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Polymer Adsorption At Solid/Liquid Interface From Solution

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Polym er Adsorption at Solid/Liquid Interface from Solution
by
P an g L iu
A Thesis Presented to the
FACULTY O F T H E SCHOOL O F ENGINEERING
UNIVERSITY O F SOUTHERN CALIFORNIA
In P artial Fulfillment of the
Requirem ents for the Degree
M ASTER O F SCIENCE
(Chem ical Engineering)
Dec. 1995
Copyright 1995 F a n g L iu
UMI Number: 1378423
UMI Microform 1378423
Copyright 1996, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
This thesis, written by
Fang Liu
under the guidance of 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
Master of Science
Chemical Engineering
Date...S3./.l5j.93.
['acuity C om m ittee
Chairman
Dedication
TO M Y GRANDM OTHER
for her love toward life and devotion to her family
Acknowledgment
I wish to express my deepest appreciation to my advisor Dr. K atherine Shing and
Dr. Victor Chang, for their support and encouragement. Working w ith them was
a pleasure and I will always relish this experience. I would also like to thank Dr.
Yortsos, Dr. Salovey and Dr. Shaffer, for their valuable suggestions, fellow students,
for their constant discussions, my friends, for their affections. All of these m ade my
stay in USC a memorable one. Finally, I would like to thank Karen and Min for
their kindness and help.
Contents
D e d ica tio n
■ •
li
A ck n o w led g m en t iii
L ist O f T ables vi
L ist O f F ig u re s vii
A b s tra c t v iii
1 In tro d u c tio n 1
1.1 General fe a tu re s....................................................... ......................................... 3
1 .2 Theoretical d e v elo p m en t...................................... ......................................... 8
1.3 Experim ental o b se rv a tio n s.................................. ......................................... 11
2 C ritic a l A d so rp tio n E n e rg y a n d Q u a n titie s a t C e r ta in E x c e ss A d
s o rp tio n E n e rg y 18
2.1 In tro d u ctio n .......................................................................................................... 18
2 .2 Critical adsorption energy fo rm u la................................................................... 2 1
2.2.1 Roe’s derivation of critical adsorption energy for the case of
atherm al s o lu tio n .................................................................................. 2 1
2.2.2 Our derivation of critical adsorption energy including solvent
e ffe c t.......................................................................................................... 23
2.3 Simulation results and d is c u s s io n .................................................................. 27
2.3.1 Gong’s m o d e l.......................................................................................... 27
2.3.2 Surface excess at critical a d so rp tio n ................................................. 33
2.3.3 Sharp transitions at critical adsorption e n e r g y ............................. 33
2.3.4 Effects of chain length and c o n ce n tra tio n ....................................... 44
2.3.5 Properties at critical a d s o rp tio n ............................................................ 46
2.3.6 Excess adsorption energy and adsorption properties at fixed
excess adsorption e n e r g ie s ................................................................. 47
2.3.7 Segmental anisotropic factors for non-critical and critical ad
sorption 53
iv
Conclusion and future outlook
List Of Tables
2 .1 Characteristics of lattice s tr u c tu r e s ............................................................. 2 1
2.2 Characteristics of lattice models and critical adsorption energy . . . . 26
2.3 List of ten cases calculated....................................................................................46
2.4 Adsorption quantities at critical adsorption - Compare the new crite
rion for critical adsorption energy x"c w ^h the old one Xac- different
X same x?c / X-c.................................................................................................... 47
2.5 Range of adsorption quantities at strong adsorption (r=1000)............... 49
2.6 Range of adsorption quantities at weak adsorption (r=1000).................. 50
2.7 Range of adsorption quantities at depletion (r=1000)............................... 50
2.8 Range of adsorption quantities at strong adsorption (r = 50)................. 51
2.9 Range of adsorption quantities at weak adsorption (r= 50).......................... 51
2.10 Range of adsorption quantities at depletion (r= 50)........................................52
vi
List Of Figures
1.1 Schematic diagram of polym er adsorption at in te rfa c e ........................... 5
2.2 The surface excess of polym er r exc plotted against polymer bulk con
centration (f> * ....................................................................................................... 34
2.3 Surface tension versus adsorption e n e rg y .................................................... 36
2.4 Free energy versus adsorption energy............................................................. 37
2.5 Surface excess versus adsorption energy. ................................................... 38
2.6 Bound fraction versus adsorption energy...................................................... 39
2.7 Fraction of tail versus adsorption energy....................................................... 40
2.8 Root-M ean-Square thickness of adsorbed layers versus adsorption en
ergy. 41
2.9 Root-M ean-Square thickness of loops versus adsorption energy............. 42
2.10 Root-M ean-Square thickness of tails versus adsorption energy. . . . . 43
2.11 Chain length effect and bulk concentration effect to critical adsorption
energy value.................................................................................................................45
2.12 Three dimensional plot of segmental anisotropic factors at critical
adsorption energy-new criteria applied........................................................... 5 5
2.13 Three dimensional plot of segmental anisotropic factors at critical
adsorption energy-old criteria applied.................................................................56
2.14 Compare segmental anisotropic factors at x« = 0.312 w ith th a t at
(Xt)old — 0.312. Com pare m ade for two x values........................................ 57
vii
Abstract
Theoretical studies of polymers confined at solid/liquid and liquid/liquid inter
faces have been carried out. The Flory lattice model in the self-consistent mean-
field approxim ation has been adopted in this study. The critical adsorption energy
for non-atherm al solution, including solvent effect and chain length effect has been
form ulated as x*c = — (1 — 7 )ln (l — <^i) ~ ^iX (l — 2 < A £ )> where x is Flory-Huggins
interaction param eter, r is chain length, Aj is lattice param eter and (f> * is bulk concen
tration. To compare the new criterion for critical adsorption energy with the old one,
properties of polymer confined at solid/solution interface at critical adsorption have
been sim ulated. Simulation results show th a t the adsorption properties such as free
energy at interfacial layers, surface tension, surface excess, bound fraction, fraction
of tail, root-mean-square thickniss of adsorbed layers, root-mean-square thickness
of loop and root-mean-square-thickness of tail all exhibit sharp transition at critical
adsorption. The transition point (corresponding to critical adsorption energy) is a
function of polymer-solvent interaction param eter x and bulk concentration as well
as chain length. The critical adsorption energy obtained through sim ulation is well
consistent w ith the value given by our formular for critical adsorption energy. This
might provide a novel method to measure adsorption energy of polymer on solid
surface which is difficult to get through experim ental m easurem ent, i.e. by m easur
ing the transition point of the adsorption properties (for example root-m ean-square
thickness of adsorbed layer), knowing x, r and < f> * (which are easy to obtain), using
above formula, we can get polymer-surface interaction param eter x* = Xac at critical
adsorption.
Another interesting phenomenon under the new criterion for critical adsorption
energy is, given chain length r, the adsorption properties all were found to rem ain
unchanged at critical adsorption while the Flory-Huggins interaction param eter x
and < f> * change substantially.
viii
For non-critical adsorption, an excess adsorption energy is proposed to measure
the adsorption energy deviation from the critical adsorption point. Sim ulation re
sults show th a t through excess adsorption energy the range of properties of polym er
confined at solid/liquid interface can be roughly predicted. This can be used to
avoid sometimes tedious experiments or simulations.
Chapter 1
Introduction
The system of polymer adsorption from solution at solid/liquid interface has great
practical importance. Its applications can be found in colloid science [1, 2, 3], petro
chemical technology, coating [4, 5], adhesion [6 ], chromatography, bio-compatibility,
etc. Its im portance becomes increasingly evident as the usage of polymer m aterials
continue to rise.
One of the m ajor application areas for polym er adsorption at interface is in colloid
dispersions, in which the polymer is added either as a stabilizer or as a de-stabilizer.
In this application, the m icrostructure and thickness of adsorbed polym er layers are
key properties. Suspensions of small particles in solution often form aggregations
due to the attraction between particles. For dispersions in organic solvent systems,
polymer is usually added as a stabilizer to inhibit particle coagulation, in th at the
adsorbed layers of polymer can act as steric barriers between two particles. Although
an electrostatic stabilizer can also be used, a polym eric stabilizer is usually prefered
over an electrostatic one for the following reasons: (1 ) in some aqueous solutions,
electro-viscous effects and the existence of electrolytes m ay be undesirable; (2 ) a
polymer stabilizer can provide stability for a longer tim e at high solid concentration;
(3) even when aggregation of grains does occur in polym er stabilized system, it could
be re-dispersed by simply changing the solvent condition.
Because of its long chain character, one polym er molecule can be adsorbed onto
more than one solid particle, a process known as bridging. Polym er bridging of
suspension can cause grains to aggregate into larger clusters which is prone to sed
im entation. This process has a num ber of im portant applications. In the area of
water purification, the particles in water could be removed by adding oppositely
1
charged, water-soluble polymers. An encouraging application would be to remove
coal particle and tar from crude oil through a precipitation process induced by bridg
ing rather than a more expensive and more time-consuming distillation process.
W hen polymer is used to hold two solid m aterials together, it acts as an adhesive
[7]. Polymeric adhesives have been widely used to bind similar or dissim ilar surfaces,
which include metals, woods, plastics, etc. The bonding strength of polym er adhe
sives depend largely on the structure, boundary properties and interfacial energy of
the adsorbed polymer layer between two surfaces.
Polymer adsorption is also involved in wetting phenomenon [8 ], which describes
the process of spreading of a liquid on a solid substrate. The liquid can be polymeric
surfactants, paints, inks or dyes. The solid surface can be paper substrates, fibers or
finely divided suspensions. Since polymer can be adsorbed strongly at solid-liquid
interface, it is expected th at they will wet the substrate m ore effectively th an small
molecules do. In wetting process, surface tension and polymer-surface interaction are
im portant param eters and have a strong effect on the w etting process and polym er
structure at interface.
The polymer adsorption also finds applications in the biochemical process and
bio-m aterials [9, 10]. For example, adsorption of biopolymers such as proteins,
polysaccharides and lipids, onto artificial organs is the critical factor th a t determ ine
the usability of artificial m aterials in hum an body.
The relevance of polymer adsorption to the diversity of applications described
above has attracted intensive research activities during past four decades [11]-[15].
New developments in both experim ental and theoretical approaches have substan
tially advanced our knowledge in this field. First, m odern experim ental technology,
especially small angle neutron scattering technology provides b etter physical insight
to equilibrium and dynamic behaviors of polymer adsorption. Secondly, high speed
computers facilitate the Monte Carlo and molecular dynamics sim ulations in this
field. Thirdly, theoretical models are getting more sophisticated. However, despite
these efforts, the mechanism of polymer adsorption and how it relates to practical
applications quantitatively are still far from being understood. This encourages us
to do more investigation in this area. In this study, we propose a new criteria for
critical adsorption energy to sim ulate the adsorption properties at critical adsorp
tion energy, and to compare them with the results based on the old one. We shall
2
also define a new param eter: “excess adsorption energy”, to describe the energy
deviation from the critical adsorption energy. From the value of excess adsorption
energy, we are able to predict the adsorption behavior and the range of adsorption
properties. This prediction will provide a useful guidance to experim ental studies.
1.1 General features
W hen polym er solution is brought into contact with a solid surface, there are m any
forces in action at the same tim e: the interaction between polymer segment and
solid surface usp, between solvent and solid surface us0, between segment and seg
m ent hpp, between segment and solvent /iop, and between solvent molecules hoo.
There is also an entropy loss due to the presence of the wall. The concentration
profile and m icro-structure of polym er at interfaces is the result of the com petitive
adsorption of polymer and solvent, governed by the principle of free energy m ini
m ization. There are m any factors th a t influence this process, which include polymer
characteristics, solvent power and surface characteristics. The polymer could have
geometric an d /o r construct differences: geometrically it could be branching or lin
ear, flexible or rigid, chemically it could be non-ionic polymer or polyelectrolytes,
homopolymer or block polymer. The solvent can be good or poor, dependent on its
solubility to the polymer. The surface can be geometrically and/or chemically differ
ent too: Geometrically, it could be flat, spherical, cylindrical or irregular; physically
it can be sm ooth or rough; chemically it can be isotropic or heterogeneous. How
ever, m ost theoretical studies so far are restricted to flexible, linear, homopolymer
or block polym er adsorbed on a sm ooth flat surface.
The adsorption of polym er a t solid/liquid interface is quite different from th at
of small molecule adsorption. Some attem pts have been m ade to apply Langm uir’s
equation and Freudlich’s equation, which are used for small molecule adsorption, to
polymer adsorption. Langm uir’s adsorption isotherm agrees well with experim ental
data only at low concentrations [16], Freudlich’s equation agrees rather poorly at all
concentrations, especially at low concentrations [17]. The big difference between the
small molecule and polymer adsorption lies in the fact th at the polymer has large
num ber of configurations, both in the solution and at interface. Upon adsorption, the
entropy loss per polymer chain due to adsorption is much greater than th a t of small
3
molecules. The number of configurations of a flexible chain increases tremendously
with the increase of chain length. Some other differences are: (1 ) the adsorption rate
of polymer is very low because of it small diffusion coefficient. (2 ) once the polymer
is adsorbed, it is difficult to desorb it by dilution. This is due to the m ultiple
attachm ents of a single chain on the surface. The total adsorption energy is high for
a chain even though the segm ental adsorption energy is usually comparable to th at
of small molecules. This leads to some people conclude th at polymer adsorption is
irreversible. But in reality polym er adsorption is therm odynam ically reversible, the
desorption can be achieved by adding different solvent or more strongly adsorbed
polymers.
W hen polymer solution is brought into contact with a solid surface, due to the
com petition of adsorption between polym er and solvent, It could result in positive,
critical or negative adsorption, corresponding to w hether the polymer concentration
at the interface is greater than, equal to or smaller th an the bulk concentration. See
Figure 1.1.
Positive adsorption: (a)
The polymer is favored by th e surface over the solvent, resulting in accumula
tion of polymer near or on the surface. The concentration of polymer at the
interfacial region is higher th an th a t of bulk, and it decreases with the distance
away from the surface, reaching the bulk concentration at some point.
Critical adsorption(b):
The energy change upon adsorption is balanced by the entropy loss due to the
presence of the surface. The concentration of polym er at the interfacial region
is equal to th at of the bulk.
Negative adsorption(depletion):(c)
The solvent is favored by the surface over the polymer, resulting in driving
polymer segments away from the surface. The concentration of polymer at
the interfacial region is lower than th a t of the bulk, and it increases with the
distance away from the surface, reaching the bulk concentration at some point.
The factors th at determ ine the properties of polym er adsorbed at interface are:
4
a: Positive A dsorption
bulk concentration Is 0.1
0.6
£
7 9
0.3
0.2
0.1
16 25 30
D istance from the Wall
20
Distance
40 46 35 50
b: Critical Adsorption
0.2 -----------------1 -----------------1 -----------------1 - ----------------1 -----------------1 -----------------1 - - - - - - -- - - - - -'i i ■ — r —
0.18 -bulk concentration Is 0.1 -
0.18 - -
| ° . 1 4 .
£
W 0.12
E rt 1
- -
qT _ i
E 0-08 - -
g 0.06 - -
0.04 - -
0.02
n ■ - ■ - - « - ■ « » — | — - ■ ■
O 5 10 15 20 26 30 36 40 45 50
Distance from the Wall
c: Negtlve Adeorptlon(Depletlon)
0.14 - bulk concentration Is 0.1
0.12
0.1
0.08
0.08
0.04
0.02
lO 16 20 26
Layers Away from the Wall
30 35 40 45 60
the Wall
Figure 1.1: Schematic diagram of three types of behavior of polymer adsorption
at interface, a: positive adsorption; b: critical adsorption; c: negative adsorp-
tion(depletion).
5
1)Segment adsorption energy Xs-
2)Flory-Huggins interaction param eter x-
3)Polymer bulk concentration < f> * .
4)Molecular weight and molecular weight distribution.
5)Surface shape and regularity.
Adsorption energy Xs is the key param eter that determines the outcome of poly
m er adsorption. It is defined as the interaction energy difference between the sur
face/solvent uao and surface/polym er uap. The dimensionless adsorption energy Xs
is w ritten as:
x . = 2 = ^ * (1.1)
If Xs is positive, it means polymer segment is favored over the solvent molecule
by the adsorbent. It is vice versa if Xs is negative. High value of Xs results in
strong adsorption, as Xs decreases, adsorption becomes weaker. Adsorption does
not occur until the adsorption energy exceeds a critical value Xsc- This critical
adsorption energy is needed to compensate for the free energy increase due to the
conformational loss of polymer molecules in the presence of the solid surface, and
the free energy change caused by segment-solvent interactions. Below the critical
value of Xsc, depletion (or negative adsorption) occurs. This critical value is called
critical adsorption energy.
The nature of solvent is another im portant factor th at has profound effects on
polymer adsorption. Polymer-solvent interaction can be characterized by Flory-
Huggins interaction param eter x [18], which is defined as potential energy change
when an unlike contact between polymer segment and solvent occurs as follow:
v — z(^°p ~ 2^°° hP P )) . .
X - kT (L.t)
where the hop, hpp and h00 are the interaction energy of polymer-solvent, polymer-
polymer and solvent-solvent respectively. Polymer in a poor solvent are adsorbed
more easily than th at in a good solvent because of the weaker interaction with the
solvent. The adsorbed amount from a poor solvent can be three times bigger than
th a t from a good one. Theoretically, x value can be negative for very good solvent, 0
6
for atherm al solvent, 0.5 for 0 solvent. Experim entally, typical x values lie between
0.35 to 0.5.
At most bulk concentrations, the translational entropy is inversely proportional
to the chain length. Though low concentration favors adsorption of longer chain over
th at of short ones, at high bulk concentration, conformational entropy become more
im portant than the transitional entropy, therefore the adsorption of short chain is
favored.
Molecular weight also has strong influence on the adsorption properties. For
example the root-mean-square (RMS) thickness is directly proportional to the square
root of molecular weight. Molecular weight distribution influence both adsorption
kinetic and adsorption equilibrium. Polymers of shorter chains adsorb faster at the
beginning due to larger diffusion coefficients, but they will gradually be replaced by
longer chains due to stronger adsorption energy of longer chains. The adsorption
amount is strongly dependent on the molecular weight distribution too.
Surface roughness and configuration influence the adsorption greatly as well.
The same adsorbent but with different preparation process m ay results in different
adsorption properties. Surface factor is often very complex, most existing theories
are lim ited to flat and smooth surfaces.
In addition, there is an entropy loss due to configuration decrease when polymer
is adsorbed at surface.
Tem perature affect adsorption by changing the solvent polym er quality x 7 the
adsorption energy Xs, and increase the macromolecule m obility of polymers.
All of the above factors can vary independently and influence the process of
polymer adsorption in a complicated way. Some effects may reinforce each other
and some may cancel each other. Exact results have only been obtained for in
finitely long chain adsorbed from the ideal systems th at the interaction of segments
with each other and with solvent molecules can be neglected. W ith consideration
of segm ent/segm ent and segm ent/solvent interaction, only num erical solution has
been obtained. It seems th at no system atic studies have been designed to explore
the combined effect of all these param eters. In chapter 2, we shall propose a new
param eter, excess adsorption energy xt, to combine all the factors. By using the
value of Xg, we can roughly predict the range of adsorption param eter values.
7
1.2 Theoretical developm ent
Theoretical development in this field can be divided into three stages: In the first
stage, an ideal infinitely long chain adsorbed on a flat solid wall, w ith one of its end
attached on the solid surface, and the tail effect was ignored. M ost typical approach
used a random-walk model. The excluded-volume effect and segm ent/solvent inter
action effect were ignored. In the first stage, the exact solution was obtained, and
only concentration profile and properties derived from concentration profile were cal
culated. In the second stage, an ideal infinitely long chain adsorbed at interface was
studied, and one of its end attached to the solid surface, so tail effect was ignored
too. In this stage the excluded-volume effect was taken into account by employing
a self-avoiding random-walk model, however, the segm ent/solvent interaction effect
was still ignored. Exact solutions were obtained, and the properties calculated were
the same as those in the first stage. Third stage was mainly represented by a group
of self-consistent mean field theories, based on Flory-Huggins lattice m odel of poly
m er solution. W ithin the realm of mean-field approxim ation, theory of adsorption
of finite, interacting polymer chains can be derived, thus effects of tails, excluded
volume effect and segment-solvent interactions effect can be incorporated. P roper
ties including concentration profile and distributions of segments can be obtained.
However, due to the complexity of resultant equations, only num erical solutions are
obtained so far.
The first model for theoretical study of polymer adsorption a t interface was
proposed by Simha et al [19]. An isolated chain with its one end adsorbed on a
surface was treated. The interaction between polymer segments and solvent was
ignored. The significance of his contribution is th at he established the basic picture
of long flexible polymer chain adsorbed on a solid surface, as a collection of loops
extending into the solution, trains lying on the surface and tails dangling from the
end of the chain to the first surface contact. But the model is far from reality, since
the segments do not feel each other, there is no lim it of surface accom m odation, the
adsorbed am ount increase linearly with the increase in bulk concentration, w ithout
bounds.
Silberberg [2 0 ] first adopted quasi-crystalline lattice model for an interacting
polymer adsorption system, and derived analytical theories. He assum ed th a t loops
8
had a sharp distribution, and ignored the tail effect, polymer-solvent interactions
were represented by Flory-Huggins interaction param eter. Hoeve [21] modified Sil-
berberg’s theory by assuming a Gaussian distribution for the loop segments, but
he still neglected the tail effects. Their approaches were quite similar, both started
from the statistics of an isolated chain. Later on, Roe [22], and Scheutjens-Fleer
[23] separately modified the theory. Both of them took account of tail effect. Their
approach were different from th a t of Silberberg’s and Hoeve’s in a way th at they
did not start from an individual chain but calculated the partition function for
the system, i.e. the solution, by calculating the total ways of polymer and solvent
molecules to arrange themselves in the lattice. The equilibrium concentration profile
was obtained by the m axim ization of the partition function, i.e. minim ization the
free energy of the system. Flory-Huggins model was used to account for segment-
segment and segment-solvent interaction. The difference between Roe’s model and
Scheutjens-Fleer’s model is th a t the latter gave not only the concentration profile,
but also the distribution of trains, loops and tails, consequently their com putation
tim e was much longer than Roe’s com putation time.
However, Scheutjens and Fleer did not consider the connectivity of segments
and the orientational bias of segment advancement in different layers. The model
developed by Gong et al fully [38, 39] incorporated the connectivity of segments
and the orientational bias of segment advancement in different layers. It was based
on Helfand’s mean-field model [40] for infinitely long polymer chains [40]. Gong
extended Helfand’s model into finite chain, by introducing a new set of anisotropic
factors: segmental anisotropic factor t( i ,j ), which accounts for the segmental distri
bution dependence on the ranking num ber of the chain. Comparing Gong’s model
with Scheutjens-Fleer model, the former has advantages in the following areas: first,
it takes full account of chain connectivity, indicated by the layer anisotropic factors
g t i 9^i and 97 • Secondly, its com putation is much simpler, since the introduction
of the two sets of anisotropic factors: t(i j ) and g f , < 7°, and g f greatly reduces the
com putational difficulty in deriving numerical solutions. Furthermore, Gong’s model
can be easily adapted to m any different systems.
New development of analytic theory of polymer adsorption are largely based on
the mean-field model and Dolan-Edwards [24] diffusion equation approach. The ad
sorption of ring-liked polymers [25], star-liked polymers [26], copolymers [27, 28, 29],
9
polyelectrolytes [30, 31, 32] and adsorption between two surfaces [33, 34, 35] have
been studied as extensions of Scheutjens-Fleer’s mean-field model. The contributions
by Russel at al [36, 37] revived the use of diffusion equation approach to the ad
sorption of interaction system. T he modified diffusion equation was solved through
eigenfunction expansion. A variety of results, including the adsorption am ount,
bound fraction, surface coverage, and adsorbed layer thickness were obtained. The
diffusion equation approach is lim ited to infinitely long chain (tail effect was ignored),
and analytic solutions of higher than first order is not viable at present time.
Mean field theories have been quite successful in treating high concentration
systems. However, in the system of the polym er coil swelling in good solvents and
overlapping each other, or in extrem ely dilute solution systems, the spatial segment
density fluctuates, and the m ean field assum ption is invalid. For such systems,
scaling m ethod has been used.
Scaling theories of adsorption are an extension of scaling treatm ent of polymer
solution which was pioneered by de Gennes [41]. The adsorbed layer is divided into
three regimes (i) proximal regime: which is in the close vicinity of the surface, where
the segment concentration is sensitive to the details of segment-surface interaction,
(ii) central regime: where the concentration profile and the number of monomer “g”
_ 4 _ i
in a subunit of sizes £ follow the scaling law, < f> p = a and g = < f> p 5. (iii) dis
tal regime: where the concentration profile decays exponentially towards the bulk
concentration. Scaling theory give fast predictions of general trends. It is particu
larly good at information for long chain molecule where self-similarity are significant.
Scaling theory is only qualitative in a sense th a t it does not yield definitive numerical
results. However scaling theory is only qualitative, and is lim ited to linear, flexible
and neutral chains in a good solvent. Scaling theory in conjunction with mean-field
assum ption has been applied into different systems, such as term inally anchored
polymer chain, polymer adsorption between two plates and in pores [42, 43], onto
colloidal particles [44], and adsorption of block copolymers [45, 46].
More recently, Monte Carlo sim ulation and molecular dynamic sim ulation [47]-
[56] have also been widely used for the study of polymer adsorption. The former is
easier to apply but lim ited to static properties; while the latter can provide tim e-
dependent as well as tim e independent information.
10
1.3 Experimental observations
Tremendous progresses have been made on experim ental m easurem ent of adsorption
properties, and experimental d ata are often compared with theoretical predictions.
For the experimentally measurable quantities such as adsorption isotherm , bound
fraction and adsorption thickness and concentration profile, qualitative agreement
has been found between theory and experiment. Q uantitative agreements between
experim ental results and theoretical predictions have not been achieved. The dis
crepancies may be attributed to several aspects: the inherent lim itations of lattice
model itself, the polymer polydispersity, characterization of the adsorbent, etc.
In a lattice model, the same lattice size is used to characterize bulk polymer, pure
solvent and polymer solution of all interm ediate compositions. This is often not jus
tified, since in m any cases the polymer segment and solvent size varies as the polymer
concentration varies due to the difference in polymer-polymer and polymer-solvent
interactions. In addition, most theories are lim ited to only monodisperse homopoly
mers adsorbed on smooth and flat surface. In experim ental m easurem ents, polymers
used all often have some range (wide or narrow) of molecular weight distribution,
and the surface are often not sm ooth and homogeneous. All of these factors m ake it
extrem ely difficult to quantitatively compare experim ental results with theoretical
predictions.
Among all the measurable quantities, adsorption isotherm , which is the plot of
the adsorbed am ount at interface as a function of bulk concentration, is the easiest to
be measured, other quantities are more difficult to get due to m any reasons discussed
below.
l)A dsorption Isotherm:
The most common experimental m ethod to m easure adsorption isotherm is to
bring a polymer solution into contact with an absorbent, and the adsorbed am ount
can be determ ined by measuring the residual polymer solution concentration.
Existing theories predict an initial increase of adsorption am ount w ith the in
crease of molecular weight, followed by a plateau region for mono-dispersed polymer
adsorption. A sharp transition to the plateau region is predicted by theory. Experi
m ental results are in qualitative agreement with experim ent. However, round-shaped
11
isotherm instead of sharp transition is often observed [57] experimentally due to the
polydispersity effects of polymers.
2)Bound Fraction “p”:
Bound fraction is defined as the fraction of train segments over the total adsorbed
segments, including trains, loops and tails. Available experim ental measurements
are spectroscopic measurements such as infrared (IR) [58], electron paramagnetic
resonance (EPR ) [59] and nuclear magnetic resonance (NMR) [60] and therm ody
namic m ethod like micro-calorimetry [58]. None of them is quantitative.
The infrared technique relies on the frequency shift of IR band induced by inter
action between the adsorbed functional groups and active sites. EPR and NMR use
spin-labeled species to chemically attach to a polymer at random. They are based
on the change (longer) in the relaxation times of protons (’H NMR) or free electrons
in trains due to the reduction of mobility. Micro-calorimetry is based on the as
sum ption th at a proportionality between heat of adsorption Htr and bound fraction
“p ”, which m ay not be justified. The bound fraction measured by IR usually has
much lower values compare to th a t by EPR, NMR and calorimetry [58, 59, 60] for
the same system.
The explanation for the contradictions could be as follows: the infrared m ethod
detects only those segments which have specific bonds with the surface. Dispersion
force has no consequence for the infrared spectrum, although they may contribute
the the adsorption energy, so it is very likely that infrared m ethod underestim ates
“p” . However, E P R and NMR also have limitations. One of them is th at the spin-
labeled species m ight influence adsorption behavior, the other is that the mobility
of un-adsorbed segments is often affected by the adjacent adsorbed segments, this
would lead to overestim ate “p”. In micro-calorimetry, the proportionality between
Htr and “p” may not be justified, which may either overestim ate or underestim ate
V-
Com pare to these different techniques, some people [61] believe th at magnetic
spectroscopy (NMR, EPS) seems a more reliable way to determ ine the bound frac
tion, micro calorim etry also yields reasonable result, but only for relatively short
polym er chains.
12
3)Adsorbed Layer Thickness 8:
Thickness of adsorbed layers is an im portant quantity in practical applications.
The ’staying’ character of the dangling portion (loops and tails) is the basis for dis
persion and emulsion stabilization. The thickness of adsorbed layers has been mea
sured both hydrodynamically and statically. The available static m ethods are ellip-
sometry [62,63,64], Fourier Transform infrared attenuate to tal reflection (FT IR /A T R )
[65], and small angle neutron scattering (SANS) [66]. Both ellipsom etry and F T IR /A T R
methods detect the refractive indices of the adsorbed layers and th a t of the bulk so
lutions. This requires highly smooth surface which can give strong optical reflection,
which is difficult to accomplish in most cases. For th a t reason, hydrodynam ic m eth
ods are commonly used, and the result is often referred to as hydrodynam ic thickness
8}i. Hydrodynamic thickness is usually regarded as being the effective thickness of
adsorbed layers.
The most frequently used methods to m easure hydrodynam ic thickness 8h can
be divided into two categories. One is based on the reduction of flux of solvent when
polymer is adsorbed on the wall of flow channel. This category includes sintered
glass disks, porous membranes, single glass capillary [67, 68, 69]; The other is to
measure the increase of radius of colloidal particles upon adsorption of polymer
on surface, which include viscosimetry, sedim entation coefficient m easurem ent [70],
microelectrophoresis [71], and photon correlation spectroscopy (PCS) [72, 73].
In the m ethod using viscosimetry, the adsorbed layer thickness is determ ined by
measuring the viscosity of suspension, since the increase of suspension viscosity is
related to the increase of particle radius in the following way
where 77/770 is the relative viscosity, 7 7 is the suspension viscosity, 770 is the supernatant
viscosity, $ 0 is the volume fraction of dispersed phase w ithout polym er, k is the
Einstein coefficient of the suspension, r 0 is the particle radius, and 8 is the thickness
of the adsorbed polymer layer.
Sedimentation coefficient measurem ent explores the property th a t the sedimen
tation coefficient s of a particle is related to its radius as follow:
13
where s° is obtained by extrapolation of the s curve, as a function of the particle
concentration to zero particle concentration, r is the radius of the particle and 8 is
the adsorbed polymer layer thickness.
In microelectrophoresis (ME), when a non-ionic polymer is adsorbed on a particle
surface, a displacement of the shear plane takes place, and the distance of displace
m ent of the shear plane can be taken as the thickness of the adsorbed polym er
layer.
tan h ^ ~ 4 & f l = “ A )1 ( L 5)
where X is the normal distance to the surface and 'k(X ) is the potential in the
diffuse phase. 2 is the valence of the counter ion, e is the elem entary charge, A is
the thickness of the Stern layer, i.e. approxim ately 3-4 angstrom , 'k j is the potential
m easured in the absence of polymer, and k is the Debye-Huckel param eter. The
thickness of the adsorbed polymer layer 8 equals X when 'k(X’) is substituted by 'k
which can be measured directly by ME, so 8 can be calculated from this equation by
measurem ent of the zeta potential of the suspensions with and w ithout the presence
of polymer.
According to the Stoke-Einstein equation [74], PCS determ ines the diam eter
of particles through their diffusion coefficients which are m easured by analysis of
fluctuation of the diffused light intensity of the particles w ith tim e by means of a
photom ultiplier.
Siffert and Li [75] had compared the adsorbed thickness at solid-liquid interface
by different techniques: photon correlation spectroscopy (PCS), m icroelectrophore
sis, sedim entation coefficient m easurem ent and viscosimetry. Their results indicate
th at the uncertainty of the results by sedim entation m easurem ents and by viscosime
try are very high, since it is difficult to m aintain the suspension stable during the
m easurem ent, while the result obtained by PCS and ME are relatively significant.
The results given by PCS is slightly higher than those by ME, this m ight be caused
by the fact th at ME attenuates the contribution of tails, so it is possible th a t ME
underestim ates the thickness of the adsorbed polymer layer. Among these four tech
niques, PCS was most recommended.
For th eta solvent (x = 0.5), existing theories [38] [84] [21] [23] predict th at the
RMS thickness S of adsorbed layers is proportional to the square root of molecu
lar weight, i.e. S ~ M 2 , which is in good agreement with experiments [76]. For
good solvents (x = 0), m ean field theories [23] predict the exponent dependence
of thickness is 0.8 [77], i.e. S ~ M 0-8, which has agreement with the experim ental
d ata on polystyrene lattices covered by PO F (in water) [73], while the scaling theory
predicts the exponent as 0.6 [73], and was supported by another experim ent on the
same system [78].
The hydrodynam ic thickness is often much larger than the thickness m easured by
ellipsom etry and neutron scattering. The reason might be th at the volume fraction
of tails at the periphery of adsorbed layer is very low, which can not be detected by
ellipsom etry and SANS, while hydrodynam ic techniques are more sensitive in this
regard. Another reason is th at certain experim ental m ethod is based on a special
model and uses a special m ethod of data analysis, therefore it is quite possible th at
different experim ent techniques m ay yield different thickness values.
4) Concentration Profile:
T he polym er concentration profile can be investigated by small angle neutron
scattering (SANS) [79], evanescent wave ellipsometry [80], and neutron reflectivity
[81]. M ost of the experim ents are conducted on adsorption of copolymer at solid
interface.
SANS has been used to detect the polymer concentration profile at interface.
SANS has been a powerful technique for it can determ ine not only the overall size
but also the spatial distribution of loops and tails at the interface. Q ualitative com
parison of the experim ental d ata using SANS with theoretical predictions has been
m ade [61]. The shapes of the profile are similar. However experim ental concentra
tion gives lower value at large distance [78]. The reason could be th at the SANS
may not detect the tail segments in the peripheral layers, so the tail effect may not
be seen.
Evanescent wave ellipsom etry can also be used to measure polymer concentration
profile near liquid/solid interface [80]. If the thickness of the interfacial layer is small
compared to the optical wavelength, then the critical angle of the total internal
15
reflection 0* is determ ined solely by the dielectric constant of the bulk phases. The
phase difference at critical incident angle 0 < is given by equation 1 .6 .
9 A t * t°° 1
A»(e,) - ( e i _ e i ) 1 / 2 1 ^ [ £J-£,*))[£(*)-£ ,]< fe (i.6)
where ko is 2tt/X, A is the wavelength of laser beam in vacuum, and 62, £1, and
6(z) are the dielectric constants of the dense, lean, and interface region at distance z
from the interface, respectively. The sign of A $ ( 0 f) gives a qualitative description of
whether the interfacial layer is an adsorbed (positive) or a depletion layer (negative).
T he concentration profile for positive adsorption is given by equation 1.7 [82].
$(z)/$b = coth2(z/R + cr) (1.7)
The polymer concentration for depletion layer is given by equation 1.8 [83].
$ ( z ) /$ t = tanh 2(z/R) (1.8)
where $ 6 is the bulk concentration, R is the characteristic length, and 1/cr is the
polym er concentration enhancem ent factor at surface. This technique has a limited
spatial resolution in determ ining the concentration profile because of the finite op
tical wavelength. Also it has large error when it is used to measure depletion layer
due to smaller signal-to-noise ratio.
Recently new developments in both experim ental and theoretical approaches have
substantially advanced our knowledge in this field. First, m odern experim ental tech
nology, especially small angle neutron scattering technology provides better physical
insight to equilibrium and dynam ic behavior of polymer adsorption. Secondly, high
speed com puters facilitate the M onte Carlo and molecular dynamics simulations in
this field. Thirdly, theoretical models are getting more sophisticated. This encour
ages us to do more investigation in this area. In this study, we propose a new criteria
for critical adsorption energy to sim ulate the adsorption properties at critical ad
sorption energy, and to compare them with results based on the old one. We shall
also define a new param eter: “excess adsorption energy”, to describe the energy
deviation from the critical adsorption energy. From the value of excess adsorption
16
energy, we are able to predict the adsorption behavior and the range of adsorption
param eters. This prediction will provide a useful guidance to experim ental studies.
17
Chapter 2
Critical Adsorption Energy and Quantities at
Certain Excess Adsorption Energy
2.1 Introduction
Critical adsorption is an im portant phenomena in polymer adsorption on a solid
surface. At critical adsorption, the free energy change caused by entropy loss due to
the presence of the wall is counterpart by that caused by interactions. As a result,
the interfacial concentration of polymer is equal to th at of the bulk concentration,
and surface excess of polymer at interfacial layers equal to zero, so is surface tension.
In the vicinity of critical adsorption , quantities such as root-m ean-square thickness
of adsorbed layers, root-mean-square extensions due to loops and tails, and bound
fraction undergo a sharp transition.
Critical adsorption of polymer at solid/liquid interface has been theoretically
studied by Roe [84], E. A. DiMarzio and F. L. McCrackin [85], and Rubin [8 6 ] as
early as 1965. Rubin and DiMarzio independently developed a random-walk model
for an isolated polymer chain adsorbed at interface with one end attached on solid
surface. The solvent effect and exclusive volume effect were excluded in their model.
The analytical solution for critical adsorption energy was only determ ined in the lim it
of infinitely long chain. Roe used the partition function pertaining to an isolated
chain, and the solution was obtained for r — > • oo in the absence of solvent effect.
Both their studies arrived at the same result, th at is the critical adsorption energy
Xac = — ln (l — Ai), (Ai is lattice param eter. Assume every lattice has Z nearest
neighbors, the fraction of lattice sites at the layer above or at the layer below is
18
Aj, the fraction of lattice sites at the same layer is A 0, and 2 Ai + A o = 1.)- Their
conclusions are only valid for atherm al solutions of an infinitely long chain. This
criterion for critical adsorption energy without solvent effect, Xac = — ln (l — A),
has been used since then, even for non-atherm al solutions. Scheutjens and Fleer
used it as adsorption criteria in 1979 for non-atherm al solutions [23]. They plotted
the adsorbed am ount T and surface coverage 0 as a function of adsorption energy
Xa. If the transition point (which correspond to critical adsorption energy) is read
from their plot Figure 9, Xac can be found to be about 0.288 for atherm al solution
(x = 0)» but 0-17 for theta solution (x = 0.5). They used Xac = — ln (l — A)
as general criterion for critical adsorption in their paper [8 8 ] in 1982 again. In
1986, Scheutjens-Fleer and Cohen Stuart [89] plotted the fraction of tail segments
as a function of adsorption energy Xa for th eta solvent. They used the 0.288 value
for critical adsorption energy for hexagonal lattice model, but the transition point
of hydrodynam ic thickness in Figure 9 is corresponding to a value of 0.17 rather
than 0.28. In 1988, Fleer, Scheutjens and Cohen S tuart realized th a t the critical
adsorption energy is a function of solvent quality [90], but they did not give detailed
derivation and did not consider the concentration and chain length effect either. No
further studies about the critical adsorption behaviors have been carried out.
In this work, we have derived the approxim ate form ula for critical adsorption
energy for finite length chain adsorbed from non-atherm al solutions. The solvent
effect, the chain length effect and the bulk concentration effect have been taken into
account. Flory’s quasi-crystalline lattice model was adopted here and mean-field
assum ption was made. Two assum ptions were m ade in this study to simplify the
m athem atical treatm ent: First, the bulk concentration assum ption, i.e. the con
centration at interface is the same as the bulk concentration, which is a reasonable
assum ption for critical adsorption case. Secondly, interaction energy assum ption, at
some stage of the derivation, we assume the interaction between polym er/polym er
and sol vent/sol vent are similar, this is somehow not always true, however, the energy
difference contributed to the total free energy caused by this assum ption is insignif
icant compared to the total free energy. Sim ulation results show th at this is a good
approximation. The chain length effect was added in an em pirical but plausible way.
We have proposed a new param eter: “excess adsorption energy”, as a mea
surem ent of energy deviation from critical adsorption energy. We have shown th at
19
excess adsorption energy greater than zero, equal to zero, and sm aller than zero
corresponds to positive adsorption, critical adsorption, and negative adsorption re
spectively. Furthermore, from the value of excess adsorption energy, we can roughly
predict the value range of adsorption quantities without perform ing the sometimes
tedious experiments or simulations.
Simulations were conduct for the following properties to check the correctness of
the new criteria for critical adsorption energy.
(a) The surface excess at the new critical adsorption energy were compared with
th at at the old one. Four cases x = ~2, x = — 1, X — 0-2, and x = 0-5 were chosen
for this study, (see Figure 2.2). Simulation results show th at the critical adsorption
is a function of polymer-solvent interaction param eter Xi because the zero surface
excess point varies with x values.
(b) Thermodynamic properties and m acrostructure properties were plotted against
adsorption energy. Several concentrations ((f)* — 1 0 " 2, < f> * = 10-3 , 4>* = 10-5 ) and
polymer-solvent interaction param eters (x = — 2, -0.5, and 0.5) were chosen. The
results show th at adsorption properties have sharp transition at critical adsorption,
and the transition points is a function of the polymer-solvent interaction param eter
X-
(c) Surface excess was plotted against bulk concentration at a adsorption energy
near critical adsorption energy Xac Xa = 0.2. Several chain lengths, r=10, r=100,
and r=1000, were calculated. Since the critical adsorption energy is a function of
bulk concentration and chain length, we expect th at for a different chain length, zero
surface excess would occur at a different bulk concentration. This is indeed the case
as we shown in Figure 2.11. The concentration resulted from zero surface excess by
com puter simulation agrees very well with the value calculated based on our model.
(d)The various adsorption quantities were sim ulated at critical adsorption cases.
Several combinations were used: < f)* = 10- 3 and < f> * = 10“5, and polymer-solvent
interaction param eter x = — 2, -1, 0, 0.2, 0.4. The sim ulation results show th at
as long as the combination leads to critical adsorption energy, the value of adsorp
tion param eters remain unchanged even the bulk concentration and polymer-solvent
interaction change substantially.
20
lattice Z Zo # of sites at the adjacent layer Ai A o
simple cubic 6 4 1
i
6
2
3
hexagonal 1 2 6 3
I
4
_Y _._
?
face-centered cubic 1 2 4 4
I
3 3
Table 2.1: Characteristics of lattice structures
2.2 Critical adsorption energy formula
We will first give brief introduction of Roe’s derivation of critical adsorption energy
in the absence of solvent, then give detailed description of our derivation for critical
adsorption energy in the presence of solvent.
2.2.1 R o e’s derivation o f critical adsorption energy for the
case o f atherm al solution
The space is modeled as lattice model w ith each layer has So lattice site. Each lattice
has Z nearest neighbors. A o and Ai are the site fractions in the same layer and in
each of the neighboring layers respectively. The common lattice types are listed in
table 2 .1 .
Roe [22] derived partition function Zn pertaining to a single isolated polym er
chain, solvent effect was not considered there. A microscopic state of an adsorbed
polymer chain was given as they occur along the chain:
S — (do, O j, dj,..., dm ? dm, am +i, (2.1)
The partition function Zn was evaluated, with generating function r(£ ) = %n(n
m ethod due to Lifson [91]. The equation for the partition function Zn for the finite
length was obtained from r ( ( ) by the use of Cauchy’s residue theorem , w ith the
contour integration being evaluated by the steepest-descent method. The resulting
saddle-point equation was
. . 1 - = u (() ^ CM + i / a
V(C) {U n - l - 2 ( A 2 { }
where the U, V are sequence generating functions for sequence partition function of
desorbed loop u(d) and adsorbed sequence v(a).
21
The sequence partition function u(d) equates the probability th at a segment
starting from the origin in a given direction returns to the origin for the first tim e
on the (d + l)th step, i.e.
u(d) = I , d I (2.3)
v 2,i U W + 1 ) / 1 ’
and corresponding sequence generating function U is
O O
> d
u({) =
= ( l 1 - A)/C (2.4)
where A = (1 — C 2)*.
The equations for partition function of adsorbed sequence v(a) and corresponding
generating function V are
v(a) = 7 sa (2.5)
and
V ( 0 = 7 sC /(l - »<) (2.6)
where 7 is the sequence end correction factor. It represents the relative tendency of
a segment to term inate the adsorbed sequence -~2 Za - as compare to its tendency to
continue the sequence Z0. So 7 = z ~^° (Zo is the num ber of neighbor lattice site at
the same layer and Z is the total num ber of neighbor lattice site per lattice), s is
the segmental adsorption constant and was expressed as
s = aexp(—A c /k T ) (2.7)
where a accounts for entropy effect, it was calculated [85, 8 6 ] from lattice models
in the absence of solvent molecules as The critical transition point sc is given
as the value of s at which the partition function Zn for a molecule of infinite chain
22
length equals unity. That is, sc is the solution for s of equation U(1)V(1) = 1. The
solution for sc was
Therefore the critical adsorption energy was given by
- A ec/k T = ln(sc/cr)
= -ln[(l + 7)Z0/Z] (2.9)
from which follows that — A ec/k T is equal to ln (|) for the simple cubic, ln ( |) for
the hexagonal, and ln (|) for the face-centered cubic lattice.
In their paper, numerical solution for the saddle-point equation for the face-center
cubic lattice type was plotted against s, as the value of s varies past the critical value
sc, the relative number of segment belongs to loops and to tails vary drastically, and
consequently either loops or tails predominates except when s is very near sc.
2.2.2 Our derivation of critical adsorption energy including
solvent effect
Flory’s quasi-crystalline lattice model was adopted in our study [18], where the
lattice is divided into L layers parallel to the surface, w ith S q sites in each layer.
Each site has Z nearest neighbors, A 0 and A i are the site fractions in the same layer
and in each of the neighboring layers respectively, and A o + 2AX = 1 . The system
is assumed to be a mixture of np polymer molecules of length r and n 0 solvent
molecules. The solution is assumed to be homogeneous. Each polymer segment or
solvent molecule is allowed to occupy only one lattice site, and each lattice site is
occupied either by a polymer segment or a solvent molecule. The volume fractions
of segments and solvent in layer i are denoted by < f > % and < f > % 0 respectively.
The forces in the system are interactions between polymer segm ent/solid surface,
between solvent/solid surface, between segm ent/segm ent, between segm ent/solvent
and between solvent/sol vent, represented by interaction energy u3p, ui0, hpp, hop,
hoo respectively.
23
The interaction energy of polym er/surface uap represents the interaction en
ergy change of the transferring a segment from bulk polym er to the surface, sol
vent/surface interaction uao is the interaction energy change of the transferring a
solvent molecule from pure solvent to the surface.
The dimensionless segment adsorption energy Xa is defined as:
X. = (2-10)
The Flory-Huggins interaction param eter x is defined as the follow [18]:
_ z (hop ~ 2 ( ^ 0 0 + hpp)) /o 11 \
x - kT
Let (f> p represents the polymer bulk concentration, < / > * represents bulk concentra
tion of solvent, and — 1. < / > * represents the polym er concentration in layer i at
interfacial region. Two assum ptions are m ade in the following derivation to simplify
the formula for critical adsorption energy. First assum ption is the concentration at
interface is the same as the bulk concentration, i.e. < f > x p = < j > * which is a good assump
tion at critical adsorption case. The second is interaction energy assum ption, to get
equation 2.19 from equation 2.17, we assume hpp = h00, this is somehow not exactly
true but energy difference caused by this assum ption give very small contribution to
the whole energy, simulation results show th at this is a good approxim ation.
The process of a polymer segment in solution replacing a solvent molecule at
interface can be illustrated in the following picture:
interface s interface s
0 0
© ©
before adsorption after adsorption
The free energy change of a solvent adsorbed at interface replaced by a polymer
segment from solution is calculated as the follows:
A F = A H - T A S (2.12)
24
where
A H — usp -f z( \ q 4- Ai)(l — (f>p)hp0 + z(Ao 4- ^i)(f>phpp
~i~z(l < f> p)h00 zcj)ph0p
U go z(^X q -f* A j)(l < f> p)hop z(Ao 4" Xi)(f>ph0p
~ z ( 1 “ <%)hpp ~ Z(f > * P (2.13)
Each segment in the surface layer loses a fraction Ai of its possible orientations
as com pared to the bulk solution, so the conformational entropy loss for per segment
is
A S = - k l n ( l - X 1) (2.14)
where k is the Boltzmann constant, so
A F — uap 4- z{Xq 4- A i)(l — < f> p)hpo 4- z(Ao 4- Xi)*hpp
+z(l - (f> *)h0 0 4- z 4 > * ph0 p
w ao z(X0 4" Ai)(l < l> p )h 0 p z{Xq 4” Xi)phop
- z ( 1 - < l> ;)hpp - z< Fp - (kT ln (l - AO (2.15)
A F = U ap ua o 4” Ai2 ( 1 $p)h00 4" X\zphp0
- X lZ(l - < f> ;)h po - XlZ(f> ;hpp - k T ln (l - Ax) (2.16)
f p = -X . + 2flA ,x + ^ f T - W - ln (l - A,) (2.17)
assume hpp = h00, we get
z
kT
and
X umihpo h00) (2.18)
25
15r = - X . - - ' i X ( l - 2 ^ ) - l n ( l - A 1) (2.19)
At critical adsorption, the entropy loss due to the presence of the solid surface
is offset by the enthalpy change of replacing one solvent by a polymer segment, so
A F = 0 and xa = Xac, so we get
Xsc = - ln(l - Ai) - A x x(l - 24Q (2.20)
As we know, critical adsorption energy Xac decreases as chain length r decreases,
for r = l atherm al solution, Xac — 0. To incorporate chain length effect, we put an
empirical factor (1 — A) in front of the entropy term ,
x « = - ( 1 - - ) M 1 - A,) - AxX( i - 2 ^ ) (2 .21 )
r
Lattice z A o -^1 new XsC old x»c
Close-packed hexagonal 12
1
2
1
4
(l-±)ln (4 )-S (l —2&) I n i
Simple cubic 6
2
3
1
6
ln(|)
Body-centered cubic 8 0
1
2
( l - I ) l n 2 - * ( l - 2 « ) In 2
Face-center cubic 12
1
3
1
3 ( l - l ) I n ( 2 ) - ? ( l - 2 « )
In I
Table 2.2: Characteristics of lattice models and critical adsorption energy
We got the critical adsorption energy counting the solvent effect and chain length
effect is x"c = - ( l - i ) l n ( l - A 1) - A 1x ( l - 2 <^;), which is x"c = ( 1 - J ) In § - f ( 1 - 2 $ ;)
for close-packed Hexagonal lattice. This critical adsorption energies for four types
of lattice are listed in Table 2.2.
The above derivation is sim ilar to Cohen, Fleer and Scheutjens’s derivation in
their “displacem ent of polym ers” [92]. They described a method to desorb polymers
through displacem ent by a low molecular weight component. The critical displace
concentration, above which the polym er desorption is complete, was derived using
a m ethod which is sim ilar to our m ethod: they assumed that close to the critical
point the polym er surface excess is very small and the segment density equals the
bulk value except the first layer. They proposed a way to estim ate the adsorp
tion energy, which involved a lot of param eters: param eters for polym er/solvent,
26
displacer/solvent and polym er/displacer interactions, as well as for polym er/solid,
displacer/solid and solvent/solid interaction energies. All of them are difficult to
obtain and cause more uncertainty. Another thing needed to notice is th a t they still
did not include polym er/solvent interaction into the critical adsorption energy even
in non-atherm al solvent case. In their experimental part [92], they assumed th a t the
polym er/displacer interaction x pd — 0 an(l polym er/solid interaction energy equal
displacer/solid interaction energy, and got negative critical adsorption energy. T h at
was not consistent with their criterion Xsc = — ln (l — Ai), which should be positive.
Even though they attributed this disagreement to the rotational entropy difference,
the experim ental result still did not fit their formula. In our criterion, th e critical
adsorption energy could be negative for good solvent.
2.3 Simulation results and discussion
All the simulations were conducted using the program originally developed by Dr.
Xuefeng Gong, who was an earlier researcher in our group. Some modifications have
been made.
2.3.1 G ong’s m odel
Gong [38] used Flory lattice statistics in which the excluded volume and the nearest
neighbor interactions are taken into account in the self-consistent mean-field approx
imation. The main feature of his theory was characterized by two sets of anisotropic
factors. The first set, g f , gf, and g j accounts for the orientational preference of
packing successive segments along the chain advancing to the layer above, to the
same layer and to the layer below respectively. The second is a set of segm ental
anisotropic factors t(j,i) accounting for the dependence of individual segment distri
bution over the layer i on the ranking number of (t)th segment along the chain in
an inhomogeneous field. By adopting the grand canonical ensemble form alism , the
free energy of the polymer solution is obtained.
The following notations are used to describe his model: c j > * represents the poly
m er bulk concentration, < f > * represents bulk concentration of solvent, and < ^ > * + < ? ! > * = 1 .
< f > ' p represents the polymer concentration in layer i at interfacial region, < f > * 0 represents
27
Solid Surface
Train Loop
Adsorbed Polymer Chain Free Polymer Chain
Figure 2.1: The schematic plot of polymer adsorption at planar
solid / solution interface fron solution
28
the solvent concentration in layer i at interfacial region, and < j > p -f- = 1 . (ftp)a
represents the polymer concentration in layer i contributed by adsorbed polymer,
(< fipY represents the polymer concentration in layer i contributed by free polymer
(see Figure 2.1), and (ftp)a + (ft0Y = ftp- (ftp)1 represents the polymer concentra
tion in layer i contributed by loop, (ftp)ir represents the polymer concentration in
layer i contributed by train, (ftpY represents the polym er concentration in layer i
contributed by tail, and (< f > p)' + (ftp)tr + ftpY = (ftp)a.
The total free energy of mixing A F mtx is the result of A H mtx minus T A S mtx.
T he entropy of mixing of the interfacial system was evaluated by counting all the
possible configurations of the solution.
A S m ,x = — S0kB ]T) 2 ln9? + (! “ 2Ai )g° ln£?° + A x gT In f ir ," ]
i=l *=1 r
- S .* f l£ [ i# i( M ) lii( 4 < ( M ) ) + £ M j l (2.22)
1=1 r
The enthalpy of mixing was approxim ated by Gragg-W illiams random mixing
approximations in each layer
A H * 6 = S ,k BT \x £ + (1 - 2 A ,)^ + (2.23)
i=l
where Xi X «p> and Xso are the Flory-Huggins interaction param eter, solid-polymer
interaction param eter and solvent-solvent interaction param eter, respectively. The
combining equation of free energy of mixing is
P 'm ix Jjfm ix q m ix
S0kBT S0kBT S0kB
= ^-9i + (1 ~ 2Ai)5'°ln^° + Aig f In # -]
« 3= 1 “
+ Y\ftoln fto + \ftpt(l , 0 ln (ftpt( i, 0)]
+ x £ # [ W p+1 + (1 — 2Ai)<^p + Ai^p-1] + Xwftp + X»oft0 (2.24)
29
To determ ine the concentration profile and other properties, the interfacial free
energy is minimized with respect to < j ) J , and g f , < 7°, and g l subject to following
constrains.
Xl91 + (1 - 2At)g? + Xygr = 1 (2.25)
and
£ i)gf = £ * + 1 )^r i+i
a=l «=1
M inimization yields:
g f = exp(—ai - /?,•)
S ', 0 = e x p ( - a . )
g - = ex p (-a i + /3,_i)
and
_ a . g i ( M _ l n ( 1 _ ^ + ? M (ln ^ t ( i, 0 + 1)
«=i r T
-2 A ,x (4 + 1 - 2 4 + 4 ' 1) - x.<(« - 1)
+ l n ( l - ^ ) - ! ^ + 2 x ( 4 - 4 ) - i = 0 (2.30)
The resultant system of simultaneous equations 2.25 and 2.30 is solved to yield
the concentration profile and anisotropic factors. O ther adsorption quantities can
be expressed in term s of the concentration profile.
The quantities th at are generally used to describe the adsorption are:
1. Concentration Profile: The concentration profile is the polym er concentration
in the interfacial region as a function of distance from the solid surface. Strong
adsorption gives high concentration near the surface.
2. Total Free Energy of Mixing
The total free energy of mixing is defined as A F m,x = A H m,x — T A S mtx
30
(2.26)
(2.27)
(2.28)
(2.29)
3. Interfacial Tension 7 :
Interfacial tension is defined by subtracting the free energy of the uniform so
lution from the total free energy of the interface. The interfacial tension is
an im portant therm odynam ic param eter for any interface. Liquid/liquid in
terfacial tension can be experim entally measured, while solid/liquid interfacial
tension 7 can not be experim entally measured.
7 n„a = A F mtx — n0A /i0 - npAfip (2.31)
where a is the cross-section area of a cell.
Afi0 — fj,0 — fj®, where (i0 is the chemical potential of solvent in the solution,
and n°0 is the chemical potential of pure solvent. Similarly, Afip = fj,p — //°,
where fip is the chemical potential of polymer in the solution, and fi® is the
chemical potential of pure polymer melt.
T he form ula for A / / 0 and Afxp are given by Flory-Huggins theory [18] as:
A ^ / k T = In f t - (1 - i)fip + x ( ^ ) ! (2.32)
A p r/ k T = i In £ - (1 - V + X(I? (2.33)
r r
4. A dsorption Amount T:
The dimensionless adsorbed am ount is expressed as the sum m ation of the
interfacial concentration of adsorbed polymers. The stronger adsorption,
the big adsorption am ount.
r = E ( 4 ) “ (2-34)
t'=l
5. Surface Excess Tei;c:
Positive surface excess means polymer favorable adsorption (positive adsorp
tion), while negative surface excess means polymer unfavorable adsorption
31
(negative adsorption or depletion). Zero surface excess means critical adsorp
tion, where the energy change upon adsorption is offset by the entropy loss
due to the presence of the solid surface. Surface excess is defined as:
r ™ = £ « - 4 > ’ r )
(2.35)
1 = 1
6 . Surface Coverage:
Surface Coverage is the fraction of surface site occupied by polymer segments.
It is given by the first layer concentration (f> x.
7. Bound Fraction p:
Bound fraction is the fraction of segments in direct contact with the surface
among the total adsorbed segments.
f t
(2.36)
8 . Root-m ean-square thickness of adsorbed polymer:
Root-m ean-square thickness of adsorbed layers can be calculated from the
concentration profile due to adsorbed polymer segments as follows:
( £ ) * =
\
e l w
(2.37)
Root-m ean-square extensions due to loops (Sf)? or tails (£*2) 2 can be formu
lated in a sim ilar fashion:
m
E fai »*(#)'
Y t i m
(2.38)
and
( i? ) i =
\
(2.39)
32
The root-mean-square extensions are im portant quantities for determ ining the
effect of steric barriers in colloidal stabilization by polymers.
2.3.2 Surface excess at critical adsorption
Another feature of critical adsorption phenom ena is the zero surface excess at critical
adsorption. In order to compare the new criterion for critical adsorption energy
w ith the old one, the surface excess r eic was plotted as a function of polym er bulk
concentration, which is shown in Figure 2.2.
From Figure 2.2, we can immediately see th a t the critical adsorption energy is
a function of polymer-solvent interaction param eter x • Using old criterion, when x
is not equal to zero, the surface excess is no longer zero at all bulk concentration
4 > * p.{see *). The most obvious case (d) (x = 0.5) in Figure 2.2, when x — 0.5 and
< f > * =0.01, the surface excess is greater than 1. This plot is sim ilar to Figure 12 in
Roe’s paper [22].
Using the new criterion, we can notice th a t when x changes from a very good
solvent (x = — 2) to a poor solvent (x = 0.4), the surface excess is negligible at
all bulk concentrations (see solid line). We conclude th at the new criteria properly
include the x effect, which is missing in the old one.
2.3.3 Sharp transitions at critical adsorption energy
Adsorption properties such as free energy of interfacial layers, surface tension, sur
face excess of polymer, bound fraction, fraction of tail, and RMS thicknesses all
exhibit interesting behavior at critical adsorption energy, i.e. they all have a second
order transition. Simulation results show th at the transition point is a function of
adsorption energy x«> Flory-Huggins interaction param eter x, bulk concentration < f > *
and chain length r.
The plots (Figure 2.3 to 2.10) dem onstrate the critical adsorption energies for
X = — 2, x = ~0.5 and x = 0.5 are 0.79, 0.41 and 0.17 respectively. T he critical
adsorption energy calculated from equation x ”c = (1 — ^)ln§ — |x ( l “ 2 < f)*) are
0.787, 0.412 and 0.162 corresponding to x = — 2, x = — 0.5 and x = 0.5. They agree
w ith each other quite well.
33
e x c
0.05 a:X = - 2
-0.05
0.1
- 0.1
b :x = - 1
1
*
— ---------5
5 K
*
X
10
exc
0.05
-0.05
c:x = 0 .2
d:y = 0.5
< f > ; 10
Figure 2.2: The surface excess of polymer Texc plotted against polym er bulk
concentration ( f> * for critical adsorption cases. Solid lines represent using x?c =
(1 — i ) ln | — * ( 1 — 2 (/> * ) as criterion for critical adsorption energy, and (*)s represent
using Xac = ln | as criterion for critical adsorption energy.
34
From Figure 2.6 to Figure 2.9 we can see th at, at critical adsorption, the rel
ative num ber of segments belonging to loop and train and to tail sequences varies
drastically, and either the num ber of segments belong to loop and train or th at to
tail predom inates except Xac region. The bound fraction and surface excess of poly
mer have a sharp increase from a very small value at critical adsorption energies.
The transition point (corresponding to critical adsorption energy) shifts from 0.16
to 0.788 as the Flory-Huggings interaction param eter x changes from -2 to 0.4.
Scheutjens, Fleer and Cohen Stuart also plotted the fraction of tail against the
adsorption energy [89] for a th eta solvent, which is sim ilar to our plot (c) in Figure
2.7. If the transition point for fraction of tail is m easured from their plot, a value
of 0.16 instead of 0.288 for critical adsorption energy Xac would be found, which is
exactly the same value obtained from our sim ulation results as well as criteria for
critical adsorption energy. This is another proof of the suitability of the new criteria
for critical adsorption energy.
35
Surface Tension
•0.04
•0.5 0 0.5 1.5
x = -0 .5
•0.5 0 0.5
Xa 1-5
X = 0.5
•0.5 0 0.5 Xa 1.5
Figure 2.3: Surface tension 7 / K T per lattice site versus dimensionless adsorption
energy, plotting at different x and bulk concentrations < /> * . Solid line: < f > * = 10-5 ;
dash dot lines: < f > * = 1 0 “3; dash lines: < / > * = 1 0 - 2 .
36
Free Energy
A F /fc r J ^
•1 -
----------------------------------------------------------
____________ I ____________I _____________ I _____
-0.5 0 0.5 1 1.5
Xs
X = _ “ ^-§
-1
0
x» 1-5
0.5
X = 0.5
-1
- 2 -
-0.5 0 0.5 x* 1.5
Figure 2.4: Free energy per surface site A F /k T plot at different x values and bulk
concentrations. A F /k T is dimensionless. Solid line: < j > * = 10-5 ; dash dot lines:
< f > * = 1 0 -3 ; dash lines: < f > * = 1 0 -2 .
37
Surface Excess
exc
•0.5 0 0.5 1.5
exc
X = -0 .5
■ 0 .5 0 0.5
x* 1-5
exc
X — 0.5
-0.5 0 0.5
x» 1.5
Figure 2.5: Surface excess at interfacial layers per lattice site versus dimensionless
adsorption energy, plotting at different x values and bulk concentrations. Solid line:
< f > p = 1 0 -5 ; dash dot lines: < p * = 1 0 -3 ; dash lines:<£* = 1 0 -2 .
38
Bound Fraction
•0.5 0 0.5 1.5
X = -0 .5
-0.5 0 0.5
X* 1-5
-0.5 0 0.5
x» 1.5
Figure 2.6: Bound fraction versus dimensionless adsorption energy, plotting at dif
ferent x values and bulk concentrations. Solid line: 4 > * — 10-5 ; dash dot lines:
< j > * = 1 0 “3; dash lines: < f > * = 1 0 “ 2.
39
a Fraction of tail
b
c
Figure 2.7: Fraction of tail plot against adsorption energy X a at different x values
and bulk concentrations. a)x = — 2, b )x = — 0.5, c)x = 0.5. Solid line: (j> * = 10-5 ;
dash dot lines: * = 10-3 ; dash lines: * = 10-2 .
40
Root-mean-square thickness of adsorbed layers
b
0.4120.5
c
0 0.162
Figure 2.8: Root-m ean-square thickness of adsorbed layers versus dimensionless ad
sorption energy, plotting at different x values and bulk concentrations. a)x = — 2 ,
b )x = — 0.5, c)x = 0.5. Solid line: < j) * = 10-5 ; dash dot lines: < f > * = 10-3 ; dash
lines:<£* = 1 0 -2 .
41
root-mean-square thickness of loop
0.5 1.5 -0.5 0
X = - 0 .5
-0.5 0 0.5
-0.5 0 0.5
x« 1.5
Figure 2.9: Root-mean-squaxe thickness of loops versus dimensionless adsorption
energy, plotting at different x values and bulk concentrations. Solid line: (f)* = 10-5 ;
dash dot lines: < f> p = 10-3 ; dash lines: < j)* = 10-2 .
42
root-mean-square thickness of tail
-0.5 1.5 0 0.5
x = -0 .5
-0.5 0 0.5
x , 1.5
X = 0.5
■0.5 0 0.5 x» 1.5
Figure 2.10: Root-mean-square thickness of tails versus dim ensionless adsorption
energy, plotting at different x values and bulk concentrations. Solid line: * — 10~5;
dash dot lines: * = 10-3 .
43
2.3.4 Effects o f chain length and concentration
Figure 2.11 shows th a t at fixed adsorption energy Xa and Flory-Huggins interaction
param eter x , the surface excess could be positive, zero or negative at certain con
centration and chain length. This plot is similar to the Figure 2 in Roe’s paper [22],
which he called the surface azeotrope phenomena.
Roe first noticed the phenomenon of the surface azeotrope but did not explain
how it happened. Using our criterion, we are able to offer some reasonable explana
tions.
In Figure 2.11
1. For 0* = 0.5:
X ' = 0.2 - [(1 - i) In | - M (i _ 2 0 ;)] = -0.0874.
Xl < o.
It is polymer unfavorable adsorption, negative surface excess occurs.
2. For < f > * = 0.1:
Xf = 0 .2 - [(1 - i) In § - °-f ( 1 - 2 0 ;)] = 0.0126. X e 8 > 0 .
It is polymer favorable adsorption, positive surface excess occurs.
On the other hand, at critical adsorption, we have x"c = (1 — £) In | — * ( 1 — 20;).
If x ”c = = 0-2, x = 0.5. we can calculate the corresponding concentration at which the
critical adsorption happen,
1. For r = 1000:
0 2 = (1 - lMo)ln 3 - ¥ ( 1 - 2^ )
= o .i5
2. For r = 100:
0 2 = (1 ~ jfe) I n f - f ( 1 - 2 0;)
0; = o .i6
44
:f=1000 0.35
x: r=100 exc
o: r=10
X. = o.
0.25
0.15
0.05
- 0.05
- 0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 2.11: Chain length effect and bulk concentration effect to critical adsorption
energy value: Plot of surface excess against bulk concentration for different chain
lengths: r = 1 0 0 0 , r = 1 0 0 , and r = 1 0 .
45
X = - 2 x = - i
X = o X = 0.29
X
I I
p
CO
1
O
r - H
I I
V j
1 2 3 4 5
< & = i o - 5
6 7 8 9 1 0
Table 2.3: List of ten cases calculated.
3. For r = 10
O .2 = ( l - i ) t o | - ^ ( l - 2 0 ; )
4>; = 0.264
As shown in Figure 2.11, we can see th at the sim ulation results support our
choice for the criterion of critical adsorption energy.
2.3.5 Properties at critical adsorption
Table 2.4 compared the adsorption quantities at different critical adsorption energy
criteria. Polymer-solvent interaction param eter varies substantially, from x = — 2 to
X = 0.4, and the bulk concentration ( f > * also changes from 10~ 3 to 10-5 . Table 2.3
listed 10 cases chosen to conduct the simulation. In Table 2.4, values in columns
(al) and (a2 ) correspond to new criterion x«c = (1 — r)ln f — 4 ^ ( 1 — 2 while
those in columns (b l) and (b2 ) correspond to x«c = In f •
The column (al) and (a2), give the average value of quantities in the 10 cases
listed in Table 2. we can see th at the adsorption quantities rem ain unchanged
while the Flory-Huggins interaction param eter x and < f > * change substantially. The
standard errors in these columns are less than 1%. The standard error also increases
with the increase of bulk concentration. For example, if the < f > * increases from 10- 3
to 1 0 -2 , the standard error will increase from 1 % to 2 %.
Column (b l) and (b2 ) give the value range of adsorption quantities for old cri
terion of critical adsorption energy. Since the adsorption param eters vary greatly
when the x and < / > * change, it is not meaningful to give the average values of them
here, instead, the range of quantities is given.
Using the property that adsorption quantities rem ain unchanged at critical ad
sorption, we can get the adsorption energy x«? which is difficult to measure through
46
microscopic r-=1000 r=50
param eter
al: X ?c bl: x»c a2 : Xn a c
b2 : Xac
loop num ber 13.2 1-54 2 .2 2 3-5
loop size 23.2 10-80 4.92 4-7
tail num ber 1.90 1.75-1.96 1.59 1.5-1.7
tail size 342 175-481 18.6 16-24
train size 3 1.7-3.5 2.97 2-4
RMS thickness adsorbed layer 16.4 5.5-19.5 4.19 3-5
RMS thickness of loop 8.18 4.5-10 2.84 2.6-3.2
RMS thickness of tail 19.6 1 0 -2 0 5.11 4.5-5.5
T (in unit of < /> * ) 18.3 3.4-5170 4.53 1 .2 - 1 1
r eic (in unit of < £ * ) 0 1 0 .2 (-9.8)-5410 0 1 0 .0 2 (-4)-9.4
bound fraction 0.0423 0.004-0.20 0.188 0.06-0.33
surface coverage (< /> * ) 0.77 0 .0 1 - 1 0 0 0 0.85 0.06 - 300
Table 2.4: Adsorption quantities at critical adsorption - Compare the new criterion
for critical adsorption energy x"c with the old one Xac' different x same x*c / Xac-
experim ent, by properly adjusting x value so that the critical adsorption can occur.
X value can be adjusted by changing the tem perature or choosing different solvents.
2.3.6 Excess adsorption energy and adsorption properties
at fixed excess adsorption energies
Using x and Xa as param eters separates the effect of characteristics of the solution
and th at of the surface, so th a t it is not apparent what combinations of the pa
ram eters Xa and x result in positive, critical or negative adsorption. Since the net
adsorption of the system is a result of the combined effect of both the solvent and
the surface, it seems th at a suitable combination of the param eters Xa, X) segment
length r and bulk concentration < f > * can be used as a convenient indicator of the gen
eral trend of adsorption. We here propose a new param eter, excess surface energy
Xg, which measure the adsorption behavior at the non-critical adsorption point.
We define
x! = X. - X" = X, - [(1 - \ ) In | - |( 1 - 2#)] (2.40)
47
In contrary we define
M ° “ = X. - 1 - 4 (2.41)
We show th at x t = 0 corresponds to critical adsorption, x t > 0 and x t < 0 cor
respond to positive and negative adsorption respectively. From the above equation,
it is clear th a t an infinite combinations of Xa and x and < f) * and r can result in x t = 0
the critical adsorption. Furtherm ore by looking at the value of x*j we can roughly
predict the range of adsorption properties.
Simulations were carried out at different cases to compare the adsorption quanti
ties at fixed (Xj)0,d and at corresponding x,- Three types of adsorption were chosen:
(a)strong adsorption (x* = 0.712 and (xt)°ld = 0.712).
(b)weak adsorption (x* = 0.312 and {xt)°ld — 0.312).
(c)depletion (x« = — 0.288 and (x t)old = — 0.288).
Table 2.5 to Table 2.7 give the range of adsorption quantities for chain length
r=1000. Table 2.8 to Table 2.10 give the range of adsorption quantities for chain
length r = 50. In Table 2.5 to Table 2.10, the m iddle columns give the range of
quantity values for cases 1 to 5 in Table 2.3; The right columns give the range of
quantity values for cases 6 to 10 in Table 2.3.
In all the tables, column (a l) and column (a2) list the range of adsorption quan
tities at fixed Xa value, w ith varies combination of x and x« (x range from - 2 to
0.4) while column (b l) and column (b2) list the range of adsorption quantities at
fixed corresponding ( x e s)°ld value, with x varies from -2 to 0.4. From those tables, we
can see th at column (al) and (a 2 ) are always give much narrower range than th a t
in column (b l) and (b2 ), which suggest th at excess adsorption energy x t is always
a better criterion to predict the adsorption than adsorption energy is. The m ost
obvious comparison is shown at Table ?? and Table 2.9, which compare the range
of values at x* = 0.312 with th at at (x*)o W = 0.312. At x t — 0.312, the adsorp
tion type is always polymer favorable adsorption, and it has m uch narrower range
of values than th at of at (x t ) old — 0.312. At { x e s)old = 0.312, the adsorption types
change from polymer favorable adsorption to polymer unfavorable adsorption when
the polymer-solvent interaction param eter x changes from a good solvent (x = — 2 )
to a poor solvent (x = 0.4).
48
microscopic
I I
* a
1 0 " 3
< ! > ; =
1 0 “ 5
param eter
Xa (x\)oid Xa { x i r *
loop number 69-78 54-78 80-89 70-90
loop size 3.6-6.2 4.3-7.3 3.4-5.8 4-6.5
tail number 1.53-1.66 1.57-1.75 1.45-1.60 1.5-1.7
tail size 155-190 154-220 100-125 105-150
train size 4.5-6.4 4.1-5.3 4.5-6.4 4-5.5
RMS thickness adsorbed layer 3.9-4.7 4-8.5 2.8-3.8 3-4.7
RMS thickness of loop 3.4-4.1 3.5-4.9 3.2-3.8 3.3-4.1
RMS thickness of tail 11-13 11-16.1 9-9.3 9-11
T (in unit of < £ * ) 230-1430 75.8-1500 21400-132000 5350-140000
r exc (in unit of < £ * ) 291-1460 59.1-1530 21600-133000 5490-141000
bound fraction 0.4-0.6 0.3-0.55 0.4-0.65 0.36-0.56
surface coverage (in unit of < £ * ) 141-570 20-618 13800-57000 2000-61400
Table 2.5: Range of adsorption quantities at strong adsorption r = 1000. Com pare
the quantities at excess adsorption energy x« = 0.712 w ith th at at (Xg)old = 0.712.
However, at fixed x*, comparing the range of adsorption quantities at non-critical
adsorption case (x , ^ 0 ) with th a t at critical adsorption case (x« = 0 )i the range
of quantities for the non-critical adsorption cases is much wider than th a t for crit
ical adsorption cases. The reason is th a t the bulk concentration assum ption th a t
polymer concentration at interfacial layers equals to the bulk concentration does
not hold well in non-critical adsorption cases. For strong adsorption, the adsorption
energy is big, the interfacial polym er concentration is m uch higher th an the bulk
concentration. While for depletion, the concentration at interface is sm aller than
the bulk concentration. This means th a t x t alone can’t quantify the adsorption
characteristics for non-critical adsorption and th a t both x and Xa m ust be defined.
49
microscopic 1 0 ~ 3
IQ-5
param eter
xS (.x e a)otd Xa ( x T d
loop number 60-68 2 - 6 8 73-79.6 3-80
loop size 5.7-7.7 6-52 5.2-7 5.5-13
tail number 1.67-1.72 1.68-1.95 1.61-1.67 1.62-1.95
tail size 170-205 165-435 115-140 110-410
train size 4-4.6 2.3-4.4 4-4.6 2.3-4.4
RMS thickness adsorbed layer 5.3-6.9 5-23 3.9-4.3 3.8-16.5
RMS thickness of loop 4.1-4.5 4-12.5 3.8-4.1 3.8-9
RMS thickness of tail 11.5-15.1 11-24 9.6-10.2 9.4-18
T (in unit of < £ * ) 107-915 6-1080 8630-79900 8-97400
r exc (in unit of 0 *) 91-933 (-13)-1100 8000-80800 (-9)-l 10000
bound fraction 0.3-0.4 0.026-0.35 0.36-0.46 0.01-0.35
surface coverage (in unit of (f> * ) 40-300 0.01-400 4000-30000 0.1- 37700
Table 2.6: Range of adsorption quantities at weak adsorption for r = 1000. Compare
the quantities at excess adsorption energy x t = 0.312 w ith th at at (x»)0,d = 0.312
microscopic = 1 0 “ 3 = 1 0 " 5
param eter
(al):Xae bl:(Xa)°< d a2 :xS b2 :(Xa) 0 < (1
loop number 1.47-1.65 0.48-2.6 2.99 1-4.5
loop size 48-56 45-75 60 50-95
tail num ber 1.95 1.95 1.95 1.94-1.96
tail size 463-475 440-500 416 400-465
train size 2.07 1.5-2.3 2.16 1.5-2.5
RMS thickness adsorbed layer 18.5-20 18.9-21 17.4 16-17
RMS thickness of loop 9.9-10 10-10.5 9.1 8.8-9.1
RMS thickness of tail 19-21 19.2-21.6 18.7 16-19
T (in unit of < £ * ) 2.5-5.3 2.2-4.3 8.4 6-13
Texc (in unit of < £ * )
(-8M-18) (-9M-18) -14 (-4M-64)
bound fraction 0.0044 (1.7-5.5)E-3 7.64E-3 (3E-3)-0.012
surface coverage (in unit of (f> * ) 0 .0 1 -0 .0 2 0.0001-0.023 0.064 0 .0 1 -0 .1 2
Table 2.7: Range of adsorption quantities at depletion for r=1000. Compare quan
tities at excess adsorption energy Xa = — 0.288 with th a t at (x t)old = — 0.288.
50
microscopic = 1 0 ~ 3
4> l = l o - 1 4
param eter
(al):
(b l): (x=)o W a 2 : x e . b2 : (x T d
loop number 3.8-4.3 3.4-4.4 4.2-4.8 0.4-4.8
loop size 2.4-2.85 2.5-3.7 2 .2-2 .5 2.3-3.7
tail number 1.18-1.26 1.2-1.45 1 .1 1.05-1.45
tail size 10.3-11.1 10-14.5 7.3-8.3 6.9-14.1
train size 4.7-5.8 3.8-5.3 5.3-6.1 3.8-5.8
RMS thickness adsorbed layer 1.72-1.86 1.75-2.75 1.52-1.59 1.5-2.6
RMS thickness of loop 2.23-2.28 2.2-2.5 2.18 2.1-2.5
RMS thickness of tail 3.23-3.43 3.2-4.2 2.9-2.97 2.8-3.9
T (in unit of < j> * ) 166-788 26-877 11000-55200 79-65800
r exc (in unit of < j> * ) 168-811 24-901 11800-55600 85-66300
bound fraction 0.63-0.70 0.4-0.65 0.68-0.74 0.4-0.72
surface coverage (in unit of < £ * ) 100-480 10-540 8000-38000 30-450000
Table 2.8: Range of adsorption quantities at strong adsorption for r = 50. Compare
the quantities at excess adsorption energy x l = 0.712 with th a t at (x f) old = 0.712.
microscopic = 1 0 ~ 3
# =
= i o - f e
param eter
al: X e a (bl): (x T d a2 : x t b2 : (Xe ,) o td
loop number 3.7-4.0 1.5-4.2 4.1-4.2 1.7-4.6
loop size 3.22 3.09 3-6 2.7-5.7
tail number 1.36 1 .2 - 1 .6 6 1.32 1.2-1.65
tail size 12-13.2 1 1 -2 1 11.1-11.5 8 .8 - 2 0
train size 4.18-4.26 2.4-4.4 4.36 1.7-2.4
RMS thickness adsorbed layer 2.13-2.37 2-4.9 1.98-2.04 5-5.4
RMS thickness of loop 2.36-2.4 2.3-3 2.33 2.2-3
RMS thickness of tail 3.5-3.91 3.3-5.5 3.34-3.41 3.05-5.40
T (in unit of < £ * ) 50-230 2.2-405 720-1210 2 - 1 1 2 0 0 0
r exc (in unit of < j > * v) 49-239 (-30)-419 755-1260 (-3)-11400
bound fraction 0.5-0.52 0 .1-0 .6 0.55-0.56 0.12-0.64
surface coverage (in unit of < /> * ) 25-120 0.2-230 400-700 0.3-7200
Table 2.9: Range of adsorption quantities at weak adsorption for r = 50. Compare
quantities at excess adsorption energy at x l = 0.312 w ith th a t at ( x e a)old = 0.312.
51
microscopic
1 1
1 0 ~ 3 = 1 0 “ 5
param eter
(al): X^ (b l): ( x T d a2 : xS b 2 : (x t) old
loop num ber 1 .2 1 0.60-1.4 1.14 1-1.75
loop size 6.38 6 - 8 6.23 5.5-6.8
tail num ber 1 .6 8 1.65-1.71 1.67 1.65-1.7
tail size 2 2 .2 21-25 22.7 19-25
train size 2.27 1.7-2.4 2.28 2 .1-2 .6
RMS thickness adsorbed layer 5.06-5.11 4.9-5.4 5.3 4.7-5.4
RMS thickness of loop 3.11 3.0-3.3 3.19 3.0-3.4
RMS thickness of tail 5.55-5.6 5.50-5.65 5.76 5.42-5.8
T (in unit of < f > * v) 1.57-1.69 0.9-1.9 1.34 1-3.4
Texc (in unit of < £ * ) (-3.5M-3.9) (-3.5M-4.3) -4 (-2.2)-(-4)
bound fraction 0.077 0.037-0.091 0.0685 0.025-0.119
surface coverage (< /> * ) 0.12-0.13 0.03-0.17 0.092 0.02-0.4
Table 2.10: Range of adsorption quantities at depletion for r = 50. Com pare the
quantities at excess adsorption energy x t = -0.288 with th a t at ( x t) old = — 0.288.
52
2.3.7 Segm ental anisotropic factors for non-critical and critical
adsorption
A nother im portant factor is th a t the distribution of the individual segments among
the layers dependents on their ranking number on the chain. This is because seg
m ents near chain ends have larger mobility than segments near the middle. Seg
m ental anisotropic factors are defined to account for this effect. If < f > * represents
the concentration of polymer on layer i, and ( f > ' p(j) represents the concentration of
segment of ranking num ber j in layer i, the segmental anisotropic factor t(j, i) is
defined as:
(2-42)
%
t(j, i) equals unity in the homogeneous case.
The three dimensional plot of segmental anisotropic factors against ranking num
ber j and layer num ber i gives a three visual dem onstration of adsorption types. For
polym er favorable adsorption, polymer chain lays flatter at the surface, we have big
ger loop num ber, train num ber and smaller tail number, and the plot will look like
(a l) in Figure 2.14, where the m iddle segment has higher chance to lie on the surface
and the end segment has low probability to attach to the surface. For polym er un
favorable adsorption the polymer chains are more extended to the bulk solution, we
expect bigger tail num ber, tail size and smaller train size, and the plot would look
like (b l) in Figure 2.14, where th e end segments have higher probability attaching to
the surface while the middle segments would be driven away from the solid surface.
The bigger segmental anisotropic factors of end segments are, the m ore unfavorable
the polym er adsorption is. At critical adsorption, the polymer concentration at in
terface equals to the the bulk concentration which corresponding to homogeneous
case, we expect th at the plot of segmental anisotropic factors would have uniform
distribution around value 1 , except the end fluctuation.
The segmental anisotropic factors under new criterion for critical adsorption
energy were sim ulated for four cases: a: x = ~ 3; b: x = “ 2 ; c:x = — 1 ; d:
X = 0.5. The three dimensional plot of t(j,i) is shown in Figure 2.12. From the
plot we can im m ediately see th a t the segmental anisotropic factors indeed have
53
uniform distribution except the end fluctuation. The m axim um t ( j , i) happens at
end segment, which is about 1 .1 , the rest are about unity.
To compare the new criterion w ith the old one, the segm ental anisotropic factors
under old criterion for critical adsorption energy were sim ulated for the sam e four
cases. The corresponding three dimensional plot is shown in Figure 2.13. From
Figure 2.13, we can see th at t( j,i) varies from 6 to 0.5 when the x varies from -3 to
0.5, the rest is not uniformed unity either.
In Figures 2.12, 2.13 and 2.14, “r # " represents the ranking num ber j and "I# "
represents the layer number i.
On the other hand, for non-critical adsorption case, the distribution of segmen
tal anisotropic factors would greatly dependent on the ranking num ber and layer
number. In Figure 2.14, we compared the segmental anisotropic factors at fixed
X* (Xa = 0.312), various X ( X = — 2 and x = 0.4), w ith th a t at fixed (x t)°ld
((Xa)°W = 0.312). Compare plots (al) and (a2) with (b l) and (b2) in Figure 2.14,
one can see th at at fixed Xai the adsorption type is always polym er favorable adsorp
tion, while at (x e B )old — 0.312, the adsorption type changed from polym er unfavorable
adsorption to polymer favorable adsorption when x change from -2 to 0.4.
2.4 Conclusion and future outlook
Theoretical studies of polymers confined at solid/liquid and liquid/liquid interfaces
have been carried out. The Flory lattice model in the self-consistent mean-field
approximation has been adopted in this study. The critical adsorption energy for
non-atherm al solution, including solvent effect and chain length effect has been for
m ulated as Xac = — (1 — ~ )ln (l — Aj) — A ix(l — 20*), where x is Flory-Huggins
interaction param eter, r is chain length, Ai is lattice param eter and 0 * is bulk con
centration. To compare the new criterion for critical adsorption energy w ith the old
one, properties of polymer confined at solid/solution interface at critical adsorption
have been simulated. Simulation results show th at the adsorption properties such
as free energy at interfacial layers, surface tension, surface excess, bound fraction,
fraction of tail, root-mean-square thickness of adsorbed layers, root-m ean-square
thickness of loop and root-mean-square-thickness of tail all exhibit sharp transition
at critical adsorption. The transition point (corresponding to critical adsorption
54
a b
(r#-1)/40+1 0 0 (|#-1)/2+1 (r#-1)/40+1 0 0 (|#-l)/2+1
Figure 2.12: Segmental anisotropic factors at new critical adsorption energy Xac =
(1 — £ •) In | — *(1 — 2 < f> * ). Chain length r=1001, bulk concentration < £ * = 10-2 . Four
different x value were calculated, a: x = — 3, b: x = — 2 ; c:% = — 1 , d: x = 0 -5 .
55
a b
3
2
0
40
0 0
40
0 0
Figure 2.13: Segmental anisotropic factors at old critical adsorption energy Xac =
l n |. Chain length r=1000, bulk concentration ( f > * = 10-1 . Four different x value
were calculated, a: x — — 3, b: x = — 2; c:^ = — 1, d: x ~ 0.5.
56
(f#-1)/40+1 0 0 (|#-l)/2+l (i#-1)/40+1 0 0 (|(M)/2+i
b l b 2
Figure 2.14: segmental anisotropic factors at < j > * = 10“3, Chain length r=1000. (al):
Xt = 0.312 and x = - 2 , (a2): Xe . = 0.312 and X = 0.4; (bl): {Xe a)0,d = 0.312 and
X = - 2 , (b2): (x.e)o W = 0.312 and X = 0.4.
57
energy) is a function of polymer-solvent interaction param eter x and bulk concen
tration as well as chain length. The critical adsorption energy obtained through
sim ulation is well consistent with the value given by our formula for critical adsorp
tion energy. So we conclude th a t our criterion for critical adsorption energy take
account the solvent effect and chain length effect properly.
The character of sharp transition of polymer properties at critical adsorption
suggest a novel m ethod to measure adsorption energy of polymer on solid surface
which is difficult to get through experimental m easurement, i.e. by measuring the
transition point of the adsorption properties (for example root-mean-square thick
ness of adsorbed layer), knowing Xi r and < / > * (which are easy to obtain), using above
form ula, we can get polymer-surface interaction param eter Xa — Xac at critical ad
sorption.
A nother interesting phenomena under the new criterion for critical adsorption
energy is, for given chain length r, the adsorption properties all were found to rem ain
unchanged at critical adsorption while the Flory-Huggins interaction param eter x
and < f > * change substantially.
For non-critical adsorption, an excess adsorption energy has been proposed to
m easure the adsorption energy deviation from the critical adsorption point. Sim
ulation results show th at through excess adsorption energy the range of properties
of polym er confined at solid/liquid interface can be roughly predicted. This can be
used to avoid sometimes tedious experiments or simulations.
In the future, we may aim to adapt our model to different systems such as polymer
m ixtures, block polymers, adsorption on spherical surface, adsorption between two
surfaces. As far as the model concern, the existing models m ade m any assumptions,
such as polym er solution is incompressible, polymer chain is totally flexible. Advance
in the m odel itself would be removing one or more assumptions mentioned above,
which will m ake the model more practical.
58
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