also indicated that the c r itic a l ve lo city does not depend s ig n ific a n tly on the in te rfa c ia l temperature gradient. Because of the foregoing evidence and s im ila r conclusions based on the organic system, we did not pursue th is subject fu rth e r. Theoretical studies o f the factors influencing the in te rfa c ia l temperature gradient were also made during th is research. A paper was published based on part o f th is study and is attached as Appendix A. The Influence of the length and radius o f a crysta l on the in te rfa c ia l temperature gradient was estimated by use o f a one dimensional heat tra n sfe r calculatio n. I t was found th a t the tempera ture gradient continues to change fo r quite long crystals when the Biot number H = hR/k is small (where h is the heat tra n sfe r coef fic ie n t from the crystal surface, R is crystal radius, and k is the thermal co n d u ctivity), especially fo r good conductors, but does not depend on crysta l length s ig n ific a n tly when the Biot number is large. For our Bridgman growth o f naphthalene, H was about 1 and the c r itic a l length was then equal to R. Therefore, a length o f tube equal to its 72 TABLE 6 EFFECT OF TEMPERATURE GRADIENT ON CRITICAL VELOCITY (N aphthalene-C arbon S y s te m ) ^ Run No. BN-5 BN-3 BN-7 BN-8 Bath Tem perature, °C -20 0 10 20 H eater In p u t, w a tts 6.8 5 .0 4 .5 4 .5 Tem perature G ra d ie n t, °C/cm<2 ) ~ 2 7 ~ 6 C r it ic a l V e lo c ity , mm/hr. -1 2 -1 2 - 1 2 -1 2 ^ C a r b o n added d ir e c t ly in to the m e lt. (21 ' 'Measured in te r f a c ia l tem p eratu re g ra d ie n t in th e m e lt a t the in te r fa c e . 73 diameter was submerged in the cooler at the beginning o f each experi ment, so that constant temperature gradients were maintained a t the freezing in te rfa ce during con trolled s o lid ific a tio n . 4. S ti r r i ng I t was hoped th a t s tir r in g o f the m elt would increase the c r itic a l v e lo c ity . S tirrin g should enhance the supply o f m elt to the contact region between the p a rtic le s and the in te rfa c e , move p a rtic le s along the in te rfa ce g ivin g them less time to be trapped, and keep many more of the p a rtic le s in suspension and away from the in te rfa ce . Because o f its very low vapor pressure, s a lo l was o rig in a lly used fo r the s tir r in g experiments with carbon. The experiment w ithout s tir r in g gave V c = 7 mm/hour. However, the runs w ith a g ita tio n did not give s a tis fa c to ry re s u lts . W hen carbon was added d ire c tly in to the melt (BSA-9A), most of the carbon was immediately incorporated in to the growing sa lo l and made a dark band as shown in Figure 18 (three additions were made in th is run). W hen the carbon was dispersed in to the melt u ltra s o n ic a lly (BSA-12A), the in te rfa c e trapped p a rtic le s quite uniform ly, re s u ltin g in a nearly uniform ly dark s o lid . Other kinds o f p a rtic le s , FegOg (BSA-13-1), Si (BSA-14) and Zn (BSA-13-2), were also used which were reported to be e a s ily separated from salol by Uhlmann e t a l. [6 ]. However, we did not obtain the expected results even w ithout s tir r in g (see Table 4). Therefore, the naphthalene-carbon system was used henceforth in the study o f s tir r in g , due to the ready separation. Inse rtin g a Figure 1 8 C arbon particles trapped immediately after each addition during Bridgm an grovth of salol with stirring (BSA-9A). Three additions m a d e at V* 6 m m /hr. • > a •P - 75 s tir r e r in the growth tube made operation q u ite d i f f ic u lt . I t was impossible to keep the distance between the s t ir r e r and the in te rfa c e constant. I f the distance was la rg e , s tir r in g would not be expected to influence V c . I f the s t ir r e r and the In te rfa ce contacted each other, e ith e r the growth tube o r the s tir r in g rod were damaged. Although we used a stepless jack to a d ju st the p o s itio n o f the s tir r e r fre q u e n tly during changes in the growth ra te , th is distance was very d i f f ic u lt to c o n tro l and made the e ffe c t o f s tir r in g uncertain. However, one s tir r in g run (BNA-8A) gave V = 30 mm/hour, which was q u ite encouraging. Another phenomenon was observed as fo llo w s . When s tir r in g was in itia te d , the shape and p o s itio n o f a steady sa lo l in te rfa c e changed d ra s tic a lly . The planar In te rfa c e became convex in to the s o lid due to higher heat tra n s fe r rates through the glass w a ll. P a rtic le s moved up the bulge and c irc u la te d in to the m elt {motion p ictu re s were taken of th is aspect). A fte r the in te rfa c e reached ste a d y-sta te , i t was concave in to the s o lid making d ire c t m icroscopic observation o f p a rtic le pushing d i f f ic u lt . The m elt was decanted (BSA-6A, 7A) and the in te rfa c e was observed thereby to con sist o f many la rge fa c e ts , as shown in Figure 19. Some grains protruded up to 1.5 m m . The deepest p o in t in the concave in te rfa c e was about 9 m m from the periphery. 5. M a trix -P a rtic le Dependence Previous data [ 6] in d ica te d th a t there were v a ria tio n s in Vc 76 Figure 19 The concave interface of growing salol containing many large facets (30 mm OD tube* BSA-7A). 77 am ong the p a rtic le m aterials in a given m atrix m ate rial, and among the m atrix m aterials fo r a given p a rtic le . W e used a d iffe re n t, improved experimental method and observed that carbon p a rtic le s were readily separated from naphthalene, but were very d if f ic u lt to remove from camphor (V less than 2 mm/hour). A summary of resu lts is lis te d w in Table 7. 6. Gas Bubbles and Surface-Driven Flows The s o lu b ility of gases in solid s is normally much lower than in the corresponding melts. Segregation takes place at the freezing interface and increases the gas concentration next to the in te rfa ce . A s a re s u lt, gas bubbles often nucleate and grow on the in te rfa ce during s o lid ific a tio n [49]. Because temperature varies in the m elt, the surface tension varies along the g a s-liq u id in terface and can produce vigorous c irc u la tio n . In the present experiments, bubbles gave ris e to vigorous flows which circu la te d the p a rticle s around the bubbles. Figure 20 shows vigorous c irc u la tio n o f carbon p a rtic le s around a bubble growing on the in te rfa ce o f naphthalene (frame from motion p ic tu re ). These gas bubbles had a large influence on the separation of foreign p a rtic le s from the m atrix m aterials. Gas "worms" were often created in the s o lid behind the bubbles on the in te rfa ce , as shown in Figure 21. These bubbles sometimes flo a te d away. Sometimes a bubble moved ra p id ly back in to its gas worm o r in to a sudden crack in the solid and deposited a large number o f p a rtic le s in the s o lid . In such 78 CRITICAL O rganic M a trix Naphthalene S a lo l Camphor TABLE 7 TRAPPING VELOCITIES IN VERTICAL BRIDGMAN CRYSTAL G RO W TH P a r tic le Cu Vc mm/hour 21 16 12 18 14 Remarks U ltr a s o n ic a lly d is p e rs e d . T rapping next to glass w a ll. Agglom erate p a r tic le s . T rapping ne xt to glass w a ll. Fe2°3 Zn Si C no t pushed a t 6 n o t pushed a t 6 no t pushed a t 6 not pushed a t 2 79 Figure 20 Surface-driven flows circulating particles around a gas bubble during freezing of naphthalene. 8 0 Figure 21 A large gas worm and a crack across the gas worm in solidified naphthalene (10X, BNA-4). 81 cases i t was very d if f ic u lt to know whether the p a rtic le incorpora tio n was due to a true c r itic a l ve lo city or to the Influence of bubbles. S om e in te resting observations, not related to p a rtic le pushing, were also made on bubble nucleation and growth. In the carbon- naphthalene system, bubbles occurred at a change of crystal growth rate, or when the growth rate exceeded 16 mm/hour (BNA-4). W hen the growth tube was evacuated at 0.3 T orr, bubbles did not appear u n itl V = 26 mm/hour (BNA-6V, 7V). However, no bubbles formed when copper foreign p a rticle s were employed in naphthalene with a 0.3 Torr evacuation even at V * 32 mm/hour (BNA-9V Cu). Hence, nuclea tion o f bubbles depended on the type o f p a rticle s present and the amount o f gas in the matrix m aterial. S tirrin g the melt reduced bubble formation (BNA-1A, 2A, 3A), but did not elim inate the problem (BNA-8A). Bubbles also formed on the freezing in te rfa ce of salol under some conditions. A theoretical treatment o f bubble nucleation during s o lid ific a tion is presented in Chapter VI. 7. Grain Surfaces The s o lid ifie d m aterial in th is study normally consisted of 2 to 4 grains fo r naphthalene and about 12 grains fo r salol when using a 13 m m O.D. glass tube. Several single grains were separated from one another and examined under a microscope. In s a lo l, grain boundaries contained many steps, c a v itie s , gas worms and p re fe re n tia lly 82 tra p p e d p a r t ic le s , as shown in F ig u re 22. However, th e s u rfa c e s o f naphthalene g ra in s p r im a r ily e x h ib ite d steps w ith o u t p r e fe r e n tia l tra p p in g o f p a r tic le s (F ig u re 2 3 ). The l a t t e r appears to c o n firm th e re s u lts o f C isse and B o llin g [ 9 ] , in which th e y found p a r tic le s on g ra in boundary grooves a re pushed to h ig h e r v e lo c itie s than th o se a t f l a t g ra in s u rfa c e s o f an ic e w a te r in te r fa c e . 8 . Growth O rie n ta tio n E xam ination o f th e s o lid if ie d p ro d u cts o f many runs in d ic a te d th a t one sid e o f the in te r fa c e began tra p p in g p a r tic le s much sooner than th e o th e r s id e , as shown in F ig u re 24A. This le a d s one to b e lie v e th a t c e r ta in g r a in o r ie n ta tio n s may push p a r tic le s , w h ile o th e r g ra in o r ie n ta tio n s in c o rp o ra te p a r tic le s a t th e same grow th ra te . The tra n s v e rs e c u t o f th e s o lid if ie d In g o t (BNA-8A) showed th a t one g ra in trap pe d p a r t ic le s , w h ile th e o th e r g ra in s re je c te d p a r tic le s (see F ig u re 24B). In c id e n ta lly , th e r e je c tio n o f s o lu b le im p u ritie s was a ls o observed to depend on gro w th o r ie n ta tio n . In z o n e -re fin e d camphor one g ra in was l i g h t y e llo w , w h ile the a d ja c e n t g ra in was c o lo rle s s . 9. S urfaces o f S o lid if ie d P rod ucts L ik e th e s u rfa c e s o f the g r a in b o u n d a rie s , th e c y lin d r ic a l s u rfa c e s where th e s o lid c o n ta cte d th e g la s s tu b e were n o t always smooth. F re q u e n tly , th e y were rough and co n ta in e d s te p s , bumps, t in y c ry s ta ls (F ig u re 2 5 ), and gas bubbles and worms (F ig u re 2 6 ). As (A) Displaying many bumps and waves (100X) (B) Containing small gas worms (or cavities) and carbon particles (50X). Figure 22 Surface of salol grains (BSA-10A). Figure 23 Surface of naphthalene grains containing many steps (50X, BNA-8A). 85 Vertical Bridgman growth of naphthalene. Upper side trapped carbon earlier than lower side (The arrow at the right hand indicating the beginning of the run( BNA-oV)• Transverse section of solidified naphthalene (BNA-8A). One grain trapped carbon (dark grain with a gas worm) while the other grain did not (light grain). Figure 24 Effect of growth orientation on particle pushing. (A) Containing many steps and waves, 100X (B) Showing humps and tiny crystalsv 50X Figure 25 Surfaces of solidified naphthalene at tube wall. 87 Figure 26 Gas bubbles trapped between the solidified naphthalene and tube wall, 32X. 88 mentioned before, carbon p a rtic le s were trapped between these surfaces and the glass w a ll a t a lower c r it ic a l v e lo c ity than a t the advancing g ra in surfaces, e s p e c ia lly w ith convex in te rfa c e s . This in dica te s th a t the re s u lts o f Uhlmann, e t a l. [ 6] obtained w ith th in layers o f m elt held between microscope s lid e s m ight have been s tro n g ly in flu enced by the glass s lid e s and m ight not have been c h a ra c te ris tic o f a fre e in te rfa c e . 10. Cracking and Healing Because the c o e ffic ie n t o f thermal expansion d iffe r s fo r a s o lid and it s growth co n ta in e r, thermal stresses form during the period o f cooling a fte r s o lid ific a tio n . Sometimes thermal stresses were s u ffic ie n t to cause cracking o f s o lid ifie d m ate rial both in the h o rizo n ta l and v e rtic a l d ire c tio n s . The cracks did not reach the freezing in te rfa c e . Therefore, p a rtic le s were not p re fe re n tia lly trapped a t these lo c a tio n s . A v e rtic a l crack was sometimes d i f f i c u l t to d is tin g u is h from a g ra in boundary. O ccasionally a h o rizo n ta l crack healed to form a v e il o f m icroscopic gas bubbles, as shown in Figure 27. S im ila r observations were made in movement o f in clu sio n s 1n KI c ry s ta ls [5 0 ]. 11. Bouncing P a rtic le s An In te re s tin g phenomenon was observed during zon e-re fin in g o f camphor. Some o f the small fo re ig n p a rtic le s o r ig in a lly present in the reagent bounced back in to the m elt as soon as they touched the 89 Figure Unhealed Veil of microscopic gas bubbles in healed portion portion 27 Crack (black area) and partially healed crack (a veil of microscopic gas bubbles) in solidified naphthalene (50X* NBA-4), 90 fre e z in g in te r fa c e . O thers bounced in to th e m e lt fo llo w in g a ra p id approach to th e In te rfa c e w ith o u t a c tu a lly to u c h in g i t . T h is 1s s u g g e s tiv e o f e le c t r o s t a tic e ffe c ts o r o f ir r e g u la r c o n v e c tio n c u rre n ts . F u rth e r stu d y is w a rra n te d . 12. Summary In a d d itio n to p re v io u s ly e s ta b lis h e d param eters such as p a r t ic le s iz e and shape, p a r t ic le roughness, and g r a in b o u n d a rie s, we have fou nd th a t b u b b le s, in te r fa c e shape, grow th c o n ta in e r and th e grow th o r ie n ta tio n a l l s ig n if ic a n t ly in flu e n c e th e c r i t i c a l v e lo c ity in the carbon>naphthalene system . Because o f th is complex dependence, i t was never observed th a t a ll p a r tic le s were c le a r ly in c o rp o ra te d a t one lo c a tio n in the s o lid if ie d p ro d u c ts . In f a c t , th e p a r tic le s were trap ped o v e r a f i n i t e d is ta n c e . The measured v a lu e s o f Vc v a rie d from 21 to 35 mm/hour (BNA-7V) w ith o u t s t ir r in g th e m e lt. The grow th o f bubbles a t th e fre e z in g in te r fa c e was e v e n tu a lly p re ven ted by eva cu a tin g th e grow th c o n ta in e r. The fre e z in g In te r fa c e had to be ke p t s li g h t l y concave in to the grow ing s o lid to p re v e n t p a r tic le s from being trapped between th e c r y s ta l s u rfa c e and th e c o n ta in e r. The In t e r f a c ia l tem p eratu re g ra d ie n t d id n o t a f fe c t th e c r i t i c a l tra p p in g v e lo c ity w ith in th e range in v e s tig a te d . The Vc f o r tra p p in g o f carbon by s o lid if y in g s a lo l was 7 m m /hour, b u t p a r tic le s o f Zn, S i and were n o t r e a d ily pushed. These r e s u lts a re c o n tra ry to d a ta by o th e rs [ 6] in w hich a h o riz o n ta l th in f ilm h e ld between two g la s s s lid e s w ith a v e r t ic a l fre e z in g In te rfa c e was 9 1 employed. S tirrin g might have increased V , but a new method to perform s tir r in g was needed to replace the unsuccessful technique o f in se rtin g a s tir r e r . B. Horizontal Zone-Refining with Rotation 1. Interface Shape Tube ro ta tio n averaged out ra d ia l asymmetries. The shapes of the freezing and m elting in te rface s in a horizontal ro ta tin g zone depended on the growth ra te . The va ria tio n s o f in te rfa ce shapes are shown in Figure 2 8 . Planar in te rface s were obtained only a t zero or very slow growth rates. W hen the growth rate was increased, the la te n t heat lib e rate d at the freezing in te rfa c e had to be conducted through the cry s ta l. Since cooling took place only a t the ro ta tin g tube w a ll, the freezing in te rfa ce gradually became more and more con cave. Meanwhile, the m elting in te rfa ce moved closer to the heater (heat power kept constant) and became increasingly convex, likew ise because o f la te n t heat consumption. At high growth rates ( >90 m m / hour), in s u ffic ie n t time was a va ilab le to completely m elt through the feed and the m elting in te rfa ce began contacting the freezing in te rfa c e , as illu s tra te d in Figure 28D. Because o f th is , the c r itic a l v e lo c itie s o f ce rta in p a rticle -o rg a n ic systems w ith high V were d if f ic u lt to determine even when heating power was continuously increased during the experiment (fo r example, carbon-naphthalene in a 19 m m I.D . growth tube, H R L - 5 Cu). As the freezing rate o f naphthalene was increased, small 92 Resistance { heater (A) V = 0 (B) V = 3 0 m m /hr (C ) V = 5 0 mrn/hr (D ) V > 9 0 m m/hr Figure 28 Change of zone shape with travel rate in horizontal zone-refining of naphthalene with rotation (50 RPM, 19 mm I.D. tube). 93 amounts of carbon p a rticle s gradually accumulated on the central portion of the concave interface and adhered there. These p a rticle s were then trapped in to the growing crystal when the growth rate reached V . The length of crystal over which p a rticle s were entrapped depended on the growth rate a fte r exceeding V . The trapping length was shorter as the growth rate was increased. Periodic trapping was observed over a f in it e length when the growth ra te was kept at Vc . In other words, accumulation of p a rticle s on the concave interface and trapping o f p a rticle s in to the crystal took place a lte rn a tiv e ly at V u n til a ll p a rticle s in the ro ta tin g zone were incorporated. Figure 29 shows tha t carbon p a rticle s were pushed from the le f t end to the rig h t end of a naphthalene ingot by the ro ta tin g zone-refining method. R ed iro n oxide p a rticle s gave trapping characte ristics s im ila r to carbon. Both were entrapped along the axis of growing naphthalene. However, the pushing and trapping phenomena were d iffe re n t fo r copper p a rticle s. At Vc the p a rticle s did not accumulate a t the central part of the in te rfa ce , but were incorporated in to the crystal rather uniform ly. The e ffe c t of interface shape on p a rtic le pushing was also studied. W hen the freezing interface was kept planar by means of a second heater, the results were s lig h tly d iffe re n t from those obtained fo r a concave interface. Carbon p a rticle s were trapped uniform ly in to the growing naphthalene and V was increased from 30 to 35 mm/hour I# (HRT-6 , 7 and 8). A planar interface was not easily obtained because Figure 29 Horizontal zone-refined naphthalene with carbon particles pushed from the left to the right end <VC*30 mm/hr and 5 RPM in a 13 mm OD tube)• 9 5 each heater required a d iffe re n t heat input dependent on growth rate and ro ta tio n . 2. Rotation A minimum speed o f ro ta tio n was f i r s t established in order to keep most o f the p a rticle s suspended 1n the ro ta tin g zone fo r each pa rticle -organic m atrix p a ir. The ro ta tio n was then Increased to determine the e ffe c t of ro ta tio n speed on V c. Tables 8 , 9 and 10 are summaries of results fo r carbon, copper and red iro n oxide in a naphthalene m atrix. Two kinds of V are shown 1n these tables. One is the fo r c c p a rtic le trapping in to the s o lid , the other is fo r trapping between the tube wall and the s o lid . Copper p a rticle s gave a strong dependence of Vc on ro ta tio n speed. Increase o f ro ta tio n increased the c r itic a l trapping rate of copper in to the s o lid . O n the other hand, increase of ro ta tio n increased the cen trifug al force causing some of the larger p a rticle s to s tic k to the tube w a ll. These p a rticle s were then engulfed next to the tube w a ll. Figures 30, 3 1 and 32 indicate the variations o f V c in the two positions with rotation speed fo r copper, iro n oxide and carbon. For example, in the copper- naphthalene system V in the crystal increased with ro ta tio n , w hile V c next to the wall decreased with ro ta tio n . The intersection o f these two curves gives an optimum ro ta tio n speed under which a maximum Vc at the crystal is reached with no trapping next to the w all. The optimum ro ta tio n speeds and corresponding Vc are tabulated 1n Table 11. 96 TABLE 8 EFFECT OF ROTATION RATE O N CRITICAL FREEZING RATE V FOR TRAPPING OF CARBON IN NAPHTHALENE c USING HORIZONTAL ZONE REFINING Tube Size R o ta tio n C r it ic a l V e lo c ity , .ran/hour Run m m I.D . R P M C ry s ta l Wall HR- 6 10.5 1.5 29 30 HR-8 10.5 1.4 28 30 HR-2 10.5 4 34 no HR-5 10.5 4 30 no HR-7 10.5 16 29 30 HR-3 10.5 33 32 32 HR-4 10.5 33 30 31 HR-15M 10.5 45 33 28 HR-16M 10.5 45 34 29 HR-23 10.5 80 30 20 HRL-1 19 10 24 28 HRL-2 19 30 36 36 HRL-3 19 60 37 trapped 97 TABLE 9 EFFECT O F ROTATION RATE O N THE CRITICAL FREEZING R ATE Vr F O R TRAPPING O F C O PPER IN NAPHTHALENE c USING HORIZONTAL ZO N E REFINING R otation Cri g -1 - M ° c i ty.,. m / h o u r Run m m I.D . R P M Crystal W al HR-11 10.5 44 47 no HR-12 10.5 44 5 1 no HR-13 10.5 44 48 no HR-19 10.5 50 66 80 HR-18 10.5 58 70 68 HR-17 10.5 70 77 65 HR-14 10.5 100 82 61 HRL-5 19 44 < 90 53 H R L-8 19 50 < 90 90 98 TABLE 10 EFFECT OF ROTATION ON Vc FOR IRON OXIDE-NAPHTHALENE IN HORIZONTAL ZONE REFINING Tube S ize R o ta tio n W P S M l x mm/.hour Run m m I.D . R P M C ry s ta l Wal 1 HR-21 X 10.5 8 28 33 HR-20X 10.5 40 35 34 HR-22X 10.5 80 36 trapped HRL-7X 19 8 32 no HRL-6X 19 40 46 20 mm/hr 99 Center Wall o > 50 40 50 60 70 80 90 100 Rotation Speed, RPM Figure 30 The influence of tube rotation on incorporation of copper particles by naphthalene during horizontal zone melting (10 mm I.D. tube). m m/hr 10 0 4 0 Center 30 ^ Wall 20 o O 10 20 3 0 4 0 50 60 70 80 Rotation Speed, RPM Figure 31 The influence of tube rotation on incorporation of iron oxide particles by naphthalene during horizontal zone melting (10 mm I.D# tube)# mm/hr 101 40 Center Wall 20 o > 0 10 20 30 40 50 60 70 80 Rotation Speed, RPM Figure 32 The influence of tube rotation on incorporation of carbon particles by naphthalene during horizontal zone melting (10 mm I*D* tube)* 102 TABLE 11 OPTIMUM ROTATION RATES FOR SEPARATION OF PARTICLES FRO M NAPHTHALENE BY A HORIZONTAL ZONE REFINER WITH ROTATION P a rtic le s Carbon Copper Iro n O xide Tube S ize I. D. 10.5 19 10.5 19 10.5 19 Optimum R o ta tio n _______R P M ________ 33 30 55 50 40 ~ 25 V mm/hour 30 36 70 < 90 35 - 3 8 103 3. Tube Diameter Tables 8 , 9, 10 and 11 also give the experimental re su lts obtained fo r a la rg e r growth tube (19 m m I.D ., 22 m m O .D .). By increasing the tube diameter by a fa c to r o f two, V was increased V about 10% fo r iro n oxide, 20% fo r carbon and more than 30% fo r copper under the respective optimum ro ta tio n ra te s. Furthermore, the optimum ro ta tio n rates fo r the la rge tube were reduced a t le a s t 10% from th a t o f the small tube. 4. Bubbles and Void The dissolved gases in the organic compound are continuously rejected in to the m elt during s o lid ific a tio n . These gases grew in to bubbles a t the advancing in te rfa c e in the v e rtic a l Bridgman experiments. In h o rizo ntal zone re fin in g w ith ro ta tio n , however, the gases were transported in to the m elt s u ffic ie n tly to avoid nucleatlon and growth a t the in te rfa c e . These gases were lib e ra te d in a gas space o r void located a t the top o f the ro ta tin g zone. The void helped in observation o f p a rtic le s a t the in te rfa c e . Carbon and Iro n oxide p a rtic le s always adhered a t the center o f the in te rfa c e before they were incorporated in to the c ry s ta l. Sometimes, adhesion of carbon p a rtic le s on the in te rfa c e gave a s p ira l p a tte rn . Bubbles appeared a t the in te rfa c e only a t slow ro ta tio n rates and high growth ra te during separation o f carbon from naphthalene. They did not give d if f ic u lt ie s in determ ination o f Vc because they began occurring a t about 30 mm/hour. Bubbles were not encountered a t 104 th e in te r fa c e in any o th e r h o riz o n ta l zone m e ltin g exp erim en ts. 5. Summary H o riz o n ta l zone r e fin in g w ith r o ta tio n achieved the goal o f in c re a s in g th e c r i t i c a l tra p p in g v e lo c ity o f p a r tic le s . S u ff ic ie n t d a ta were gath e re d to p e rm it d e sig n o f a s e p a ra tio n process and to make la b o ra to ry -s c a le p ro d u c tio n o f p a r t ic le - fr e e o rg a n ic s . T h is new s e p a ra tio n process is sim p le and e ffe c tiv e and g iv e s s ig n ific a n t improvements and advantages over th e v e r tic a l Bridgman method. The advantages are lis t e d as fo llo w s : (1 ) V is increased about 50% f o r carbon p a r tic le s and a t le a s t 300% f o r copper in n a p h th a le n e ; (2 ) c le a r c u t s e p a ra tio n is p o s s ib le ; (3 ) tra p p in g between th e c ry s ta l and the tube w a ll can be p re ve n te d ; (4 ) bubbles do no t grow on th e fre e z in g in te r fa c e , so th a t th e grow th tube does n o t have to be evacuated; and (5 ) o p e ra tio n is s im p le , e ffe c tiv e and econom ical. The economics o f th is s e p a ra tio n process w i l l be discussed in the n e x t c h a p te r. An optimum r o ta tio n speed was determ ined f o r each p a r ti c le - naphthalene system , above w hich a c o n s id e ra b le amount o f p a r tic le s were trap pe d n e x t to th e g la s s w a ll and below w hich tra p p in g was p re fe rre d in th e c r y s ta l. F u rth e r d is c u s s io n on h o riz o n ta l r o ta tio n is g ive n in C hapter V I. C HAPTER V POTENTIAL APPLICATIONS A N D E C O N O M IC S In addition to in ve stig a tio n of p a rtic le pushing and trapping by a freezing in te rfa c e , the fin a l goals o f th is work were to make p a rtic le -fre e organic chamicals, especially those which are s o lid at room temperature. In a d d itio n , several po te n tia l applications were discovered during the course o f th is work. Som e of them were fu rth e r studied experim entally. The re s u lts and economics of the developed process are given as follow s. A. Separation of Mixed P a rticle s The re su lts shown in Chapter IV and previous data [ 6] have demonstrated th a t each kind o f p a rtic le has a d iffe re n t freezing rate V in a given m atrix m aterial. In other words, a m ixture o f d iffe re n t kinds of p a rtic le s may be separated in a properly selected organic compound i f an increasing growth rate is employed during controlled s o lid ific a tio n . The re su lts o f horizontal zone re fin in g w ith ro ta tio n gave Vc = 30 mm/hour fo r carbon and V c = 70 mm/hour at 55 R P M fo r copper in naphthalene. Because o f th e ir wide difference in V , i t should be possible to separate a m ixture o f carbon and copper from one another by freezing o f naphthalene. Run HR-15M accomplished th is goal. W hen the freezing rate was progressively increased from 25 to 75 mm/hour at 44 R P M in a period of 4 hours, carbon p a rtic le s started to be trapped a t 34 mm/hour. A ll o f the carbon was incorporated 105 106 In to th e c e n tra l p o rtio n o f th e c ry s ta l o v e r a le n g th o f 2 cm. Copper p a r tic le s were then trapped u n ifo rm ly a t 54 mm/hour, u n t il th e end o f the run (75 mm/hour). F ig u re 33 in d ic a te s th a t carbon p a r tic le s were c le a r ly separated from copper in s o lid if ie d naphthalene. The v e r tic a l Bridgman grow th method was a ls o used to separate m ix tu re s o f p a r tic le s . Carbon and copper p a r tic le s were u ltr a s o n lc a lly d isp e rse d In to m olten naphthalene (BNM-3). The growth tube (1 0 .5 m m I . D . ) was evacuated and sealed a t 0.05 T o rr. The growth ra te s were Incre ase d from 10 to 40 mm/hour. The s o lid if ie d c ry s ta l was about 7 cm long a t the end o f a 3-hour ru n . Most o f th e carbon was trapped in the upper p o rtio n o f th e c r y s t a l, w h ile copper was in c o rp o ra te d in th e m id d le p o rtio n . A lthough the d iffe re n c e 1n t h e ir in d iv id u a l Vc was s lig h t (about 21 mm/hour f o r carbon and 18 mm/hour f o r copper) in n a p h th a le n e, they were s t i l l separated from each o th e r by programmed s o lid if ic a t io n . A m ix tu re o f red iro n o x id e and carbon was no t separated a p p re c ia b ly in naphthalene (BNM-1). However, the grow th tub e was n o t evacuated and a grow th o f bubbles on th e fre e z in g in t e r fa c e a d ve rse ly a ffe c te d th e s e p a ra tio n . Both grow th methods have shown e x p e rim e n ta lly th a t a m ixtu re o f p a r tic le s can be separated from one a n o th e r by programmed s o lid if ic a t io n o f a suspension in an a p p ro p ria te liq u id . W e c a ll o u r in v e n tio n " p a r t ic le chrom atography". T h is new se p a ra tio n technique may be used to id e n t if y p a r tic le s from a m ix tu re o f va rio u s kinds o f p a r t ic le s , d e rive d from drugs and p a r tic u la te p o llu ta n ts , as examples. 107 Figure 33 Carbon and copper particles were separated during programmed solidifi cation of naphthalene in horizontal zone-refining with rotation (HR-16M). 108 B. S ize C la s s ific a tio n Screen a n a ly s is is a sim p le and a c c u ra te method o f fr a c tio n a tin g m a te ria ls in to v a rio u s s iz e s . I t is w id e ly used to determ in e the s iz e d is tr ib u tio n s and to y ie ld fra c tio n a te d p ro d u cts which may be s tu d ie d s e p a ra te ly from th e w hole m a te ria l. However, th e method is a p p lic a b le o n ly to about 44 pm o r 325 mesh. Below th is s iz e th e re are se ve ra l methods to analyze th e d is t r ib u t io n o f s iz e s , b u t i t appears th a t o n ly th e method o f e lu t r ia t io n may be employed to o b ta in fra c tio n a te d p ro d u cts in th e s iz e range o f 1 to 50 ym [5 1 ]. The b a s is o f th e e lu t r ia t io n method is th a t an upward v e lo c ity o f f l u i d su p p o rts o n ly p a r tic le s s m a lle r than a g ive n s iz e . The s m a lle r p a r tic le s are c a rrie d upward, w h ile la rg e p a r tic le s s e t t le in the stream . The c o l le c te d p a r tic le s may be f u r t h e r e lu tr ia te d a t d if f e r e n t v e lo c itie s to make f in e r s iz e fr a c tio n s and thus i t is d i f f i c u l t to s t a r t w ith sm all amounts o f p a r t ic le s . Here we have'a method to make s iz e fr a c tio n s f o r a sm all q u a n tity o f p a r tic le s . P re vio u s s tu d ie s [ 5 , 6 ,7 ,9 ] in d ic a te d th e dependence o f the c r i t i c a l v e lo c ity on p a r t ic le s iz e . L a rg e r p a r tic le s have lo w e r V , s m a lle r p a r tic le s have h ig h e r V . T h e re fo re , p a r tic le s w ith V # V v a rio u s s iz e s co u ld be sie ve d in a m a trix m a te ria l by g ra d u a lly in cre a sin g th e fre e z in g ra te s d u rin g c o n tro lle d s o lid if ic a t io n . A m ix tu re o f s ilv e r powder and naphthalene (BNSC-3) was s o lid if ie d by the Bridgman grow th method. The fre e z in g ra te s were in cre a se d from 6 to 26 mm/hour. The s o lid if ie d c r y s ta l (5 .2 cm) was tra n s v e rs e ly s lic e d in to fo u r equal s e c tio n s . The s ilv e r p a r tic le s in each s e c tio n 109 were collected on a M illip o re f i l t e r , and were examined under a microscope. The size d is trib u tio n s varied from the top to bottom sections. The micrographs of the top and bottom sections are shown in Figure 34. Small p a rticle s were dominant in the top section, but some large p a rticle s were also present in th is section, which might have been pushed by grain boundary grooves and grain boundary tr ip le points [9 ]. The bottom section contained many large p a rtic le s as well as small p a rtic le s . These small ones were probably incorporated in the grooves between the crysta l and glass tube where V£ was found to be su b sta n tia lly reduced 1n th is study. The method could probably be improved by growing a single crystal with a s lig h tly concave freezing in te rfa ce so th a t the e ffe cts of grain boundary and grooves next to the wall can be elim inated. C. Economics The c r itic a l trapping v e lo c ity w ill have a major influence on the cost o f making p a rtic le -fre e organic compounds since p a rtic le s can only be separated from the m atrix m aterial at freezing rates below V . Because the value o f V is d iffe re n t fo r each p a rticle -o rg a n ic system w p a ir, the cost w ill be strongly dependent on the system being p u rifie d . An organic compound usually contains several d iffe re n t kinds of insoluble p a rtic le s , the Id e n titie s of which may or may not be known. This makes estim ating the cost of producing p a rtic le -fre e ultrapure chemicals very d if f ic u lt unless one knows (a) the kinds of p a rticle s and th e ir values of Vc , and (b) the methods o f determining (A) Larger particles dominated in the bottom sectionv 200X. * « (B) Smaller particles dominated in the top section* 200X. Figure 3^ Size classification of spherical Ag particles (less than 5 0 >um) by vertical Bridgman growth of naphthalene• I l l the amount o f p a rticle s 1n organic compounds. Although carbon is a com m on insoluble im purity in s o lid organic chemicals, some other kinds of p a rticle s are also frequently present. To supply the f ir s t Information one has to conduct experiments to determine the minimum value of Vc among the various kinds o f particles present before a p a rtic le -fre e compound can be produced. Prediction o f V c might become possible i f the foreign p a rticle s were id e n tifie d . The second problem of determining the p u rity (in terms of insoluble im purity) may be q u a lita tiv e ly solved e ith e r by an optical microscope or by a f ilt r a t io n method as described 1n Chapter I I I . There is no accurate and convenient method available at the present time fo r q u a n tita tive measurements o f particles in organic chemicals. With these lim ita tio n s , we attempted to make p a rtic le - free naphthalene. The "native p a rticle s" present in naphthalene were d if f ic u lt to remove by a Fisher zone-refiner as mentioned in Chapter I I I , but most of these p a rticle s were separated by means o f a horizontal zone-refiner a t 28 mm/hour w ith 5 R P M tube ro ta tio n . This Vc is close to the one fo r separation o f carbon p a rticle s from naphthalene (see Table 8). With our present knowledge o f p a rtic le pushing and trapping, a crude cost estimate fo r producing p a rtic le -fre e naphthalene 1s made. Because the demand is not ye t s u ffic ie n t to warrant in d u stria l-sca le production, a laboratory-scale production is more re a lis tic . Further more, i t is recommended th a t a small group with one engineer and one technician be created 1n a chemical company which has established the market fo r ultrapure organic chemicals. With these assumptions, the 112 d e ta ile d economic c a lc u la tio n s a re shown in Appendix B. The t o ta l c a p ita l in ve stm e n t is about $68,000 and th e to ta l p ro d u c t c o s t is about $65,000 f o r an annual p ro d u c tio n o f 300 Kg o f p a r t ic le - fr e e u ltra p u re n a p h th a le n e . T h e re fo re , the p ro d u ct c o s t is 22 cen ts per gram, i f a re a g e n t grade chem ical is used as th e feed m a te ria l. T his c o s t w i ll be reduced i f la rg e r s c a le p ro d u c tio n is employed [5 2 ]. I t is in te r e s tin g to compare the p re se n t c o s t e s tim a te w ith th e p ric e o f z o n e -re fin e d o rg a n ic chem icals (rem oval o f s o lu b le im p u r itie s ) . Over 100 z o n e -re fin e d o rg a n ic p ro d u cts are a v a ila b le co m m e rcia lly. A few examples a re given in Table 12 [5 3 ], The p ric e o f zone re fin e d naphthalene is about $1.42 pe r gram in c lu d in g p r o f i t , p u r if ic a tio n c o s t and the m a te ria l c o s t (a b o u t 0 .5 c e n t p e r gram f o r the reagent g ra d e ). One does n o t know the amount o f p r o f i t made by the producers and s u p p lie rs , b u t th e p u r if ic a tio n c o s t appears to be a s u b s ta n tia l amount and is much h ig h e r than th e p re s e n t e s tim a te . Zone m e ltin g o f naphthalene [5 3 -5 5 ] is t y p ic a lly conducted in a v e r tic a l z o n e -re fin e r w hich n o rm a lly re q u ire s s lo w e r grow th ra te s and numerous zone passes in o rd e r to reach u ltr a p u r e p ro d u c ts . The reason is th a t the boundary la y e r n e xt to th e fre e z in g In te rfa c e is much th ic k e r in v e r t ic a l zone m e ltin g than in h o riz o n ta l zone- r e fin in g w ith r o ta tio n . T h e re fo re , the developed new s e p a ra tio n process is econom ical to p ro d u c t p a r t ic le - fr e e u ltr a p u r e o rg a n ic chem icals. 113 TABLE 12 PRICE O F SELECTED ZONE-REFINED ORGANIC C O M P O U N D S FO R PRIMARY STANDARDS [5 3 ] P rice Compound ( fo r 2- 1/2 g r . ) A c e ta n ilid e $ 3.55 Anthracene 4.70 Anthraquinone 3.55 Benzoic Acid 3.55 p-Brom oacetanilide 3.55 p -C hloroa ceta nilide 3.55 p-Fluorobenzoic Acid 12.25 o-Iodobenzoic Acid 4.70 Naphthalene 3.55 S a lic y lic Acid 3.55 Succinic Acid 3.55 S ulfanilam ide 3.55 V a n illin 3.66 CHAPTER VI INTERPRETATION A. Pushing Mechanisms 1. Mass Transport T h e o re tic a lly , the steady-state pushing o f a p a rtic le by an advancing sol id - 11 quid in te rfa c e demands a force to prevent in co r poration o f the p a rtic le In to the growing c ry s ta l, and a supply o f liq u id to the contact area between the p a rtic le and the freezing in te rfa c e . The rep ulsive force must equal the re s u lta n t o f viscous drag [ 6 ,8] , surface energies [ 6] , e le c tro s ta tic in te ra c tio n [ 6] , thermal forces [8 ,9 ], and external forces [ 6,8]. C ry s ta lliz a tio n pressure (Eq. (2 -3 )) is the maximum fo rce th a t can be exerted by a growing c ry s ta l on the p a rtic le . On the other hand, Corren's phase- boundary force (Eq. (2 -4 )) is the minimum force needed to prevent the p a rtic le from adhering to the c ry s ta l. N either o f these two lim its considers the tra n sp o rt o f liq u id which may lim it the rate o f pushing. In fa c t, the supply o f m aterial p ra c tic a lly determines whether a growing cry s ta l traps o r repels a fo re ig n p a rtic le . The viscous drag 1s apparently a major force when a p a rtic le is In contact w ith a ho rizo ntal c ry s ta lliz a tio n fro n t. I f the freezing In te rfa c e is regarded as s ta tio n a ry , a c ry s ta lliz a tio n flow [39] toward the in te rfa c e is generated by the crysta l growth. The drag fo rce then re s u lts from the flow o f flu id to the contact area between 114 115 p a rtic le and in te rfa c e . This force in te ra cts w ith the advancing in te rfa ce through a th in film , and produces a repulsive force on the in te rfa ce . In other words, the crysta l growth rate determines the transport process which, in tu rn , produces a pushing force (Eq. (2-24)). Their re la tio n s h ip is derived in Chapter I I . Equation (2-37) describes V , below which the p a rtic le s are pushed by the freezing in te rfa ce . The c a lcu la tio n o f Vc fo r the present work is shown in the next section. 2. Heat Transfer Theoretical developments fo r V c described in Chapter I I assumed that thermal co n d u ctivitie s o f p a rtic le , m elt and c ry s ta l are the same, and thus th a t there is no e ffe c t of heat tra n s fe r on the in te r face shape behind the p a rtic le . However, the thermal con du ctivity o f the p a rtic le is in general d iffe re n t from those o f m elt and c ry s ta l. For instance, Figure 35 shows th a t a p a rtic le ( a ir bubble) is present in fro n t of an ice-w ater in te rfa ce [5 6 ]. W hen the freezing in te rfa ce approached the a ir bubble fixe d under a small p la te , the in te rfa ce ju s t under i t rose up (Figure 35-1, 2, 3, 4 and 5). The thermal in sula tin g e ffe c t of the a ir bubble (kp < k^) caused the water under the bubble to become colder than other portions o f water at the same le ve l. This caused the in te rfa ce ju s t under the bubble to freeze fa s te r than other parts o f in te rfa c e , so th a t a bump was formed on the advancing in te rfa c e . The a ir bubble escaped as soon as the plate above i t was in c lin e d (Figure 35-6). This suggested th a t a th in film Figure 3 5 T h e presence of a n air bubble in front of an advancing ice-water interface. T h e interface below the bubble rose up (from Reference 56). 116 117 of water existed between the bubble and the ice surface. The schematic diagrams o f isotherms and heat flows are illu s trated in Figure 36 fo r the cases o f kp < k^, kp = k^, and kp > k^. The presence of a p a rtic le on a s o lid -liq u id in te rface with kp d iffe re n t from k^ changes the Isotherms and thus d is to rts the uniform heat flow which would e x is t fo r kp ■ k^ or w ithout the p a rtic le . Because the equilibrium in te rfa ce normally approximates an isotherm, the in te rface shape behind the p a rtic le becomes convex fo r kp < k^ and concave fo r k > k „. Among these three cases, the case w ith P * kp > k^ gives an e ffe c t s im ila r to th a t due to the presence o f a heavier p a rtic le . Both can cause an indentation on the interface below the p a rtic le and reduce the mass transport. Experim entally, copper p a rticle s gave a sm aller V than carbon p a rticle s in the Bridgman growth o f naphthalene. The e ffe c t of p a rtic le conductivity on the in te rfa c ia l tempera ture gradient is not exactly known [8 ,9 ], but the e ffe c t of in te r fa c ia l temperature gradient on the p a rtic le pushing is discussed here. The heat flu x qz across a p a rtic le through a v e rtic a l central lin e can be expressed by ’ z = kp 3 z s kp S = kp t r (6- 1] The temperature difference between the top T^ and bottom T^ o f the p a rtic le is then ‘^ V V r ' z • <6- z > - Isotherms ** Heat flows (A) kp<kjl (B) kp -k^ (C) k p > k g Figure 36 Effect of particle thermal conductivity on the shape of the equilibrium interface (assuming no gravity effect). 1 1 9 The above equation can also be used to explain the in te rfa ce shapes behind the p a rtic le s as shown in Figure 36. At a constant steady- sta te heat flu x , AT is decreased as kp increases. For the case kp > k^, T^ is higher than the temperature in the liq u id a t the same le vel and thus the in te rfa ce m elts back (Figure 36C). The reverse is true fo r the case of kp < k^ and the in te rfa c e rise s up (Figure 36A). W hen the imposed temperature gra dien t in the m elt 1s Increased, I . e . , q2 is increased, AT increases. This means Tt increases w hile T^ remains approximately constant. In order fo r q2 to increase, e ith e r T^ must increase s lig h tly , the in te rfa c ia l temperature must decrease, or the thickness o f the film must decrease. As a re s u lt, the w ell behind the p a rtic le becomes deeper fo r the case o f kp > k^, and the bump becomes sm aller fo r the case o f k < k „. The former is not p l favorable fo r p a rtic le pushing, w h ile the la tte r may not a ffe c t V . This in te rp re ta tio n is based only on the above equation, and real s itu a tio n s may not be so simple. However, published data [9 ] in dicated th a t Vc fo r pushing o f copper p a rtic le s by the ice-w ater in te rfa c e ( k > k „) was decreased from 1.0 to 0.47 um/sec. fo r 65 ym p a rtic le s , P from 1.2 to 0.6 ym/sec. fo r 40 ym p a rtic le s , and from 2.4 to 1.7 ym/ sec. fo r 10 ym p a rtic le s when the temperature gradient in the water was increased from 1 to 10°C/cm. As shown in Table 6 , Vc was not dependent on the temperature g ra d ie n t fo r carbon p a rtic le s pushed by freezing naphthalene (kp < kA). Since the heat ca rried away in to the growing crysta l equals th a t conducted from the m elt plus the la te n t heat lib e ra te d a t the 120 fre e z in g In te rfa c e In a v e r tic a l Bridgman g ro w th , th e heat balance a t the In te rfa c e may be expressed by where the s u b s c rip t s denotes s o lid , l m e lt, and 1 a t the s o lid - liq u id in te rfa c e . Both In te r fa c ia l tem perature g ra d ie n ts change w ith th e ra te o f s o lid if ic a t io n . As the growth ra te in cre a se s, heat generated by VpsAHf g ra d u a lly dominates the con du ctio n in to the s o lid . Thus (dT/dz)_ j Increases w h ile (d T /d z )„ . decreases. Our experim ental S 11 1 o b se rva tio n s In d ic a te d th a t the p o s itio n o f th e fre e z in g In te rfa c e moved downward in o rd e r to d is s ip a te th e in cre a sin g la te n t heat more e a s ily in to the c o o la n t. Because the in te rfa c e moved away from the h e a te r, (d T /d z )^ ^ decreased in a d d itio n . W e d e fin e "ra p id s o lid if ic a t io n " as the s itu a tio n when (d T /d z) 0 . approaches zero o r 9 ■ heat conduction from th e m e lt becomes n e g lig ib le 1n Eq. (6 -3 ). Under t h is c o n d itio n th e in te r f a c ia l g ra d ie n t in th e growing c ry s ta l can be d ir e c tly c a lc u la te d by T ie n 's tem perature d is tr ib u tio n s measured d u rin g the normal fre e z in g o f naphthalene [5 7 ] were re -e v a lu a te d in lig h t o f the above. As shown in Table 13, th e measured (d T /d z)^ ^ approached zero a t the fa s t growth ra te (66 mm/hour) and the measured (dT /dz) . agrees w ith S i I the value c a lc u la te d by Eq. (6 -4 ) fo r two size s o f grow th tubes. Thus, fo r ra p id s o lid if ic a t io n (dT/dz)_ _ • may be d ir e c t ly estim ated 5*1 (6 -3 ) (6 -4 ) 1 2 1 TABLE 13 °C/cm Calculated by Eq. (6- * ) ( dT/dz>s. i 106.2 106.2 ^^Bridgman growth of naphthalene, 86°C hot bath and 0°C cooling bath fo r both cases (Tien [5 7 ]). 12) ' 'G raphically measured from Tien's d is trib u tio n s by present author. INTERFACIAL TE M P E R A TU R E GRADIENTS F O R RAPID SOLIDIFICATION^) Temperature Gradient, ( 2 ) Measured' ' G r M ; ; b e G rLr < 37 66 0 107 19 66 6.3 110 122 from the la te n t heat and does not depend on the c ry s ta l dimensions ( th is is not tru e in slow s o lid if ic a t io n ; see Appendix A ). I f the c r it ic a l tra p p in g v e lo c ity f a ll s w ith in the c r ite r io n o f ra p id s o lid if ic a t io n , V may no t depend upon (dT /dz) ^ because C jC p 1 the la t t e r is a c tu a lly always sm a ll. The la te n t heat o f fu s io n 1s very high f o r naphthalene (35 c a l/g ) and w ater (80 c a l/g ) and thus one more re a d ily reaches the c o n d itio n o f ra p id s o lid if ic a t io n than w ith o th e r tra n s p a re n t m a trix m a te ria ls . Our re s u lts on the carbon- naphthalene system and some o f Cisse and B o llin g 's re s u lts on copper- w ater system in d ic a te d th a t V was not s ig n ific a n tly a ffe c te d by (dT/dz) z 3. H o rizo n ta l R o ta tio n The p a r tic le pushing o r tra p p in g process 1n h o riz o n ta l zone- re fin in g w ith ro ta tio n is d iffe r e n t from th a t in the v e r tic a l Bridgman growth method. The form er pushes p a rtic le s by a v e r tic a l fre e z in g in te rfa c e , the la t t e r pushes p a rtic le s by an upward h o riz o n ta l in te rfa c e . (The in te rfa c e is a c tu a lly s ta tio n a ry . The m otion re fe rre d to is th a t o f the in te rfa c e r e la tiv e to the in g o t.) P a rtic le m otion in the ro ta tin g zone e lim in a te s two Im portant fa c to rs . I t prevents p a rtic le s from continuous c o n ta c t w ith the advancing in te rfa c e , and also e lim in a te s g ra v ita tio n a l e ffe c ts . Both o f these a re encountered in the v e r tic a l Bridgman technique and reduce V . This is p a r tic u la r ly tru e f o r copper p a rtic le s which possess a very high d e n s ity (*'-9 g/cm ). T he re fore , V was increased a t le a s t 300% fo r copper in naphthalene 123 when the horizontal ro ta tio n method was employed. Fluid motion in the rotating zone was not a simple solid-body ro ta tio n ; i t was complicated by the presence of radial and axial temperature gradients, a ir bubbles accumulating on the top, and solid pa rticle s moving in the m elt under the influence o f g ra vita tio n a l and centrifugal forces. In addition to local s tirrin g e ffe c ts , micro scope observation revealed that two large-scale convective currents circulated from the bottom to the top in opposite directions and they met around the center of the melting zone. The flow patterns are sketched in Figure 37. These convective currents were not stable and fluctuated p e rio d ica lly. The strength and duration o f these currents seemed to depend on the temperature gradients and the size o f the molten zone. In addition to the convective currents, two flu id streams moved toward the freezing interface. The f i r s t one was a c ry s ta lliz a tio n flow [2 9 ], equal to V(ps/p £). The second one was an axial flow (or secondary flow ) re s u lt ing from the existence o f density gradients and ce n trifu g al forces [5 8 ]. Owing to fr ic tio n , the m elt next to the s o lid -liq u id interface was carried by i t and then forced to the tube wall by the cen trifugal acceleration. Thus the mass of flu id which was driven outward by cen trifug al forces was replaced by an axial flow , which was proportional to /ftn/p^, where ft is the angular ve lo city. These three flu id streams would have interacted with p a rticle s in contact w ith the interface when the growth ra te approached Vc . As soon as the p a rticle s were in contact w ith the in te rface , Convective current Gas space e a \ Resistance heater Figure 37 Flow patterns in a horizontal rotating zone (side view). 125 some o f the p a rtic le s would remain there through fr ic tio n and adhesion. The c e n trifu g a l force acting on the p a rtic le s was balanced by the shearing stress and viscous drag. A fter the p a rtic le s attached to the rota ting in te rfa c e , the pushing and trapping mechanisms would then be sim ila r to those in the v e rtic a l Bridgman method. In other words, transport is the lim itin g process. Without considering the convective currents, the rad ia l ve lo city u in the boundary layer next to the in te rfa ce is proportional to rft and the thickness of the boundary layer 6 is proportional to /n/fip^. W hen the ro ta tio n speed is Increased, u increases w hile 6 decreases. This, in tu rn , serves to improve the mass transport in the contact area between the p a rtic le and the Interface. Further, th is mass transport is expected to be enhanced when the convective currents are considered which become stronger in a la rg e r diameter tube. This explains why V was increased at higher ro ta tio n speeds (copper and iro n oxide p a rtic le s ) and in a larger diameter tube (23 m m 0 .0 .). B. Calculations o f Vc In order to compare with the experimental re s u lts , calculatio n o f VQ is made here by employing the simple Eq. (2-37) w ithout con sidering effects o f g ra v ity and p a rtic le roughness. Som e o f the constants such as o , n, aQ and < f> (c x ) needed fo r the ca lcu la tio n are not rea dily available from the lite ra tu re . These constants are thus estimated f ir s t . 126 The d ir e c t measurement o f the s p e c ific fre e energy osA o f the s o lid - liq u id in te rfa c e was re c e n tly made f o r the tra n s p a re n t m a te ria ls camphene, s u c c in o n itr ile , w a te r and w h ite phosphorous by Jones and Chadwick [5 9 ]. T h e ir experim ental fig u re s were in good agreement w ith th e values p re d ic te d from T u rn b u ll's th e o re tic a l e q u a tio n , [6 0 ,6 1 ] where AH* is the heat o f fu s io n , N is Avagadro's number, and V is the t m m o le cu la r volume. Because th e c a lc u la te d os£ using th e low er bound o f the co n sta n t 3/8 in th e equation gave b e tte r re s u lts fo r the above tra n s p a re n t m a te ria ls , th is co n stan t is used to e stim a te aS Jt o f naphthalene, s a lo l, camphor and benzophenone. The c a lc u la te d re s u lts are shown in Table 14, along w ith Jones and Chadw ick's data. The p h ysica l p ro p e rtie s o f th e supercooled th in f ilm between the p a r tic le and in te rfa c e are not known. (Data f o r p ro p e rtie s o f supercooled bulk liq u id are even m eager.) The v is c o s itie s o f the supercooled m elts f o r s a lo l [6 2 ] and benzophenone [6 3 ,6 4 ] are shown in F igu re 38. When the m elts are above the m e ltin g tem p eratu re, the p lo ts o f An n versus 1/T g ive s tr a ig h t lin e s fo llo w in g the A rrhe nius r e la tio n . But these lin e s become curved when th e m e lts are super cooled. The v is c o s ity o f the supercooled m e lt is s ig n ific a n tly h ig h e r than the m e lt a t the m e ltin g p o in t. Since the sup erco olings o f fre e z ing o rg a n ics f o r the present work were not measured, the v is c o s itie s a t t h e ir m e ltin g p o in ts are used fo r th e c a lc u la tio n s o f Vc . Experim ental o b se rva tio n s in d ic a te d th a t the su p e rco o lin g s o f (6 -5 ) TABLE 14 ESTIMATION O F V r F O R PARTICLES (1 and 10 ym) 2 °s£* er9^cm V mm/hour Matrix Materials AH^, cal/mole V . cm /mole m Theoretical Experimental 10 u m 1 u m Naphthalene 4500< lj 130 32.5 24.6 767 Salol 46 5 0 ^ 172 28 2.5 82 Camphor 1 1 2 5 ^ 154 7.3 2.7 85 Banzophenone 3 8 6 0 ^ 164 23.8 3.6 114 (41 Camphene' 7 5 5.3 (4l S u ccino nitrile' 7 26 28 W a te r^ 38 41 White Phosphorous^ 7 10.3 ^ ^Obtained from Reference 46. (21 ' Obtained from Reference 25. (31 v Estimated by present author. (41 v 'Data from Reference 59. VISCOSITY, cp 20 1 0 0 90 80 70 60 50 40 30 1 0 0 60 M.P. Benzophenone M.P. Salol 60 40 2 0 Salol / 10 Benzophenone 2.8 3.0 2.6 3.2 3.4 l/ T , °K “ ' Figure 38 Viscosities of supercooled organic melts. 129 naphthalene and camphor were very sm all, but the supercoolings of salol and benzophenone seemed to be q u ite large. The molecular volume V m was obtained by d iv id in g the molecular weight by the density of the liq u id at the m elting p o in t. The molecular diameter a^ was then estimated from the molecular volume. o A p lo t o f <f>(a) against a w ith 8 as a parameter is Illu s tra te d in Figure 39. For a given value o f 8, d>(ot) reaches a maximum value at a < 1/3 (see Eq. (2 -3 9 )). By combining Eq. (2-26) w ith < J> (ot)* the separation distance d between the p a rtic le and in te rfa c e can be c a l culated by the fo llo w in g equation. d/a - « ■ (6- 6) 0 a ( l- c )2 The calculated values o f d/aQ are tabulated in Table 15. The v a ria tio n o f d/aQ is less than a fa c to r o f two when the a is increased from 0.05 to 0.3. Since a represents the curvature of the freezing in te rfa ce which is generated due to the in te ra c tio n between the p a rtic le and in te rfa c e , i t may approach a value o f 0.2 ~ 0 .3 at the onset o f p a rtic le trapping. In th is p ra ctica l range o f a, the values o f d/aQ are almost constant fo r a th in film e x is tin g between the p a rtic le and advancing in te rfa ce . For the present calculations o f V c , 8 = 1 and <|»(a) max = 0.335 w ill be used. Thus fa r the constants needed fo r p re d ictio n o f Vc have been estimated. Calculated re su lts o f Vc are also presented in Table 14 fo r p a rtic le s o f 1 and 10 m diameter. Since the experimental re su lts of Vc (Table 7) were determined fo r the sm allest trapping v e lo c itie s , 0.6 0.3 0.2 0 . 1 O 0.1 0.2 0.3 0.4 05 Figure 39 Plot of vs ol with ^ as a parameter from <f>CeLJ = o U /-«*/(£ -JnolJ. 131 TABLE 15 VARIATION O F d/ao W ITH a A N D 3 _____________ a 3 = 1 3 = 2 3 = 3 0.05 4.0 5.0 6.0 0.10 3.3 4.3 5.3 0.20 2.6 3.6 4.6 0.30 2.2 3.2 4.2 ( o r Vc f o r la r g e r p a r t ic le s ) , they a re , in f a c t , in good agreement w ith the c a lc u la te d valu e s. C. F orm ation o f A ir Bubbles The fre e z in g in te r fa c e can r e je c t d is s o lv e d g a s , e x a c tly in th e same way as any o th e r s o lu te f o r w hich th e s e g re g a tio n c o e f fic ie n t kQ is le s s tha n u n ity . As th e gas c o n c e n tra tio n in cre a se s a t the in te r fa c e , th e m e lt may become s u f f ic ie n t ly s u p e rs a tu ra te d to n u c le a te b u b b le s. The bubbles formed a t the in te r fa c e grow by d iffu s io n o f d is s o lv e d gas from a d ja c e n t s u p e rs a tu ra te d areas o f the m e lt. The bubble n u c le a tio n may be e ith e r homogeneous o r heterogeneous. Thorough t r e a t m ent o f n u c le a tio n th e o ry f o r vapor to liq u id , liq u id to s o lid , and s o lid to s o lid tra n s fo rm a tio n have e x is te d in th e lit e r a t u r e f o r some tim e . O nly re c e n tly has such a th e o ry been pre sen ted f o r the liq u id to vapor tra n s fo rm a tio n [6 5 -6 8 ]. The purpose o f th is s e c tio n 1s to m o d ify th is th e o ry and to make i t a p p lic a b le to n u c le a tio n o f bubbles fro m th e m e lt c o n ta in in g d is s o lv e d gases d u rin g c o n tr o lle d s o l id if ic a tio n . The homogeneous n u c le a tio n ra te o f bubbles developed by H irth and Pound [6 5 ] was estim a te d by m u ltip ly in g th e c o n c e n tra tio n o f it c r it lc a l- s iz e d bubbles n by th e freq u e n cy o f adding m olecu les o j. The e q u ilib riu m c o n c e n tra tio n was c o rre c te d by a n o n -e q u ilib riu m o r Z e ld o v ic h f a c t o r Z because th e process a c tu a lly occurs a t stea dy s ta te r a th e r than e q u ilib r iu m . The n u c le a tio n ra te e q u a tio n is o f th e form 133 * The value of n is estimated from the metastable equilibrium between two phases o f d iffe rin g Gibbs free energy n* = nQ exp (-AG*/kT) (6- 8) where nQ is the to ta l concentration of molecules in the liq u id phase •k and A G is the fre e energy change fo r form ation of a c r itic a l bubble. The Gibbs fre e energy o f formation of a spherical bubble in a supersaturated m elt under a prevailin g hydrostatic pressure (the schematical diagram of bubble formation is shown in Figure 40) is AG° = 4irr3 AG y/3 + A i r r 2 a + 4ur3Ph/3 (6-9) in which A G y is the volume free energy change and o is the surface energy. Since AG y is always negative fo r a supersaturated m elt, Eq. (6-9) e xh ib its a maximum fo r a c ritic a l-s iz e d bubble r* where (3AG°/3r) = 0 and AG° = A G . The re la tio n between AG* and r* is obtained as AG* = 4 no(r*)2/3 (6-10) * where r = -2a/(AGy + P^). The condition fo r mechanical equilibrium of a bubble applied to a c ritic a l-s iz e d bubble may be expressed as P* = P h + % (6-11) r ★ in which P is the to ta l pressure in the c r itic a l bubble. An alte rna - tiv e form of A G is obtained through e lim in atio n of r , by Eq. (6-11) 134 Gas Melt V# Solid-liquid interface with a critical bubble P* Crystal Figure 40 Schematic diagram of bubble formation at a freezing interface. 135 AG* * 16t t c t3/3 (P* - Ph)2 . (6 -12 ) ★ P represents the to ta l pressure in a c r itic a l bubble. I t is the vapor pressure i f the liq u id is a sin g le component. I t is the pressure of dissolved gas i f the vapor pressure o f the melt is n e g lig ib le . I t is the sum of p a rtia l pressure o f dissolved gas p£ and p a rtia l pressure of the melt p* when both the m elt and dissolved a ir have considerable p a rtia l pressures, such as in the present experiments on naphthalene. The la s t s itu a tio n is treated here. Assuming ideal gas behavior, P = P -| " * ■ Pg • (6-13) The usual Kelvin equation [23] deals only w ith a pure liq u id and a nucleus of its vapor. The vapor pressure p* of the solvent in sid e the c r itic a l bubble was related to the properties of a d ilu te solutio n surrounding the bubble by Ward, e t a l. [6 7 ]. The re la tio n is derived here fo r d ilu te ideal so lu tio n . The chemical p o te n tia ls fo r the solvent in the so lu tio n and vapor phases are given as [6 9 ], u] = y;(P h .T) - kTx' (6-14) and Pi = Pq(P* * T) + kT X1 (6_15) where the subscripts 1 and 2 re fe r to the solvent and solute respec tiv e ly . The single prime in dicates the liq u id phase and the double prime indicates the vapor phase. vjq is the chemical p o te n tia l of 136 the pure component and x 1s th e c o n c e n tra tio n in mole fr a c tio n . The chem ical p o te n tia l o f the pure component uQ is a fu n c tio n o f pressure and tem perature. A t co n sta n t tem p eratu re, th e G ibbs- Duhem eq ua tion reduces to ndyQ = VdP . (6-1 6) I f th e pure s o lv e n t 1s assumed to be In co m p re ssib le , the above equa tio n 1s In te g ra te d as M i< ph *T > ■ W>(P~ ’ T> + W p~> • < 6- 17> Equation (6 -1 6 ) is a ls o in te g ra te d 1n the vapor phase and becomes ^ * u " (P .T ) - U o ( P „ .T ) + k T i n £ - ( 6 - 1 8 ) C O where Is th e vapor pre ssure o f the pure s o lv e n t across a f l a t s u rfa c e . S u b s titu tin g Eqs. (6 -1 7) and (6-18) in to Eq. (6 -1 4 ) and (6 -1 5 ), re s p e c tiv e ly , th e chem ical p o te n tia ls are u j - P i(P „.T ) + v 4 < P h -P J - k T x'2 ( 6 - 1 9 ) and ★ u j * ^ ( P ^ . T ) + kT i n £ - + kT An xV . (6-2 0) oo Since U g fP ^ T ) = M gfP ^tT ), the c o n d itio n f o r e q u ilib riu m across the curved in te rfa c e (p j = p^') , using Eqs. (6-1 9) and (6 -2 0 ), is ★ p x ; k T A n -p -i- - v t( P h -P J - k T xi O O The nucleatlon o f bubbles occurs a t the freezing interface where the concentration o f dissolved gases has the highest value c r itic a l bubble can be evaluated from (o r x p by the Henry's law P 2 * where H is an empirical constant. Because the segregation of a dissolved gas by the freezing in te rface is analogous to the segregation o f a so lu te , the segrega tio n c o e ffic ie n t kQ is defined here as the ra tio o f the gas concentra tio n in the s o lid to that in the liq u id a t equilibrium . Using a constant value of kQ in the quasi-steady-state stagnant film approach [7 0 ], the gas concentration at the in te rfa ce during fra ctio n a l s o lid ific a tio n is found to be where is the gas concentration in the bulk m elt, and 6 is the th ick ness o f the boundary layer. The vapor pressure o f dissolved gas at the in te rfa ce is then The importance o f the gas concentration in the bulk melt is (or x£ in mole fra c tio n ). The p a rtia l pressure of gas 1n the C 1 1 (6- 22) (6-23) 1 3 8 o b v io u s , sin ce i t d ir e c t ly a ffe c ts th e gas c o n c e n tra tio n a t the fre e z in g in te rfa c e . The va lu e o f depends on the s o lid if ic a t io n c o n d itio n s . I f the grow th tube c o n ta in in g a s o lid if ie d o rg a n ic is evacuated and sealed as d e scrib e d in Chapter I I I , the maximum number o f a i r m olecules rem aining in th e vacuum space is N q2 = PvV "/kT where Py is th e pressure and V" is the volume in the vacuum space. As soon as the o rg a n ic is m elted b e fo re b e g in n in g s o lid if ic a t io n , th e amount o f a ir in th e vacuum space increases to N2 - p2V "/kT where p2 is th e p a r tia l p re ssure o f a ir . The d iffe re n c e N2 - N q2 r e s u lts from tra n s p o rt o f d is s o lv e d a ir from the m e lt to th e gas space. Assume is the gas c o n c e n tra tio n in th e s o lid ( i t s volume VQ) b e fo re m e ltin g , and is the c o n c e n tra tio n o f the b u lk m e lt w ith a volume V ', the m a te ria l balance on the gases is P2V" T T PvV” H r r ■ cobvo - cbv (6 -2 4 ) S u b s titu tin g p2 = in to th e above e q u a tio n , th e gas c o n c e n tra tio n in the b u lk m e lt is expressed as P V" + C r t|v kT q _ v______ ob o b V* kT + H.V" (6 -2 5 ) Using Eqs. (6 -1 3 ), (6 -2 1 ) and (6 -2 3 ), the a c tiv a tio n fre e energy m f o r n u c le a tio n o f a bubble is 3 * AG = — I 16tto' " A k o + ( 1 -k o ) e x p / 6V ® s\ V 0 < •» / + P ex oo { e t < V p J - x 2 )-' (6 -2 6 ) 139 The growth process of a bubble consists of a molecule (d is solved gas o r solvent) in the surface o f the bubble evaporating in to the c r itic a l bubble. The frequency o f occurrence is equal to the frequency of condensation of a molecule from the bubble, u exp ( < & ) . * L . (6. 27) \ / ( 27rmkT) ' And the Zeldovich correction fa c to r is z=t ■ = I ~T * 4~ > <6'28> 4 iT r kTy where s is the surface area of a molecule at the in te rfa ce , v is * the Debye maximum frequency o f v ib ra tio n o f a liq u id molecule, AG yap is the a ctiva tlo n a l fre e energy fo r evaporation, a is the condensa tio n c o e ffic ie n t, and m is the molecular mass. The fa c to r (Zw) has been estimated fo r pure liq u id s [7 1 ]. H irth , et a l. [ 66] suggested th a t growth step o f dissolved gas was * the favored process due to AG„a_ fo r desorption o f gas being less than vap th a t fo r evaporation o f the m elt. W e assume here th a t the growth frequency is the same fo r the solute and solven t, and the value o f m can be estimated by the re la tio n l/*^n = 1/ * ^ + 1/v fw h e r e m g is the gas molecular mass and m ^ is the solvent molecular mass. An e x p lic it expression fo r the nucleation of gas bubbles 1s obtained by s u b s titu tin g the Eqs. (6-26), (6-27) and (6-28) in to Eq. (6 -7 ), 140 + P«, ex o o p( u f (Ph’ P» ) ‘ x2) x exp -167ra3/3kT + ex l * o + 0 - k o ) e x p A ll param eters in th e above e q u a tio n a re , in p r in c ip le , known o r e xp e rim e n ta l ly m easurable f o r a g iv e n system . The s u rfa c e te n s io n a used in th e above d e r iv a tio n may n o t be a c o n s ta n t b u t w i l l be a ffe c te d by s o lu te a d s o rp tio n a t th e in te r fa c e . H ir th and Pound [6 5 ] p o in te d o u t in t h e ir e b u llit io n (o r b o ilin g ) s tu d ie s th a t th e s iz e o f c r i t i c a l bubble ( o f the o rd e r o f a m icro n in d ia m e te r and c o n ta in ing m illio n s o f m o le cu les) was s u f f ic ie n t ly la rg e th a t m acroscopic thermodynamic q u a n titie s m ig h t be employed w ith some degree o f co n fid e n ce . The above n u c le a tio n e q u a tio n com bining w ith Eq. (6 -2 5 ) in d ic a te s t h a t in c re a s e o f (de crea sin g V' o r in c re a s in g Py ) , o r 6 V/D is fa v o ra b le f o r the fo rm a tio n o f b u b b le s. Our exp erim en ta l r e s u lts q u a lit a t iv e ly agree w ith th e above th e o r e tic a l developm ent. In o th e r w o rd s, e va cua tio n o f the grow th tu b e (d e cre a sin g Py ) , s t ir r in g o r r o ta tin g th e m e lt (re d u cin g 6 ) and slow grow th ra te V hin d e re d b u b b le o c cu rre n ce , w h ile bubbles form ed a t th e end o f th e 141 Bridgman runs due to Increase o f by decreasing V1. Gas bubbles were also found 1n sin g le crysta l growth of p a ra te llu rite TeOg [72] by the p u llin g method. Formation o f bubbles was prevented by the slow p u llin g rate (about 1.5 mm/hour) combined with high ra te of the crystal ro ta tio n (40-50 RPM). This is in agreement w ith our th e o re tica l prediction s. The in te rp re ta tio n s made 1n the referenced paper are unsound. Recently Swanger and RMnes [73] made th e o re tica l calculations fo r homogeneous nucleation of gas bubbles 1n the blood and found tha t homogeneous nucleation is impossible because the in it ia l pressure o f the gas must exceed 1900 ATM. Maeno's experimental observations [56] indicated th a t a ir bubbles did not form on a smooth ice-w ater in te r face unless the in te rfa ce was scratched w ith a glass rod. Further, a ir bubbles could develop between the p a rtic le s and the ice-w ater in te r face, depending on the types and surface conditions o f p a rtic le s . Thus, form ation o f a ir bubbles was p ra c tic a lly the re s u lt o f hetero geneous nucleation. Our re su lts revealed th a t carbon p a rtic le s could act as nucleation s ite s fo r a ir bubble form ation, but copper p a rticle s had no nucleating a b ility fo r bubbles. CHAPTER V II CONCLUSIONS A t le a s t three o b je c tiv e s have been accomplished from the research described in th is d is s e rta tio n . (1) D iscovery o f some im p o rta n t and in te re s tin g phenomena le a d in g to improved understanding o f the fundamental nature o f p a r tic le pushing and tra p p in g a t a fre e z in g In te rfa c e . (2) The development o f a new se p a ra tio n process enabling economical removal o f fo re ig n p a rtic le s from organ ic chem icals. (3 ) The In ve n tio n o f a new method to separate m ixtures o f d iffe r e n t p a rtic le s by s o lid if ic a t io n , opening up a p o te n tia lly new f ie ld — p a r tic le chromatography. The experim ental re s u lts o f the v e r tic a l Bridgman c ry s ta l growth in d ic a te th a t in te rfa c e shape, growth c o n ta in e r, growth o rie n ta tio n , and the presence o f bubbles a ll s ig n ific a n tly in flu e n c e the c r it ic a l v e lo c ity f o r tra p p in g . These v a ria b le s , plus p re v io u s ly e sta b lish e d parameters such as p a r tic le s iz e and shape, p a r tic le roughness, g ra in boundaries, d e n s ity and therm al c o n d u c tiv itie s , make the process o f p a rtic le s pushing and tra p p in g extrem ely com plicated. For example, p a rtic le s were always trapped over a f i n i t e le ngth o f fre e z in g c r y s ta l, i . e . , over a range o f v e lo c itie s (21 to 35 mm/hour). Among the fo u r organic compounds used, o n ly naphthalene can push p a rtic le s re a d ily due to it s low v is c o s ity . The e ffe c ts o f p a r tic le p ro p e rtie s on V£ appeared to be in s ig n ific e n t, although therm al c o n d u c tiv ity , d e n s ity and roughness are p o s s ib le fa c to rs fo r the same 142 143 size p a rtic le s . The measured values o f V agreed w ith those calcu- la te d by Eq. (2-37). The V fo r the carbon-naphthalene system was reduced from 21 to 18 mm/hour by trapping between the glass w all and a convex in te rfa c e . The form ation o f gas bubbles a t the free zing In te rfa c e was hindered by evacuating the growth con tain er, s tir r in g o f the m elt, and use o f a slow growth ra te . These observations are in good agreement w ith the th e o re tic a l treatm ent o f bubble nuclea tio n (Eq. (6-29)) developed here. Carbon p a rtic le s provided nucleation s ite s fo r bubbles, but copper p a rtic le s did not. The in te rfa c ia l temperature gra dien t in the m elt had no observable e ffe c t on Vc> A simple theory based on Eq. (6-2) p re d icts th a t increase o f temperature gra dien t m ight reduce V i f k > k B, and would have no L p X * e ffe c t on p a rtic le pushing i f k^ < k^. For rapid s o lid ific a tio n the g ra dien t in the s o lid can be simply estimated from Eq. (6-4) w ith small and n e g lig ib le heat conduction in the m elt next to the in te r face, regardless o f the c ry s ta l dimension. Horizontal zon e-refining w ith ro ta tio n enhanced mass tra n sfe r and elim inated the detrim ental e ffe c ts re s u ltin g from bubbles and high p a rtic le d e n sity, and thus s ig n ific a n tly increased Vc over th a t obtained w ith the v e rtic a l Bridgman technique. The measured Vc was increased 300% fo r copper (p = 9 g /cc) and 50% fo r carbon (p = 2 g/cc) p a rtic le s in naphthalene. With th is new process i t is possible to make c le a r-c u t separation o f p a rtic le s . The cost estim ate showed th a t product cost is only 22 cents per gram fo r p a rtic le -fre e naphthalene. P a rtic le and flu id motion in the ro ta tin g zone was 144 found to be com plex. Three f lu i d stream s flo w toward the fre e z in g in te r fa c e . They are c r y s t a lliz a tio n flo w , secondary flo w , and con v e c tiv e c u rre n t. The ra d ia l v e lo c ity in the boundary la y e r should have a d ir e c t e ffe c t on tra n s p o rt o f liq u id in th e c o n ta c t re g io n between th e p a r t ic le and in te rfa c e . T h is v e lo c ity is th e o r e tic a lly increa sed by In c re a s in g r o ta tio n speed and c o n v e c tiv e c u rre n t. The p re se n t re s u lts in d ic a te d th a t the Vc was increa sed a t h ig h e r r o ta tio n ra te s f o r copper and Iro n o x id e p a r tic le s and 1n a la rg e r d ia m e te r tub e. P o te n tia l a p p lic a tio n s suggested by th is work were a ls o e x p lo re d . Two a p p lic a tio n s were found to be fe a s ib le . S eparation o f a m ix tu re o f d if f e r e n t p a r tic le s ( p a r tic le chrom atography) and c la s s if ic a tio n o f p a r tic le s were e x p e rim e n ta lly te s te d . I t is hoped th a t the re s u lts o f these s tu d ie s w i l l c o n trib u te to b e tte r unde rstand in g o f p a r t ic le pushing and tra p p in g by grow ing c r y s ta ls , th a t th e new s e p a ra tio n process w i ll be re a liz e d commer c ia lly and th a t the p o te n tia l a p p lic a tio n s w i l l be w id e ly accepted. The suggested fu tu r e work is b r ie f ly o u tlin e d in the n e xt ch a p te r. C H A P TE R V III S U G G E S T E D F U TU R E W O R K Based on the results o f th is d is s e rta tio n , fu rth e r work on p a rtic le pushing and trapping might be continued along the follow ing 1in es. A, Czochralski Crystal Growth In addition to the Bridgman crystal growth and zone m elting, the Czochralski technique is widely used fo r the growth o f single crysta ls. A seed is attached a t the end of a rotating rod. The crysta l grown on the seed is pulled v e rtic a lly from the melt. Since the seed controls the o rien ta tion of growing c ry s ta l, the study of the e ffe c t o f growth o rien ta tio n on V c is possible. This technique does not need a growth container which lim its the ro ta tio n speed in horizontal zone-refining due to p a rticle s being trapped next to the tube wall a t high rota tion rates. Further, sectioning of s o lid ifie d products w ithout containers 1s easier fo r the purpose o f microscopic in vestig a tio n . Like horizontal zone-refining w ith ro ta tio n , the Czochralski method also eliminated s e ttlin g effects which produce continuous contact of p a rticle s with the in te rface during the process of p a rtic le pushing. However, the former pushes p a rticle s by a v e rtic a l interface the la tte r w ill use a horizontal interface to re je ct p a rticle s. The ro ta tio n speed is an important fa cto r in Czochralski crystal growth, 146 and can s ig n if ic a n t ly a f f e c t th e shape o f grow ing in te r fa c e and flo w p a tte rn s in the m e lt [3 ,7 2 ,7 4 ,7 5 ]. A t slow ra te s o f r o ta tio n , n a tu ra l c o n v e c tio n c u rre n ts dom inate in th e m e lt and the in te r fa c e is n o rm a lly convex. The in te r fa c e becomes more concave a t h ig h e r r o ta tio n ra te s owing to an a x ia l flo w (secondary flo w ) r e s u ltin g from c e n tr ifu g a l a c c e le ra tio n . I t would be in te r e s tin g to stu d y how r o ta tio n in flu e n c e s V in th is geom etry. A m ix tu re o f carbon w p a r tic le s and s a lo l appears good m a te ria ls to s t a r t in v e s tig a tin g p a r t ic le s e p a ra tio n by th is p o te n tia l p ro ce ss. B. P a r tic le Chromatography As d e scrib e d 1n C hapter V, h o riz o n ta l z o n e -re fin in g w ith r o ta tio n has proved to be a p ro m isin g te ch n iq u e f o r s e p a ra tio n o f m ixed p a r tic le s . For in s ta n c e , carbon p a r tic le s were r e a d ily separated from a m ix tu re o f carbon and copper p a r tic le s . T h is new te ch n iq u e should be e xp lo re d fu r th e r f o r a m ix tu re c o n ta in in g s e v e ra l d if f e r e n t kinds o f p a r t ic le s , e s p e c ia lly f o r m etal o r sem iconductor m a te ria ls . Since q u a n tita tiv e analyses o f these m a te ria ls have been w e ll e s ta b lis h e d , d e te rm in a tio n o f s e p a ra tio n e ffic ie n c y is p o s s ib le by a n a ly z in g each tra p p e d b a rd . The s e p a ra tio n e ffic ie n c y is expected to depend on the fre e z in g r a te , th e speed o f r o ta tio n and th e m a trix m a te ria l s e le c te d . C. C ry s ta l Growth from S o lu tio n T h is d is s e r ta tio n so f a r has been co n ce n tra te d on p a r t ic le 147 separation fo r crystal growth from the m elt. However, c ry s ta lliz a tio n from solutio n is a very com m on and important separation technique in the inorganic chemical process in du stries. As mentioned in the chapter on "L ite ra tu re Studies", the evidence has shown th a t certain crystals growing from solution have a tendency to re je c t foreign p a rtic le s . N o inform ation is available a t the present time on what kinds o f crystals tend to re je c t foreign p a rticle s more than the other kinds of crysta ls. I t has been known tha t some inorganic sa lts can c ry s ta lliz e in d iffe re n t forms and habits under varied c ry s ta lliz a tio n conditions. Sodium su lfa te c ry s ta lliz e s NagSO^'lOHgO from the aqueous solution by cooling i t below 32°C, w hile anhydrous NagSO^ 1s formed above 32°C by evaporation o f water. I f the solution is mixed with carbon p a rtic le s , the results w ill indicate whether NagSO^lOHgO or anhydrous NagSO^ rejects more p a rtic le s . A s im ila r experiment can be made fo r Na2C03 solution. Sodium carbonate can c ry s ta lliz e in several d iffe re n t forms as Na2C 03*10H 20 , Na2C03*7H20, Na2C 03*H20 , and anhydrous Na2C03. The form incorporating the le ast amount o f pa rticle s should be c ry s ta lliz e d during p u rific a tio n . Further study may be carried out w ith a complex s o lu tio n , such as the Great S a lt Lake brine. Several companies are Interested 1n recovery o f saline minerals as well as magnesium from the brine o f the Great S a lt Lake by the solar evaporation processes [7 6 ]. Fractional c ry s ta lliz a tio n by solar evaporation is under dynamic conditions. Certain s a lts are very d if f ic u lt to c ry s ta lliz e , although they should be formed according to the phase equilibrium . Minerals such as NaCfc, epsom ite (MgSO^-7H20 ) , sc h o e n ite (K2S04 *MgS04 *6H20 ) , k a in lte (KCJt* MgS0^*3H20) and c a r n a l!ite (KC£,-MgCfc2 *6H20) may be c o n s e c u tiv e ly c r y s ta lliz e d in the In d iv id u a l s o la r ponds. However, i f a con s id e ra b le amount o f SO^- is removed from b rin e a t the b e ginn in g o f s o la r e v a p o ra tio n , a d iff e r e n t c r y s t a lliz a tio n path is fo llo w e d . S a lts o f NaC£, MgS0^*7H20 , KC& and c a r n a llit e are dropped o u t 1n the s o la r ponds. These s o la r s a lts a re harvested and are fu r th e r proces sed in the m a n u fa ctu rin g p la n ts to produce KC& (p o ta s h ), K2SO^, Na2S04 , Mg, e tc . However, a c o n s id e ra b le amount o f the In s o lu b le m a te ria ls (such as sands, c la y s , muds, e t c . ) trapped in the s a lts have to be removed by se ve ra l p ro cessing steps in c lu d in g th ic k e n in g , f lo t a t io n , c e n tr ifu g a tio n , and f i l t r a t i o n . These s e p a ra tio n steps in c re a s e th e c a p ita l and o p e ra tin g c o s ts , and thus in flu e n c e the o v e ra ll process economics and p r o f i t a b i l i t y . I f one knows some c ry s ta ls ( li k e ly k a in ite ) tra p more fo re ig n p a r tic le s than o th e rs , these c ry s ta ls should be avoided in process developm ent. An experim en t a l stu d y is needed to id e n t if y these c r y s ta ls . N O M E N C LA TU R E 2 A Area, cm a Molecular diam eter, cm a^ Interatom ic distance, cm o B Weight of load, dyne b Height of a monoatomic la y e r, cm 3 -3 C C oncentration, g/cm o r cm 3 -3 C^ Saturated concentration, g/cm o r cm c A constant in Eq. (2-6) 2 D D iffu sio n c o e ffic ie n t, cm /sec. d Separation distance o r thickness o f a th in film , cm E (r) Distance between a concave in te rfa c e and it s lowest point as a fu n ctio n o f r , cm F Force, dyne Fc Viscous drag a t a plane s in k , dyne F(a) Viscous drag a t a curved s in k , dyne Fq Force needed fo r d iffu s io n , dyne G Gibbs free energy, erg H B io t number (= hR/k) H Henry's constant, dyne cm/g AHf Latent heat o f fu sio n , ca l/g o r erg/mole h Separation distance between the p a rtic le and in te rfa c e at any value o f r , cm 2 h Heat tra n s fe r c o e ffic ie n t, cal/sec. cm °K -3 -1 J Nucleation ra te , cm sec 149 T o ta l c u rv a tu re , cm“ ^ Botlzmann co n sta n t, erg/m olecule °K Thermal c o n d u c tiv ity , c a l/s e c . cm °K E q u ilib riu m segregation c o e ffic ie n t L a te n t heat o f fu s io n , erg/atom Mass o f a m olecule, g/m olecule 3 Slope o f liq u ld u s lin e in a phase diagram ,°C cm /g Number o f molecules (o r a number) Avagadro's number, m olecules/m ole A co n sta n t in Eq. (2 -5 ) _ 3 C oncentration o f c r it ic a l bubbles, cm _3 C oncentration o f m olecules in the liq u id phase, cm 2 P ressure, dyne/cm 2 H y d ro s ta tic pressure a t the fre e z in g in te rfa c e , dyne/cm 2 T o ta l pressure in a c r it ic a l bubble, dyne/cm 2 S aturated pressure, dyne/cm 2 Pressure in the evacuated space, dyne/cm 2 P a rtia l pressure, dyne/cm 2 Heat flu x in the z d ir e c tio n , c a l/s e c . cm Radius o f a p a r tic le o r c r y s ta l, cm Gas c o n s ta n t, e rg /°K mole D istance in a ra d ia l d ir e c tio n , cm E ffe c tiv e co n tact ra d iu s between a p a r tic le and an in t e r fa c e , cm Mean displacem ent, cm 151 s Entropy, erg/°K AS 3 Volume entropy o f fu s io n , erg/°C cm ASf Molar entropy o f fu sio n , erg/°K mole s Surface area of a molecule a t the in te rfa c e , cm T Temperature, °C o r °K Tb Temperature a t the top o f the p a rtic le , °C Tt Temperature a t the bottom o f the p a rtic le , °C * Tc Temperature depression due to curvature e ffe c t, °C ATi Temperature change due to In te ra c tio n o f a p a rtic le and In te rfa ce , °C * T S Supercooling fo r growth o f c ry s ta l, °C U Flow v e lo c ity 1n the contact area, cm/sec. u ‘ Radial v e lo c ity in a boundary la y e r, cm/sec. V 3 Volume, cm V Crystal growth v e lo c ity , cm/sec. Vc C ritic a l trapping v e lo c ity , cm/sec. V m 3 Molar volume, cm /mole V 3 volume, cm /molecule W Work, erg w p A parameter (= r (l-a )/2 R ) X Concentration in mole fra c tio n Z Non-equilibrium or Zeldovlch fa c to r z Distance in a z-axis d ire c tio n , cm Greek Symbols ct A param eter f o r the shape o f In te rfa c e . a Condensation c o e ffic ie n t e A c o n s ta n t In Eq. (2-25) 6 Thickness o f a boundary la y e r, cm n V is c o s ity , g/cm sec. e Angle X Mobi 11 t y , cm /sec. u Chemical p o te n tia l, erg /m o le cu le ^0 Chemical p o te n tia l f o r a pure substance, erg/m olecule V Debye maximum fre q u e n cy, sec- ^ p 3 D e n s ity , g/cm a 2 In te r fa c ia l fr e e energy, erg/cm < p ( < x ) p A param eter ( = a ( l- a ) ( 6- An a )) CO Growth fre q u e n c y , sec"^ CO A ngular v e lo c ity , ra d ia n /s e c . S u b scrip ts 1 S olvent 2 S olute b Bulk liq u id i In te rfa c e g Gas phase z L iq u id phase 0 O rig in a l c o n d itio n p External body or p a rtic le s Solid phase Superscripts * C ritic a l nucleus conditions 1 Liquid phase " Vapor or gas phase REFERENCES (1) K. A. Jackson, P h il. Mag., 7 (1962) 1615. (2) H. E. Buckley, "C rysta l Growth," W iley, New York, 1951, p. 468. (3) M. Z ie f and W . R. W ilcox, e d ito rs , "F ra c tio n a l S o lid ific a tio n ," Dekker, New York, 1967. (4) Y. Y a Khaimov-Mal'kov in "Growth o f C ry s ta ls ," A. V. Shubnikov and N. N. S h e fta l1 , e d ito rs , Consultants Bureau, In c ., New York, Vol. 2, 1959, p. 14. (5) A. E. C orte, J. Geophysical Research, 67 (1962) 1085. ( 6 ) D. R. Uhlmann, B. Chalmers, and K. A. Jackson, J. A ppl. Phys., 35 (1964) 2896. (7) P. Hoekstra and R. D. M ille r , J. C o llo id . & In te rfa c e S c i., 25 (1967) 166. (8 ) G. F. B o llin g and J. C isse, J. C rystal Growth, 10 (1971) 56. ~ (9) J. Cisse and G. 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(21) V. Y a Khaimov-Mal'kov in "Growth of Crystals," A. V. Shubnikov and N. N . S h e fta l', ed itors, Consultants Bureau, In c., N ew York, Vol. 2, 1959, p. 3. (22) C . W . Correns, Faraday Society, Discussion, £ (1949) 267. (23) A. W . Adamson, "Physical Chemistry of Surfaces," Wiley, N ew York, 1967, Chapter 7. (24) M . V. Plkunov, Metalloved, i Obrabotka Tsvetnykh M etal. i Splavov, Sbornik State i , 1957, 55; through CA, 52 (1958) 16826c. ~ (25) D. R . Uhlmann, Ph.D. D issertation, Harvard U niversity, Cambridge, Massachusetts, 1963. (26) R. B. Bird, W . E. Stewart and E. N. Ligh tfo ot, "Transport Phenomena," Wiley, N ew York, 1960, Chapter 3. (27) W . Jost, "D iffusion in Solids, Liquids, Gases," Academic Press, N ew York, 1960, Chapter 1. (28) J. W . Cahn, W . B. H lllig and G. W . Sears, Acta M etallurgica, ] 2 (1964) 1421. (29) K. A. Jackson, D. R . Uhlmann and J. D. Hunt, J. Crystal Growth, 1 (1967) 1. (30) K. A. Jackson, in "Liquid Metals and S o lid ific a tio n ," American Society fo r Metals, Cleveland, 1958, 174. (31) K. A. Jackson and J. D. Hunt, Acta M etallurgica, 13 (1965) 1212. ~ (32) J. W . Cahn, Acta M etallurgica, 8 (1960) 554. (33) F. D. Rosi, R C A Review, 19 (1958) 349. 1 5 6 (34) C. Hulse and J. B a tt, Tech. Report K910803-5, United A ir c r a ft C o rp., May 1971. (35) G. A. Chadwick, in "F ra c tio n a l S o lid ific a tio n ," M. Z ie f and W . R. W ilcox, e d ito rs , Dekker, New York, 1967, Chapter 4. (36) W . R. W ilcox, J. C rysta l Growth, £ (1970) 203. (37) J. W . R u tte r and B. Chalmers, Can. J . Phys., 3]_ (1953) 15. (38) W . A. T ill e r , K. A. Jackson, J. W . R u tte r, and B. Chalmers, Acta M e ta llu rg ic a , 1 _ (1953) 428. (39) W . R. W ilcox, J. C rysta l Growth, £2 (1972) 93. (40) W . A. T ill e r and J . W . R u tte r, Can. J . P hys., 34 (1956) 96. (41) T. S. P la s k e tt and W . C. Wlnegard, Can. J. Phys., 37 (1959) 1955. (42) W . G. Pfann, C. E. M ille r and J. D. Hunt, Rev. o f S clent. In s t . , 37 (1966) 649. (43) S. Ruben, "Handbook o f the Elem ents," 2nd E d itio n , Howard W . Sams and C o ., Ind ian a, 1967. (44) R. H. P e rry, e d ito r , "Chemical Engineering Handbook," 4th E d itio n , M cG raw -H ill, New York, 1963. (45) J. S. G lass, Chem. Engr. Progress, 68 (1972) 59. (46) R. C. Weast, e d ito r, "Handbook o f Chem istry and P hysics," 53rd E d itio n , The Chemical Rubber C o., C leveland, 1972. (47) P. G. S techer, e d ito r , "The Merck Index o f Chemicals and Drugs," 7th E d itio n , Merck and Co., New Jersey, 1960. (48) N. B. Tsederberg, "Thermal C o n d u c tiv ity o f Gases and L iq u id s ," the M .I.T . Press, Cambridge, Massachusetts, 1965, through Reference 25. (49) B. Chalmers, "P rin c ip le s o f S o lid ific a tio n ," W iley, New York, 1964, Chapter 6 . (50) W . R. W ilcox and P. J. S h lic h ta , J. A p pl. P hys., 42 (1971) 1823. (51) J. M. D a lia v a lle , "M ic ro m e ritic s ," Pitman, New York, 1948, Chapter 4. 157 (52) M . S. Peters and K. D. Tlmmerhaus, "P lant Design and Economics fo r Chemical Engineers," 2nd E d itio n , McGraw-Hill, N ew York, Chapters 2, 3 and 4, 1968. (53) Fisher S c ie n tific Co., Product B u lle tin 5-712-100. (54) M . J. Joncich and D. R. B a ile y, Anal. Chemistry, 32 (1960) . 1578. (55) E. F. G. Herington, R. Handley and A. J. Cook, Chemistry and Industry, (London) (1956) 292. (56) N. Maen%, in "Physics of Snow and Ic e ," Proc. o f In t. Conf. on Low Temp. S c i., 1966, The In s titu te o f Low Temp. S c i., Hokkaido U n ive rsity, Sapporo, Japan, Vol. 1, p. 207. (57) L. Tien, "Freezing o f a Convective Liquid in a Crystal Growth Tube," Ph.D. D issertatio n, Univ. o f Michigan, Ann Arbor, 1968. (58) H. S ch lich tin g , "Boundary Layer Theory," McGraw-Hill, H ew York, 4th E d itio n , 1960, Chapter 5. (59) D. R. H. Jones and G . A. Chadwick, P h il. Mag., 22 (1970) 291. (60) D. T urnb ull, J. Appl. Phys., 2 1 _ (1950) 1022. (61) D. T u rn b u ll, "Liquids: S tructure , P roperties, S olid In te r a c tio n s ," T. J. Hughel, e d ito r, E lse ive r, N ew York, 1965, p. 16. (62) 0. Jantsch, Z e it. K ris t., 108 (1956) 198. (63) R. R. Dreisbach, "Physical Properties o f Chemical Compounds," Advances in Chemistry Series, No. 15, American Chemical S ociety, Washington, D.C., 1950, p. 354. (64) F. G iordani, in "In te rn a tio n a l C ritic a l Tables,", E. W . Washburn, e d ito r, McGraw-Hill, New York, Vol. 7, 1930, p. 211. (65) J. P. H irth and G. M . Pound, "Condensation and Evaporation," in Progress in M aterial Science, Pergamon Press, Oxford, Vol. 11, 1963, Chapter F. (66) J. P. H irth , G. M. Pound and G. S. St. P ie rre , Met. Trans., 1 (1970) 393. 1 5 8 (67) C. A. Ward, A. B a la krish n a n and F. C. Hooper, J . Basic E n g r., 695, December 1970. (68) U. D. W eatherford, J . C o llo id and In te rfa c e S c1 ., 34 (1970) 197. — (69) L. D. Landau and E. M. L 1 fs h 1 tz , " S t a t is t ic a l P h y s ic s ," Addison-W esley, Reading, M assachusetts, 1969, Chapter IX. (70) J . A. B u rto n, R. C. Prim and W . P. S lic h te r , J . Chem. P h ys., 21. (1953) 1987. (71) 0. T u rn b u ll and J . C. F is h e r, J . Chem. P hys., 17 (1949) 71. (72) S. Miyazawa and H. Iw a s a k i, Japanese J . A p p l. P h ys., 9 (1970) 441. ' ~ (73) L. A. Swanger and W . C. R hlnes, J . C ry s ta l G rowth, 12 (1972) 323. — (74) T. A rizum i and N. Kobayashl, Japanese J . A ppl. P h ys., 8 (1969) 1091. (75) N. Kobayashi and T. A riz u m i, Japanese J . A ppl. P h ys., 9 (1970) 361. “ (76) G. F lin t , in K irk-O th m e r's "E ncyclopedia o f Chemical Technology," 2nd E d itio n , W ile y, New Y ork, S uppl. V o l., 1971, p. 438. I APPENDIX A INFLUENCE O F CRYSTAL DIMENSIONS O N TH E INTERFACIAL TE M P E R A TU R E GRADIENT 159 R rp rin te d fro n t J o u rn a le f C rystal G row th 12 (1972) 191-194 O N orth -H o lla n d publishing Co. P rin te d In the Netherlands IN FLU EN C E OF CRYSTAL D IM EN SIO N S O N TH E INTERFACIAL TEM PERA TURE G RADIENT V IN C E N T I I . S. K U O and W IL L IA M R . W IL C O X Chemical Engineering and M a te ria ! Science Departments, Unlcersity o f Southern C alifornia. Las Angeles. C alifornia 90007. U.S.A. Received 26 February 1971; revised manuscript received 29 November 1971. T h e influence o f the length and radius o f a crystal on the interracial temperature gradient w a s estimated by use o f a one-dimensional heat transfer calculation. It was found that the temperature gradient is sub stantially independent o f crystal length only when the Biot number hR /k is large (where h is the heal transfer coefficient from the crystal surface, R is crystal radius, and k is thermal conductivity). 1. Introduction 2. Theory T he temperature gradient at the so lid -tiq u id inter face o f a solidifying crystal is an im portant parameter. It influences the microscopic shape stability' o f the interface, the generation o f dislocations, the macro scopic shape o f the interface, eutectic structures, and free convection in the adjacent melt. In directional solidification techniques such as Czochralski and Bridgm an growth the geometry o f the crystal changes d uring grow th, i.e., the crystal grows longer. As the crystal grows the interfacial temperature gradient changes as well. In th is paper we estimate the interfacial temperature gradient and in particular examine its dependence on crystal geometry. A cylindrical crystal is assumed which loses heat to its surroundings which are at constant tem perature Tm . The heat transfer coefficient h fo r the cylin d rica l surface is assumed constant, but different from th a t at the cold end o f the crystal l i L* . The prob lem is solved fo r a non-growing crystal, but the error caused by neglecting the heat carried by the m oving crystal is estimated. A one-dimensional approxim ation is em ployed, w ith the geometry and heat transfer parameters as shown in fig. I. * T he heat transfer coefltcicnt is defined as the race o f heat toss per unit area from the surface divii.ed by the tcirpcrature differ ence between the surface and ihe surroundings (am bient gas for convective heat iransport, heat shields o r cooling surfaces for heal iransport by radiation). The non-dimcnsionalized differential equation go verning heat transfer in a m oving th in rod is e * o e o e o - a — 2 H 0 = a d l l 3r (*) where 0 = (T -T ,)/(T t-Tt). » ) = zjR . a - V p C ,tt/Jt (Pcclet number), H = h R jk (B iot number fo r cylin d ri cal surface), and r = tV jR = L /R . Here T is Ihe tem perature at distance ; from the interface. T t is the interface temperature, R is the radius o f the crystal, V is the grow th rate, p is the crystal density, Cp is the heat capacity, k is the therm al conductivity, and L is t-L .T .T ftJ MELT Fig. I. Geometry and parameters o f problem. 191 160 161 192 V IN C E M T H. S. KUO AND W IL L IA M It. W ILCOX the length of the crystal. The boundary conditions arc it i| ** 0 (the solid-liquid interface), 0 > l ; i t if » L/R - t (the cold end o f the crystal), M /A l- - ~ 0 = —Hl 0 ; (2) (3) (4) at t *» 0 (infiniicsnially short crystal). 0=1. When H l = H the thermal conditions at the cold end of the ingot are the same as on the cylindrical surface, while H l = 0 corresponds to an insulated crystal end and / / L -* co to a perfectly cooled end. T(L) ** T,. We have been unable to solve these equations ana lytically because of the moving boundary condition, eq. (3). Although a numerical solution could be ob tained this is thought to be unprofitable since the equations only approximate reality. Therefore, we first examine the solution for a « = 0, which is*. f2\* 0 - |^ s in h (2 H )* (t—if ) + Q ) cosh (2H )*(r—»;)J x J~t sinh(2H)*T+ cosh (2H)’ r | . (5) From this the dimensionlcss interfacial temperature gradient is found to be + ( ^ ) * sinh (2//)*rj x [lS in h ( 2 « ) » r + ( i) c o s h ( 2 H)*e] . (6) Curves calculated from this equation are shown in fig . 2. It is seen that the predicted temperature gradient is independent of length only for U = / / L « = 2. As the crystal length increases, the dimcnsionless tempera ture gradient approaches -< 2 //)f . The fractional deviation from this asymptotic value is A “ [1(2 W) *— (c0/of)01](2//)” * - [ | £ - ( 5 ) 1 “ p(- (2 '','° ] x sinh (2//)‘ t+ Q J cosh (2 //)‘ t ] . (7) * Reed considered this problem for radiation into zero tempera ture with* perfectly cooled end (H i = * . T, Oand A - o i f ’, where c is the cmissivity and a is the Stefan-Boltzmann con- slant), > . as = C t2 •I H^/M *0 H /H * ■ h J I ___ _l 1 aot L/R Fi*. 2. Predicted dependence of the dimcnsionless interfacial temperature gradient on tnc length of the cr>stal and on the Biot number H ■ = hRIk for the cylindrical surface for H l ^ P (cold end insulated), Hl = H, and Hl ** » (perfectly cooled end, 7 \I) - T.); eq. (6>. 2 0 - 1 0 - ,l« - — - ” a- Fig. J. Length of the crystal for which the interfacial tempera ture gradient deviates by I % from that for a long crystal, for H J H ■ I and 0 or so (same result for 0 and x j; eq. (S). 162 IN F L U E N C E O F C R Y S T A L D IM E N S IO N S O N T H E IN T E R F A C IA L T E M P E R A T U R E G R A D IE N T 193 n » too O O I O . l . v n c ,R Fig. 4. P tK tn M ie error in the interracial temperature gradient o f i long crystal caused by neglecting the heat carried by the crystal motion due to growth; eq. 111). F ro th this wc may find the length o f crystal Tor w hich the interfacial temperature gradient differs from the *sympi.:>n‘c value by any chosen am ount. I f )\_ < Z 1, we note that sinh (2 H )*t a: cosh (2 //)* r as i exp [( 2 H ) * r ]. and we find that the length at w hich the error is / L is given by r. - (L/K)e - b W ' - M S * © ! ■[R?-S)'!ri' This is plotted in fig. 3 fo r a I deviation from the asym ptotic gradient, i.e., / L = 0 .0 1 . rc may be re garded as the dimcnsionlcss length below which the interfacial temperature gradient depends on length. The influence o f crystal m otion on the intcrfncial temperature gradient can o n ly be determined analyti cally fo r a very long crystal (L — * ac). The steady state solution (cOjdz — 0) o f eq. ( I ) fo r L -* oo is1) 0 = exp (1C*- ( x ’ + S H )*].,}. (9) w hich yields the dimcnsionlcss interfacial temperature gradient <d9/d*)a • 4 [ * - (x* + 8 //)*3 ■ (10) Thus the fractional error in the interfacial temperature gradient o f a long crystal caused by neglecting the m otion o f the crystal is found to be , _ - ( 8/ / ) * - * + ( a 2 + 8/ / ) * j * - ( . * + 811)* j ’ U U which is plotted in fig. 4. 3. Discussion These results show that when a crystal's length ex ceeds its diameter, its interfacial temperature gradient depends on its length only fo r small values ( < I) o f the B io t number //. This is reasonable since the B iot number is the ra tio o f the surface to the longitudinal therm al conductivity, or the ra tio o f the ease o f losing heat fro m the surface to the ease o f conducting heat down the crystal. Thus, fo r example i f a crystal's length is twice its diameter, then fo r large / / ( ^ 0 .IS) its interfacial temperature gradient is influenced o n ly by the thermal conditions at the cylindrical surface, while fo r small t ! 0 .IS) the therm al condition? at the other end o f the crystal also influence the in icr- facial temperature gradient. Values o f H can be es tim ated fo r typical crystal grow th situations. F or good T a b l e I Comparison o f Ihe present results (tisinc HOH = so, T, ** 0 ‘ K. X = 1 cm, and h = at T ,*\ with those o f Reed1) M aterial Ge C r,0 , W Melting point, T, ( K> Thermal conductivity, k 1210 2336 3640 (W /cm °K ) 0.4 0.04 1.4 Emissivity, c 0.2 0.8 0.36 Heat transfer coefficient, A (W /cm * ' K) 0.002 0.073 0.099 Biot number. H = hR!k 0.003 1.87 0.071 Present estimate o f critical length L, beyond which interfacial gradient con stant (fig. )| term 26 1.4 7.2 Reed's critical length. /.« (cm) 13 0.8 3.9 Deviation o f the interfacial gradient from value for semi-infinite crystal using Reed's L, in eq. (St, tl (" .) IO 9.1 10.8 Present estimate of interfacial gradient for semi-infinite crystal. (dr,d.-)0 --- K/cm t 121 4880 1363 Reed’s estimate of (J r/d.-|» ( K /cm ) 76 3000 •6 3 Experimental (dF/dcfo from ref. 6 ( K/cm) 113 163 V IN C E N T H. S. KUO A N D W IL L IA M k. W IL C O X 194 conductors wc generally find1) H < 0.1. For an organic compound with It = 1 0 '2 W/cm2 5 C for free convection lo air9), k = 1 0 '9 W/cm C C. and R — 4 cm, we find / / = 4. A t this point it is worthwhile to examine the validity o f the various assumptions that have been made. It has previously been shown2'4) that the one-dimensional approximation used here is valid for H $ 0.2. For tunately it is for these low values o f I I that our results are significant in that a dependence o f interfacial tem perature gradient on length is predicted then. For H 2 0.2 wc expect our results to b: inaccurate pri marily in the sense that the interfacial temperature gradient w ill be a function o f radial position. Our con clusion that the interfacial thermal conditions are substantially independent of crystal length is expected to remain valid. We have assumed h is constant, whereas in almost all real crystal growth systems It will be a function of r and T, both for convective heat transfer and for radiative heat transfer. When one estimates h, it is done for conditions near the interface, which have the primary influence on interfacial temperature gra dients. It is o f interest to compare the present results with those o f Reed, as shown in table I. The error caused by neglecting the heat carried by the movement o f a growing crystal is estimated in cq. (II). The worst possible case is that o f a rapidly growing insulator. Thus for an organic with V = 5 cm/hr. R — 2 cm. H = 2 and a *= 1.3, the error is about 20°', i.e., the interfacial temperature gradient o f a long organic crystal is 20°,j tower than the quasi- steady state estimate. Acknowledgement The authors arc grateful to the Petroleum Research Fund (administered by the American Chemical Society) for its financial support. References 1) H. C. Carslaw and J. C. Jaeger. Conduction of Heat I n S o l i d s , 2nd cd. (Clarendon Press, Oxford, 1959) p. 14?. 2) W. R. Wilcox and R. L. Duty, J. Heat Transfer 88c (1966) 45. 3) T. B. Reed, in: Crystal Growth, Ed. H. S. Peiscr (Pcrcamon Oxford, 1967) p. 39. 4) W. M. R olm rnow and II V. Choi. H tul, Mass and Momen tum Transfer (P rcnticc-H all, Englewood d ills . N.J., 1961) ch. 6. 3) W. H. McAdams, Heat Transmission (McGraw-Hill, New York. 1954) ch. 7. 6) J. C. Brice and P. A. C. Wiffin, Solid-State Electron. 7(1964) 1 1 3 . APPENDIX B ECONOMIC CALCULATIONS F ollo w in g are th e d e ta ile d c a lc u la tio n s fo r the co st e stim a te f o r producing p a r tic le - fr e e naphthalene. The assumptions made fo r th e c a lc u la tio n are as fo llo w s : (1 ) The p ro d u ctio n group c o n s is ts o f one engineer and one te c h n ic ia n . This group works in d e p e n d e n tly in a la rg e company. (2 ) The engineer is re s p o n s ib le fo r research , developm ent, and a d m in is tra tio n . The te c h n ic ia n is re s p o n s ib le fo r p ro d u ctio n and m aintenance. (3 ) The feed m a te ria l is a reagent grade chem ical, such as B a ke r's analyzed grade naphthalene. A. Annual P roduction In the two-man p ro d u c tio n group, the te c h n ic ia n can handle f iv e o f s ix u n its o f h o riz o n ta l z o n e -re fin e rs w ith ro ta tio n in ro u tin e p ro d u c tio n . Each u n it is equipped w ith au tom atic c o n tro l, not re q u irin g much a tte n tio n . The grow th tube is 25 m m O.D. and 40 cm lo n g . (The p u r ifie d in g o t is 22 m m in diam eter and about 30 cm lo n g .) The fre e z in g ra te is assumed to be 25 mm/hour ( <28 mm/hour). Because two heaters are on each r e f in e r , two zone passes are assumed to ensure p a r tic le - fr e e naphthalene. Travel tim e = ^ ^ o u r - 16 hours 164 165 Zone-refining operation can be conducted overnight so that most of the technician's time 1s spent in preparations fo r zone-refining and handling o f zone-refined products. Thus, d a lly production o f this group 1s about fiv e p u rifie d Ingots. Assuming 250 working days per year, the annual production 1s 5 x (u x 2.2 x 30 cm^) x (1.145 gr/cm^) x 250 = 5 x (207)(1.145) x 250 = 300 Kg/year . B. Total Capital Investment (1) Fixed-Capital Investment (a) Six units o f horizontal zone-reflners with ro ta tio n . Design and fa b rica tio n by the production group o r m odification of the Fisher zone-reflner ($800/unit): $12,000 (b) Building w ith a u x ilia ry fa c ilitie s : 30,000 (c) Laboratory equipment: 10,000 (d) In d ire ct costs (design, engineering and contingency): 10,000 Fixed-Capital Cost: $62,000 (2) Working C apital, 10% o f the Fixed-Capital cost: $ 6,000 Total Capital Investment (1) and (2): $68,000 C. Total Product Cost (1) W ages with 100% overhead (fo r one engineer and one technician): $50,000 (2) Depreciation (10 years) $62,000 x 0.1: $ 6,000 166 (3 ) Feed m a te ria ls (400 Kg): $ 3,000 (4 ) U t i l i t i e s ( e l e c t r lc lt y , w a te r and f u e l) : $ 1,000 (5 ) Maintenance and re p a irs : $ 1,000 ( 6 ) General expenses (a d m in is tra tiv e c o s ts , research and developm ent): $ 2,000 (7 ) M iscellaneous ( In te r e s t, in s u ra n c e , ta x e s , e t c . ): $ 2,000 T o ta l Product C ost: $65,000 0. Cost o f P roduction $65,000/300 Kg = 2 2 */g .

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Non-Darcy Flow Of Gas Through Porous Media In The Presence Of Surface Active Agents

Asset Metadata

Creator Kuo, Vincent H. S. (author)

Core Title Removal And Separation Of Particles (In Solid Organic Chemicals And Metals) By Crystallization

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Wilcox, William R. (committee chair), Copley, Stephen M. (committee member), Rebert, Charles J. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-797107

Unique identifier UC11364279

Identifier 7314421.pdf (filename),usctheses-c18-797107 (legacy record id)

Legacy Identifier 7314421

Dmrecord 797107

Document Type Dissertation

Rights Kuo, Vincent H. S.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA