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Computer simulation of thermodynamics and dynamics of diblock copolymers in lamellar phase

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Computer simulation of thermodynamics and dynamics of diblock copolymers in lamellar phase

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Content CO M PUTER SIMULATION OF THERM ODYNAM ICS AND DYNAM ICS OF DIBLOCK COPOLYMERS IN LAMELLAR PHASE by Xiaohong Pan A Dissertation Presented to the FACULTY OF THE ENGINEERING SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree M ASTER OF SCIENCE (Chemical Engineering) August 1996 Copyright 1996 Xiaohong Pan UMI Number: EP41852 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. U M T Dissertation Publishing UMI EP41852 Published by ProQ uest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United S tates Code ProQ uest LLC. 789 East Eisenhow er Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 This thesis, written hy X iaohong Pan under the guidance of his/her Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of M aster o f S c ie n c e C hem ical E n g in e e r in g June 2 8 , ]996 Faculty Committee D ed ication Dedicated to my parents Naizi Pan and Kemei Wang for their love and for being special parents pushing their lim its for me to have education when I was young. A cknow ledgm ents I would like to express my sincere respect and indebtedness to my advisor Dr. J. S. Shaffer for his guidance, vision, encouragement, support and the substantial time and effort provided throughout the course of this research. Financial support from Sun M icrosystems, the S. C. Johnson Wax Company, and the James H. Zumberge Research and Innovation Fund (University of Southern California) is gratefully ac knowledged. C ontents D e d ic a tio n ii A c k n o w le d g m e n ts iii L ist O f T a b les v i L ist O f F ig u r es v iii A b s tr a c t x ii 1 In trod u ction 1 1.1 Introduction to diblock cop olym ers:.......................................................... 1 1.2 T herm odynam ics:.............................................................................................. 1 1.3 D y n a m ics:............................................................................................................ 4 1.3.1 Diffusion: ..................................................................................................... 4 1.3.2 Relaxation Tim e Distributions: .......................................................... 5 2 M od el and M eth od 8 2.1 Introduction to the Simulation Model: ...................................................... . 8 2.2 Model D e sc r ip tio n :.......................................................................................... 9 2.3 M e th o d :................................................................................................................ 11 3 T herm odynam ics 15 3.1 Introduction to th erm o d y n a m ics:............................................................. 15 3.2 M e t h o d : ............................................................... 16 3.2.1 Get the stable two lamellae structure in one simulation box: . 17 3.2.2 External Field Energy: ................................................................. 60 3.2.3 Crossing and Non-crossing:............................................................... 62 3.2.4 How to calculate the radius of gyration of diblock copolymers which is parallel and perpendicular to the lamellar phase? . . 65 3.3 Results and D iscu ssion :................ 65 IV 3.3.1 R e s u l t s : ......................................................................................................... 65 3.3.2 Discussion: .............................................................................................. 6 8 4 D yn am ics T O 4.1 Introduction: 70 4.2 M e t h o d : ....................................................................................................................... 71 4.3 Results and D iscu ssion :................................... 73 5 C onclusion and Im provem ent 85 5.1 C o n c lu s io n :................................................................................................................ 85 5.2 Future W o r k : ............................................................................................................ 8 8 v List O f Tables 3.1 The lamellar length estim ated from Larson’s simulation respecting to various degree of polym erization of diblock copolymers . ................... 17 3.2 The sign of external field energy which is put into the simulation box 61 3.3 Simulation parameters: A is the number of monomers per chain in the sym m etric monodisperse diblock copolymers; N P is the number of chains in the simulation cell; L is the edge length of the primary lattice that forms the cubic periodic simulation cell; eAB/k T is the interaction energy for A-B monomer contacts in units of the thermal energy; Xeff A is the product of the effective Flory-Huggins parameter and the chain length. The lamellar period is given by L /2 .................................................... 6 6 3.4 The simulation result of lamellar length from Larson’s and ours d'h d'. 6 6 3.5 The simulation results of average square radius of gyration com ponents which is parallel and perpendicular to the lamellae, < R2^ > and < Rg x > ■ > the square radius of gyration of polymer m elts in random walk R2 gfl, the ratio of < R2 X > and < R 2^ > , that is r and the ratio of < R 2 1 | > and R2 0, that is r' for c r o ssin g situation............................... 67 3.6 The simulation results of average square radius of gyration com ponents parallel and perpendicular to the lamellae, < R2^ > and < R2yX the square radius of gyration of polym er m elts in random walk R 2 g-Q , the ratio of < R2 x > and < R2 ^ > , that is r and the ratio of < R2^ > and R20, that is rf for n on — c r o ssin g situation.......................................... 67 4.1 Average segment mobility, /zav, for crossing and noncrossing chains as a function of chain length. The average m obility is obtained from the sim ulations as the overall fraction of accepted Monte Carlo moves. . . 73 4.2 Diffusion coefficients paralleling to the lam ellae, D\$(D\\ ±8D\$ * 104), versus degree of polym erization A for the lamellar phase with y A ss 45 and f = 0.5................................................................................................................... 76 4.3 Diffusion coefficients paralleling to the lam ellae, D|j in crossing simula tions, versus chain length A for the lamellar phase with x A « 45 and / = 0.5. . ................................................................................................................ 76 VI Diffusion coefficients paralleling to the lamellae, Dy in noncrossing sim ulations, versus chain length N for the lamellar phase with \ N ~ 45 and / = 0.5................................................................................................................... List O f Figures 2.1 There are six nearest neighbor sites denoted by P ( 1 , 0 , 0 ), twelve sec ond nearest neighbor sites denoted by P ( l , 1, 0) and eight third nearest neighbor sites denoted by P ( l , 1 ,1 ) in the lattice model. . . . . . . . 10 2.2 The flow chart of simulation c o d e ............................. 12 3.1 In one particular Monte Carlo step, we cut the simulation box along the directions which are perpendicular to x, y, 2-axis and get the slices of yoz at x — x l, xoz at y = y 1 , xoy at z = z l corresponding to x, y, 2 -axis............................................................................................................................... 19 3.2 We choose the simulation box length L = d \ one lamellae is formed along x-axis. The following is the slices cut perpendicular to x, y and 2 directions. Here, N = 20, d' = 13.5 and L = 24......................................... 22 3.3 We choose the simulation box length L = cf, where d' is the lamellar length for polym erization N of diblock copolymers. One lam ellae is formed in the sim ulation box along x axis. The left part is A-rich region and the right is B-rich region................................................................... 23 3.4 Compositional density profile of monomer A and B along simulation box length direction. Em pty is for monomer A, filled for monomer B. Circle is along x-axis, triangle up and down are along y -axis and 2 -axis for N = 2 0 and L = 14............................................................................................. 24 3.5 Distribution of m idpoints of diblock copolymer chains in lamellar phase. Circle is along x-axis, triangle up and down are along y-axis and 2 -axis respectively for N = 20 and L = 14.................................................................... 25 3 .6 (cos 6X, cos Oy, cos#*) is the unit direction of face diagonals, where dx, 0y, 6Z are the angles between the face diagonal and x, y, 2-axis respec tively. the twelve face diagonals are : (-0.7071,0,-0.7071), (0.7071,0,0.7071), (0.7071,0.7071,0), (-0.7071,-0.7071,0), (0,-0.7071,-0.7071), (0,0.7071,0.7071). 3.7 Two lam ellae formed along one face diagonal (0.707,0,0.707) when we choose sim ulation box L = 1.414d',where d' is the lamellar length for this particular length of polym er chains. Here, A ^ = 20, L = 19, df = 13.5. * and o are for monomer B and A respectively, we gives some slices of yo2 , x< ? 2 and xoy .............................................................................. 34 3.8 We choose simulation box length L = 1.414<f, and get two lamellae formed along one face diagonal (0.707,0,0.707). Two A-rich regions and two B-rich regions.............................................................................................. 35 3.9 Compositional density profile of monomer A and B along one face diagonal (0.7071,0,0.7071). Empty is for monomer A and fiiled for monomer B .......................................................................................... 37 3.10 Distribution of midpoints of diblock copolymer chains in lamellar phase along one face diagonal direction (0.7071,0,0.7071) for N — 20 and I = 19............................................................................................................................. 38 3.11 (co s0Xy c o s c o s # * ) is the unit direction of body diagonals, where #r are the angles between the body diagonal and x, y, 2 -axis respectively, the four body diagonals are: (0.5773, 0.5773, -0.5773), (- 0.5773, 0.5773, 0.5773), (0.5773, 0.5773, 0.5773) and (0.5773, -0.5773, 0.5773)................................................................................. 39 3.12 We choose the sim ulation box length L = 1.732d', three lamellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d' = 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0, 0.707, -0.707). We define the directions as the following: Where x is (1,0,0), yz is (0, 0.707, -0.707).................................... 40 3.13 We choose the simulation box length L = 1.732c/', three lamellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d! — 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0, 0.707, 0.707). We define the directions as the following: x is (1,0,0), yoz is (0, 0.707, 0.707).......................................... 41 3.14 We choose the sim ulation box length L = 1.732d', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d! = 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0707, 0, -0.707). We define the directions as the following: y is (0,1,0), xz is (0707, 0, -0.707)............................................ 42 3.15 We choose the simulation box length L = 1.732d', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d' = 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0707, 0, 0.707). We define the directions as the following: y is (0,1,0), xoz is (0707, 0, 0.707)........................................... 43 3.16 We choose the sim ulation box length L = 1.732*/', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, dl — 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0.707, -0.707, 0). We define the directions as the following: z is (0,0,1), xy is (0.707, -0.707, 0 ).................................................. 44 3.17 We choose the simulation box length L — 1.732d', three lamellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d! = 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0.707, 0.707, 0). We define the directions as the following: z is (0,0,1), xoy is (0.707, 0.707, 0 )................................................. 45 3.18 We choose the sim ulation box length L = 1.732c/'. Three lamellae are formed along body diagonal (0.5773,0.5773,0.5773). There are three A-rich and three B-rich regions separated by five planes perpendicular to the body diagonal. Here, N = 20, d! = 13.5, L — 24...................... 46 3.19 We choose the sim ulation box length L = 1.732c/', three lamellae are formed along one body diagonal whose direction is (0.5773,0.5773 0.5773). Here N = 20, L — 24, d! = 13.5. The following is series of slices selected cut along yoz,xoz and xoy planes........................................... 54 3.20 Com positional density profile of monomer A and B along one body diagonal direction (0.5773,0.5773,0.5773). Em pty is for monomer A and filled for monomer B for N = 20 and L = 24......................................... 56 3.21 Distribution of midpoints of diblock copolymer cahins in lamellar phase along one body direction (0.5773,0.5773,0.5773) for N = 20 and L = 24. 57 3.22 Compositional density profile of monomer A and B along simulation box length directions. Em pty is for monomer A and filled for monomer B. Circle is along x-axis, triangle up and down are along y-axis and 2-axis respectively fo N = 80 and L = 45......................................................... 58 3.23 Distribution of m idpoints of diblock copolymer cahins in lamellar phase along sim ulation box length directions. Circle is along x-axis, triangle up and down are along y-axis and 2-axis respectively fo A = 80 and L = 45................................................................................................................... 59 3.24 We introduce external energy field by making region 1 and 3 favored by monomer A and region 2 and 4 favored by monomer B in our simulation box whose length L = 2c/', where d' is the lamellar length.................. ... . 61 3.25 Compositional density profile of monomer A and B along simulation box length direction appliedby external energy field. Em pty is for monomer A and filled for monomer B. Circle is along x-axis,triangle up and down are along y-axis and z-axis respectively for N = 26 and L =28...................................................................................................................... 63 3.26 Compositional density profile of monomer A and B along simulation box length directions. Empty is for monomer A and filled for monomer B. Circle is along x-axis, triangle up and down are along y-axis and 2-axis for N = 26 and L = 28...................................................................... 64 x 4.1 mean square displacem ent corresponding to sim ulation box length di rection for both crossing and noncrossing sim ulations. Empty and filled are for crossing and noncrossing sim ulations respectively. Cir cle is along x-axis, triangle up and down are along y — axis and 2-axis respectively for N = 50 and L = 39.................................................................... 75. 4.2 The product of diffusion coefficient of diblock copolymers in lamellar phase which is parallel to the lamellae D\\ and degree of polym erization of diblock copolymers N as function of degree of polym erization N. Filled is for crossing simulations and em pty for noncrossing sim ulations. 77 4.3 The ratio of product of diffusion coefficient of diblock copolymers in lamellar phase which is parallel to the lamellae £)|| and degree of poly merization of diblock copolymers N and average segm ent m obility p av). Filled is for crossing sim ulations and em pty for noncrossing simulations. 78 4.4 Autocorrelation function o f Rouse model Cp(t ) as function of simula tion tim e t (or N = 20 and p = 19 in the direction of paralleling to the filled is fitted by W illiam s and Watts equation.............................................. 80 4.5 The ratio of relaxation tim e rp and square of wavelength N /p as func tion of wavelength N /p for degree of polym erization of diblock copoly mers for crossing sim ulations and filled for non-crossing. Triangle up and down are for the direction of paralleling and perpendicularing to the lamellar phase.......................................................................................................... 81 4.6 The ratio of relaxation tim e t p and square of wavelength N /p as func tion of wavelength for N = 80. Empty is for crossing simulations and filled for non-crossing. Triangle up and down are for the direction which are parallel and perpendicular to the lamellar phase...................... 82 4.7 Combing Figure ?? and Figure ?? . Empty is for crossing and filled for noncrossing. Circle and diamond are for the direction which are parallel and perpendicular to the lamellar phase for N = 20 . Triangle up and down are for N = 80 corresponding to circle and diamond. . . 84 xi A bstract Monte Carlo sim ulations of the therm odynam ic and dynam ic properties of diblock copolymers in lamellar phase are carried out on a new lattice-based model of polymer dynamics in which the polymer chain topology can be altered without perturbing any static properties and local segm ent m obility introduced by Shaffer [19] with the chain length N as large as 80 lattice sites with ~ 45. The lattice model can control chain entanglem ent by allowing or forbidding the overlap of midpoints of second cube lattices which are called crossing and noncrossing simulations. The results of simulations show that the chain topology does not have any significant effects on the structure of the copolymer chains in lamellar phase and also the magnitudes of radius of gyration both in the direction of paralleling and perpendicular to the lamellae are very near both in crossing and noncrossing simulations for same length of diblock copolymers. We also report that the ratio of square of lamellar length d! and chain length N (d'2/ N ) is almost constant for various lengths of diblock copolymers in lamellar phase and the stronger stretch of diblock copolymers in lamellar phase in the direction which is perpendicular to the lamellar phase than the direction which is parallel to the lamellar phase. Besides the therm odynam ic properties, we report our primary results in dynam ics, the variation of the diffusion coefficients paralleling to the lamellar phase £)|| within lam ellae with x N « 45 for both crossing and noncrossing chains. Entanglem ent caused by noncrossing slow D\\ in the lamellar phase as in the homogeneous, disordered bulk phase at the same chain length. Micro-phase does not alter the critical molecular weight for the onset of entanglem ent. We also study comparison of rheological functions for entangled polymeres m elts in the bulk phase - with the lamellar phase by giving the relaxation tim e dipstribution {rp} for different modes p for the region of chain length 2 0 to 80. C hapter 1 In trod u ction 1.1 In trod u ction to diblock copolym ers: Diblock copolymers are macromolecules which consist of two sequence homoge neous blocks of different chemical units. They becom e more and more attractive to scientists and engineers due to their special properties which can produce coexisting phases by undergoing a type of micro-phase separation. In order to get more useful diblock copolymer products, we need to'study their spe cial properties from two aspects. One aspect is therm odynam ics, which controls the micro-phase structures formed by diblock copolymers. The other aspect is dynam ics, which governs the rheological properties and the processability of the materials. 1.2 T herm odynam ics: There are three im portant parameters which determ ine therm odynam ics of A-B type diblock copolymers: the degree of polym erization N, the m agnitude of the Flory- Huggins segm ent-segm ent interaction parameter x a^d the volum e fraction of type A / . We can regulate N, f values by polym erization stoichiom etry in the process of preparing diblock copolym ers, x is m ainly determ ined by the interaction of A-B monomer pairs which cannot be controlled by synthetic process and has a relationship 1 with tem perature T as x ^ a /T + 6 , where a > 0 ,b are constants respecting to the special value of / . From the therm odynam ic aspect, the entropy term (N) and the enthalpy term (x ) in free energy density determ ine diblock copolymer phase behavior. We will dem onstrate these two term s in the following paragraph. [1] M onodisperse diblock copolymers favor being arranged in minimum free energy configurations at equilibrium. The free energy density is dom inated by two param eters. One is the A-B segm ent-segm ent interaction parameter x? which governs the enthalpy of diblock copolymers. The other is polym erization N, which controls the entropy of diblock copolymers. Generally, the opportunities of A-B monomer pair contacts of diblock copolymers are reducing by lowing the tem perature T, that is, increasing the interaction parameter x- Sufficiently large N will have relative small translational and configurational entropy. Macroscopic phase separation is impossible but micro-phase separation can be formed with sufficiently large x N value in virtue of dom inating enthalpy factors (long polymer chains N and low tem perature T). Al ternatively, small XN value (short polymer chains N and high tem perature T)will lead to a com positionally disordered phase. Diblock copolymers in disordered states are not different from homogeneous polymers whose therm odynam ical and dynamical properties are studied by Shaffer [19]. In this thesis, we will only concentrate on ther m odynam ics and dynamics of diblock copolymers in microscopic phase. Bates and Fredrickson [1] divided the micro-phase separation formed by diblock copolym ers into three regimes according to value: the weak, strong and interm ediate segregation lim its (W SL, SSL and ISR) corresponding to certain value of / . The first lim iting regime is weak segregation lim it corresponding to relative low XN value. W ith sufficiently low x N value, the individual diblock copolym er chains are Gaussian distributed just like homogeneous polym er m elts with characteristics of random walk Rg ~ N 1^2, where Rg is the gyration radius of diblock copolymers. As XN increases to the order of 10, a disorder-to-order transition(O D T ) appears. In this region of x N value, the diblock copolymer chains are also unperturbed, the 2 m icrodomains formed by diblock copolymers are scaled as N 1^2 and the com position density profile along the direction which is perpendicular to the microdomains is ap proxim ately sinusoidal. One thing that needs to be em phasized is that almost all disorder-order transition theories depend on WSL theories although no strict experi m ental proof. W ith increasing x N , diblock copolymer chains extend significantly from the un perturbed topology in WSL. This signals appearing of the second segregation lim it, that is, interm ediate segregation regim e(ISR). In this situation, both theory(Fredrickson and Helfand [2]) and experim ent (Alm dal et al [3] have given the proof of an ISR scal ing regime characterized by d" ~ iV0 8, where d' is the order scale of the microdomains. Continuing to increase x N , the last segregation lim it (i.e. strong segregation lim it(SSL )) finally reaches accom panying diblock copolym er chains’ extrem e extend ing and narrow interfaces’ forming between pure A and B in m icrodomains which are scale as d! ~ N 2/3. As discussed above, we know that the diblock copolymers can form microdomains due to micro-phase separation in a special range of x N value. But the value of x N cannot tell us the order sym m etry of m icrodomains. The order sym m etry of microdom ains can be determ ined by volum e fraction of type A / . There are six different kinds of microstructures: spheres, hexagonally-packed cylinders, lamellae, hexagonally m odulated lam ellae, hexagonally perforated layers and bicontinuous Ia3d “gyroid”. [4, 5, 6 , 7, 8 , 9] Lamellae is the sim plest microstructure formed by diblock copolym ers at / = 0.5. In this thesis, we m ainly focus on diblock copolymers in interm ediate segregation regions with x N * = » 45, lamellar microstructures, and a wide range of polym erization N (20 to 80). 3 1.3 D ynam ics: Diffusion coefficients and relaxation tim e distributions are two im portant aspects of dynam ics. We will focus on these two dynam ical properties of diblock copolymers in lamellar phase in this work- 1.3.1 D iffusion: As hom ogeneous polym er m elts, diblock copolym ers are divided into two regimes according to molecular weight, “un-entangled “ and “entangled” , which have qualita tively different chain dynamics and transport properties. The transport coefficients of short polym ers are: T j ~ M I } for M « Mc (1.1) D ~ M ~l J W here rj is the shear viscosity and D is the diffusion coefficient. In long polymer chains, the transport properties are much more sensitive to molecular weight. T f ~ A/ 3'4 D ~ M ~2 > for M » Me ( 1.2 ) Therefore un-entanglem ent of sufficiently short polym er chains leads to molecular m otion just like small molecules which can be approxim ately described by Rouse Model [10]. As increasing polym er chain length, polym ers have a tendency of repta- tion due to their entanglem ent and we can describe this kind of molecular m otion by R eptation Model. [18] The critical molecular weight M c is a very im portant parameter separating these two kinds of regimes. A typical kind of diblock copolymers has a certain value of Mc. Many theoreti cal [17] and experim ental [12, 13, 14, 15, 17] papers have shown that D\\, that is, the diffusion coefficient in the direction of paralleling to the lam ellae is unaffected by the 4 potential and D x, that is, the diffusion coefficient in the direction of perpendicular- ing to the lam ellae is slowed by the potential respecting to short diblock copolymers which form lam ellae, however, both D\\ and D±_ are slowed by the potential with a little bit faster m otion in the direction of paralleling to the lam ellae than that in the direction of perpendicularing to the lamellae for long diblock copolym er chains. Supposing that we can determ ine the critical m olecular weight Mc correctly, we can apply Rouse Model [10] to short diblock copolymers and R eptation Model [11] to long diblock copolymers. We can also use the results from experim ental and theoretical papers for different molecular weight regimes. From the above, it is necessary to attain the critical molecular weight Mc of diblock copolymers in lamellar phase , to relate it to homogeneous polymers which was studied by Shaffer [19] [18] in order to develop quantitative theories of chain dynam ics and rheology for diblock copolymer micro-phases and to apply theoretical treatm ents to specific material system s. In this thesis, we specifically address to the following question: Does the formation of the block copolymer lamellar phase alter the critical molecular weight for the onset of entanglem ent constraints? It is natural to say that the critical molecular weight will change due to two characteristics of lamellar phase. One is the stretch of individual diblock copolym er chains which might reduce the extend of entanglem ent compared to hom ogeneous polym er m elts, the other is the hindrance of m onom ers’ m oving into different kinds of m onom ers’ region. In later chapters, we will give the computer simulation results concerning the alteration of critical molecular weight in lamellar phase formed by diblock copolymers at / = 0.5. 1.3.2 R ela x a tio n T im e D istrib u tion s: In theoretical, numerical and experim ental analysis, the polym er chain dynamics can be decom posed into a set of relaxation m odes, labeled by the index p, with characteristic relaxation tim es rp. [18] We can extend the results of bulk polymer m elts to diblock copolym ers in lamellar phase and conclude that the com plete distribution 5 of relaxation tim es {rp} of diblock copolymers in the lamellar phase provides to test com pleting theories of diblock copolymer dynam ics. By applying Rouse Model [10] and R eptation Model [1 1 ], we can get the relationship between relaxation tim e rp and the ratio of polym erization N and m ode index p just like the results gotten from sim ulations and experim ents. rp ~ (N /p)2 for p iV and N < N c (1-3) for short polym er chains whose molecular weight is smaller than the critical molecular weight Mc, and rp ~ (N /N e)(N/p)2 for N > Nc (1.4) for long polym er chains whose molecular weight is larger than the critical molecular weight Mc, where Ne denotes the number of chain segm ents between entanglem ent. For m odes with wavelengths N /p above an effective entanglem ent spacing, the above equation becom es the following: T p ~ (N /p)3 (1.5) V iscosity p is one of im portant parameters in polym er dynam ics. Generally, we calculate rj using the following formula: J rco ' G(t)dt ( 1 .6 ) 0 and G(t) is gotten from G(t) = EspCp(() (1.7) P where gp is the strength of m ode p and Cp(t) is the individual relaxation function for that m ode. Many theories predict exponential relaxation for each m ode with: G(t) = e x p (- * /r p) ( 1.8) 6 although we can observe non-exponential relaxation. Combining all the above, we can also get the relationship between rj and N. 7 7 ~ N for N < Nc (1-9) 7 ? ~ J V 3+* for N > Nc (1.10) where x = 0.4 for experim ents on many polym eric species, however the Reptation Model [11] predicts x = 0. The discrepancy between the experim ental and theoretical scaling probably is due to that entangled polym er chains do move in a manner that is qualitatively similar to reptation, but additional relaxation processes are present and contribute to stress relaxation, especially in weakly entangled system s for which N /N e is of order 10. In this thesis, we will give com plete distribution {rp} in the lamellar phase both perpendicularing and paralleling to lam ellae and provide test analysis for dynamics theories of diblock copolym ers in lamellar phase. Before ending this chapter, we will give the outline of my thesis as follows. Chapter 2 focues on computer simulation m odel and m ethod. Chapter 3 describes therm odynam ical properties of diblock copolymer in lamellar phase by com puter sim ulation. Chapter 4 discusses the crossover from un-entangled to entangled chain dynam ics in the lamellar phase of diblock copolym ers, using computer sim ulation m ethod and gives the com plete distribution of relaxation tim es {rp} in the lamellar phase of diblock copolymers both in parallel and perpendicular to the lam ellae for a range of polym er chain length crossing over the critical molecular weight. Chapter 5 concludes the above results and explains the future work in this field. 7 C hapter 2 M o d el and M eth od 2.1 In trod u ction to th e S im u lation M odel: The therm odynam ics and dynamics of diblock copolymers in lamellar phase are sim ulated using a lattice-based M onte Carlo dynam ical model developed from Shaf fer. [19] The general features of this m odel are borrowed from bond fluctuation models of Carmesin and Kremer [20] and Deutsch and Binder [21]. The main developm ent of the m odel comparing to old m odels is that chain entanglem ent can be switched “on” and “off” without perturbing any other properties. In this thesis, we conducted two different kinds of sim ulations corresponding to “un-entangled” and “entangled” separately to study the crossover from un-entangled to entangled chain dynam ics of diblock copolym ers which form lamellar microstruc tures. B y virtue of this reason, we adopt the m odel which can apply these twr o different sim ulations without disturbing any static properties and the local segment m obility. In the following of this chapter, we will give detailed descriptions of the m odel and the m ethod which carries out the com puter sim ulations using the model. 8 2.2 M o d el D escription: The m odel consists of two kinds of cubic lattice: the primary and the secondary cubic lattice. The primary cubic lattice is occupied by polym er m onomers with lattice constant a.i = 1. The connectivity condition can be m aintained by allowing bond length 1, \ / 2 and \/3 while excluded volume condition is enforced by forbidding overlap of the primary cubic lattice sites by different m onomers. T he secondary cubic lattice w ith lattice constant a2 = 1 / 2 is involved by virtue of m idpoints of all bond vectors. The polym er topology can be im plem ented using the secondary cubic lattice which allowing or forbidding double occupancy of m idpoints of bond vectors for crossing and non-crossing polymer motion respectively. As shown in Figure 2.1, there are six nearest neighbors, twelve second nearest and eight third nearest neighbors which allowing the bond length 1 , y/2 and \/3 respectively. b e P ( 1 ,0 ,0 ) U P ( 1 ,1 ,0 ) U P ( l , 1 ,1 ) (2.1) W here b is the set of bond vectors, P ( 1 ,0 ,0 ) , P ( 1 ,1 ,0 ) and P ( l , 1 ,1 ) denote the set of nearest, second nearest, third nearest neighbors of the m onom er respectively. P ( l , 0 , 0 ) = { ( 1, 0 , 0 ), (0 ,1 ,0 ), ( 0 ,0 , 1 ), (—1 ,0 ,0 ), (0 , —1 ,0 ), (0 ,0 , —1)} P (l, 1,0) = {(1,1,0), (1,0,1), (0,1,1), (—1, —1,0), (—1, 0, —1), ( 0, —1, —1), (—1, 1, 0), ( 1, —1, 0), ( - 1, 0, 1), (1, 0, - 1), (0, - 1, 1), (0, 1, - 1)} 9 (0,1,1) (1,1,1) (0,0,1 (-l,0,n (1,0,1) (1,-1,1 (-1,1,0) (0,1,0) (1,1,0) - I ----------- I (1,0,0) | . ^ I (0,0,0) 1 (1,-1,' i ( - 1,0,1 (-1,1,-U) / "> (0,o7-l) (-1,0,-U (-1.-1.-1) Figure 2.1: There are six nearest neighbor sites denoted by P ( 1 ,0 .0 ) . twelve second nearest neighbor sites denoted by P ( 1 , 1 , 0 ) and eight third nearest neighbor sites denoted by P ( l , 1 ,1 ) in the lattice model. 10 Figure 2 (from Shaffer [19]) illustrates how entanglem ent constraints are enforced in the secondary cubic lattice. By using this m odel, the chain topology can be mod ified easily by allowing or forbidding the overlap of bond m idpoints. 2.3 M ethod: We have described the m odel in 2.2, In the following, le t’s give the answer of the question: “How can we apply the m odel into the com puter sim ulations of diblock copolymers in lamellar phase?” . As illustrated in Figure 2.2 - the flow chart of the com puter sim ulation code, there are seven steps in Monte Carlo dynamical sim ulation, it one by one. STEP 1: Before doing the sim ulation, we need to select suitable sim ulation box length L and volum e fraction of polym er monomers to total sim ulation box sites p. Large sim ulation box benefits getting more accurate therm odynam ical and dy nam ical properties, but it also causes very long sim ulation tim e for one Monte Carlo step. Small sim ulation box needs shorter sim ulation tim e but provides low accuracy of therm odynam ical and dynam ical properties because periodic images may interact w ith each other. There are no boundaries because of periodic boundary conditions. So we need to choose an optim al simulation box length which can provide satisfied sim ulation results and needs relative short sim ulation tim e. Here, we set simulation box length L — 2d', where d' is the lamellar length formed by diblock copolymers under this sim ulation condition, which is the sm allest sim ulation box length offering global therm odynam ical and dynam ical properties of diblock copolymers in lamellar phase because of two lamellar planes in one sim ulation box. In this work, we set volum e fraction of polym er m onomers to total sim ulation box sites p = 0.5 to compare the sim ulation results of diblock copolym ers in lamellae phase with those of hom ogeneous polymer m elts published by Shaffer. [19] 11 STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 STEP 6 STEP 7 select L and f set initial coordinate set attempted move No excluded volume is satisfied? Yes No connectivity? Yes No energy judge? Yes No end? Yes end Figure 2.2: The flow chart of sim ulation code STEP 2: Get the initial coordinates of polym er monomers in the sim ulation box. First, we calculate the number of polym er chains in the sim ulation box npoly by virtue of volum e fraction of polymer monomers to the simulation box sites p, simula tion box length L and polym erization N using the formulae: npoly = ceiling^L3/(pN)) Second, we fill the first cubic lattice sites with the first beads of all polym er chains by randomly choosing the coordinates while enforcing excluded volum e conditions. Third, we grow all the polym er chains by successively filling the lattice sites with the second, third, ... Nth monomers. In this procedure, we allow the possible bond length 1, \/2 while enforcing excluded volum e conditions. Fourth, if this procedure fails in the m iddle steps, we need to start it again until it is successful. STEP 3: Randomly choose one monomer in the sim ulation lattice and attem pt to m ove it to one of its nearest neighbor by randomly choosing coordinate. STEP 4: Check the excluded volum e conditions which are satisfied or not, if they are, go to step 5, otherwise go back to step 3. If the new site of attem pted move of the selected monomer has already been occupied by other monomers, the move should be rejected, the M onte Carlo step fails and we need to go to step 3, otherwise, we just proceed. STEP 5: Check bond length which is 1, \/2 or \/3 ? If it is 1, \/2 or \/3 , then continue to step 6, otherwise returns step 3. STEP 6: Calculate the difference energy SE between the old configuration and new configuration induced by interaction energy of A-B monomer pairs. If SE < 0 or exp(—5E/kt) < randomly generated number between 0 and 1, the attem pted move is successful, otherwise, the sim ulation returns to step 3. In order to get S E , we need to calculate E 0a and E n e w , where E 0id and E new are the energy of the selected monomer which interacts with its nearest, second nearest and third nearest neighbor monomers except the monomers which connected with it 13 by chem ical bonds in old configuration and new configuration separately and then apply the formula SE — Enew — E0i& to get SE. Steps 1-6 are repeated until the sim ulation is concluded. 14 C hapter 3 T h erm od yn am ics 3.1 In trod u ction to th erm od yn am ics: Therm odynam ics is one of two fundam ental factors which m ust be understood and quantified in the process of producing of marketable diblock copolym er products. It establishes the state of molecular association and order and then determines the structures of diblock copolym ers. ( The other fundam ental factor, dynam ics, will be studied in Chapter 4.) As we discussed in Chapter 1, there are three im portant parameters which de term ine the therm odynam ics of diblock copolymers. T hey are: 1) x (Flory-Huggies interaction param eter), 2) N (degree of polym erization) and 3) / (volum e fraction of monomer A ). We can distinguish three regions according to increasing m agnitude of x N value as W SL(weak segregation lim it), ISR(interm ediate segregation region) and SSL(strong segregation lim it), the structure order sym m etry can be determ ined by /■ In this thesis, we consider the following situations: xNf*4 5 (3.1) / = 0.5 (3.2) 15 where N can be selected as 20, 26, 32, 40, 50 and 80 and X = pZefjtAB/kT (3.3) where p is the volum e fraction of polym er monomers in one simulation box which is chosen p = 0.5 here. zej j is the effective coordinate number for the monomer- monomer interaction which approximates 22 in this sim ulation model. [22] This is due to the fact that there are on average about four contacts per monomer which should not contribute to the interaction parameter according to Flory lattice theory. [36] cab is the interactive energy between monomer A and B (€As/fcT is the dim ensionless term ). By virtue of fixed p = 0.5, zejf = 22, we can get required x by m odifying the dim ensionless value of CAB/kT. x N ss 45 m eans that the diblock copolymers we sim ulate here are in the range of ISR (interm ediate segregation region), / = 0.5 means that the microstructures formed by diblock copolymers are lamellae. 3.2 M ethod: In order to study the therm odynam ics of diblock copolym ers in lamellar phase, we need to get the stable lam ellae microstructures of diblock copolym ers which can give the global therm odynam ical and dynamical properties under the sim ulation condition of XN ~ 45 and / = 0.5, then analyze d! (lam ellar length), /?<jr(radius of gyration of diblock copolym ers in lamellar phase) and the comparing results with bulk polymer m elts in various lengths of diblock copolymers. 16 N 4/Rgfl Rg, 0 dr 20 5.2 2.583 13.43 26 4.9 2.945 14.42 32 4.8 3.267 15.02 40 4.7 3.652 17.17 50 4.6 4.084 18.78 80 4.4 5.165 22.73 Table 3.1: The lamellar length estim ated from Larson’s sim ulation respecting to various degree of polym erization of diblock copolym ers 3.2.1 G et th e sta b le tw o lam ellae stru ctu re in on e sim u lation box: We need to estim ate the lamellar length d! in various lengths of diblock copolymers N. As shown in the second column of Table3.1 (from Figure 1 in Larson [22]), we get the radio of d\ (lamellar length estim ated from Larson’s sim ulation) and R g < o (radius of gyration of bulk polym er m elts in random walk) in various lengths of diblock copolym ers N. We also use the following formulae for bulk polym er m elts to get Rg< 0 corresponding to the same length of polym er chains N : R,.0 = ( , / A W . (3.4) where a is the average bond length gotten by: a = (1 * 6 + >/2 * 12 + \/3 * 8 )/2 6 = 1.4164 (3.5) 17 In this formula, there are six nearest, twelve second nearest and eight third nearest neighbors contribute to the bond length of 1, \/2 and \/3 respectively, so we can give the rough values of Rg and d! listed in the third and fourth colum n of Table'S.1 Determ ine the rough sim ulation value of lamellar length by analyzing the one lam ellae m icrostructures formed by diblock copolymers. According to the sim ulation m ethod we discussed in Chapter 2, we choose the sim ulation box L roughly as {d\ — 2, d\ + 2) in various degree of polym erization of diblock copolym ers and then do the following. 1. C alculate the relaxation tim es for various sim ulation conditions. As described in Chapter 2, after getting the initial coordinates, we run the Monte Carlo sim ulation, record all coordinates of polym er monomers and calculate the prop erty - end-to-end distance correlation function < (R(f)-R(O)) > / < (R 2) > , where R (t) is the end-to-end distance vector at sim ulation tim e <, then plot < (R(t)-R(O)) > / < (R 2) > as sim ulation tim e t. We plot end-to-end distance correlation function < (R(t)-R(O)) > / < (R 2) > as function of sim ulation tim e t and can get approxim ate value of relaxation tim e (r) by selecting the tim e point which corresponds to enough sm all m agnitude of < (R (i)-R (O )) > / < (R2 ) > along tim e axis t. 2. After getting relaxation tim e r, we choose equilibrium tim e period 5 ~ lOr in order to guarantee that the properties we calculate from the coordinates after equilibrium tim e period are stable and not affected by different initial coordinates. W hen the sim ulation system reaches equilibrium, we record all coordinates of monomers for enough long period of sim ulation. 3. After finishing the sim ulations, we do the following: (1) Select one characteristic Monte Carlo step, cut the sim ulation box along the direction which is perpendicular to x, y and z axis (see Figure 3.1) and get the slice of yoz, xoz and xoy (see Figure 3.2) respectively. 18 Z=Z1 Y=Y1 (0,0,0) X=X1 C Z I represents yoz plane at x=xl. Q represents xoz plane at y=y 1. d represents xoy plane at z=zl. Figure 3.1: In one particular M onte Carlo step, we cut the sim ulation box along the directions which are perpendicular to x\ y. 2-axis and get the slices of yoz at x = xl. xoz at y — y 1, xoy at z = z i corresponding to x , y, 2-axis. 19 * • ★ * * 0 * ■ k k k 1 o k * * k * 1 k ★ ★ * k k k | ★ o 1 o * * ★ * 1 o o * it it o 1 k o ★ ★ * k 1 ★ ★ 0 * o 1 * o 0 * • o o o 1 o o * o o o o o * o o o o o 1 o k o * o o 1 * ★ ★ it it o 1 * ★ o 1 Y D X 1 o * * o o 0 1 k o o o o o 1 o o it o o o k 1 o o o o 1 0 o o o o o O I 0 o o o o ° I 0 o o k ★ o o O I o ★ o o o o o o o 1 0 o o o o o o O I o o o o o o o o o o o 1 0 o o o o ★ 0 o 1 0 o o o 0 o ° 1 o o o o o ° 1 k o o o o 1 jx = 2 o o o o o o o o o ° 1 o 0 0 o o o o O I o o o o o o o o o 0 o 1 o o o o o o o 1 o o o o o o 0 o O I 0 0 o o o o 0 o o 0 0 1 0 o o o o o o o o 0 ° 1 o o o o o o o o j o o o o o o 1 o 0 o o o o 0 0 I o o o o o o 0 o j o 0 o o o 0 O I o o o o o o 0 o O I o o o o o o o o o 1 r j x = 5 20 j x - 7 O o o o o ★ k k o 1 O o o k O I o k k o 1 k k k o 1 * o k k O I o k * k k 1 * o o ° 1 o o k * k o ° 1 o o o k * o o 1 o ★ k k k o * 1 k ★ k i k k k 1 o k * k ★ ★ k k 1 o ★ k k k k ★ k o o ★ 1 o k k k ° 1 jx = 8 k k k k 1 ★ k k k k k 1 k * * k k k * 1 k * k k * ★ k * • 1 k k k k k k k ★ k 1 * k k k k k k ★ k * 1 k ★ k k k k ★ ★ j ' k k k k k * k k k ★ 1 k k k k k k 1 k k k k k k k k k I * k k k k k k k i k k k k k * 1 k k k k k k 1 ★ k k k k 1 ' j x = 11 ★ k k k k k k ★ k I k k k k k k * * 1 k k k k I k k k k k k k k k * 1 k k k k k k k k 1 * k k k ★ * 1 k k o k k k k ★ 1 k k k k k ★ 1 k k k k 0 o 1 k k k o k 1 o * • k k k k o k k k k k o k o O I k k k k k k k k 1 k k k k k 1 c j x = 14 21 ★ o 0 o o 0 k k k 1 * * o o o o o k k ★ 1 o o o o k k k * 1 o o o o k k 1 o 0 o o ★ k k k 1 o k k k k 1 o o o o * k k I ★ o ★ k k 1 o 0 k k k 1 0 o o o k k 1 o o o 0 k k I 0 o o k k k k * 1 o o o k k k k I o o o 0 k k k k I jy = 8 o o 0 o o 1 ★ o o o o o o k k * 1 * k o o o k k k 1 o o 0 k k ) * o o 0 o k k k 1 o o o o k k 1 o o o o o o k k k k 1 o o o o k k k k 1 o o o k k 1 o k k k k 1 o o o o k k 1 o o o o o k o | o o o o k k 1 o o o o 1 r r j z = 14 Figure 3.2: We choose the simulation box length L = d', one lam ellae is formed along x-axis. The following is the slices cut perpendicular to x, y and z directions. Here, N = 20, d! = 13.5 and L = 24. B - r i c l j i A - r i c h o o Figure 3.3: We choose the sim ulation box length L = d', where df is the lamellar length for polym erization N of diblock copolym ers. One lamellae is formed in the sim ulation box along x axis. The left part is A-rich region and the right is B-rich region. 23 0.80 1 0.60 o Q. W tz < D 3? 0.40 to c o (0 o CL E 8 0.20 0.00 15.0 10.0 0.0 5.0 simulation box length direction Figure 3.4: Com positional density profile of monomer A and B along sim ulation box length direction. Em pty is for monomer A, filled for monomer B. Circle is along a>axis, triangle up and down are along y-axis and 2-axis for N = 20 and L = 14. Figure 3.2 and Figure 3.3 shows the one lamellar structure formed in one simu lation box along x axis. We anneal the lamellar structure for a period of sim ulation tim e giving sufficiently small value of , it will disappear and form a new lamellar structure along y or 2 axis. We plot com positional density function of monomer A along x , y and 2 directions at som e particular M onte Carlo step. As shown in Figure 3.4, the curve is sinusoidal along the direction which is perpendicular to the lam ellar phase showing that diblock copolym ers are in interm ediate segregation region and the curve is flat w ith a little 24 / c 1 0.10 o w C D E .> « “ ■ 0.08 o VI c 'o Q. 1 0.06 o c o 0.02 15.0 5.0 10.0 simulation box length direction 0.0 Figure 3.5: Distribution of m idpoints of diblock copolym er chains in lam ellar phase. Circle is along a>axis, triangle up and down are along y-axis and z-axis respectively for N = 20 and L = 14. bit fluctuation around 0.25 along the direction paralleling to the lamellar phase. One thing we need to em phasize is the m axim um point value in sinusoidal curve is 0.5 because of p — 0.5 which means that there are only monomers A occupying this sim ulation box slice. Figure 3.5 shows the distribution of m idpoints of polym er chains along x, y and 2 directions which is also the interface curve which separating segregation region of monom er A and B along the three directions. We observe that there are two peaks 25 occurring showing that the sim ulation box can be divided into two equal parts: A-rich region and B-rich region. (2) Get the correct lamellar length values for various degree of polym erizations of diblock copolym ers. As discussed before, we have the rough sim ulation values of lamellar length of diblock copolym ers for different degree of polym erization. In order to verify the correctness of these rough values of lam ellar length, we need to do more sim ulations. For exam ple, choosing sim ulation box length as \/2 d ', \/3 d! or 2d' and observe whether two periods of lamellar microstructures can be formed along one face diagonal, three periods along one body diagonal or two periods along one sim ulation box length respectively. We will discuss these three situations individually. Supposing that we choose the sim ulation box length L as \/2 d \ the face diagonal length of the sim ulation box is \/2 * y/2d' — 2d!. We postulate that we can get two lamellar m icrostructures along one o f six face diagonal directions in the sim ulation box. Figure 3.6 gives the unit direction (cos 6X, cos0y, cos0z) of twelve face diagonals, where 9X, 6y ,6Z are the angles between the face diagonal and x, y , z axis respectively. Figure 3.7, 3.8 gives two lam ellae formed in one face diagonal (0.707,0,0.707) for degree of polym erization N = 20 and sim ulation box length L = \/2d! = y/2 * 13.5 « 19. There are two A-rich regions and two B-rich regions along the face diagonal. In Figure 3.7 we choose slices of x = 1 ,6 ,1 0 ,1 6 ,1 9 which are parallel to yoz planes, y = 1 ,6 ,1 2 ,1 9 which are parallel to xoz planes and y = 1 ,1 0 ,1 4 ,1 9 which are parallel to xoy planes. Combining Figure 3.7, 3.8, we conclude the following: The slices of x = 1 and x = 19 show that the square slice can be separated into two equal parts along z axis, the left is occupied by m onom er B and the right by A. The slice of x = 12 - the m iddle slice of x-axis is opposite to the slices of x = 1 and x = 19 whose left part is occupied by monom er A and right part by m onom er B. The slice of x = 6 show that the square slice can be separated into three parts along z 26 Figure 3.6: (cos#*, cos#y, co s# z) is the unit direction of face diagonals, where #*. 9y, 6Z are the angles between the face diagonal and x. y, --axis respectively, the twelve face diagonals are : (-0.7071,0,-0.7071), (0.7071,0,0.7071). (0.7071.0.7071,0), (-0.7071,-0.7071,0), (0,-0.7071,-0.7071), (0,0.7071,0.7071). 27 [ * k ★ ★ * * * o o o o * i 1 * O k ★ ★ o o o o o o | 1 k k o * k o o o o 1 I * * ★ o o 0 o o o 1 ! ★ k * O o o o 1 1 k * ★ o o o 1 1 k ★ * o k 1 1 * o * * k k k * o 0 O I 1 * k k k 0 o o ★ 1 1 * * k k o o o * 1 1 k o o o o o o o 1 k k k o o o 1 1 ° k o o o o O I 1 * k ★ k k o o 0 o o k ] 1 k k * k k o o o o 1 1 k o o o ° 1 1 ★ k o o 1 1 k * k k k ★ 1 1 ° k k ★ k k * o o o O I X 1 1 k o o ★ * 1 1 ★ 0 o * * k k j 1 * * 0 o 0 o o ★ k 1 1 * o o o o k 1 1 o k k k 1 1 0 o ★ * 1 1 * o o 0 k k k k 1 1 o k k * 1 1 * o o k * k k 1 1 o o o o o o o 1 1 o o o o k 1 1 ★ k * o o o o o o k o k 1 1 * k o o 0 o o o k k 1 1 ★ k * • o o o o 1 1 * o o o o k ★ 1 I * k ★ o o o k 1 1 o o o o o o 1 1 k k o o o o 0 o o * k k 1 I ★ k k o o o o ★ ★ 1 x = 6 Z Y 28 o o o * * k k 1 1 o o k k ★ k 1 1 o o o o o k k k j 1 o o o k o 1 1 o o o k k ★ k k [ 1 ° o o o o k k k k k I 1 0 * o o k k ★ k k 1 1 o o o o + k k * ! 1 o o o o o * ° 1 1 o k ★ k k k 1 1 o o * k k k 1 I 0 o o o o * k k k k k k ° 1 1 k o o o ★ k k k k k k * 1 1 • k k k k i 1 0 0 k k k I 1 o o o o o k k k k k t 1 o o o k k ! 1 o k k o | 1 o o o k 0 1 X 12 1 ° o o o * k k k O 1 1 0 o o k * k k * k o o j 1 o o k k o o 0 1 1 o o o k * k k o O 1 1 o k o o o 0 ° 1 1 o k k k k o o o 1 1 ° k k k k o o o i 1 o k o o k k k k o O 1 I o k k k k o o o 1 1 o k k k o o o o 1 1 o o k k k o o o O 1 1 o o o ★ k k k o o 1 1 o ★ o ★ k k k k k o o ★ o o 1 1 ° o ★ k k k k o 1 1 ° ★ k k k k * 0 o O 1 i ° ★ k k k k k o 1 1 ° 0 k k k * ■ o o 1 1 ° * k k k o o i 1 ° o k k k * o O 1 X = 16 29 1 ° o k ★ k * k k o a 1 1 0 ★ * ★ * k o o o o o o 1 o o k k k o o o o o o o j 1 * k * o _ k o o o t 1 4 c ★ * k k 0 o o 1 1 * 4 c * ■ k ★ k o o 1 1 * o * ★ k 4c o o k 1 1 ° * * k o o o ° ! 1 ° k * k o o i 1 o ★ * k k k k 0 o o o i 1 * k k k k o o o i 1 * k o 0 o o o o 1 1 o o o O 1 1 * ■ k 0 o O 1 1 ° ★ ★ k J f c - 0 o o I i ★ ★ o o 1 1 ★ ★ ★ k o o 1 1 k ★ * * k 1 I o ★ * ★ it k k o o o ° 1 X = 1 9 1 * o o ■ k k 0 0 o o ° 1 1 * * ir k ★ o 0 o O I 4c * * o o o 0 o o 1 * ★ ic o o o 0 o * k 1 j * * * k o k * k * 1 I * * o o 0 0 ★ 1 1 * * o o o o o o 0 * 1 1 * 0 o O 0 k | 1 * o ★ 1 1 o o 0 o o k 1 1 ° o o o o k k k * 1 1 k k 1 1 o * ★ 4 c k o | 1 ° o * • 4 c * 1 1 ° o o * • k * k o O 1 1 ° o ★ * k * o o 1 1 * o ★ k k k o 1 1 o o • k ★ k * o 0 0 o o 1 1 * o k ★ k o o o 0 o 1 1 r z 30 1 ★ ★ ★ * k o ★ o o o * if | 1 ★ * O o o o o o if * 1 + o o o o o o 1 1 ★ k ★ o 1 1 ★ o o o k * * 1 1 * o o o o o o k k 1 1 * o o o o o o o * k k I 1 * o o o o o o k k * 1 1 o o o o o o o k k k * 1 1 o o o o ★ ★ k k 1 1 o ★ k 1 1 o k ★ k k 1 1 ° o o k * • ★ k k k 1 1 o * * ★ k ★ k k o Q 1 1 ° * * ★ ★ k * 0 o ° 1 1 ° * * k ★ k k o o 1 1 k k ★ i t k k k o o O 1 i ★ * ★ ★ k o o o O 1 1 * * k * o o 1 Y = 6 1 ★ * o 0 o 1 1 ★ * * ★ o o 1 1 * ★ o 0 0 o o 1 1 * * 0 o o o 0 o 1 1 ★ o o o 0 o 1 1 * o o o o k O 1 1 * o o o o * * o * 1 1 * * o o o k k 1 1 * 0 o o if k k 1 1 o o o o k k k * * 1 1 0 o o o o k k ★ * k k i f 1 1 o k * k * k * 1 1 o o o k * k * o O ° 1 1 ° o ★ o * k k * ic k o o 0 1 1 o o k k o * 0 O ° 1 1 o o o ★ k * 0 0 0 1 1 o o ★ k k * o o 0 o o O 1 I ° o k k o o o o O 1 1 k ★ O 0 o o O 1 Y = 12 31 1 O ★ k k o o o o o ★ 1 1 ★ k k k o O 1 1 * k k k o 0 o o 1 I * ★ * ★ o o * I 1 * * k o o o o k ★ 1 1 * k o O o o k * * 1 1 * k o o o o o o * 1 1 * ★ o o o o o o o * ■ ★ * 1 1 * o o o o o o k * 1 1 * ★ o o o o 0 o o o 1 1 o 0 o * k 1 1 o * k ★ O 1 1 ° * 1 k O 1 1 ° o o k * k I 1 o 0 o k k * k * o k k 1 1 ° o o o ° 1 1 k k k k o o 1 1 k o k k * o * o o o o o i 1 ° o o k k * o o o O 1 Y 1 9 1 ^ o o k * o o 0 o O 1 1 * * ★ k ★ o o o o 1 1 ★ * k k ★ o o o 0 o 1 1 * * * k k o o 1 1 k * • o o 1 ★ * ★ ★ * o * o 0 o ★ k [ 1 k * * k o o 0 * 1 ' I * k k ★ * 0 o o O 1 1 * k * k k o o o o 0 o | 1 * ★ * o o o 1 1 * k o o o o 1 1 * k o o 0 1 1 o o * k k o o. o 1 1 * o k k • k o o o 0 1 o k k * o o o 1 1 k o o o o o 1 1 o o 1 1 o o o o 1 1 ° k k k 1 1 o 1 I o o 0 o * 1 Z — 1 — ► ▼ Y 32 1 0 o o o o * 1 1 o o o o 0 o k 1 1 o o 0 o o o o o k * 1 1 o o o o o o o k k k A 1 1 o o o o k * 1 1 o o 0 o k k k * 1 1 o o o o o * k ★ k [ 1 o o k k k k k * I 1 o o k k 1 1 ° 0 k k k k k k k * I 1 ° o o o o k k * k k ★ k ★ I 1 o o o o k ★ k * ★ 1 1 o 0 o * ★ 1 1 o o o o * - * k 1 1 o o o o k ★ ★ r k k 1 1 o k ★ k k 1 1 o o k k * 1 1 * o o * k 1 1 * ★ o o o o o 1 o 1 l 0 o 1 z = 1 0 1 ° o ★ k k 1 1 O o o k k ★ O 1 1 ° o 0 k k + O 1 1 ° o k k o 1 1 ° o o * * ★ ★ k ★ o o 0 1 1 o k * * * • j t r k k o 0 1 1 ° o 0 * ■ k • k •k k k k o O 1 1 o o * o o k * k k o o ° 1 1 o k k k o o 1 1 o - k k k k o 0 O 1 1 ° o o * k k k ★ o 0 1 1 ° o ★ o k k k k k ★ o o o 1 1 o o o k k k k o o 1 1 0 o o k 0 o O 1 1 O o o o 0 o k k k * ★ O 1 1 ° o o o o o k k o 1 1 o o 0 k k k k * 0 ° 1 1 o o o k k k k * 1 1 0 o o ★ k ★ 1 z = 14 33 * o * k k o o o o o o ★ ★ k k o o 0 o * k k o o o o ★ * k k k k o o o o o 0 ★ * k k o o o o o * ★ k k o o k llf k k k o o k o ■ k k k o o 0 ■ d r ★ k k k o ★ o o o ★ ★ k k k o o k o o ★ i k o o o o o o k k * o o o o o k k k k k k o o * o o k k ★ o o o o 0 o o o k * o o o o o k k k o o k k o k o o o o o o k k k o o o o z = 19 Figure 3.7: Two lamellae formed along one face diagonal (0.707,0,0.707) when we choose sim ulation box L = 1.414c?',where d' is the lam ellar length for this particular length of polym er chains. Here, N = 20, L — 19, d! — 13.5. * and o are for monomer B and A respectively, we gives som e slices of yoz , xoz and xoy. 34 Figure 3.8: We choose sim ulation box length L = 1.414c?', and get two lam ellae formed along one face diagonal (0.707,0,0.707). Two A-rich regions and two B-rich regions. 35 axis, the left most and right most quarter regions are occupied by m onom er B and the m iddle half region is occupied by monomer A. However, the slice of x = 16 shows the opposite way as the slice of x = 6. The slices of y = 1, y = 6, y = 12, and y = 16 show directly the two lam ellae formed along this face diagonal (0.707, 0, 0.707). Similar to the above, z — 1 and z = 19 show that the square slice can be separated into two equal parts along x axis, the left is occupied by monomer B and the right by A. The slice of 2 = 10 - the m iddle slice of 2 -axis is opposite to the slices of 2 = 1 and 2 = 19 whose left part is occupied by monomer A and right part by monomer B. The slice of 2 = 14 show that the square can be separated into three parts along 2 axis, the left m ost and right m ost quarter regions are filled by monomer A and the middle half region is filled by monomer B. We also plot the com positional density function of monom er A and B along face diagonal (0.7071,0,0.7071) (Figure 3.9) and observe a sinusoidal curve which shows that two lam ellae formed along this diagonal direction in the sim ulation box. The distribution of interface between segregation region of monomer A and B {Figure 3.10) having four peaks shows the sim ulation box can be separated four regions (two A-rich domains and two B-rich dom ains) along the face diagonal which verify further two lam ellae formed along this face diagonal direction in the sim ulation box. We also choose the sim ulation box length L = \/3 d' and get three lam ellae along one of four body diagonals in one sim ulation box because the length of body diagonal is 3d1. Figure 3.11 gives the unit direction (cos 6X, cos 6y, cos$z ) of four body diagonals, where 0X, 0y ,$ z are the angles between the body diagonal and x , y , 2 axis respectively. As shown in Figure 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, we choose the sim ulation box length L = s/3d’, run the sim ulation for som e tim e period and get 36 / 0.50 c « e z o re1 ® 0.40 c a < u sz O ) § 0.30 c o o Q . 0.20 CO c CD T J *c0 c o *= 0.10 C O o C L E 8 0.00 30.0 20.0 10.0 face diagonal direction(0.7071,0,0.7071) 0.0 Figure 3.9: Com positional density profile of monomer A and B along one face diagonal (0.7071,0,0.7071). Em pty is for monomer A and fiiled for monomer B. 37 distribution o f midpoints o f polymer chains 0.40 0.30 0.20 0.10 0.00 30.0 10.0 20.0 - 10.0 0.0 face diagonal direction(0.7071,0,0.7071) Figure 3.10: Distribution of m idpoints of diblock copolym er chains in lamellar phase along one face diagonal direction (0.7071,0,0.7071) for N — 20 and L = 19. Figure 3.11: (cos0x, cos#,,, cos Bz) is the unit direction of body diagonals, where Bx. 0y. Bz are the angles between the body diagonal and x, y, 5-axis respectively, the four body diagonals are: (0.5773, 0.5773, -0.5773), (-0.5773, 0.5773, 0.5773), (0.5773. 0.5773, 0.5773) and (0.5773, -0.5773, 0.5773). 39 X o k O k k k o 0 o o o o o O o o o o 0 . o o o o o o o O o o o o o o o o o o o ■ o o o o o o o o o o o o o o o o o o o o o o o o o o o k o o o o o o o o o o o o o o o o o o o o o o o o 0 o o o o o o o o o o o o o o k o o o o o o o o o o o k 0 o o * o k o o o k k o k * k o 0 o k * k o k k o o k ★ k k k k k * k k k k k * o k k ★ * * * * k k k * k • * k k k k * * k k k k k k k k k k * k k k k k k k k k k k k k k k k k k k k ★ * k k ★ k k k * k k k k * * k k k k ★ k k k k k k k ★ k k k k k ★ k k k o o k k * k o ★ o o o k k * k o o k k k k k o o o o ★ o o k k k k o o o o k o o o o k k k k o o o o o o o Figure 3.12: We choose the sim ulation box length L — 1.732<f', three lam ellae are formed along body diagonal (0.5773,0.5773.0.5773). Here, we select N — 20, d' = 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0, 0.707, -0.707). We define the directions as the following: W here x is (1,0,0), yz is (0, 0.707, -0.707). 40 X o o * • ★ * o o * k k o o o 1 f c k o o o o k k k ★ * o * k k k o ★ k k ★ o o ★ o k k k o k * o o k k ★ k k k k k k k k o o o k k k k k o o k k k o k k k o o k k k O k o o o o ★ o o k o O o k o o o o o o o o * k k o o k o o o o o o k ★ k o o o ■ k o o o o o o k k k o ★ it o o o + ★ k k o o o o k k k o o o o ★ k k o o o o k k o o o o o o k k k o * k o o k k k o k * o o k * k o o ★ o o ★ * k k k o o o k o o o o k ★ k o o o ★ o o o o k o 0 o o o o o o * * k k k o o o o o k k o o o o * * k k o o o o o o o o ★ o o k o o o k k k * o k o k k ★ o Figure 3.13: We choose the sim ulation box length L — 1.732c/', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, cf = 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0, 0.707, 0.707). We define the directions as the following: x is (1,0,0), yoz is (0, 0.707, 0.707). 41 o o o o o O o o o o o 0 o o o o o o 0 0 o o o o o o o o o o 0 o 0 o 0 o o o o o o o o o o o o o o o o o o o 0 0 o o o o o o o o o o o o 0 o o o o o 0 0 o o o o o o o o o o o o o 0 o o o o o o o o o o o o o 0 o o 0 o k o o o o o o o o ★ k k o o o k o o ★ o o o k o k o o k k k k o k k ★ k k k k k * k * k k k k k ★ * k k ★ o ★ k k k • k k k k k ★ k k ★ k k k k k k k k k k k k k k k k k k * k ★ k k k ★ k k k k k k k k k * * k k ★ k * k k k k k k k k k * k k k k k o k k k ★ k k k k k k o k o k k k o k o * o k o o o o o o o o o o o o 0 o o o o o o o o o o o o o Figure 3.14: We choose the simulation box length L = 1.732d", three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N — 20, d! — 13.5, L= 24. The slice cut along the two face diagonals whose direction is (0707, 0, -0.707). We define the directions as the following: y is (0,1,0), xz is (0707, 0, -0.707). 42 XO Z r Y o k k k o o 0 k k * * k k o ★ k k k O o o k * * k k k ★ k k ★ k o o ★ k k k k k k o o * • k * o o o * ★ k k k k + k o o o o k k k k k * k k k k k k k k ★ k k o o o o k k k ★ * o o o o o o ★ k o o k o o o o o k k O o k k o o o o o k k o Q o o o o o o o ★ k o o k k o o o o ■ k k k o o o k k k k o o ★ k k k o o k k k k ★ * k k k k o k k o o ★ k o o O ★ k k k k k o o o o o k k k k k k ★ k o k k k k k * k o o o o k k k k k k k o o k o o k k k o k k o o o k k k k k o o o o k k k o o o o o k k k o o o o o k k k k k o o o o o o k o o o k k k ★ k o o o * o Figure 3.15: We choose the simulation box length L = 1.732cf, three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d! = 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0707, 0, 0.707). We define the directions as the following: y is (0,1,0), xoz is (0707, 0, 0.707). 43 2 o o o o o o o o o 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o k o o o o 0 o o o o o o o o o o o o o o 0 o o o o o o o o o o o o * o * o * k k o o k ★ o o o o o k o o o o * k k k k * ★ ★ ★ ★ * k k o o k * * k k k k ★ ★ ★ k k k * k k ★ * * ★ k k k k k ★ k k k k k k ★ k ★ * * k k k k * k k * k * k ★ k k k k k ★ o ★ k ■ k ★ k k k - * k ★ ★ ★ k k k k k k o k k * k k ★ k k k k k k * ★ k k k o k k k o k k ★ ★ k k * o 0 o k k o k ★ k * k o o o o k * ★ o * o o o o o o o o * ★ o o o k o o o o o o 0 o Figure 3.16: We choose the sim ulation box length L = 1.732d', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, we select N = 20, d! = 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0.707, -0.707, 0). We define the directions as the following: z is (0,0,1), xy is (0.707, -0.707, 0). 44 ► -x o y z o ★ o o o o * ★ k o o ★ ★ k o o o o k o o o o o o * * * o o o o o * k o o o ■ k * * k k o o o o o * * * k o o k k k o o o o ★ ★ k 0 o k * k k o o o o ★ k * 0 k k k o o ★ * k o k k k k o * * o k k k k o o * k o o k k k o o o o k k o o o k o k k o o o o k k k o o k o o o o k * k o o k k o o k ★ k k o o 0 o k ■ ★ k o o o k k k ★ k k k k k a o o k k k ★ * o o o ★ o k k k k k o o * k o o o k k k k o o * • * o o o k k k k k o o o o * k o o o o * * k k k o o o o o k o o o o k k ★ k o o o o ★ k o o o o ★ k k k o k • k o o o k * k o o o o * • k o * + ★ o o o o k k o o o o o o k k k k ★ o ★ k o Figure 3.17: We choose the sim ulation box length L = 1.73*2d', three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). Here, vve select Ar = 20, d1 = 13.5, L = 24. The slice cut along the two face diagonals whose direction is (0.707, 0.707, 0). We define the directions as the following: z is (0,0,1), xoy is (0.707, 0.707, 0). 45 o Figure 3.18: We choose the sim ulation box length L = 1.732d!. Three lam ellae are formed along body diagonal (0.5773,0.5773,0.5773). There are three A-rich and three B-rich regions separated by five planes perpendicular to the body diagonal. Here, N = 20, d! = 13.5, L = 24. 46 c * 1 ° o k k k * * k o k o 1 1 o k o * k k k k k k k * 1 1 o * k k k k k k o k 1 1 ° * k k k k k k k * k k k o 1 1 k k k k k k k ★ o o o o 1 1 k k k ★ k o o o o O I 1 * it k k k * o ★ o o o o o 1 1 * ★ k k o o o o o o I 1 k k k k k k o o o o o o o o O 1 1 ★ * k k ★ k k o o o o o o o 1 1 k k * o o o o o o 1 1 * k * * • o o o o o o * 1 1 * ★ * * o o o o * * * 1 1 * * * k o * k o o - i t • * ° 1 1 * * o o o * • * ★ * 1 1 k * k o o o 0 ★ * ★ ★ 1 1 o o o o o o o o o o ★ k k * ★ I 1 * o o o o o o o o o k k k * * i 1 * o o o o k k k ★ k * 1 1 k o o o o o * k k k k 1 1 o o o o k k ★ ★ k o k 1 1 o o o o o k * o k o o 1 1 o o o o k o k o 1 1 O o o o o o k ★ k k k k k k o o o 1 X = 1 1 * ★ ★ o o o o o o i 1 * ★ k k k o o o o o o o o o o 1 1 ★ * o o o o o o o o k o ★ * 1 1 * ★ o o o o o o o * * 1 I * * o o k o o * o 0 o o o o 1 1 * o o o ★ k o o o o o o * * 1 1 * o o k o o o o * k k * 1 1 o o o o o k * k k k I 1 o o o o o o o o o o o k k k k I 1 * o o o o o o o ★ o k k k k 1 1 ° o o o o k k k k 1 1 o o o o 0 o k k k k 1 j o o o o o o o o o ★ k k k O 1 1 o o o o k o k ★ k k k k o 1 1 ° o o o o o o k * k k k o o 1 1 ° o o o o o o k k ★ o o o 1 I ° o o o k ★ k k o o o 1 1 ° o o o k * ★ k k o o 1 1 o o ★ k k k ★ k k k o o o 1 1 o o k k k k * * o o 1 1 o k k k k k k k o o o o 1 1 k k k k k k k o o o o O 1 1 * * * k k o o o o o o O 1 1 k * o o o o o o o 1 j X = 9 47 * * ★ k at O o o o o o O O at at ! * ★ at at o o o o at at 1 i t * o o at at at at at at ! * ★ O o o o o o at at at at k at at 1 * at o o o k at at at at at at 1 at at o o o o k at at at at 1 k o o o o o o o at at at at at at at at k 1 o o o o o at at at O 1 o o o o k at k o O O | o o o o o o at at O O O | o o o k o o at at k k at at o O 1 o o o at at at k k at at O 1 o o o o at k it at at at k at o 1 o o o o k at at at ★ at at k at o 0 1 o o o o o at at at at at at o 1 o o o o at at at at o O 1 o at k at it at O O o O o 0 1 * at at at at * at at O o I o * ★ at at at at at at o O o O 1 o ■ k k * at at at at at O at o o o I o o * k k k at at at at o o o o O 1 o ♦ at at at at o o o o O 1 ★ * k at at at at O O O o o O o o o k 1 ★ ★ k at k at O o O O o O o at at k 1 X 14 ★ o o o O k k at at at at 1 o o O O o at at at at at at j O o at at at at at at at 1 o o o o at at at at at at at at O o 1 o o o o O at at at at at O O 1 o o o o O at at at at at at at o ° 1 0 o o o o o o at at at at O o O o 1 o o o at * at at at at at at at o o o o ° 1 o at at at at at at at at o o o ° 1 o o O at at at at at O o o o o 1 o * * at at at at at 0 o o o ■ O 1 o * at at at at at o O ° 1 o at at at at at at o O o 1 k * at at at at O o o o o 1 o ★ k at at at o o o 1 o * k k at at at o o at 1 * at at O at o o o 1 o ★ * at k at O o at o at at 1 at k at O O O o at 1 * k O o 0 o o at at at 1 * * * O o o o O O o o o at 1 ★ O O o o o O 1 * o o o o o o O 1 * ★ o O O o o o o O at at at at at at 1 x = 18 48 o o o o ★ k o * k k k k k k o O 1 o o o * k k k k k k k k k k 1 o o o k k k k k k k * k k O 1 o o o k * k k k k o k 1 o o * k k k o o o 1 * o k k k k k o o o o 1 o k k k o o o o O i * k k k o o o o o o o O 1 ★ k k k * k k k k o o o o o o o O 1 o k k k k o o o o o o o o 1 k k k * o o o o o 1 k k k k o o o o o o o o 1 k k k k k k o o o * o * 1 k k * o Q o o o o o * 1 k k k k k ★ 0 o o o o k 1 k k k k k o o o k 1 k k o o o o o o o * k k * k I O o o o o o o o o ★ * k ★ * 1 k k o o o o o o o o o k * k * k k * 1 k o o o o o o o o o k o k k 1 o o o o 0 o k k k k 1 o o o o o o o o k * k k ★ 1 o o o o o o o o o o k k O 1 o o o o ★ k ★ o ★ k k k o O 1 X = 2 4 o o * o k k k k k 0 1 ★ k k * ★ * k k k k o o o 1 o ★ ★ * k k k o o o O 1 * * * * * * k o o o o 1 k ★ k k k k k o o o o o O 1 k k k k ★ ★ k o o o o o O 1 ★ ★ k k k k o o o o i ★ k k o o o o o o 1 * k k o o o o o o * 1 k k k k k o o o o o o k k 1 k k k k k o o o o o o k k k O 1 k o o o o o k k k ] o o o o o o o o o o k k 1 k k o o o o o ★ k k k I k k k k k o o o o o o o o * k k k 1 k k k o o o o o o o o k k k 1 k k k k o o o o k ★ * k k 1 k k k o O o o o * k k 1 k k O o o o o o o o o k k k ★ k k | o o o o o o o k k k k k k k k * 1 o k O o o o k k k k k * i k o o k ★ k k k k k k k ★ 1 o o o k ★ k k k k k O I o o k k o k k k k k O 1 r X k k * k * k k o o o o o O 1 * k k o k k o o o o o o k I * k k k o D o k o o o o o 1 k k k k k o o o o o o o k 1 * k k k 0 o o o o I ★ k k k k k o o o o o 1 ★ k k ★ o o o o o o o o ★ k 1 ★ k k o o o o o o o o ★ 1 k k o o o o o o ★ k k 1 k k k k o o * * k 1 k k o o o k k 1 0 o o k k * * k k k k k 1 o o o o o k k k k ★ k I o o o o * k * k k k k ★ k 1 o o o o o o o o ★ k k k k k k k * - 1 o o o o o ★ k k * ■ k k k k 1 k o o o o k k k k ♦ k k k * 1 o o o o k * k k O 1 o o o o o o k k k k o o o o o o o 1 o o o o * k k k • * k k o o o O 1 o o k ★ k k k k k 0 0 o O 1 o k * * k k k o o o o o o 1 o * k k o k o o o o o O 1 k * k k k k k o o o o o o j y = 7 k o o o k 1 * o o o o o o o o o k I k ★ k k o o o o o o o o o k k * k k k ( * k o o o o o o o o * k k k * 1 o o o o o o o * k k 1 k ★ k k 0 o 0 Q o o * k k k 1 k o o o o o ★ k k k k k k 1 ★ o o o o o o k ★ k k k k * 1 ★ o o o o o o o ★ k k 1 k o o o o o * * k k 1 o o o o o o o o o ★ k k o o 1 o o o o o k k k k k k O 1 o o o o k k k ★ k o o 1 o o o o o k * k k k o O 1 o o o k * * k k o o 1 o o k k k k ★ o o 1 o o k k k k k k k k o o o t * * k k k k o o o o ° 1 * k ★ ★ o o 1 ★ k k * k ★ ★ o k o k o o 1 k k * k * ★ k o o o o o 1 ★ k k * k o o o o o o k | k k o o o o ° 1 k k k o o k o k I y = 13 50 ★ * o o o o o k k * k k k 1 o o o o o o o k k 1 o o o o * ■ i r k 1 o o o k k * k k k o o 1 o o o o o o o k + * k k k ★ k 1 o o o o o k * k k k k ★ ★ O 1 o o o o k * - k k k k o I o o o o k k k k k k o O 1 o o o k k k o o o o 0 O 1 o k k * k k o o o o o o i o o k k k * k k k k k o o o o 0 \ o o * k k k k o 0 o o o O 1 k ★ ★ * k k k k k o o o o o O 1 * k k k k * k k o o o o o o ° i k ★ k k k k k k * o k o o o i o * • k k k o o o * 1 * k k k k o k k k * k o k o 0 o o ★ 1 k ★ ★ k k k o o 0 o o o * * i k * k k o k o o o o * ! k ★ k o o o o o o 0 o 0 0 o A I k k k o o o k k * i k o o o o * * * i k k o o o o o o k + ★ j y 5 = 19 o o o * k k * k ★ o 1 o * o k o k k k k ★ k o ° 1 o o * ★ k k k k k k k k k k o 1 o k k k k k k k k o 0 o o o o 1 o ★ o k k k k k k k k o o o ° 1 * k k * k k k 0 o o 1 o k k k k k k o o o o o o 1 k k k ★ k k k * k k o o o O 1 k k k o o o I k k k k o o o o k k o o ° 1 k ★ k k k k o o o o o o o o 1 k k k k k o o o o o o o o o o k k o * 1 k k k o o o o * o o k t k k k o o o o o o o k 1 k k o o o o o o o o o o * 1 k k k o o o o o o o ★ o o ° 1 k k k o o o o o o o o o 1 k k k o o o o o ★ * ★ 1 k o o o o k * k I o o ★ * k * * 1 k o o o o o o o ★ k ★ * * * 1 o o o o o * ★ * k * o 1 o ★ o o k k k k * * o 1 o o o o o ★ k k k O 1 y = 24 51 f T o o ★ o k k k k k o ★ i t k ★ ★ * k k k o o o o o k ★ * k k k k k o o o o o o o ★ * ★ * k k k o o o o o o * * * ★ * k k k k o o o o o o o ★ ★ k * ★ i t * k k k o o o * ★ ★ * k k k o o o o o o * ★ ★ * ★ o o o o o o o * * ★ ★ * o o o o o o o o o o * k ★ k o o o o o o 0 o o o o o k o o o o o o o o k * o 0 o o o o k o o o * ★ ★ o o o o o o o k ★ k ★ * k o o o o o o o ★ k k * * o o o o o o k * k k * * k o o o o o o o ★ ★ k * ★ * o o o o o k k ★ ★ o o o o o k ★ k k k k o o o o o o o o o k k o o i t k o k o o k o o o o * o k k k o o o o o o o o o o k i t k o o o o o o • i t k k k * k o z = k ★ k o o o o o o k k k ★ k o o k ★ o o o o k k o o o o o o o o o o k k k o o o o o o k o o o o o o o k k k o o o o o o k I t k k k o o o o o o o o k o o o k ★ k o o o o o o o i t i t k k k o o o o o o o i t i t k * k o o o o o o k k i t k o o i t i t k k * k * i t k o o o o o o i t * • i t k k k k k k o o o o o o o o i t k k k i t k ★ o o o o o i t o k k k o o o o o o 1 t o k k k o o o o o o i t k k k k k o o o o o o i t i t k k k k o o o o o k k i t k i t k o o o o o o * k i t * k k i t i t k o o o o o o k * k i t o o o o o k ★ i t o o o o o o * k k i t o o o k i t ★ k k k k i t i t k o o o o z = 8 o o o o o o o o o o k k 1 * k * o o o o o o o o o k ★ * 1 * k ★ o o o o o o o o k k k [ * i k o o o o o o o o ★ k k ★ ★ 1 o o o o o o o k k 1 o o o o o k * k k k k * k 1 o o o o o k k k k k ★ 1 o o o * k k 1 o o o o o o ★ ★ ★ k k k 0 O 1 o o o o k k k k o o O 1 o 0 o o o o o o k * k k k k o o o o 1 o o o o o k ★ k k ★ o o O 1 o o o o k k k k * o o 1 o o ★ o k k * k o 1 o ° o ★ k k * k k o 1 o o * * k k ★ k k k k k k * ■ o o 0 O 1 0 * ★ k k k k k k o o o 0 ° 1 o * ★ k ★ k * k k o 0 0 o o 1 k ★ ♦ * * • k k * k o o o o o o I k ★ * * k k k k k k o o o o o o o o O 1 * ★ * k k k k k o o o o o o o 1 k ★ k k o o o o o o 1 ★ k k o o o o o 1 k ★ k o o o 0 k o o * 1 z = 13 ★ * ★ ★ o o o o k ★ * k k 1 ★ o o o o o k * * * k ★ * 1 * o o o o o * k * * k * k 1 * o o o o o o * ★ k k * k I * ★ o o o o o o o o ★ k k ' k 1 * o o o o o o + * * * k ★ * I ★ o o o o * ★ * k k k k k k 1 o o k ★ * ★ o o i o o o o o ♦ k k k k k k k o o o o o 1 o o o o k o k k o o o o O 1 o o o o * k k ★ o o o 1 o o ★ k k k k k ★ k k o o O 1 o o * * k k k k k k o o o 1 * k k k k k k o o o o o 1 ★ ★ k k k k k o o o o 1 k k ★ k k k o o o o * ★ k k k k o o o 1 ★ • k k k * k k o o o ★ I ★ ★ k k k o ★ 1 k ★ k o o o o o 1 o ■ k ★ k ★ k o o o o o o o 1 o ★ * k o o o o o o o o o o 1 * k o o o o o o 1 * ★ k o o o o o o o o o 1 z = 17 53 o o k k o ★ k ★ k k o * o k * ★ k ★ k o o o o o o o o ★ * ★ k k k * o o o o G o ★ * k k k ★ o o o o o o o k k ★ * k k o o o o o o k k k ★ * k o o o o o ★ k ★ ★ k * k o o o o o o o k * * k k k ★ k o o o o o o o k ★ k k o k o o o o o o o o o o k k o o o o o o o o k + k k k o o o o o o o * k k k k o o o o o o o o • k * k o o k 0 k o k k o o o o o k o * * * k k k o o o o o k k k k o o o o o o o o k k k * * k k o o o o ★ * ★ k o o o o o k k k * ★ k o o o o o o ★ k ★ * o o o * o o * k k k o o o o o o o o * k k * o o o o o k * o k ★ o o o o o o k o ★ k o o o o o o k *• k k o j z = 24 Figure 3.19: We choose the sim ulation box length L = 1.732c/', three lam ellae are formed along one body diagonal whose direction is (0.5773,0.5773 0.5773). Here iV = 20, L — 24, d! = 13.5. T he following is series of slices selected cut along yoz,xoz and xoy planes. 54 three lam ellae structures formed along one body diagonal (0.5773, 0.5773,0.5773). Here N = 20, L = 24 and d' = 13.5. Figure 3.18 gives the three-dim ension structures we get from the sim ulation re sults. There are three A-rich regions and three B-rich regions, all these regions are separated by planes which are perpendicular to this body diagonal. Figure 3.12, 3.13, 3.14, 3.15, 3.16, 3.17 and Figure 3.19 are our sim ulation results. Figure 3.12, 3.13, 3.14, 3.15, 3.16, 3.17 shows the slices cut through two parallel diagonals. Figure 3.12 describes the slice which is cut through two parallel diagonals whose direction are (0, 0.7071, — 0.7071). The body diagonal (0.5773, 0.5773, 0.5773) is not in this plane. Com bining with Figure 3.18,we can divide the slice into two equal parts along x axis, one is A-rich region and the other is B-rich region. Figure 3.13 describes the slice which is cut through two parallel diagonals whose direction are (0,0.7071,0.7071). The body diagonal (0.5773, 0.5773, 0.5773) is in this plane. Com bining with Figure 3.18, we can see clearly from this graph that six regions formed along the body diagonal, three A-rich and three B-rich regions. Figure 3.14, Figure 3.15, Figure 3.16 and Figure 3.17 are the slices which are cut through two parallel diagonals whose direction are (0.7071, 0, — 0.7071), (0.7071, 0, 0.7071), (0.7071, -0 .7 0 7 1 , 0) and (0.7071, 0.7071, 0). Sim ilar to Figure 3.12 and Figure 3.13, if the body diagonal (0.5773,0.5773,0.5773) is in this plane, we can see six regions are distributed along the body diagonal. Otherwise, two equal regions are formed. Figure 3.12, 3.13, 3.14, 3.15, 3.16, 3.17 and Figure 3.19 Figure 3.19 gives the slices which are perpendicular to x, y and z directions (yoz , xoz and xoy respectively). There are four regions in these plane, two for A-rich and two for B-rich. As described in Figure 3.18, the four regions are sliding along one diagonal. That is, four regions are m oving along face diagonal (0, 0.7071, 0.7071), (0.7071, 0, 0.7071) and (0.7071, 0.7071, 0) in yo z , xoz and xoy planes respectively. 55 0.80 CO c 0 o> CO ' ■ 0 > N 1 0.60 JD 0 > J C O ) c: o CO CD 0.40 H — > O t » » Q. & CO c CD " O I 0.20 o w o Q . E 8 0.00 40.0 50.0 0.0 10.0 20.0 30.0 body diagonal direction (0.5773,0.5773,0.5773) Figure 3.20: Com positional density profile of m onom er A and B along one body diagonal direction (0.5773,0.5773,0.5773). Em pty is for m onom er A and filled for monomer B for N = 20 and L = 24. 56 c n c 'ea j= o k _ C D E .> > o Q. O v> -4 — » c o Q . T 5 O C o ‘5 5 3 ja < / > T J 0.030 0.020 0.010 0.000 0.0 10.0 20.0 30,0 40.0 - 10.0 body diagonal direction (0.5773,0.5773,0.5773) Figure 3.21: Distribution of m idpoints of diblock copolym er cahins in lamellar phase along one body direction (0.5773,0.5773,0.5773) for N = 20 and L = 24. 57 0.0 10.0 20.0 30.0 40.0 sim ulation box length direction 50.0 Figure 3.22: Com positional density profile of m onom er A and B along sim ulation box length directions. Em pty is for monomer A and filled for monom er B. Circle is along x-axis, triangle up and down are along y-axis and 2-axis respectively fo iV = 80 and L = 45. Just like L = y/2d', we also plot the com positional density function of monomer A and B along this body diagonal (0.5773, 0.5773, 0.5773) Figure 3.20 which gives a three periods of sinusoidal curve and the distribution of m idpoints of polym er chains Figure 3.21 which have six peaks showing that three lam ellae formed along this body diagonal direction. Two lam ellae form along one sim ulation box direction when we choose sim ulation box L — 2d'. As shown in Figure 3.22 and Figure 3.23, four regions form along one sim ulation box length w ith two A-rich regions and two B-rich regions. 0.20 ® E I 0.15 o o o . g * * — o to 0.10 c o Q . T J E .§ 0.05 to T O 0.00 50.0 10.0 20.0 30.0 40.0 0.0 sim ulation box length direction Figure 3.23: D istribution of m idpoints of diblock copolym er cahins in lam ellar phase along sim ulation box length directions. Circle is along x-axis, triangle up and down are along y-axis and 2 -axis respectively fo N = 80 and L = 45. 59 3 .2 .2 E x tern a l F ield E nergy: The relaxation tim e r for long polym er chains is very long. If we begin the sim ulation at random coordinates, it takes very long tim e, for exam ple, about one m onth for N = 50 to form lam ellae structures in sim ulation box L = 2d' « 39. Assum e we have order structures-lam ellae which may not be the equilibrium structure and we anneal it for relatively short sim ulation tim e period, for exam ple, 5r, we can guarantee that the lam ellae structures gotten through annealing are the structures that we sim ulate very long tim e from the disorder state which will be used for studying therm odynam ical and dynam ical properties later. T he m ain problem becom es how to get initial order microstructures. In this thesis, we solve this problem by putting external field energy cexi (dim ensionless term is e6xt/ k T )into the simulation. As described in flow chart of sim ulation code in Chapter 2, we accept one Monte Carlo m ove by satisfying the excluded volum e, connectivity and energy with a prob ability given by: p = m m { l,e x p ( —SnABcAB/k T )} (3.6) where Stiab is the change in number of A-B contacts, if we put external field energy, the above equation becomes: p = m i n { l , e x p ( - ( 6 n . A B t A B + Seext)fkT )} (3.7) where Seext is the external field energy difference (8eext/k T is dim ensionless term ). How can we put the external field energy in the sim ulation box whose length is 2d'? As shown in Figure 3.24, we can divide the whole sim ulation box into four equal parts 1, 2, 3, 4 along one axis, say, x direction, and then choose a suitable value eAB /k T and involve the external field as shown in Table3.2. 60 .B - - - ■ * A r Figure 3.24: We introduce external energy field by making region 1 and 3 favored by- monomer A and region 2 and 4 favored by monomer B in our sim ulation box whose length L = 2d', where d' is the lamellar length. region monomer A monomer B 1 - + 2 + - 3 - T 4 + - Table 3.2: The sign of external field energy which is put into the sim ulation box 6 1 where -f and — means that we put positive and negative field energy eext in this region for particular kinds of monomers. Sum m arizing all the above, we know that region 1 and 3 favor the existence of monom er A and region 2 and 4 favor that of monomer B, After long enough sim ulation tim e period, we can get two lam ellae with A-rich in region 1 and 3, B-rich in region 2 and 4 respectively. Figure 3.25 is the sim ulation result for N = 26, d! — 14 and L = 2d!. After getting the initial order structures, we anneal it for 3 ~ 5r and attain equilibrium two lam ellae structures as shown in Figure 3.26 3 .2 .3 C rossin g and N on -crossin g: As discussed in Chapter 1, we have two m odels describing polym ers, Rouse Model [10] for short polym er chains and Reptation Model [11] for long polym er chains which are separated by critical molecular weight Mc. Short polym er chains whose molecular weights are sm aller than the critical molecular weight move just like single m olecules, however, long polym er chains whose m olecular weight is bigger than the critical m olec ular weight can not m ove freely with constraints forbidding crossing each other and prefer m oving through reptation especially in high concentrations. In order to study the change of therm odynam ical and dynam ical properties influ enced by the m oving constraints imposed to long polym er chains, we adopt a m odel discussed in Chapter 2. In this m odel, we can turn “on” and “off” the constraints by allowing or forbidding the overlap o f second cubic lattice sites to carry out crossing or non-crossing respectively. 62 0.8 i 1 r sim ulation box length direction Figure 3.25: Com positional density profile of monomer A and B along sim ulation box length direction appliedby external energy field. Em pty is for m onom er A and filled for monomer B. Circle is along x-axis,triangle up and down are along y-axis and z-axis respectively for N = 26 and L=28. 63 0.80 < u 0.60 CL >* to c o 0.40 c o '5 5 o Q. E ^ 0.20 0.00 20.0 30.0 10.0 0.0 simulation box length direction Figure 3.26: Com positional density profile of monomer A and B along sim ulation box length directions. Em pty is for monomer A and filled for m onom er B. Circle is along z-axis, triangle up and down are along y-axis and 2-axis for N = 26 and L = 28. 64 3 .2 .4 H ow to calcu late th e radius o f g y ra tio n o f d iblock cop olym ers w h ich is parallel and p erp en d icu lar to th e lam ellar phase? Sum m arizing all the above, we get the accurate lam ellar length of diblock copoly mers in various degree of polym erization shown in Table'S A. We also get the sta ble lamellar structures which are used for studying therm odynam ical and dynamical properties later for various length of polymer chains at the condition of yjV « 45 and / = 0.5. We anneal the stable lam ellae structures for sufficiently long tim e, record all co ordinates of polym er chains and calculate the radius of gyration paralleling and per- pendicularing to the lamellar phase. The formula for both crossing and non-crossing sim ulations are in the following. ^9,11 = (l / N ) - Vcm)2 + {Zi - z c m )2} (3.8) t ' = i = a w £ { ( * . - *C«)2} (3.9) - t=l where x-axis is perpendicular to the lamellar phase, y -axis and 2-axis are parallel to the lamellar phase, (x cm ,U c m , zc m ) and (x,-, y,, 2;)is the coordinates of center of mass of polym er chains and ith monomer along x, y, 2 axis respectively. R 2 g^ and ^ 9 A are radius of gyration which is parallel and perpendicular to the lamellar phase respectively. 3.3 R esu lts and D iscussion: 3.3.1 R esu lts: TableS.S gives sim ulation parameters. The first colum n of the table is N (degree of polym erization) in the sym m etric monodisperse diblock copolym ers, the third column 65 N N P L Cab/k T XeffW 20 492 27 0.200 44 26 422 28 0.167 48 32 381 29 0.130 46 40 491 34 0.100 44 50 593 39 0.080 44 80 570 45 0.056 49 Table 3.3: Sim ulation parameters: N is the number of m onomers per chain in the sym m etric m onodisperse diblock copolymers; NP is the number of chains in the sim ulation cell; L is the edge length of the primary lattice that forms the cubic periodic sim ulation cell; eAB/k T is the interaction energy for A-B monom er contacts in units of the therm al energy; XeffN is the product of the effective Flory-Huggins parameter and the chain length. The lamellar period is given by L j 2. N d\ d’ d'2/ N 20 13.43 13.5 9.113 26 14.42 14.0 7.538 32 15.02 14.5 6.570 40 17.17 17.0 7.225 50 18.78 19.5 7.605 80 22.73 22.5 6.328 Table 3.4: The sim ulation result of lamellar length from Larson’s and ours c l'h d!. is the sim ulation box length L which is twice of lam ellae length for particular degree of polym erization of diblock copolym er, the second colum n is the number of polymer chains in one sim ulation box Np, the fourth is the dim ensionless term of interactive energy of monom er A and B which is chosen to keep the condition of XejfN ~ 45 listed in colum n 5. Tabled A lists d'h the sim ulation results of Larson in the second colum n and the third colum n is put our sim ulation result of <f. T he fourth gives the value of d'2/N . Table3.5 and Table3.6 show the average square radius of gyration paralleling and perpendicularing to the lamellar phase < R2 ^ > and < R2 x > in colum n 2 and N < *;,n > < * L l > R2 n 9,0 r r' 20 4.71 5.45 6.68 2.31 1.06 32 8.03 7.37 10.67 1.84 1.13 40 9.77 9.18 13.34 1.88 1.10 50 12.50 12.05 16.68 1.93 1.12 80 20.16 17.69 26.68 1.75 1.13 Table 3.5: The sim ulation results of average square radius of gyration com ponents which is parallel and perpendicular to the lam ellae, < > and < R^tl_ > , the square radius of gyration of polym er m elts in random walk R2 0, the ratio of < R2^ L > and < R2 || > , that is r and the ratio of < R2^ > and R2$, that is r' for c r o ssin g situation. N A O S V < ^ , x > *J.o r r' 20 4.63 5.39 6.68 2.33 1.04 32 7.86 7.32 10.67 1.88 1.10 40 9.80 9.25 13.34 1.89 1.10 50 12.33 11.88 16.68 1.93 1.11 80 20.52 17.64 26.68 1.72 1.15 Table 3.6: The sim ulation results of average square radius of gyration com ponents parallel and perpendicular to the lam ellae, < R2^ > and < R2 x > , the square radius o f gyration of polym er m elts in random walk R 2 0, the ratio o f < R 2 x > and < R2^ > , that is r and the ratio of < R 2^ > and Rg Q, that is r' for n o n — c r o ssin g situation. 3 respectively for various degree of polym erization of diblock copolym ers. We also calculate Rg0, r and r' listed in the colum n 4, 5 and 6 using the following equations. 4,0 = (\/AW> r = < 4 .i i >■ / < 4 . x >■ r' = < R l u > , / ( 4 , 0), (3.10) (3.11) (3.12) where the subscript 1 means that average value calculated is scaled in one dim ension. For example: < R l h > , = < > /2 (3.13) < R2 g,± > , = < R]i± > (3.14) (R 2 ,,o) i = «2,o/3 (3.15) where Table3.5 and Table3.6 are for crossing and non-crossing respectively. 3.3.2 D iscu ssion : The comparison of our sim ulation results of lam ellar lengths in various length of polym er chains with those of Larson’s are sum m arized in Tabled A and we conclude that these two sim ulation values are very similar. The results of sim ulations show that the chain topology does not have any signif icant effects on the structures of the diblock copolym er chains in the lamellar phase, for exam ple, the com positional density function along the sim ulation box, the slices cut along three dim ensions and the distribution of m idpoints of diblock copolymers are very similar for crossing and non-crossing sim ulations. Besides these, the m ag nitude of < Rl'X > and < R2 ^ > are very near in both crossing and non-crossing sim ulations for same degree of polym erizations of diblock copolym ers. From the last colum n of Table3A, we know d!2 fN is alm ost constant (approxim ate 7) for various lengths of diblock copolym ers in lamellar phase. The reason why the value (9.113 for N = 20) is higher than 7 is probably that the periodic boundary condition applied to the sim ulation box has highest influence comparing to other degree of polym erizations of diblock copolymers. As shown in Table3.5 and 3.6, one dim ension < R2 x > bigger than one dim ension < || > (r > 1) describes the stronger stretch of diblock copolym ers in lamellar phase in the direction of perpendicularing to the lamellar phase than the direction of 68 paralleling to the lamellar phase. We also observe that one dim ension < R2 j | > a little bit longer than < R2Q > (r‘ fa 1.1) shows that the diblock copolym ers in lamellar phase along the direction which is parallel to the lam ellae a little bit stretched than bulk polym er m elts. 69 C h a p t e r 4 D y n a m i c s 4 . 1 I n t r o d u c t i o n : Dynam ics is one of two fundam ental factors in the process of producing of mar ketable diblock copolym er products. It governs the processibility and ultim ate phys ical properties of the m aterials. Diffusion coefficients of diblock copolym ers in lamellar phase, for exam ple, can be analyzed by putting a diblock copolym er chain into a spatially periodic potential field produced by the rem aining chains. [16, 26] Detailed studies have reported two basic m echanism s of polym er chain m otion, e.g., Rouse-like and reptative due to entanglem ent of long polym er chains. Barrat and Fredrickson [26] reported the Rouse like m otion of sym m etric diblock copolym ers which can be applied to un-entangled diblock lamellar phases. There is one im portant param eter, the critical molecular weight M c, which can separate the polym er chain m otion into two regimes, e.g.,Rouse- like and reptative by the molecular weight of diblock copolym ers. Shaffer [19] has given detailed studies of therm odynam ic and dynam ic properties of bulk polym er m elts. In evaluating the range of applicability, it is necessary to determ ine if the onset of chain entanglem ent in the lamellar phase occurs at the sam e m olecular weight as in the bulk disordered phase. The number of chain entanglem ent between layers in 70 i the lamellar phase must be known to apply m olecular theories of stress relaxation in high m olecular weight diblock copolym ers. [28, 29] In this thesis, we have conducted com puter sim ulations of chain dynam ics of di block copolym ers in lamellar phase, previous detailed studies have focused on the order-disorder transition, [30, 31, 22] chain structure in selective solvents, [32] and dynam ical properties at a fixed degree of polym erization. [33, 34] Here, we em ploy a special lattice m odel developed by Shaffer [19] in which topological restrictions can sim ply be turned “on” and “off” without perturbing the chain structure [19, 18] and show the diffusion coefficients of sym m etric diblock copolym ers in the lamellar phase for a wide range of degree of polym erizations which span crossover from un-entangled to entangled regime. Crossing and noncrossing sim ulations are accom plished by allow ing or forbidding overlap of m idpoints of monomer bond vectors respectively. We also extend the relationship between relaxation tim e rp and wavelength N /p from polym er m elts to diblock copolym ers which form lamellar microstructures and conclude that the com plete distribution of relaxation tim es {rp} in the lamellar phase of diblock copolym ers is provided to test com pleting theories of diblock copolymer dynam ics. Here, we will give com plete distribution {rp} in the lam ellar phase both perpendicular and paralleling to lam ellae and provide test analysis for dynam ic theories of diblock copolym ers in lam ellar phase at condition N = 20 and IV = 80. 4 . 2 M e t h o d : Local m obility can be represented by average segm ent m obility /rav which can be calculated by the acceptance of M onte Carlo steps. Self diffusion coefficients in one dim ension are calculated from the mean-square displacem ents of polym er chains’ center o f mass using the Einstein formula. D = < ( R ( t ) —R (0 ))2 > /2 1, where R (t) denotes the center of mass of a given polym er chain at tim e t. 71 By sim ulating twice for each condition, we can get four values of diffusion coeffi cient in the direction of parallelling to the lamellar phase at each condition D x, D2, D 3 and D 4 and calculate the error bar of D\\ (<5/D||) by SDn = 1 .9 6 ( £ ( A - D av) 2 / n ) ^ 2) (4.1) 1 = 1 where average value of D\\ (D av) are calculated by Dav = Di/n (4.2) 1 = 1 and 1.96 is for statistical probabilities of 96% Di in the range of (D av ± 5D\\) and n = 4. We can transfer the Cartesian coordinates for all polym er beads in the sim ulation box into Rouse coordinates for all polym er chains using the following formula: X p(t) = (1/N ) J2 rj (*) cos(7 rp(j ~ 1 /2 ) /N ) (4.3) j=x where X p(t), rj(t) are the Rouse coordinates for polym er chains and Cartesian coor dinates for polym er beads respectively. After getting X p(t), we can gain Cp(t) by the following: - Cp(t) = < X p(t) ■ X p(0) > / < X 2 p > for p = 1, 2, ...N - 1 (4.4) B y sim ulation , Shaffer [18] find that the autocorrelation function can be described by the stretched exponential form of W illiam s and W atts [37] for bulk polym er m elts. In this work, we will show later that it is also true for diblock copolym ers in lamellar phase. CP{t) = exp{ —( i/r * ) ^ } (4.5) 72 N Crossing p av Noncrossing p a v 20 0.131 0.121 26 — 0.128 32 0.147 0.135 40 0.155 0.143 50 0.160 0.148 80 0.168 0.152 Table 4.1: Average segm ent mobility, pav, for crossing and noncrossing chains as a function of chain length. The average m obility is obtained from the sim ulations as the overall fraction of accepted Monte Carlo moves. where the tim e constant r* and the stretching parameter 0 P depend on the mode number p, chain length N and topological condition. From the above, we give the relaxation tim e rp rco t p = I exP { —( t / Tp) } (4.6) J o 4.3 R esu lts and D iscussion: We calculate average segm ent m obility p av which are the acceptance of Monte Carlo steps for variation of degree of polym erizations of diblock copolym ers in lamellar phases for crossing and non-crossing sim ulations at the condition of ~ 45 and / = 0.5 and list the results in Tafe/e4.1. Shaffer [19] gave the sim ulation results of average segm ent m obility p av for differ ent lengths of bulk polym er m elts. As shown in that paper, p av is approxim ate to 2.0 for crossing and 1.8 for non-crossing polym er chains whose degree of polym erizations are higher than 20 (N > 20). All the values in tableAA are lower than the simula tion values of bulk polym er m elts due to the interaction energy of different types of monom er A and B (c^b/AtT). The lam ellae which restrict the local m obility of diblock copolym ers are formed due to the interactions between different types of monomers. 73 W hen the interaction energy (e^g/fcT) decreases, the local m obility is increased by showing increasing average segm ent m obility ^av. We also conclude that the local m obility /xav for crossing is a little bit higher than the value of /xav for non-crossing. This shows that the topology restriction has a little influence to diblock copolym ers in lamellar phases. In contrast, the chain topology has a dramatic effect on the diffusion dynamics of diblock copolym ers in lamellar phase. We will show the above conclusion using the sim ulation results of diffusion coefficients D for both crossing and non-crossing conditions. As shown in Figure 4.1, mean-square displacem ent gi(t) in the direction which is perpendicular and parallel to the lam ellae as a function of sim ulation tim e t for N = 50 and L = 39. The filled circles correspond to the crossing sim ulations, the em pty circles to the non-crossing sim ulations in x-axis which is perpendicular to the lam ellae. The straight lines formed by triangle up and triangle down show the difference in scaling behavior and also the diffusion coefficients paralleling to the lam ellae are larger in crossing D\\ than in non-crossing D±. The difference in the scaling behavior between the two chain topology should indicate that different microscopic mechanism in the two cases. Supposing there are same m echanism in this two cases, the difference in scaling for diffusion coefficients in two cases are due to only 10% difference in average segm ent m obility, ^av, this is not reasonable. A more likely explanation is the following. In the crossing situation, entanglem ent are absent at all chain lengths and follow Rouse m odel, [10] therefore it satisfies D ~ N ~ x , that is, D N should be constant for all N. For non-crossing sim ulations, when the m olecular weight is lower than the critical m olecular weight, the polym ers m ove just like in crossing situation, when the m olecular weight is higher than the critical m olecular weight, polym ers’ m oving is controlled by entanglem ent of polym er chains showing that D N decreases as increasing N. 74 :V. T - v iiA ^:ie^ e )6 8 < m W m » W W E K X X X X X X X X X X > 0 0 0 <T0 100000.0 200000.0 300000.0 400000.0 sim ulation box length direction Figure 4.1: mean square displacem ent corresponding to sim ulation box length direc tion for both crossing and noncrossing sim ulations. Em pty and filled are for crossing and noncrossing sim ulations respectively. Circle is along x-axis, triangle up and down are along y ~ a x is and z-axis respectively for N = 50 and L = 39. N Crossing Dy SD] } Noncrossing Dy <5D ,| 20 4.20 0.375 1.82 0.230 26 — — 1.45 0.14.3 32 2.85 0.550 1.17 0.128 40 2.60 0.420 0.859 0.090 50 2.13 0.178 0.571 0.020 80 1.52 0.291 0.238 0.275 Table 4.2: Diffusion coefficients paralleling to the lam ellae, Z)y((.Dy ± £Dy) * IQ4), versus degree of polym erization N for the lamellar phase with x N ~ 45 and / = 0.5. N ((D„iV) * 104 S(D\\N) * 104 (DnN / ^ v) * 104 6{DnN / v w ) * 104 20 84.0 7.5 641.22 57.25 32 91.2 17.6 620.41 119.73 40 103.9 16.8 670.32 108.39 50 106.7 8.9 666.88 55.63 80 121.2 23.3 721.43 138.69 Table 4.3: Diffusion coefficients paralleling to the lam ellae, D\\ in crossing sim ulations, versus chain length N for the lamellar phase with \ N « 45 and / = 0.5. Table4.2, TableA.3, TableAA present our m ain results. As shown in TableA.2, TableAA, TableA A, the variation of diffusion coefficients Dq (diffusion coefficient of bulk polym er m elts in random walk) and D\\ (diffusion coefficient of diblock copoly mers paralleling to the lamellar phase) for diffusion m otion w ithin lam ellae at condi tion « 45 and / = 0.5. As shown in Figure 4.2, the term D\\N of diblock copolym ers in lamellar phase for crossing condition decreases as increasing N instead of constant D N w ith N for bulk polym ers m elts in disordei'ed state due to decreasing interaction energy 76 N ((Z>l| A T ) * 104 S(DnN) * 104 (D ,|A 7//av) * 104 6(D\\N/(iw ) * 104 20 36.4 4.6 301.0 38.0 26 37.7 3.7 295.0 28.9 32 37.4 4.1 277.0 30.4 40 34.3 3.6 240.0 25.2 50 28.5 1.0 193.0 6.8 80 19.0 2.2 125.0 14.5 Table 4.4: Diffusion coefficients paralleling to the lam ellae, D|| in noncrossing simu lations, versus chain length N for the lamellar phase w ith ~ 45 and / = 0.5. 0.0016 Z Q 0.0002 15.8 N Figure 4.2: The product of diffusion coefficient of diblock copolym ers in lamellar phase which is parallel to the lam ellae D\\ and degree of polym erization o f diblock copoly mers N as function of degree of polym erization N. Filled is for crossing sim ulations and em pty for noncrossing sim ulations. 77 / 0.0100 0.0010 15.8 Figure 4.3: The ratio of product of diffusion coefficient of diblock copolym ers in lam ellar phase which is parallel to the lam ellae and degree of polym erization of diblock copolym ers N and average segm ent m obility //av)- Filled is for crossing sim ulations and em pty for noncrossing sim ulations. L 78 I param eter(eJ 4jg/^7’) in order to keep constant & 45 as increasing N. To al low for a quantitative comparison between crossing and noncrossing sim ulations , we plot D N / /xav as N in crossing and non-crossing conditions in Figure 4.3. As our expectation, the curves are overlap for both ordered and disordered states (from Shaffer [19]) in crossing condition which verifies further that diblock copolymers in lamellar phase obey Rouse Model [10] in crossing situation. T he shapes of curves are similar in non-crossing condition for both ordered and disordered states. The values of D N/ a r e higher in disordered state than those in ordered state due to the restriction of polym er m oving in lam ellae. Surprisingly, we observe that almost sam e critical molecular weight M c for both ordered and disordered states although we expect the different values of M c. In this thesis, the Rouse m odel [10] is modified to include excluded volum e inter actions and topological constraints, so the autocorrelation functions of Rouse coordi nates can be described by the stretched exponential form of W illiam s and W atts [37] (see equation 4.5). Small values of (3P, that is, greater deviations from purely exponen tial relaxation indicate stronger kinetic constraints and slower relaxation. figureAA show the perfect fitness of the autocorrelation function of Rouse coordinates and the curve described by equation 4.5. We plot the ratio of relaxation tim e tp and the square of wavelength N / p (rp/(N /p )2) as function of wavelength N /p in Figure 4.5 and Figure 4.6 for N — 20 and N = 80 respectively. Diblock copolym ers in lamellar phase with chain length of 20 are below the critical m olecular length, so the main differences between crossing and noncrossing simula tions are due to local m obility /iav. From Figure 4.5, we get the following conclusions. The values of Tp/(N /p )2 in noncrossing sim ulations are higher than those in cross ing sim ulations. The difference of the values of Tpj ( N / p )2 between crossing and noncrossing simu lations is below due to a little bit difference of local m obility p av. 79 0.9 Q. O ■g 2 0.7 0.5 0) CO 3 o tr H — o c o 0.3 o c 3 0.1 c o 3 2 o o 2 - 0.1 -0.3 3 C t ) -0.5 150.0 200.0 100.0 sim ulation time t 0.0 50.0 Figure 4.4: Autocorrelation function of Rouse m odel Cp(t ) as function of sim ulation tim e t for N = 20 and p = 19 in the direction of paralleling to the filled is fitted by W illiam s and W atts equation. 80 Q . 4.0 10.0 N/p Figure 4.5: T he ratio of relaxation tim e rp and square of wavelength N /p as function of wavelength N /p for degree o f polym erization of diblock copolym ers for crossing sim ulations and filled for non-crossing. Triangle up and down are for the direction of paralleling and perpendicularing to the lamellar phase. 8 1 39.8 Q. 2 4.0 100.0 10.0 N/p 1.0 Figure 4.6: The ratio of relaxation, tim e rp and square of wavelength N /p as function of wavelength for N = 80. E m pty is for crossing sim ulations and filled for non-crossing. Triangle up and down are for the direction which are parallel and perpendicular to the lamellar phase. 8 2 In crossing and noncrossing sim ulations, the values of rp/( N /p ) 2 ard approxim ately sam e in both directions for p > 5 because of same local m obility in both directions. In crossing and noncrossing sim ulations, the values of rp/( N /p )2 have increas ing tendency in the direction which is parallel to the lamellar phase and decreasing tendency in the direction which is perpendicular to the lamellar phase. In crossing and noncrossing sim ulations, there are autocorrelation functions which are not decayed to zero and can not be described by equation 4.5 for m odes p = 1 ,3 ,5 and relative high values of rp/(N /p )2 for modes p = 7 ,9 ,1 1 ,1 3 in the direction which is perpendicular to the lamellar phase probably due to macroscopic im m obility of diblock polym er chains in this direction. In both directions, the shape of curves in crossing sim ulations are very similar to that in noncrossing sim ulations. Diblock copolym ers in lamellar phase with chain length of 80 are above the criti cal m olecular length 40, the entanglem ent m ust be introduced and different topology shows dram atic difference. From Figure 4.6, we get the following additional conclu sions besides above. In noncrossing sim ulations, the values of Tv/{N /p )2 produce dram atic increase as increasing wavelength N /p just like in bulk polym er m elts (see Figured [18]) due to entanglem ent between long polym er chains in noncrossing sim ulations. Combining these two figures in Figure 4.7, we observe the overlap curves in both directions for both crossing and noncrossing sim ulations for sm all p (p < 5) and a little bit difference for big p (p > 5) which explains that the local m obility /iav is not contribute to m acroscopic m obility (sm all p) and only influence the local polymer beads’ moving. 83 39.8 Q. 2 4.0 100.0 10.0 N/p 1.0 Figure 4.7: Combing Figure 4.5 and Figure 4.6 . Em pty is for crossing and filled for noncrossing. Circle and diam ond are for the direction which are parallel and perpendicular to the lamellar phase for N = 20 . Triangle up and down are for N = 80 corresponding to circle and diam ond. 84 C hapter 5 C on clu sion and Im provem ent 5.1 C onclusion: After com m ercialization of the class (diblock copolym er) of soft m aterials, engi neers and scientists have been attracted by diblock copolymers for over thirty years. The particular property, therm oplastic elasticity formed by m icro-phase separation in m ulti-blocks, and the surfactant-like behavior of diblocks and graft blocks when blended w ith hom opolym ers have provided im portant application. A fundam ental quantity of polym er dynam ics is M c, the critical molecular weight because of its connectivity to different chain dynam ics and transport properties, ”en- tangled”, delineated by Rouse M odel [10] for short polym er chains whose molecular weight is smaller than Mc, ”un-entangled”, predicted by R eptation Model [11] for long polym er chains whose molecular weight is more than M c. The previous work done by Shaffer [19] has presented the polym er dynam ics in bulk phase. We can im agine that if we know the difference between the critical m olecular weight for the onset of entanglem ent constraints of bulk polym ers and that of hom ogeneous diblock copolym er which forming lamellar micro-phase structure, we can qualitatively even quantitatively apply the results of bulk polym ers to homogeneous diblock copolymer. 85 In this thesis, we try to answer the question: Does the formation of the block copoly mer lamellar phase alter the critical molecular weight for the onset of entanglem ent constraints? In this thesis, com puter sim ulation results of diblock copolym ers for a range of chain lengths spanning the crossover in the lamellar phase at the therm odynam ical condition x N « 45 from un-entangled to entangled regimes have been presented. A m odel created by Shaffer [19] whose general features were inspired by the bond- fluctuation m odels of Carmesin and Kremer [20] and Deutsch and Binder [21] and the structural characteristics specific to block copolym er from previous work of Lar son. [22, 23, 24, 25] By virtue of this lattice m odel, chain connectivity is m aintained by only accepting m oves which generate bond lengths taken the values 1, \/2 , and y/3. Excluded volum e interactions for the monomers are enforced by rejecting any attem pted moves which would result in double occupancy of a primary lattice site, the effective chain topology can be controlled through rejecting or accepting any attem pted moves which would result in double occupancy of a secondary lattice site. Different kind monomer- monomer interactions are quantified through the energy parameter cab for contacts between monomers of types A and B , external energy cext is introduced for long polym er chains conducted in sim ulation boxes with L = 2d' to get stable ordered lamellar structures in short sim ulation tim e. We present the variation of the lamellar thickness, d', with chain length while m aintaining « 45, we also show values reported by Larson [22] at similar therm o dynam ics conditions. As expected, we get similar results as the values o f d1 obtained in ref [22]. We concluded that the ratio d'/Rgfi is nearly independent of N over the range of the chain lengths studied. We also quantify the degree of chain stretching by reporting the radius of gyration com ponents paralleling and perpendicular to the lam ellae, (-R^n) and {R2 g j_), and the ratio r = 2{R2 x )/ {R2 for crossing and noncrossing chains. T he values of r w 2 in 86 the lamellar phase with « 45 indicate that the copolymer chains are elongated som ewhat in the direction which is perpendicular to the A-B interface. The chain length dependence of the local m obility in the bulk phase is due entirely to the increased m obility of chains ends. In our sim ulations of the lamellar phase, however, besides the increased m obility of chain ends, additional variation of /iav with N is introduced through changes in the A-B interaction energy cab, combining these two factors leads to an overall increase in the average segm ent m obility with chain length in the lamellar phase when x N ^ 45 is fixed. Lastly, we report our primary result, the variation of the diffusion coefficients D\\ for diffusion within lamellar in the ordered phase with » 45 for both crossing and noncrossing chains. Crossing chains can be characterized by Rouse model [10] prediction D ~ p & v/N . Entanglem ent caused by noncrossing begin to slow diffusion at the sam e chain length in the lamellar phase as in the hom ogeneous, disordered bulk phase. M icro-phase ordering leading to m oderately segregated lam ellae therefore does not alter the critical molecular weight for the onset of entanglem ent. B y virtue of the effects of micro-phase separation on Mc, we can develop quanti tative theories of chain dynam ics and rheology for block copolym er micro-phases and to apply theoretical treatm ents to specific m aterial system s. We also study comparison of rheological functions for entangled polym er melts in the bulk phase with the lamellar phase by giving the relaxation tim e distribution { tp} for different modes p for both diblock polym er chains which are smaller than the critical molecular weight (for exam ple N = 20) and diblock polym er chains which are bigger than the critical molecular weight (for exam ple N = 80) to verify further that no m atter bulk polym er chains or diblock copolym er chains in lam ellar phase, the dynam ical properties can be divided into two different region by the critical molecular weight due to entanglem ent of long polym er chains. 87 5.2 F uture Work: In the future, we need to study the details of another im portant dynam ical pa rameter viscosity rj to verify further the above results. Probably we need to study therm odynam ic and dynam ic properties of diblock copolym ers in lamellar phase in the region of strong segregation lim it (SSL) and diblock copolym ers in other ordered microstructures. 88 R eferen ce List [1] Bates, Frank S.; Fredrickson, Glenn H.Annu. Rev. Phys. Chem. 1990, 41, 525. [2] Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697. [3] Alm dal, K.; Rosedale, J.H.; Bates, F.S.; W ignall, G.D.; Fredrickson, G.H Phys. Rev. Lett. 1990, 65, 1112. [4] Hamley, I.W; Koppi, K.A.; Rosedale, J.H.; Bates, F.S.; Alm dal, K.; Mortensen, K.; Macromolecules 1993, 26, 5959. [5] Hamley, I.W; Gehlsen, M .D.; Khandpur, A.K.; Koppi, K.A.; Rosedale, J.H.; Schulz, M .F,; Bates, F.S.; Alm dal, K.; Mortensen, K.; J. Phys. 2 (France) 1994, 4, 2161. [6] Disko, M.M; Liang, K.S.; Behai, S.K; Roe, R.J.; Jem , K.J.; Macromolecules 1993, 26, 2783. [7] Schule, M.F; Bates, F.S.; Alm dal, K.; M ortensen, K.; Phys. Rev. Lett. 199 4 , 73, 86. [8] Forster, S.; Khandpur, A.K.; Zhao, J.; Bates, F.S; Hamley, I.W; yan, A.J.; Bras, W.; Macromolecules 1994, 27, 6922. [9] Hajduk, D.A.; Harper, P.E.; Gruner, S.M.; Honeker, C.C.; Kim , G.; Thom as, E.L.; Fetters, L.J.; Macromolecules 1994, 27, 4063. [10] Rouse, P. E. J. Chem. Phys. 1953, 21, 1272. [11] de Gennes, R -G . J. Chem. Phys. 1971, 55, 572. [12] Milner, S.T; W itten, T.A .; Cates, M.E. Macromolecules 1988, 21, 2610. [13] Dalvi, M. C.; Lodge, T. P. Macromolecules 1994, 27, 3487. [14] Eastm an, C. E.; Lodge, T. P. Macromolecules 1994, 27, 5591. [15] D alvi, M. C.; Eastm an, C. E.; Lodge, T. P. Phys. Rev. Lett. 1993, 71, 2591. 89 [16 [17 [18 [19 [20 [21 [22 [23 [24 [25 [26 [27 [28 [29 [30 [31 [32 [33 [34 [35 Fredrickson, G. H.; Milner, S. T. Mater. Res. Soc. Syrup. Proc. 1 9 9 0 , 177, 169. Lodge, T. P.; Dalvi, M. C. Phys. Rev. Lett. 1995, 75, 657. Shaffer, J. S. J. Chem. Phys. 19 9 5 , 103, 761. Shaffer, J. S. J. Chem. Phys. 19 9 4 , 101, 4205. Carmesin, I. and Kremer, K. Macromolecules 1988 21, 2819. Deutsch, H. P. and Binder, K.J. Chem. Phys. 199 1 , 94, 2294. Larson, R. G. Macromolecules 199 4 , 27, 4198. Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411. Larson, R. G. J. Chem. Phys. 1992, 96, 7904. Larson, R. G. Chem. Eng. Sci. 1994, 49, 2833. Barrat, J.-L.; Fredrickson, G. H. Macromolecules 1 9 9 1 , 24, 6378. Barrat, J.-L.; Fredrickson, G. H. J. Chem. Phys. 1 9 9 1 , 95, 1281. W itten, T. A.; Leibler, L.; Pincus, P. A. Macromolecules 1 9 9 0 , 23, 824. R ubenstein, M.; Obukhov, S. P. Macromolecules 199 3 , 26, 1740. Fried, H.; Binder, K. J. Chem. Phys. 1991, 94, 8349. Weyersberg, A.; Vilgis, T. A. Phys. Rev. E 19 9 3 , 48, 377. M olina, L. A.; Friere, J. J. Macromolecules 199 5 , 28, 2705. Haliloglu, T.; Balaji, R.; M attice, W. L. Macromolecules 1 9 9 4 , 27, 1473. Ko, M. B.; M attice, W. L. Macromolecules 1995, 28, 6871. The term topology is used in this paper to denote only the presence or absence of chain crossing constraints, not to specify the chain architecture, e.g., linear, branched, ring, or star. [36] Flory, P. J. Principles of Polymer Chemistry, Cornell University Press: Ithaca, N Y , 1953. [37] W illiam s, G. and W atts, D.C Trans. Faraday Soc. 197 0 , 66, 80. 90

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Creator Pan, Xiaohong (author)

Core Title Computer simulation of thermodynamics and dynamics of diblock copolymers in lamellar phase

Degree Master of Science

Degree Program Chemical Engineering

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

Tag chemistry, polymer,engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Shaffer, J. Scott (committee chair), Tsotsis, Theodore T. (committee member), Yortsos, Yanis C. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-317148

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