Title: Equilibrium properties of polymer solutions at surfaces
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Title: Equilibrium properties of polymer solutions at surfaces Monte Carlo simulations
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Language: English
Creator: De Joannis, Jason, 1973-
Publisher: State University System of Florida
Place of Publication: Florida
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Publication Date: 2000
Copyright Date: 2000
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Subject: Polymers   ( lcsh )
Monte Carlo method   ( lcsh )
Mean field theory   ( lcsh )
Surface chemistry   ( lcsh )
Chemical Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF   ( lcsh )
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Summary: ABSTRACT: Knowledge of conformational features of polymer solutions at interfaces is a prerequisite for understanding and controlling important technological processes such as colloid stabilization, biopolymer adsorption, corrosion inhibition, lubrication, adhesion, membrane separations and aggregation-induced separations. In this exposition, some examples are shown of the unique contributions that a carefully designed simulation study can make towards an improved interpretation of experimental and theoretical work. One section is devoted towards the improvement of an existing Monte Carlo algorithm to enable better tests of theoretical results. Large-scale end-swapping configurational bias moves provide the best chance of simulating high chain lengths and concentrations. The main focus of this paper is on polymers adsorbing from a good solvent onto a solid surface that has a weak (reversible) attraction on monomers. We analyze the concentration gradient that develops when the interface is in equilibrium with a bulk solution, and the force induced by adsorbed chains when subject to confinement. Strong evidence is presented supporting the scaling prediction for long chains that the loop and overall concentration decay as z-4/3 when adsorption occurs from semi-dilute or dilute athermal solutions. Mean-field theories predict a concentration decay that is too rapid at all chain lengths, consequently they underestimate adsorbed amount and layer thickness. Simulations provide the only quantitative description of finite-length random walks with excluded volume subject to adsorption conditions.
Summary: ABSTRACT (cont.): Adsorbed chains under compression exhibit lower pressures than predicted by mean-field theory as a result of its lower adsorption capacity. Even when interpreted with respect scaled variables, quantitative discrepancies persist in the compression of saturated / semidilute layers. A different comparison shows that the simulation results presented have substantial experimental implications. The compression of two adsorbed layers against each other is quantitatively similar to the compression of a single layer, when the surfaces bear high adsorbed amounts. A study also is made of the concentration gradient and confinement energy of a single chain between athermal walls. It is found that good agreement exists with theoretical predictions based on the "magnetic analogy" for a universal correlation between compression force and the depletion of segments near the walls. Finally several remarks are made concerning further research on problems including polymer depletion at interfaces and adsorption of polymers from a bidisperse distribution of molecular weight.
Summary: KEYWORDS: polymers, interfaces, Monte Carlo simulation, mean field theory
Thesis: Thesis (Ph. D.)--University of Florida, 2000.
Bibliography: Includes bibliographical references (p. 232-241).
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Statement of Responsibility: by Jason de Joannis.
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General Note: Vita.
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EQUILIBRIUM PROPERTIES OF POLYMER SOLUTIONS AT SURFACES:
MONTE CARLO SIMULATIONS



















By

JASON DE JOANNIS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2000




























Copyright 2000

by

Jason de Joannis





























This work I dedicate to my wife Sabrina. How little it is in comparison to her dedication
to me.















ACKNOWLEDGMENTS

I would like to recognize loannis Bitsanis for introducing me to this exciting field

of research and his constantly insightful advice and guidance. My other advisor, Chang-

Won Park, has always been a great source of stability and encouragement. I have

appreciated the efforts made by Raj Rajagopalan in my personal development as a

researcher, numerous technical discussions and also in some of the administrative

requirements of this degree. I am grateful to Anthony Ladd and Sergei Obukhov for

sharing their expertise with me through several discussions. Special thanks goes to Jorge

Jimenez for an infinite number of interesting discussions on random subjects and for the

fruitful collaboration of the past year.

Also I would like to acknowledge my parents Eugene and Diane, who have given

me constant encouragement and the foundation from which to think independently. My

brother Jeff and two sisters Alexa and Erica, have also been curious and supportive of my

research. Finally, I would like to recognize Mehmet Bayram Yildirim, for helping me

with some computational work, and Adriana Vasquez-Jimenez, through whom I have

finally reached a full understanding of "the physisorbed layer".
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S .............................................................................................. iv

A B ST R A C T .............................................................................................. viii

1. IN T R O D U C T IO N ................................................................................. 1

2. ELEMENTS OF POLYMER STATISTICAL MECHANICS.............. .................... 5
P olym er Solutions ...................................... ............................................. 5
Historical Perspective .................................... ............................ ......... 6
Ideal C hains ....................................................................... ......... 9
E excluded V olum e ........................................... .... .............................. 20
The Flory-Huggins Theorem A Fundamental Equation of State ..........................27
Scaling L aw s ............... ...... ............................ ......... ... ............ 29
P hase D iagram ......... ................................................... ............................ 32
Polymer Solutions at Hard Interfaces................ ............ .....................34
Mean-Field Theories of Adsorption on a Lattice............................... ...............34
The Ground State Dominance Solution to Continuum Mean-Field Theories...........37
Scaling D description of Polym er Adsorption ........................................ ................ 38
Square-Gradient Mean-Field Free Energy Functional .........................................39
B eyond the G round State ...................... .. ................................... .............. 41
3. EXPERIMENTAL CHARACTERIZATION OF POLYMER INTERFACES .........43
M easurement of Polymer Structure at Surfaces ...................................... ............... 43
Global Properties of Physically Adsorbed Layers............................... .............. 44
Sm all A ngle N eutron Scattering.................................. ..................................... 47
N eutron R eflectivity ...................................... ..................... ............... 49
Scattering and Reflectivity Measurements on Adsorbed Polymers .........................49
M easurement of Polymer Induced Forces ........................................ .............. 50
Surface F orce A pparatu s......................................................................... ...... 5 1
SFA M measurements in Good Solvent ......................... ......... ..................... 53
A tom ic Force M icroscopy .................................................................. ... 57
AFM Measurements on Elasticity of Single Chains and Forces Between
P olym er L ayers ................................................................... .............. 59
4. MONTE CARLO SIMULATION OF POLYMER CHAINS ON A LATTICE........61
Enumeration, an Exact Theoretical Treatment for Model Oligomers..........................62
M onte C arlo M ethods ............................................................................. .......... 64









Static M onte Carlo ............. .............................................................................64
D ynam ic M onte Carlo ................................................................ .............. 66
Configurational Bias Monte Carlo .............. ........................ ..............69
Sim ulation of B ulk Properties ...................... .. ............. .................... .............. 70
Sim u nation of A d sorption ...................... .. ............. .. ............................................ 72
5. REPTATING CONFIGURATIONAL BIAS MONTE CARLO FOR LONG CHAIN
MOLECULES AND DENSE SYSTEMS ..............................................82
Introduction ..................................... ....... .... .... .. .................... 82
Proposal and Detailed Balance of Configurational Bias Extensions ........................... 85
End-Switching Configurational Bias ......................................... .............85
Persistent End-Switching Configurational Bias ...................................................88
T ests of the Proposed A lgorithm s ...................................................... ................... 91
Static P properties ......................................................... ........................ 92
Dynamic Characteristics of the Algorithms..........................................................96
Su m m ary ...................................... ................................................. . 9 9
6. A POLYMER CHAIN TRAPPED BETWEEN ATHERMAL WALLS................. 115
In tro d u ctio n ........................................................................ ............... 1 1 5
Single Chains and Interfaces ........................................................... ..............116
Single C hain A dsorption .......................................................... .............. 117
Single C hain B ridging ........................ ........................................ 118
Single Chain Confinem ent ......................................................... ......... ..... 118
Sim ulation M mechanics ........................................................... .............. 122
R esults.................................................. ........... 124
C concentration Profiles .......................................................................... ............. 125
Force of Confinement and Connection with Surface Layer Concentration .......... 126
Universal Force Concentration Constant ......................................... ................ 127
Sum m ary ............................................................... .... ...... ......... 127
7. HOMOPOLYMERPHYSISORPTION.............................. 133
Introduction ............................................................... .. ..... ......... 133
Sim ulation M mechanics ............................................................ .............. 137
R esu lts............................ .. ................ .............. 140
Qualitative Picture of the Interface............................................... .................... 140
Quantitative Comparison to Finite Chain-Length Lattice Mean-Field Theory ...... 142
S u m m ary ................................................................................... 14 6
8. ADSORPTION OF POLYMER CHAINS WITH INFINITE MOLECULAR
W E IG H T ................... ............................................................... 16 3
Introduction ............................................................... ... ..... ......... 163
Sim ulation M echanics ............................................................ .. ........... ... 164
Theoretical Predictions ...... ...... .............. ..................................... 165
Analytical Solutions........................ ........................ ....... 165
Limiting Behavior of Lattice Mean-Field Calculations .................. .............. 167
R e su lts ......................................................... 16 8
0-Solv ent ......... ... ......................................................................... 168









Atherm al Solvent .................................... ....... ........... .............. 169
Further Details of the Layer Structure ......................................... .............172
C onclu sions ............................................................... ... ..... . ..... 174
9. COMPRESSION OF AN ADSORBED POLYMER LAYER OF FIXED MASS .. 195
In tro d u ctio n ................................................................................. 19 5
Sim ulation M mechanics .......................................................................... .............. 199
Sampling the Conformations of the Chains ............ ............................... 200
Calculation of Forces from Lattice Simulations ............. ..........................201
R esu lts ............ ......... ............. ............ ...... .. .............................. . 202
Compression of a Polymer Layer with an Athermal Wall ................................... 202
Interaction Between Two Polymer Layers......................................................209
Sum m ary ................................................................ .... ...... ......... 210
10. PRO SPECTS ................. ................. ...................... .. ..... ................ 221
Depletion at Non-Adsorbing Surfaces...................................... ... ..................221
Competitive Adsorption .............. ........... .............. .......... ......... .............. 222
Ring Polymers, Grafted Polymers and Other Problems...........................................223
11. CON CLU SION S .............. .............. ..................... ......... .................. 227
Extension of Configurational Bias Monte Carlo................................................... 227
Compression of an Isolated Chain ............ ............... ..................... ... ..............227
Test of Mean-Field and Scaling Theory of Adsorption....................... ............228
Compression of an Adsorbed Polymer Layer..................... ........................229
REFEREN CE S ............ ......... ............................ .......... .. ......... ........ 232

BIO GR APH ICAL SK ETCH ........................ ..................................... .............. 242















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EQUILIBRIUM PROPERTIES OF POLYMER SOLUTIONS AT SURFACES:
MONTE CARLO SIMULATIONS

By

Jason de Joannis

December 2000


Chairman: Raj Rajagopalan
Major Department: Chemical Engineering

Knowledge of conformational features of polymer solutions at interfaces is a

prerequisite for understanding and controlling important technological processes such as

colloid stabilization, biopolymer adsorption, corrosion inhibition, lubrication, adhesion,

membrane separations and aggregation-induced separations.

In this exposition, some examples are shown of the unique contributions that a

carefully designed simulation study can make towards an improved interpretation of

experimental and theoretical work. One section is devoted towards the improvement of

an existing Monte Carlo algorithm to enable better tests of theoretical results. Large-scale

end-swapping configurational bias moves provide the best chance of simulating high

chain lengths and concentrations.

The main focus of this paper is on polymers adsorbing from a good solvent onto a

solid surface that has a weak (reversible) attraction on monomers. We analyze the









concentration gradient that develops when the interface is in equilibrium with a bulk

solution, and the force induced by adsorbed chains when subject to confinement.

Strong evidence is presented supporting the scaling prediction for long chains that

the loop and overall concentration decay as z-4/3 when adsorption occurs from semi-dilute

or dilute athermal solutions. Mean-field theories predict a concentration decay that is too

rapid at all chain lengths, consequently they underestimate adsorbed amount and layer

thickness. Simulations provide the only quantitative description of finite-length random

walks with excluded volume subject to adsorption conditions.

Adsorbed chains under compression exhibit lower pressures than predicted by

mean-field theory as a result of its lower adsorption capacity. Even when interpreted with

respect scaled variables, quantitative discrepancies persist in the compression of saturated

/ semidilute layers. A different comparison shows that the simulation results presented

have substantial experimental implications. The compression of two adsorbed layers

against each other is quantitatively similar to the compression of a single layer, when the

surfaces bear high adsorbed amounts.

A study also is made of the concentration gradient and confinement energy of a

single chain between athermal walls. It is found that good agreement exists with

theoretical predictions based on the "magnetic analogy" for a universal correlation

between compression force and the depletion of segments near the walls.

Finally several remarks are made concerning further research on problems

including polymer depletion at interfaces and adsorption of polymers from a bidisperse

distribution of molecular weight.














CHAPTER 1
INTRODUCTION

In this paper, the equilibrium structure and interactions of linear homopolymer

(i.e., polymer with identical repeat units) molecules at various interfaces are examined

using a lattice Monte Carlo simulation method. Chemical and physical adsorption of

polymers on solid surfaces plays an important role in many technological processes.

Among the most important are the stabilization of colloidal suspensions (e.g., foods,

paints and inks, pharmaceuticals), adhesion, lubrication, and corrosion inhibition.

Because of the wide variety of polymers available (e.g., linear, branched, block-

copolymers, random copolymers, star, tree and homopolymers) and due to the various

surface/polymer interactions that can take place, the physical characteristics of a surface

in principle can be manipulated to any extent. One impressive example of this is in the

biological compatibilization of artificial implants.

The adsorption of linear homopolymers is the most widespread method of

stabilizing colloidal systems.1 The repulsive force that occurs when two polymer-coated

particles approach each other is termed "steric repulsion" since it arises largely from the

reduced entropy on confinement of the polymer conformations. The advantage of these

systems is that no specialized polymer synthesis or suspension preparation methods are

required. More costly methods such as the grafting of polymers onto particle surfaces are

used as a last resort and the grafting of polymers to some surfaces can be difficult.

An exact theoretical description of polymer / surface systems is overwhelming. In

principle, the equilibrium characteristics of the system depend on a multitude of factors:









monomer-monomer and monomer-solvent interaction energies (including ionic, van der

Waals forces, and hydrogen bonding forces), size and shape of the monomer and solvent

molecules, the polymer's valence angles as well as its secondary and tertiary (e.g., helical

structure of DNA) structure, and the distribution of molecular weight. Further

complications arise when the surface has a corrugated crystalline structure and can have

several kinds of long and short-range interactions with the polymer solution leading to

non-uniform adsorption characteristics.

In general no attempt is made at such a detailed description. In lieu of an exact

treatment, theories attempt to identify two or three of the most important parameters of a

problem and infer the scaling of system properties with these parameters.2 For example, a

function might be proposed, for either argumentative or observational reasons, to depend

only on two important variables x and y, f (x, y). Then the objective is to find the power

law with which these scale: f ~ x"m yn. The scaling is deduced from dimensional or

phenomenological arguments based on the relevant length and energy scales in the

system. This kind of reasoning can be surprisingly successful under asymptotic

conditions and has shed light on an amazing variety of problems in polymer physics.

When such a complex system reduces to simple functions of only a few primary variables

it is referred to as universal. That is, the results hold over a coarse grained scale

regardless of the chemical constituents of the polymer or solvent. For polymer solutions,

the chain length, N, the solvent quality, X, and the volume fraction, 0, capture most of the

information about thermodynamic properties. A discussion of theories that describe the

universal behavior of polymers at interfaces and in solution will be made in Chapter 2.









In Chapter 3 the experimental tools and observations that are relevant to the

structure and forces exhibited by polymer layers are reviewed. The most fruitful studies

have used surface force apparatus3 and small angle neutron scattering4 techniques.

Neutron scattering has been used to measure the concentration gradient within adsorbed

polymer layers. Unfortunately the limitations on this technique and the lack of atomistic

modeling capacity prevent simulations from being quantitatively compared with

corresponding measurements. Instead simulations are used to test the coarse-grained

theories of polymer adsorption which have remained unproven for two decades (see

Chapters 7 and 8).

Surface force apparatus measurements have succeeded in measuring the forces

between surfaces bearing adsorbed polymer chains. An understanding of the detailed

structure of the interface in terms of its concentration gradient and conformational

architecture is a prerequisite for making correct interpretations of these measurements. In

Chapter 9 we investigate the force vs. distance and the corresponding conformational

changes that take place. We make an interpretation based on the properties of layers of

unperturbed layers of Chapter 7 and 8. This is illustrative of the fact that simulations can

be used as a great interpretive tool.

In Chapter 6 we discuss a model problem that has found surprising application.

Theories and simulations of single chains abound in literature because they can be

studied and understood more rigorously. Such is the case with the compression of a

single chain between non-adsorbing walls. The lateral expansion of the chain has been

described by scaling and other theoretical arguments. This may appear to be a problem of

little consequence other than academic. However the lateral expansion of an adsorbed









chain into a pancake is subject to precisely the same physical arguments.2 The confining

surface is substituted by the strength of the adsorption field.

Simulations increasingly provide a number of important functions to many

scientific disciplines.5 They can be used as a phenomenological probe, suggesting new

and possibly useful directions to take in the laboratory. They can be used to test the

capability of a proposed physical model to reflect the real processes for which the model

was designed. In this paper, Monte Carlo simulations are used in order to test the

approximations used in order to obtain results from a physical model for polymer chains.

The simulations use the same model and do not require any approximations in order to

obtain results. In Chapter 4 we discuss the background of Monte Carlo simulations on

lattice models. In Chapter 5 we propose an extension to the configurational bias Monte

Carlo algorithm, an algorithm designed to obtain static properties of long chain molecules

with a high efficiency. It is interesting to note a difference between Monte Carlo and

molecular dynamics simulations. The power of molecular dynamics simulations is driven

mainly by advances in computational tools. However Monte Carlo methods also gain

performance through the development of new algorithms.














CHAPTER 2
ELEMENTS OF POLYMER STATISTICAL MECHANICS



The foundations of the statistical mechanics of polymer solutions are discussed in

the next section. In the succeeding section the modern developments in the understanding

of polymers at solid/liquid interfaces is reviewed. Particular attention is devoted to

theories of polymer adsorption.

Throughout, only the subject of linear homopolymers (i.e., non-branched and

uniform chemical composition) is treated. These objects were the first to be studied

theoretically and continue to bear relevance on more chemically complex polymers, the

study of which abounds: from dendrimers and random heteropolymers to highly

specialized helical and globular biological molecules.



Polymer Solutions



Most of the information about the conformations that polymer chains take in

solution is contained in the basic random walk problem. This problem dates to the

beginning of the century (at least). Random walks have been studied in discrete and

continuum space, however the basic results are invariant.

In this section, the discussion focuses on the equilibrium aspects of linear,

homopolymers solutions. No attempt is made to describe, for instance, the statistical









treatment of polymer dynamics, phase equilibria, or rubber elasticity, although these were

prominent and early developments in the statistical mechanics of polymers.


Historical Perspective



The first to make a complete mathematical analysis of random walks was Lord

Rayleigh6 in 1919, in the interest of describing the vibrational states of sound waves.

Einstein, Smoluchowski, and Chandrasekhar contributed to the theory of Brownian

motion through their analysis of the random walk imposed on a micron-sized particle by

thermal interactions with its solvent.' Thus the stage was set for the description of

polymer conformations in the same way. Incidentally, the phenomenon of Brownian

motion gained prominence largely because the randomness could actually be observed

with a microscope. However, it is now known that even "nanoscopic" particles such as

solute molecules diffusing through a solvent, or the particles of a gas, have the same

mechanism of motion as their microscopic cousins.' Thus it is not surprising to find that

polymer conformations possess much in common with other well-understood

phenomena. The average random walk or polymer chain has a very small size in

comparison to its fully stretched length.

In the 1930's Kuhn played a major role in the translation of the existing

knowledge about random walks into the language of polymer conformations.8 In some

sense it is an extension of the familiar ideal gas model, with the restriction that the ith

particle of the "gas" must be one unit of length apart from the (i l)th and (i + l)th

particle. Polymer models in this class are called "ideal chain" models. Such a chain is

also referred to as a "phantom chain," since the particles have no volume and can









therefore pass through each other. In the "freely jointed chain" model, the bonds are

allowed to take any angle between 0 and 360 degrees with equal probability. In 1939,

Kuhn9 pointed out that the bond angles in more realistic models for a polymer chains

(e.g., a chain with tetrahedral angles between adjacent bonds) lose all directional

correlation after only a few repeat units. Therefore, they can be "mapped" onto a freely

jointed chain by replacing the bonds with larger and fewer hypothetical bonds which

follow the true random walk. This discards information on the scale of the original bond

length and defines polymers in a very general way.

The essential feature of the ideal chain models is that they allow for the statistics

of long flexible molecules to be described in a simple way. Like the average

displacement of a Brownian particle after a certain time, the average size (e.g., distance

between ends) of an ideal chain scales as the square root of the number of steps involved:



Rend N12, (1)



where Rend is the root-mean-squared (rms) end-to-end distance. A snapshot of a 10,000

segment chain, shown in Figure 1, is reminiscent of the path of a randomly diffusing

particle.

Elastic x-ray scattering measurements on isolated (dilute) polymer chains in

solution indicated that the square root law is not completely correct.2 The average size of

the chains do scale universally regardless of the chemical units of a polymer but with a

higher exponent.

























'CONF mod' using 1:3I I
CONF10000mod' using 1:3 +


200 '
150



Figure 1. The two
walk.


200 250 300 350


dimensional projection of a 10,000-segment self-avoiding









This led the way towards an important correction to the random walk. In most

solvents, the strong repulsive interactions that prevent monomer overlap have a swelling

effect on the entire conformation. This "excluded volume" effect occurs equally between

all monomers in the chain and therefore Kuhn's method of "chain mapping" cannot be

used. Flory,8 Edwardso1 and de Gennes2 each invented entirely different methods of

accounting for the effect of excluded volume. The first two are mean-field methods and

the third comes from a parallel with magnetic systems. All three of them agreed that the

correct scaling exponent is close to v = 3/5,



Rend N35. (2)



In what follows we discuss some of the simpler models mentioned above, which

are transparent enough for presentation in some detail in the space of this paper. This will

establish the elements required to discuss more advanced subjects, primarily theories of

polymer adsorption.


Ideal Chains



A random walk has a size that is proportional to N12. There are many ways to

derive the main results of the ideal chain model. The objective is to describe the average

size of a polymer coil within a dilute solution of chains. We are interested in the average

displacement between the ends of the chain with respect to all of the spatial

conformations that are possible.








Let us use the freely jointed chain model in this example. The chain consists of N

+ 1 volume-less beads joined by N bonds. The only restriction on the position, R,, of each

bead is that the bond lengths be fixed, i.e., |R, R,+1 = 1. This length, 1, is called the Kuhn

length.

The rms end-to-end distance, Rend, is a common measure of the size of a

macromolecule in solution:



Rend = (R, RN 2) (3)



where the angle brackets are used to represent the ensemble average, i.e., the average

over all configurations allowed within the model. Another equally used characteristic of

the chain size is the rms radius of gyration:



=/Z:0(R, Ro )
R2 = (N+ 1) (R -R,)2 (4)
IN R



where Rc = (, R, (N +) is the position of the chain center of "mass." The second

equality is due to a 19th century theorem of Lagrange's (see for example, reference 8,

appendix 1) which is useful in computations.












/

R,-VR,




Figure 2: Labeling of the links in a freely jointed chain.


For the moment it is worthwhile to consider a one-dimensional freely jointed

chain. In this case the displacement in the x-direction is either I or -1. For a single-link

chain, the rms end-to-end distance, Rend, is 1, since there are only two possible

conformations. Similarly, for a two-link chain, there are 22 conformations and the rms

displacement, according to Equation 3, is /[(2(22)+0(12)+2(02))/22]1/2 = 1212

In general, for an N-step walk in one dimension, we have



Rd 2K)2 N )2NN)&) + XA
end =(x- XN Ax, I Y^ (5)




where Ax, = x, x,.i. The first term in Equation 5 obviously evaluates to NI2. The second

term is zero. The average (for i # j) of any product (Ax, ) (Ax, ), no matter how distant or

near i andj are in the chain backbone, is zero because steps in the positive and negative

direction occur with equal probability. Thus, in addition to obtaining the scaling exponent

in Equation 1 we have observed that the prefactor is exactly equal to the bond length, 1.









Next we consider a two dimensional freely jointed chain. The rms end-to-end

distance in this case is the sum of the x andy components such that




Rend = (( -x ) +( N )2N = 2 ()2), (6)



Thus, the problem amounts to finding the rms projection of a single link onto the x-axis,

A link can take any angle, 0 (measured from the x-axis), with equal probability.

Therefore,



2g
cos2 Od
(( =) Y_)=2(cos22)=2 0 2 =- (7)
SdO
0



which establishes the same relation, Rend = IN12, for the two dimensional freely jointed

chain.

Consider a random walk on a simple cubic lattice. Each step of the walk is either

in the x, y or z direction. One third of the time (on average), the step is in the x-direction,

so the average squared x-component of a step is 12 / 3, much like the case of the one-

dimensional walk. In fact the presence of the lattice allows the problem to be decoupled

in terms of three independent one-dimensional random walks. This yields for the average

end-to-end distance: Rend= N(12/3+ 2/3+ 2/3)1/2 = IN12.

The above calculations are merely for illustrative purposes and demonstrate the

utility of simple random walk arguments (as well as the advantage of dimensional









arguments). The most general and concise derivation of the freely jointed chain theorem

is as follows.




R K Ar, f Ar K (Ar, )2+ (Ar, )(Ar) (8)
( =1 j=1 1=1 \I



The dot product inside the second pair of brackets in Equation 8 is equal to 12cos,,

where the angle between any two links, 0,, can take with equal probability any value

between 0 and 27r. The factor < cos 0, > characterizes the extent of correlation between

the orientation of any two links in a chain. For the freely jointed chain, this "orientation

factor" takes a value of 1 for i = j, and a value of 0 for i j. In other words

(cos,j) = 8,. This gives the result:



Rend IN12 (9)



for all dimensionalities and for any number N.

For other ideal chain models, it is not hard to imagine that the orientation factor

decays rapidly to zero11 with the separation i j Therefore, all ideal chains obey the

square root law. Examples of other models are contained in a book by Flory.12 One of

the more well-known is the wormlike chain (WLC). Instead of discrete bonds it has a

continuous backbone with a stiffness that decays exponentially along the backbone. This

stiffness is charactized by the persistence length. The only difference between ideal chain










models is a difference in their mechanism of flexibility. This model is in common usage

to describe DNA molecules.

Another model is the rotational isomeric states (RIS) model. This attempts to

incorporate realistic rotational potentials of specific polymers such as poly(methylene).

Flory has worked out the details for a number of polymers.12 The WLC, RIS and other

ideal chain models predict different prefactors in Equation 9, but all have precisely the

same power law in the limit N >> 1.



100
1 10
0 X
0 X

O X Rg good solvent
10
S 0 Rend good solvent
4 o 0.485 A Rg near theta
Sy=0.566x X Rend near theta
L 0
S--fiti
l 1 [
S0.594 fit2
Sy=0.412x


1 0.1
1 10 100 1000 10000
N


Figure 3: Scaling of rms end-to-end distance and rms radius of gyration of an isolated
chain. Data are from simulations on a simple cubic lattice in the presence of i) a good
(athermal) solvent and ii) a 0-solvent. Equations for fitted power laws are shown. The
fits use the points i) N=200, 400, 1000 and ii) N=400, 1000, 2000.



This theoretical prediction is one that is relatively easy to test. Experimentally,

light scattering can reliably measure the radius of gyration. It is possible to show11 that

the radius of gyration, Rg, of a freely jointed chain is related by a constant of order unity

to the end-to-end distance as R= 6 2Rend. Even non-ideal chains have an end-to-end








distance that is roughly two and a half of their radius of gyration. Figure 3 shows some

results that agree with the ideal chain prediction. However, the athermal self-avoiding-

walk displayed has better agreement with experimental results in general. This

discrepancy can be rectified by the incorporation of excluded volume effects to be

discussed later.

The central limit theorem of statistics: ideal chains have Gaussian statistics. The

sum of a sequence of random events will have a Gaussian distribution when the number

of events is large. That fact is usefully applied to the sum of the N link vectors of an

ideal chain, or the end-to-end distance, RN. One can start with the definition of a

Gaussian and apply the central limit theorem to the x-component of the end-to-end

distance Rx:



1 (Rx -((R ))2] (10)
P(Rx) exp 2 (10)
(T42i 2 I



where a is the statistical standard deviation, and N >> 1. The second moment of the

distribution is known from Equation 9. The x-component of the second moment is just

N12/3 which yields the condition P(R)R2d3R = NI2 /3 and resulting in o2 = Nl2. This

ultimately yields the probability distribution for a freely jointed chain



(/3 3 3R2 1 (11)
P,(R)= (27rN/2/3)-3/2 exp (11)
I-R2NI 2









The average end-to-end vector is R=0 since chains can take positive and negative

values of R, Ry, Rz with equal probability (this explains the symmetry of the distribution).

A small inaccuracy of Equation 11 is that it gives a non-zero probability for end-to-end

distances larger than the contour length IN. According to the statistics of the Gaussian

distribution, the probability of a chain size larger than N'121 (i.e., |R| /a> (3/2)1/2) is 22%.

The probability of a chain larger than the fully extended length, NI, (i.e., IR /a>

(3/2)1/2N1/2) is about 0.3% for a 6 segment chain, and 10-5% for a 17 segment chain. It

turns out that the other properties of the Gaussian and freely jointed chain distribution

also converge swiftly with increasing chain size. For example, the simulation of one-

dimensional freely jointed walks has been directly compared with the predicted Gaussian

distributions for walks ranging from 4 to 25 steps.13 To the eye, the simulation and

theoretical distributions converge for walks larger than 25.

It is important to note the range of spatial fluctuations of the chain is quite large.

The fluctuations in the distribution of end-to-end length are of order of the end-to-end

distance itself ((2 =N 2N). This "looseness" of chain conformations is an important

characteristic that makes it difficult to model real chains mathematically. The mean-field

approaches that will be discussed later neglect these fluctuations. This is thought to be an

important shortcoming of mean-field theories. However there exists an alternative. The

correct mathematical description of strongly fluctuating magnetic systems developed by

Landau and Ginzburg, was adapted by P.G. de Gennes to describe polymer

conformations.2, 14

One interesting application of Equation 11 is towards polymer elasticity. The

elastic force due to the stretching or compressing of a single chain in solution can be









calculated. If the ends are held at a fixed separation Rx, the number of conformations the

rest of the chain can take is directly proportional to P(Rx). Therefore the entropy is

related by Boltzmann's law as S/k, = InQ -3R /2N12 +const. Since ideal chains

have no enthalpic interactions, their free energy has only the entropic contribution. The

force of stretching the chain isf= -dF/dx or



3kT
f() =-a, (12)
N1



Thus it obeys Hooke's law of elasticity with stiffness that increases linearly with

temperature and decreases as the inverse of chain length. This approach has been applied

to more general elastic phenomena in rubbers, gels, and semi-dilute to concentrated

solutions. 11, 15,8, 16

The probability "cloud" of a polymer chain. Much like the Hartree-Fock theory

of atoms, one can formulate a "cloud" that represents the probability density of finding a

bead at a certain location. The cloud is radially symmetric and densest at the center much

like the s orbitals in an atom. The contributions of each bead of the chain are also

radially symmetric. In fact, the overall cloud is formed by the superposition of the cloud

of each bead. Clearly, the middle bead of the chain will contribute most highly to the

center of the overall probability cloud. Likewise, it stands to reason that the end beads

will give the most significant contributions to the outermost layers of the overall cloud. It

is helpful to have this picture in mind as we move forward.

















pv(r) p(r) PN /2(r)
N/2< i
Figure 4: Probability distribution for a single macromolecule.



The probability, PN(R), that an N-step chain ends at R, is related to the

combination of such probabilities for two smaller chains to meet at that spot, PN(R)=PN-

M(R)PM(R). Then the probability cloud is the integral of this composition law over all

possible subchains:



N
p(R) = PN, (R)P (R)l f (13)
0



One immediate observation is that the average local monomer concentration is

proportional to the probability cloud. The concentration profile, or probability cloud, is

only partly understood for some problems.

Distribution functions are also universal over coarse scales. While the local

(atomic scale) distribution functions depend on the specific chemical shapes and

potentials, over coarser scales, they have universal limits. Information on the Fourier

transform of this function can be obtained from neutron scattering. As usual, predictions

for tertiary or higher order correlation functions are out of the question. Luckily, the

concentration of segments inside an ideal chain is low enough so that pairwise additivity









is valid. That leaves N2 pair correlation functions, N2/4 of which are non-identical owing

to the symmetry of linear chains. The average of these (overall pair correlation function)

can be obtained by a simple scaling argument.11, 2 The result is g(r) ~ 1/r for r << Rend ,

and g(r) ~ exp(-r/l) for r >> Rend. This derivation is based on the assumption that each

link has a Gaussian distribution of sizes and is thus on a coarser scale than even the freely

jointed chain link size. However the discussion at the end of the previous section

illustrated that even a subchain of only a few links is nearly Gaussian.

Ideal chain equation of state. If we wish to know the properties of multiple ideal

chains in a solution (i.e., dilute, semi-dilute and concentrated solutions), we can easily

derive them in an additive fashion. The entropy (and thus free energy) of the solution is

just the sum of the entropy of each of the individual chains in the solution. Just as the

segments inside an ideal chain do not interact, the segments of different chains do not

interact. For this reason (and others), it is clear that there will be much more severe

discrepancies than the difference in chain size between ideal and real polymer chains.

In practice, a polymer solution typically has a constant temperature, composition,

and volume. Thus the fundamental thermodynamic entity is the Helmholtz free energy,

F(T, V, N, n) where N represents the chain length and n represents the number of chains.

With all fundamental equations of state, any thermodynamic property is directly

available.

For ideal chains it is simple to find the difference in free energy between a

polymer solution and the equivalent systems of pure solvent and pure monomer. The

internal energy of the polymer solution is zero since all pair potentials are individually

zero at all distances. That leaves only the entropic contribution.









We will derive the equation of state for a lattice model, in a way analogous to the

Flory-Huggins method described in the next section. The number of conformations that n

chains each with N links can have is 2 = znN. Letting c = nN/V be the concentration of

monomers,



S / V =knNlnz /V kc. (14)



It is similar to the result that is obtained upon taking only the linear term in a virial

expansion. The introduction of excluded volume into the model will result in c2 and c3

terms. A fundamental equation of state can be a versatile tool in appropriating the

characteristics of homogeneous systems. In systems with a concentration gradient such

as adsorption, a "local chemical potential" based on the equation of state can be assigned

to describe the gradient.


Excluded Volume



The ideal chain models capture a great deal of information about polymer chains

in such a simple scheme. They successfully predict that a universal scaling exists for the

average chain size, suggesting that the two most fundamental properties of

macromolecules are their flexibility and length. The theoretical consideration of the

volume interactions in polymers adds another dimension to our predictive capabilities.

Is it worth the effort to account for the volume effects in a polymer coil, or does it

only have a minor effect due to the low concentration of monomers inside a Gaussian

coil? It can be shown15 that although the concentration inside a Gaussian coil is low, it is









enough to swell the coil to some degree. It should be expected that when the

concentration is forced to be higher, such as in situations of confinement or attraction

between chains and other objects, the difference between Gaussian and real chains will be

even more pronounced.

Flory's mean-field argument. Perhaps Flory was the first to understand the

significance of the role that excluded volume plays in polymer systems. He proposed the

following method15' 2 to determine the correct scaling of the chain size with molecular

weight. This method can be viewed as crude or elegant depending on one's perspective,

but his thoughts certainly did provoke a thorough analysis of the problem by other

researchers.

The method involves the minimization of the free energy of a single chain with

respect to its degree of swelling defined by --Rend2/NI2. For a swelling coefficient of 1,

the chain is ideal. To an approximation the internal energy is taken from binary

interactions only in a virial expansion as U/kT=B 2 and the concentration is

approximated as the number of segments in a sphere of radius Rend. The entropy is

assumed to be similar to that of the entropy of a stretched ideal chain. Essentially the

swelling or shrinking of the chain, as a result of binary interactions, is similar to the

stretching of an ideal chain with fixed ends which was quantified in Equation 12. This

results in



kTBN1 /2
F(a) = U(a) TS(a) = const + K 133 + -kTa2 (15)
/3 a3 2









where B is the second virial coefficient that is related to the amount of volume the

monomers exclude. It can be positive or negative depending on the strength of solvent-

monomer interactions. The result of minimization ofF with respect to a is



R9nd = /2N1/2 N3/5. (16)



It is useful to reconsider this problem in an arbitrary number of dimensions. If the

number of dimensions is d then the general result of minimizing Equation 15 is

Rnd IN3/(d+2) To understand this consider for the moment the 1-dimensional walk with

excluded volume. There are only two possible chain conformations, each with a length

IN which is exactly the scaling predicted above. This equation is correct for d < 4 in

fact. For the case d > 3 (of theoretical interest only), the monomer concentration

becomes so low that the ideal chain result N12 is recovered since there are no bead

interactions.

Edwards' mean-field theorem: the development of a self-consistent mean-field

that predicts the correct exponent for Rend in all dimensions. S. F. Edwardso1 derived the

correct scaling laws through a more quantitative method in 1965.

In the above discussion we made some mention of the similarity that polymer

chains have with the mechanism of mass transfer by diffusion. There is even a

mathematical correspondence between the differential equation for diffusion, and the

equation that describes the "diffusion" of the probability function: P(R, L). A

comparison of the mass transport and chain probability equations follows.









DV2C (17)
ot

and

dP
= 1V2P. (18)
(L


Without making too literal of an analogy, it is clear that the mathematical

solutions to the first equation can also be used for the second. Many of these have been

known for over a century from the work of Fick and Fourier.7 One can check that

Equation 11 is a solution to Equation 18. In fact, all of the results we have shown to this

point (and more) concerning random walks can be derived starting from Equation 18.

Edwards' innovation was to derive an equation like Equation 18 but with a term

that accounts for the effect of excluded volume. He solved it asymptotically for infinite

chain length to find the probability distribution:



P(R,L)= C(L)exp[ 27 (R-(R 2) 1/2 (19)
[ 20L1



where L is the contour length IN. The second moment of this distribution is



/ 2\1/2) 45 )3/ 5( j) 1 L315. (20)
3 31rl



He defined an "excluded volume," v, based on monomer pair potentials, u(r),

asv (1-e-u kT)dr .









The essential approximation in this framework is the replacement of a summation

of the "instantaneous" pair potentials at spatial position R with the average potential (or

probability) at that position. The polymer community has adopted the term used by

physical scientists (particularly in the Hartree-Fock theory of atoms): the mean-field

approximation.

It was Edwards' opinion (reference 10, pg. 614) that the mean-field

approximation was very nearly the highest level possible within the framework he was

considering.

It is a property of the self-consistent field for atoms that, although the first
approximation is straightforward though involving heavy computing, the
higher approximations are virtually impossible. The author suspects the
same situation here. It should be possible to evaluate the self-consistent
field numerically, but to improve on the self-consistent field is probably
very hard.

Indeed, since then it has required a completely separate approach, along the lines of the

Ising model in magnetics, in order to obtain better answers.

In what follows in this section we will review a simple derivation of the Edwards

equation proposed by P. G. de Gennes.2 This will prove useful since the modern theory

of polymer adsorption is an extension of Edwards' theory.

The derivation is in terms of the unnormalized probability function, or weight

function G(r ', r), for a chain on a simple cubic lattice. The weight for a chain starting at

the origin, GN(0, r), is proportional to the probability P(R, L) that the Nth step of a chain

starts at 0 and ends at r. For any conformation {ri}, the mean-field is realized as a

potential field U(r) that does not depend on {ri}; rather it depends on ({r,}). The









complete mean-field partition function is _l-Iexp[-U(r,)/kT]. Then our weighting


function is defined as



GN(r',r)=z N Z exp[-U(r,)/kT] (21)
Ir,}1, ,



where z is the lattice coordination number and the summation is only over the subset of

conformations that begin at r' and end at r, {ri}r'r. The factor zV is the total number of

conformations of N steps and can be thought of as a "partial normalization" which causes

G to have the same order of magnitude as the probability.

The weight of an N+1 segment chain is related to that of an N segment chain by



GN+(r', r) = z GN(r', r") exp[- U(r)/kT]. (22)
r



The linear expansion of the exponential term is valid since the local potential is small

compared to the thermal energy. An expansion of the weight function inside the

summation for which the derivatives vanish (due to symmetry) leads to the operating

differential equation.



dGN(r',r) U(r) a2
S G(r, r)+- V2G(r',r) (23)
dN kT 6









The partial derivative on the left follows from taking the limit as the lattice spacing

becomes small (i.e., large chain length). Equations 19 and 20 come from the asymptotic

solution of Equation 23 for infinite chain length with the appropriate boundary conditions

for an isolated chain (symmetric, etc.). A mean-field potential of U(r)~ oG(0, r)dN

based on hard sphere pair potentials was used. The approach used in the solution is the

dominance of the smallest eigenvalue for large N in an eigenfunction expansion.

Note the mathematical similarity of this to the standard Schrodinger equation

which is

-i f h 2
-ih V2+ V(r) (24)
dt 2m



The wave function t(r, t) is the square root of the probability of finding an electron at the

position r and time t. The derivative with respect to time is analogous to the derivative in

(20) with respect to N. In other words, the chain length is a time-like variable.

Integrating this over time gives the probability density, or electron cloud. Thus it is very

much like the function GN(r) and the mathematical methods can be borrowed. For a

hydrogen atom V(r) is quite simple; it is attractive and proportional to 1/r. For the rest of

the elements which have multiple electrons, a mean-field potential is used instead.

However this potential is different from U(r) in that is it based on long range, soft pair

potentials.









The Flory-Huggins Theorem A Fundamental Equation of State



A simple form for the free energy of mixing derives from the consideration of the

polymer on a lattice along with the common mean-field assumption.17'8 In the schematic

we show a square lattice, but the following applies to any lattice of coordination number

z.




F kT ln0 + (1- 0)ln(1 -0) + X( ), (25)




where 0 is the volume fraction (a3Nn/V) and X is a parameter that describes the

interactions (solvent quality). The first two terms are simply the entropy of the

configurations. The third term describes the internal energy of the interactions between

the solvent and monomers.

Notice that when N = 1, the regular solution result is recovered for the entropy.

The entropy is derived in the following way. What is the total number of

(distinguishable) ways that it is possible to fill a lattice with ns solvent molecules and n

groups of N connected monomers? In the usual way we call this number D. For a regular

solution this number is easy to find. Assuming the lattice is fully occupied by the nl and

n2 molecules of the binary mixture, there are n,,=nt+n2 total lattice spaces. The number of

ways to occupy no spaces is no!. Thus the number of distinguishable conformations

Q = n!/ln1!n,! which give the entropy, S / k = -01ln1i I2ln 2. There is no known way

to find for a set of chains on the lattice except by systematically counting all









configurations (e.g., with a computer) an impossible task for even small chains.

However, without much effort we can find a number, W, which approximates Q...

Start by adding the first monomer to the lattice. There are N1u+u possible ways to

do this for a system with n solvent molecules and n N-monomer chains. There are z-1

possible ways to add the next monomer in a way that is connected to the first. There are

approximately z-1 ways to add the third monomer and approximately z-1 ways to add the

fourth and so on. These approximations are too high especially as we add more and more

monomers to the system. A better way to estimate the number is to use an effective

coordination number z'=(z-1)(1- ). If we add the chains sequentially then the fraction of

occupied sites is changing depending on how many chains are already on the lattice 0.

In that case each chain has W,=(Ni+n )(1-0,) z',1 possibilities. Thus the product over all

1 n
chains is W =- W. The prefactor enforces counting of only the distinguishable
n!,=

chain conformations. The first two terms in (22) follow after substitution of factorials

into W, for convenience and from S = k In W.

Now we discuss the interaction energy associated with this lattice. We set the

pair potentials to be zero for all but neighbor interactions. For neighbor interactions u is

either %mm, Zsm or s. The quantity Z= X,, -(Zmm +Z,,)/2 describes the energy

difference for each monomer-solvent pair. The number of such pairs comes by

considering the number of (non-connected) neighbors of a single chain: (z-2)N+2 or

roughly zN. Then in the mean-field approximation, the number of unlike pairs is nzN(1-

0). The energy difference of mixing is then U = y(1-0). (on a per site basis) where we


have absorbed z into the X parameter.









Equation 27 was tested with osmometric measurements.2 As expected, it

performs poorly at low concentrations due to the large fluctuations present in dilute and

semi-dilute solutions. Under these conditions, W is much different than D, and U is much

different from the actual interaction energy. Thus a different form of the free energy

where the internal energy is just a virial expansion is more frequently used:



C 2 1 2 3
F/kT=-c nc+vc+wc 3, (26)
N



where v (excluded volume) and w are adjustable. This expression is in terms of

concentration c=O/v since it is meant for systems with variable excluded volume unlike

Equation 25 which implies an excluded volume of a3. A comparison of Equation 26 and

the expansion of Equation 25 yields v = a3(1-2y) and w=a3. Thus, Equation 26 has two

adjustable parameters. A solvent that has no change in internal energy upon mixing it

with a polymer has a X = 0 and is called athermal. For this case, the satisfying v = a3

results. This corresponds the typical case of a polymer in a good solvent. Further

modifications and extensions of the Flory-Huggins have also been proposed.18


Scaling Laws



A mathematical analogy between critical state magnetics (Ginzburg-Landau) and

excluded volume walk (de Gennes). It was realized that the strong fluctuations that are

typical of polymer conformations can be described mathematically in the same way that

certain strongly fluctuating magnetic systems near their critical temperature14 (Curie









point) are described. This theory was developed by P. G. de Gennes19 in 1972 and J. des

Cloiseaux20 in 1975. In the polymer analogy the Curie point is the chain length above

which universal scaling occurs. For example, in Figure 3 we have identified a Curie

point of approximately 50 lattice units. Based on this method, the most accurate

predictions for polymer chain are available. For example, the so-called Flory exponent v

(Ren,,NV) was found to be 0.592.11 This compares well with simulation results.21

Some more detailed information on the scaling of the probability distribution of

end-to-end distances can be derived. The probability of an end-to-end distance of zero is

zero and then increases rather steeply as R0.35. At large distances the probability decay is

steeper then Gaussian, more like exp(-R25). A schematic of the unnormalized probability

distributions for a Gaussian and exclude volume chains is shown in the diagram. Both of

these distributions should have a value of 1 for the fully extended chain W(Nl)=1 and 0

above NI.

While this theory predicted the same (and correct) scaling for the radius of

gyration as Edwards' self-consistent mean-field theory (SCMF), what gave it

considerable merit was its prediction of the scaling for the osmotic pressure in a semi-

dilute.



/ kT a3N9/ 4 (27)



which is asymptotic for N>>1. This is in contrast to the scaling obtained from the SCMF

(23) of r -2. Experiments confirm the predictions of (24). Perhaps a more important

result (at least for further theoretical progress) of the analogy with critical state magnetics









is the correct prediction of what is called the correlation length which is the basis of a the

accurate, versatile and simple "scaling theory."

Scaling theory. "Scaling theory" is a popular term used to describe the host of

problems that have been tackled using transparent arguments based on the results of the

magnet analogy.2' 14 The key component to this method is the assignment of an important

length scale that characterizes the range of spatial concentration fluctuations in a

polymer solution. For a dilute solution this length is simply the radius of gyration and

thus ~-N3/5. In more concentrated solutions, the overlap of chains reduces the

"correlation length." The solution is visualized as consisting of a number of independent

"blobs" of subchains. The size of the blobs is the correlation length. Each blob on

average has no interaction with other subchains and thus if the average number of

monomers in each blob nb is large enough it has the same scaling has the dilute radius of

gyration ~-nb3/5

The correlation length should shrink as the concentration increases. Its

concentration dependence by using the fact that the onset of chain overlap must occur at

* _-N4/5. Then the scaling law (~ ~m is deduced from the overlap condition

(0*)~(0*)m~Rg. Therefore m=-3/4 and since the correlation length should approximately

be the monomer size at high volume fraction:



S= 13/4. (28)



The SCMF was the first to suggest the importance of a correlation length in the

problem but there it was found ( = l1/2. The use of either of these correlation lengths in









the scaling of applied problems including polymer adsorption can produce markedly

different predictions.


Phase Diagram



This leads us to the discussion of the phase diagram of polymer solutions.4 The

length of polymer chains leads to an extra phase (semi-dilute) that does not exist in

regular solutions. The chain length and solvent quality will play the primary role in

determining the state of a polymer solution at a given concentration.








SEMIDILUTE


MARGINAL


0 DILUTE



CONCENT
1/2 IDEAL RATED


0 1


Figure 5: Sketch of the 5 regimes of a polymer solution.









The onset of the semi-dilute regime occurs when chains begin to overlap. This

"overlap concentration" c* is easily identifiable. At the overlap concentration the

solution will have the same concentration as the average concentration inside the

unperturbed polymer coils which is approximately c*=N /Rg3 N4/5. Therefore the

overlap concentration decreases rapidly with chain length and the width of the semi-

dilute phase increases. The overlap concentrations have been calculated in the chapter on

simulations for a self-avoiding walk on a simple cubic lattice. These calculations are

based on a slightly more accurate factor 47r/3 in the formula and are quite consistent with

data on the onset of shrinking of the average chain size.

One way of trying to quantify the various regimes of a polymer solution is starting

from the expanded mean-field free energy4 Equation 26. Differentiating with respect to

the number of solvent molecules and expanding the logarithmic term gives:

-p = / N+ u02/2+ ox 3 /3 (29)

The osmotic pressure is proportional to the chemical potential. Thus, the first term

represents the ideal (ideal gas) contribution, the second gives the binary excluded volume

interactions and the third term should be dominant for concentrated solutions. Five

regimes can be defined according to Equation 29. The ideal, marginal and concentrated

regimes are said to occur when the first, second and third terms respectively dominate.

The other two have already been mentioned: dilute regime where v causes a swollen

radius -N', and the semi-dilute regime, a situation of overlapping chains with a positive

v. There is a crossover between semi-dilute and marginal when the overlaps are strong

enough that the solution becomes homogeneous.









Polymer Solutions at Hard Interfaces



One of the more well established areas is the theoretical description of polymers

adsorbed to a solid surface. For the most part, solid surfaces have a net weak van der

Waals attraction with the monomer units of polymer chains. For simplicity, it is often

thought of as a square well attractive potential.

From a free energy standpoint three are three primary effects of adsorption:

reduction in the entropy, an decrease in internal energy for the segments that are

adsorbed, and an increase in internal energy as a result of monomer crowding near the

surface.


Mean-Field Theories of Adsorption on a Lattice



The mean-field lattice theory was first proposed by Roe22 and Helfand et al.23 and

reached its present stage at the hands of Levine et al.24 and Scheutjens and Fleer.25 The

main result of this work is a numerical prediction of the concentration profile of

monomer units next to a surface. It also gives the exact contribution to the concentration

profile of segments belonging to tails and loops.

The system is divided into M layers parallel to the surface with L lattice sites in

each layer. At z=0 there is an attractive (free) energy X,, and polymer concentration

decays to its bulk value as z approaches M. Edge effects will be small for large systems.

Conformations are distinguished only by the layers they pass through. Two

conformations are considered to be indistinguishable if their corresponding segments are

in the same layer, regardless of where they are located within the layer. The complete









specification of a conformation, c, would be (1,il)(2,i2)(3,i3)...((N,i) where the first

number is the segment rank within the chain and i, is the layer that segment is in.

The method is an extension of the Flory-Huggins method that was used for bulk

solutions, The canonical ensemble is chosen with the partition function Q({nc}M,L,T).

The set {nc} represents the set of conformations such that nc is the number of

conformations of the c type. Thus the total number of chains within the system is the

summation over all possible conformations: n = _nc. The volume fraction in layer i can
C

1 ,
be calculated from the distribution of conformations by =1- n,,c where nzc is the
L c

number of segments of conformation c that are in layer i.

The Helmholtz free energy is constructed from F= U kTln where U and DQ

are functions of the distribution of conformations {n,}. The change in free energy per

lattice site between a bulk Flory-Huggins solution and an adsorbed layer is actually used




AF = U- UF kTln (30)
QFH



The degeneracy DQ is constructed by adding one conformation at a time from an

arbitrary conformation distribution. The degeneracy of each successive conformation

depends only on the conformations that have already been added to the lattice. It depends

on these in an average way. In other words the probability for a new segment to be

placed in layer i depends only on the number of segments from previous chains in layer i

(Bragg-Williams approximation). Likewise the internal energy of each layer i is equal to









ZL(l- ,)( 1 + 1,+). (31)



The solution to this puzzle is to find the equilibrium distribution of

conformations. The variational principle of thermodynamics applies to the replacement

of N solvent molecules by a new chain with conformation d such that

dF = Pihandnd + pdn where upo and no are the chemical potential and number of the

solvent molecules. That results in the operating equation



(dF / d)M,L,T,nc, = chaw -Np . (32)



This yields an equation for the equilibrium number of conformations of type d,

however it is not readily solvable as such since the index d is not exactly a scalar. It is

recast in the form of the familiar probability p, that the ith segment of a chain will appear

N-1
in layer j. This yields a matrix equation of the form P,, = (w,,)) p,, for which a

numerical solution method was developed by DiMarzio and Rubin.26 The volume fraction

in each layer is then calculated by the same method as in Equation 13 and the individual

contributions are obtained analogously to Equation 41.

This theory gives comprehensive results for finite chain lengths and for

conformational features. Scaling theories, of course do not contain any information about

finite-sized chains. That means that the only other available method to obtain

information below the scaling limit is simulation. We will be making comparisons

between the two in Chapter 8.









The Ground State Dominance Solution to Continuum Mean-Field Theories



The Edwards' equation is still valid for confined systems, it requires only an

adjustment of the potential U(r). It can be written as U(z) =f(z) +J[(z)] where the first

term depends on the surface and the second term is a result of the average interaction

energies between solvent and monomer units. For a hard wall with short range (of order

of the monomer size) attraction, f,(z) can be taken as a constant in the region for 0
Elsewhere in the solution (z>l) it is zero, and for z<0 it is infinity so that nothing can

penetrate the surface. The virial expansion Equation 26 or the Flory-Huggins form

Equation 25 is often used to obtainA (z)].

The first attempts at the use of Edwards' equation to describe polymer adsorption

were by DiMarzio,27 de Gennes28 and Jones and Richmond.29 The starting point was the

adsorption of a single chain to a surface. This was a purely academic problem since it

was known that single chain adsorption effectively does not occur. That is, even in the

most dilute conditions the surface will be crowded with chains. This is in contrast to the

depletion interaction where very few chains are near the surface, and thus the single chain

analysis is often relevant. Jones and Richmond formulated the first realistic self-

consistent mean-field theory for physisorption. They limited their solution to the "ground

state" approximation, i.e., the first eigenfunction. Apparently this gives a poor

description of tails, and as such is valid for extremely long chains in which the tails are

greatly outnumbered by loops.









Scaling Description of Polymer Adsorption



In 1981 de Gennes30 used scaling arguments to characterize the concentration

profile at the interface. He proposed that there are three important regions calling them

the proximal, central and distal regions. The proximal region extends only a few segment

lengths D (which depends only on the adsorption strength) from the surface and its

characteristics are non-universal. In the central region the concentration decays with a

power law of-4/3 as



S )4/3
S a / D < z < b (33)
z +3D/4



and in the distal region it decays exponentially...



/ b= -1+ exp(-z/b) Z > b (34)



These predictions are meant to describe a semi-dilute solution in good solvent conditions.

The exponent in Equation 33 was derived from the scaling of the local correlation length

with local concentration 0 as (~ -3/4. This has been frequently referred to as the self

similarity principle.

The objective is to describe the scaling of the correlation length in the central

region. If that is possible then the concentration gradient is directly accessible. The main

length scales of the problem are the segment size a and the radius of gyration Rg. These

are relevant in the proximal and the distal regions respectively. In the central region,









neither of these are suitable for a description of the gradient of the correlation length in

the central region. The correlation length must have an increasing scaling law with the

distance from the surface z since for adsorption, the concentration decreases smoothly

from the surface to the bulk. Near the beginning of the central region the correlation

length will be of order of the monomer size and at the end of the central region it will

have reached the bulk correlation length or in the case of the a dilute solution, the bulk

radius of gyration. From a dimensional argument alone, the correlation length must be

linear in z since no other appropriate lengths exist.



z (35)



To summarize then the resulting predictions for the concentration profile:

DeGennes ~ -3/4 0 z -4/3
Edwards 0-1/2 o~ Z-2 (36)
0 solvent 0-1 -- 0 ~ z-1

These have not been tested conclusively. The second one listed should be appropriate for

the marginal regime. The conditions inside the central region usually fall within the good

solvent / semi-dilute regime, therefore the first part of Equation 36 is the most useful.


Square-Gradient Mean-Field Free Energy Functional



In the same paper de Gennes developed a mean-field approach that is an

extension of the Cahn-Hilliard theory for the free energy of a region of regular solution

near an interface. This theory proposes that concentration gradients in a solution add a









term proportional to the square of the gradient in the free energy expression. Thus his

expression for the free energy of a polymer solution interface, for a 0-solvent is




Y Yo= r, Y + d-- ++ dz (37)




where V is a complex number that determines the concentration profile by 0 = I V 12. The

first term is the contribution from the surface adsorption, the second is related to the

entropy of a concentration gradient (as mentioned above), and the third accounts for the

interaction energies between monomers (see Equation 26). Thus for any volume fraction

profile, (z), the difference in free energy between the interface and the bulk is fully

determined. The (z) that produces the lowest possible free energy difference

corresponds to equilibrium. It is interesting that the minimization produces the same

differential equations that would result from the Edwards equation. An approximate

analytical solution was found for the central region in a 0-solvent38 as




0 a (38)
2co(z+ D)



therefore the expectation is that a broader layer is produced in 0 -solvents. Inherent in

this solution is ground state dominance. Finally, in this comprehensive work, a slightly

different form for the free energy in Equation 37 is suggested that can produce the scaling

results Equation 33 through a minimization procedure identical to the one used in the

mean-field approach.










Beyond the Ground State



Ploehn and co-workers31 extended the mean-field theory of Jones and

Richmond.29 They attempted to accurately model the boundary proximall region) with

the idea this might have an impact on the overall structure of the interface. A second

"boundary Edwards equation" is matched with the regular Edwards equation. This

system was numerically solved for various chain lengths and solvent qualities. The

choice of the potential derived indirectly from Flory-Huggins Equation 25:



U(z)/kT= -In (-) -2X( Ob) (39)
(1 -O b)



Although solutions for various chain lengths were obtained, only those for the highest

chain lengths should be correct due to the ground state dominance approximation.

Several developments have come recently from Semenov, Joanny and co-

workers.32, 33, 34 Using the same square gradient approach as de Gennes but with an

interaction energy in (23) dominated by v, they find



O(z) = Ob th2(z / 2 + const) -o > 2/(z+ D)2. (40)



This is supposed to be appropriate for the so-called marginal solvent conditions.

Separately they start from the Edwards equation and attempt to find a solution

that takes into account the effect of tails. Two eigenfunctions are used to describe bound









and free states. The partition function G(n,z) is divided into adsorbed and free parts Ga

and Gf. The contribution that loops and tails make to the concentration can then be

calculated by




01(z) = dnG, (n,z)Ga(N -n,z)
(41)

(z) = 2b f dnGf(n,z)G (N-n,z)
N



Additionally they proposed from scaling arguments that the central region can be split

even further into two regions: call them central a and central b. Central a is dominated

by loops and central b by tails. The crossover in the concentration profiles z* scales with

the chain length as N1/3

Juvekar et al.35 followed up on the work of Ploehn et al.31 recently by extending

the numerical results to a system with variable solvent quality. Again they take an extra

boundary Edwards equation and even a boundary solvent quality in their solution. They

suggest that a simplification used previously in the mathematics was unwarranted. There

have been indications that the "effective" solvent quality can change with chain length

and concentration. They take this effective X to be of the form:



= xZoO( +c+ c202)(+ C3/ N) (42)



They suggest that the effect of this improves agreement with experimental data for bound

fraction and adsorbed amount. Also they can calculate the tail and loop profiles.














CHAPTER 3
EXPERIMENTAL CHARACTERIZATION OF POLYMER INTERFACES



Measurement of Polymer Structure at Surfaces



Knowledge of the structure formed by adsorbing polymers at interfaces is

essential to the ultimate understanding and control such layers for engineering and other

purposes. It is a demanding experimental task to measure, for example, the structure of a

layer of physisorbed homopolymers at equilibrium with a bulk solution. This requires

carrying the experiment out in a liquid environment and the use of non-invasive

techniques. These caveats limit the options considerably. For example, even x-rays exert

some physical influence on polymer layers in some reflectivity or scattering experiments.

Physical probes, even in the rare cases when they are as "gentle" as the AFM tip in

tapping-mode or hydrodynamic probes, cannot image the internal regions of a polymer

layer.

One of the few approaches that meets the requirements of such proposed polymer

measurements is the utilization of neutrons. These have a small enough wavelength to

probe the atomic scale and are completely non-destructive owing to their infrequency of

collision with sample atomic nuclei.

In the following section we discuss some of the overall, or global properties that

can be measured without resort to a neutron source. Subsequently we discuss the detailed

structural information that only neutron scattering and reflectivity can probe, and their









current limitations in sensitivity for adsorbed layers. An extensive review of all of these

techniques is contained in the Chapter 3 of the book by Fleer et al.4


Global Properties of Physically Adsorbed Layers


There are several static properties of polymer chains at an interface that can be

measured experimentally. They consist mainly of three types: measurements on the total

adsorbed mass, layer thickness, and the fraction of monomers in a chain that are directly

bound to the surface.

The adsorbed mass, F, represents the total number of chains in contact with the

(i.e., adsorbed) surface per unit area, multiplied by their molecular weight. Thus its units

are in terms of monomers per unit area. An equivalent quantity can be obtained from

theoretical or simulation data for the adsorbed-chain volume-fraction profile, 0a(z), as




F = J (z)dz= (z). (43)
0 z=1



The summation is written for use with theoretical models based on a lattice, and

consequently with a discrete concentration profile.

The adsorbed amount can be measured by flushing the solvent and excess

polymer from a system in which the chains have had sufficient time to adsorb. The

adsorbed amount is the difference between the weights before and after adsorption.

Provided desorption kinetics are slower than the flushing process, this is usually a reliable

method.4









Layer thickness is an ambiguous term as the adsorbed layer is diffuse and does

not have an abrupt "edge." Layer thicknesses can be defined in several ways. The mean

distance of polymer segments (belonging to adsorbed chains) from the surface, or for that

matter, any of the statistical moments. The most frequently used actually is the second

moment, or rms layer thickness:




SJz 20(z)dz
8.0 (44)




Again a discreet volume fraction profile obtained from simulations would require a

summation instead of the integral.

While there is no technique that actually measures this hypothetical quantity there

are some techniques which are thought to contain related information. In a dilute

dispersion of spherical particles, neutron scattering is said to measure the difference

between the squares of the second and first moments.4

One of the most common and accurate ways to measure layer thickness is by light

ellipsometry (similar to reflectometry). The sample is a macroscopically flat surface

covered with adsorbed polymer inside a fluid cell. Light is reflected off of the surface at

a low angle and the intensity distribution of the reflections give information about the

refractive index of the polymer layer as a whole. There have been several attempts to

link the quantity that is measured by ellipsometry to the theoretical volume-fraction

profile. The definition preferred by Fleer et al.4 and which we will use in succeeding

chapters is













6ell f0d2 (45)




The most intuitive quantification of thickness is through hydrodynamic methods

because these quantify the degree to which a fluid flow is influenced by the presence of

the layer (as compared with flow past the bare surface). Fluid is driven past a polymer

layer adsorbed to the surface at z = 0. The extent to which the flow is reduced is

interpreted as equivalent to the linear flow past a bare surface at z -= hyd. Typically the

polymer is adsorbed on the inside of a capillary tube. The Debye-Brinkman

hydrodynamic model for flow past a porous layer can establish the relationship of the

hydrodynamic thickness with the volume fraction profile.4 The velocity profile in the

direction normal to the surface v(z) is described by the following equation of motion:



d2v _
=v (46)
dz c(1- (a)



where the drag term on the right depends on the relative porosity of the layer (the

permeability constant c is usually taken as 1.0. The linear part of solution for v(z) is

extrapolated to zero and the z to which this corresponds is taken as the thickness.

The bound fraction p is the fraction of the segments belonging to adsorbed chains

that are in contact with the surface. Several methods exist to observe this, for example

infra-red spectroscopy can distinguish certain types of contacts with the surface such as

hydrogen bonding. Another method that can distinguish between attached segments and









non-attached segments in nuclear magnetic resonance. A magnetic pulse is applied to the

system and the relaxation of the polarization of the atomic nuclei gives information about

the thermal mobility of these atoms. Presumably the thermal motion of attached

segments is greatly reduced. The bound fraction is easy to obtain from lattice

calculations since the number of segments in layer 1 is just p = (1) / F.


Small Angle Neutron Scattering



There exist few experimental methods with sufficient sensitivity to probe the

polymer volume-fraction profile. For reasons stated above the most progress has been

made by small angle neutron scattering (SANS) and neutron reflectivity. An excellent, if

not amusing, account of neutron scatting that assumes little foreknowledge of optics is

available in the review by R. Pynn.36 Reviews of neutron reflectivity experiments on

polymers have been written by T. P. Russell.37' 38 A non-technical overview of these

techniques follows.

The length scale of neutrons and x-rays, of the order of 10-7 to 10-10 m, is

appropriate for probing the microstructure of many complex fluids. Neutrons follow the

same quantum mechanical principles as do photons and thus neutron scattering has much

in common with conventional light scattering. In fact the refractive index has a parallel in

neutron techniques which depends on the makeup of a sample's atomic nuclei the

scatterers. A measure of the strength of neutron-nucleus interaction is the scattering

length, b. For example the coherent scattering length for hydrogen and its isotope,

deuterium, are -3.74 and 6.67 fm respectively. Actually this large difference turns out to

be very important in contrast matching. By adjusting the level of isotopes in a material,









its refractive index can be controlled. Thus the feature of interest can be labeled by

adjusting the contrast of the other components in the system so that they are "invisible."

A SANS experiment on polymer layers proceeds as follows. A dilute dispersion

of colloidal particles bearing polymer layers is prepared, using deuterium isotopes to

match the refractive index of the solvent to that of the colloids. The key parameter is the

scattering vector, Q, which explores the range over which structure can be explored. The

scattering vector is varied by adjustment of the wavelength, A, and the detector angle, 0,

defined by



47y 8A
S= sin (47)
>A 2



where the neutron refractive index, p, is a derivative of the scattering lengths in the

materials.

The output of the instrument is the scattering intensity as a function of the

scattering vector, I(Q). Models have been developed to interpret the structural

information in I(Q) which describe the scattering due a thin spherical shell (i.e., the

polymer layer). Thus the scattering intensity is mathematically related to the volume

fraction profile through an integral. In one such model the limit of low Q yields4



const !
I(Q)- os (z)ezdz (48)
Q 0









At this point a difficulty in the analysis becomes apparent. The inversion of the integral in

Equation 48 to obtain (z) is not trivial. There have generally been two methods used to

proceed. The first method uses assumptions about the functional nature of (z) and a self-

consistent fitting procedure.39 The second method does not invert the integral in Equation

48 at all but rather analyzes the data directly for certain features that would be consistent

with predicted layer structure.40


Neutron Reflectivity



Neutron reflectivity is similar to neutron scattering. The major difference is that

instead of a colloidal dispersion, the experiment is carried out on a macroscopically flat

substrate bearing the polymer layer. The effectiveness of the reflectivity technique has

emerged in the last 5-10 years. Its major strength is its high depth resolution to

approximately 0.5 nm.38

The analysis is similar (broadly speaking) to that described above for neutron

scattering. Reflections at low angles off of the polymer layer and substrate can be

analyzed in terms of a sequence of Fresnel films of decreasing refractive index for the

case of homopolymer adsorption. Reflectivity has found specific merit in the analysis of

diblock copolymer interfaces.38


Scattering and Reflectivity Measurements on Adsorbed Polymers



Relatively few SANS41' 42, 43 and neutron reflectivity44' 45 studies have been

performed on physisorbed homopolymers layers in a good solvent. The SANS









experiments of Auvray and Cotton42 (polydimethylsiloxane (PDMS) with MW=270,000

adsorbed on silica) were consistent with the scaling power law prediction for the

intermediate interface (-4/3, as opposed to -2, the mean-field prediction). However,

neutron reflectivity measurements on shorter deuterated polystyrene (PS) (MW=50,000)

adsorbed on mica were consistent with a simple exponential decay of polymer

concentration.45 However the sensitivity of these techniques is such that volume-fractions

below 1% cannot be observed,39 which constitutes a substantial part of the profile from

physically adsorbing chains (unlike grafted chains).

It is notable that several attempts have been made to observe simultaneously46' 47

or in parallel48 the structural aspects and the interaction forces between polymer layers.

This is certainly a direction that will lead to a richer understanding of the cause and effect

of the response of polymer layers. A new surface force apparatus which uses neutron

reflectivity in situ is described47 and force and volume-fraction profiles for polystyrene

adsorbed on quartz in cyclohexane were obtained. In a similar experiment using block

copolymers the volume fraction profile was observed to increase due to the compression

and a collapse in the layers was witnessed when the solvent was changed from good to

poor.




Measurement of Polymer Induced Forces



The primary tool that has been used to measure forces between surfaces bearing

polymers is the surface force apparatus (SFA) developed largely by J. Israelachvili.49' 50

The SFA remains the only apparatus capable of measuring directly the forces induces by









polymer chains between semi-infinite solid surfaces. For this reason the bulk of this

section will discuss the apparatus itself, and measurement that have been made on

homopolymers physisorbed from a good solvent. Much progress in this area was made in

the 1980's, however in last decade SFA studies have focused on more complicated

systems.

Another tool that has great potential towards isolating the forces induced by

polymer chains is the atomic force microscope (AFM).51' 52 While the AFM uses an

inherently microscopic, or local, probe, continuing growth in this area has led to advances

such as the use of spherical 10 jam particles as probes.53 One advantage of AFM is the

variety of materials that can be used for the two solid surfaces in contact. In the future,

advances in AFM will complement the understanding gained with the SFA.

Other techniques for measuring surface forces have been reviewed by Claesson et

al.52 but, in general, have limited applicability to polymer systems. Total internal

reflection microscopy on a colloidal probe controlled by a laser has been used to measure

weak forces of order 10-14 N. Forces can also be inferred indirectly from phenomena such

as the swelling of clay minerals.54 Naturally prior experience with the phenomenon as a

"calibration" of the technique is prerequisite to these approaches.


Surface Force Apparatus



The surface force apparatus is an instrument that can sensitively (10 nN, 0.1 nm)

measure the force and distance between two curved mica surfaces, often enclosed in a

liquid cell. This is accomplished with a combination of spring deflection and multiple

beam interferometry. A cursory description of the apparatus is given here as a reminder









of its basic function and capability. For an extensive discussion of the development and

details of the SFA the subject is well reviewed.52' 55 3, 56

The major success of the SFA was the confirmation DLVO theory. The use of

molecularly smooth mica surfaces was instrumental in factoring out the substantial

effects of surface roughness in van der Waals forces. Mica however is also important in

the method for determining the distance between the two surfaces because of its optical

transparency. White light passed through the specially coated surfaces yields the distance

through interpretation of the interference "fringe" patterns associated with the reflection

of light off of each surface.

The fine separation between the surfaces is controlled by a piezoelectric ceramic.

Each volt of electricity applied induces a 1 nm expansion of the tube-shaped piezo.

Ultimately the force is obtained by knowledge of the spring constant and the difference

between the actual and applied displacement. Clearly then, reliable measurement of the

separation is doubly important.

The mica sheets are mounted on crossed cylinders in order to avoid the difficulty

of aligning the surfaces in parallel. Force profiles are always normalized by the average

radius of curvature of the cylinders so that they are independent of geometry.











I Piezo tube





__ Leaf spring




Light



Figure 6. Elements of the surface force apparatus. Light passes through the lower mica-
coated hemi-cylinder which is mounted on a leaf spring in order to measure the force.
The silvered surfaces of the hemi-cylinders produce an interference pattern that is
collected above. The upper cylinder is mounted onto a calibrated piezo-electric ceramic
which enables fine adjustment of the distance between surfaces.


Usually a polymer adsorption experiment will proceed as follows. A polymer

solution is introduced into the liquid cell the system is allowed to "incubate" for a period

of approximately 12 hours. When the incubation, or adsorption, is complete, the liquid

cell is washed out with pure solvent and the compression / decompression cycles begin.

Typically the rate of compression is 5 to 60 minutes. This is supposed to be a quasi-

static, or equilibrium, process.


SFA Measurements in Good Solvent



A number of measurements have been made on physisorbed homopolymers in

good solvents.57' 58, 59, 60, 61, 62, 63, 64 These experiments as well as measurements conducted









in poor solvents have been reviewed comprehensively by Patel and Tirrell,65 and also by

Luckham.66 The good solvent systems that have been studied are listed in Table I.



Table I. Experimental characteristics.


Polymer Solvent MW Reference
PEO Aqueous 40,000 57, 58, 59,
Electrolyte 160,000 60, 63
1,200,000
PEO Toluene 40,000 61, 62
160,000
PS Toluene 65


Of those that are in regular usage, there are not many polymers that adsorb onto

mica surfaces from a good solvent. Polyethyleneoxide (PEO) is the only one that adsorbs

strongly enough to make consistent measurements.61 65

In all experiments where the polymer was given sufficient time to adsorb, the

forces observed were monotonically repulsive, growing in strength with decreasing

separation. By contrast, all observations in poor solvents shown an strong attractive well

with its minimum at a separation of roughly 1 Rg.67' 68 The range of interactions in good

solvents is less exact, but generally corresponds to 5 10 Rg.59' 61 For poor solvents

interactions occur only for only 2-3 Rg, indicating the swollen layers found in good

solvents. Toluene is a better solvent than water for PEO and this results in an onset of

repulsive forces several Rg before the aqueous system.

Clearly if polymers are bridging between the surfaces, this will lead to a (probably

substantial) attractive contribution to the force. There has been some uncertainty about

the role of bridging. The simulation study by Jimenez et al.69 indicates that bridging









occurs in all systems, and the number of bridges does not vary much, except with

separation. The osmotic pressure is therefore the root of the repulsive forces observed in

good solvent systems. The monomer-solvent potentials in good solvents are such that

monomers are effectively repulsed from one another due to the entropic, and in very good

solvent, the enthalpic benefit of solvent contact. Therefore the osmotic forces which are

attractive in poor solvents, are strongly repulsive in good solvents to the extent that they

can overcome contending bridging forces.

The largest molecular weight chains used, 1,200,000 Mw PEO in water could be

sufficient70 to test asymptotic theoretical predictions provided polydispersity

(Mw/Mn=1.2) or other effects do not play a substantial role. The density functional

approach used by P.-G. de Gennes30 yields two power laws for the force profile at far and

near separations. The data are consistent with the inner power law (F D-2.25) but an

outer power law (F D-3 is difficult to discern. However the authors preferred to

interpret the results using an analogy between the tail conformations of a strongly

adsorbed layer and a grafted layer. This was motivated by the fact that the force profiles

a common qualitative feature with grafted polymer measurements when shown on a

semi-log plot. On approach of the surfaces the force rises sharply, flattens briefly and

rises steeply again. However, the role of tails should dissipate for exceptionally large

molecular weights and adsorption from a dilute solution. Furthermore, the hysteric

effects noted on decompression may be indicative of weak adsorption.

Klein and Luckham have studied the effect of undersaturation on the forces in a

PEO/aq. System.59' 63 An attractive well of depth -50 [IN / m is observed for such low

surface coverages. This is small in comparison to poor solvents (by a factor of 20) but is









probably enough to explain the flocculation of colloids. This is explained by the relative

reduction in osmotic forces since fewer monomers are present between the surfaces. The

method employed in generating undersaturated layers is as follows. The surfaces are

brought close together (several microns) before the polymer solution is introduced. This

creates a diffusion limited adsorption process. Thus, long times are required for the

layers to become fully saturated. Compression cycles are performed at several intervals

during the slow (i.e., 48-hour) adsorption process, each corresponding to a different level

of undersaturation. With increasing surface coverage, as the minimum becomes

shallower, it also shifts towards larger separations and ultimately disappears well before

full coverage is reached. This is consistent qualitatively with theoretical models

including mean-field,71' 72 scaling73 and simulation69 results, although between these there

are quantitative differences for the range of undersaturation that produces attractive

forces.

Many researchers have noticed considerable hysteresis in their measurements in

good solvents. Unresolved differences in hysteresis using the same polymer / solvent

have even been reported.59' 62 In general good solvent measurements have been less

reproducible and well behaved than poor solvent or grafted polymer measurements.65

There are several conceivable reasons for differences between decompression and

compression force profiles. Perhaps constrained equilibrium (fixed number of polymer

chains) is not plausible for slow compressions, since they could diffuse from the area of

contact. This might effect would probably be strongest for large surface coverages74 and

weakly adsorbing surfaces. Another possibility is that the compression, in compacting

the layers, forces many segments to adsorb and the desorption / relaxation of these









compact conformations is slower than the decompression. Weaker forces on

decompression have frequently been observed.63 In one experiment,59 after a rapid

compression, a constant force was applied and the surfaces decompressed at their own

speed. This produced the same force profile observed from a slow (equilibrium)

compression. Little progress has been made toward robust explanations and predictions

of non-equilibrium effects in these measurements. This is an area where dynamic

computer simulations could make a substantial contribution.


Atomic Force Microscopy



The first AFM was developed by Binnig, Quate and Gerber.75 In addition to the

impressive ability to measure topographical detail of surfaces to the order of tenths of an

angstrom (e.g., the well known imaging of the hexagonal structure of carbon-carbon

bonds in graphite76), the apparatus has been outfitted to measure forces parallel and

normal to a surface. Thus it has contributed to the growing field of tribology (atomic

scale friction) and measurement of surface interactions. Variations on the basic

instrument abound and its uses have grown to include the semi-conductor industry and

biological applications, in addition to atomic and electronic structure and interactions.

The basic instrument consists of a tip or probe attached to a cantilever-spring and

a sensor to detect the deflection of the cantilever. Supplemental components include

piezo-electric system for moving the tip laterally and normally, and a feedback system to

control the applied force, or mode of the tip. Generally there two classes of operating

modes: those that image a surface and those that measure forces between objects.










A constant force "contact mode" is used typically for imaging of hard surfaces.

The high resolution (subangstrom) of this mode is achieved by using a small loading

force between 10-7 and 1011 N.51 The "tapping mode" is typically used for imaging soft

surfaces. measurement of long range forces This uses the method of "resonance shift

detection." The cantilever vibration is driven at its resonance frequency. The vibration is

substantially dampened (or amplified) in the presence of long range repulsive (or

attractive) forces. This enables the instrument to achieve even smaller loading forces of

order 10-13 N (51) which do not disrupt the sample. Finally, and emerging method for

imaging is achieved via magnetization of the probe. An image in generated from the

response of the tip to the magnetic field of the sample.

Laser a)a
deflection
sensor b)
Cantilever





....... Piezoelectric
controlled
stage


Figure 7. AFM experiments on forces induced by polymer chains, a) An example setup
of the AFM for measuring forces between polymer layers adsorbed to colloidal particles.
b) A typical experiment on a single chain with its ends grafted to the AFM tip and the
substrate.

The above modes are used in a lateral scan of the sample. In order to measure a

the normal force profile between surfaces the technique becomes much simpler. In this

method, the voltage to the piezo is ramped (linearly) to drive the surfaces together. Often

a colloidal probe is attached to the cantilever in order to avoid geometrical complexities









in the analysis. This method was first used to measure electrostatic interactions between

colloids.53


AFM Measurements on Elasticity of Single Chains and Forces Between Polymer Layers



The strength of steric and other polymer induced forces (10-9 1011 N) has been

measured between colloids bearing adsorbed polyelectrolyte layers77 and between

colloids bearing adsorbed, aqueous PEO (MW=56, 205 and 685- 103).78, 79 In the most

recent study of Braithwaite and Luckham,79 the force profile was obtained when one or

both surfaces bear a polymer layer at a range of coverages (however both surfaces were

attractive to PEO). Mostly repulsive interactions were observed as the rate of approach of

the surfaces was relatively rapid. The range of interaction between saturated surfaces

scaled with molecular weight along the same lines as predicted theoretically

(experimental onset of interaction: N0.4, theoretical layer thickness: N0.6). No

comparison with the theoretical force profile power law30 was attempted.

Single-molecule polymer experiments are the most exciting new applications of

the AFM. These experiments started with biological molecules such as DNA and Titin

since are exceptionally large and easy to manipulate on an individual basis. In one such

experiment, a DNA molecule was covalently bonded by either end to the glass surface

and the colloidal probe.80 The adhesive force during retraction increased until a rupture

occurred between 0.8 and 1.5 nN. In a other experiments81 researchers actually observed

stretching of a DNA molecule to nearly twice its contour length indicating a rupture of

the its basic helical structure. Dextran also underwent a distinct conformational transition

during stretching at similar forces.82 Titin, a molecule found in muscle tissue, had a









sawtooth shape in its extensional force profile attributed to the unfolding of globular

units.83

Very recently, some success in similar experiments with synthetic molecules was

announced. Most notably, poly(methacrylic acid) was grafted to a gold surface treated

with surfactants in order to keep the grafting density low.81 The single molecules were

stretched by virtue of their physical attraction to the AFM tip and the elastic force profile

was fitted to the freely jointed and wormlike chain models of entropic elasticity. The

persistence length was found to be of the order of the monomer size. Also similar

experiments with polyelectrolyte84 and block copolymer85 chains have been performed.

One experiment imaged isolated adsorbed polyelectrolytes using "tapping-mode" to

obtain the lateral and normal dimensions of the adsorbed chain for different molecular

weights.86 In all, more than 30 experiments (for a summary see reference 81) have been

reported in the last 5 years that analyze the stretching of a single chain, and a new journal

has appeared entitled "Single Molecules" which takes substantial contribution from AFM

research.














CHAPTER 4
MONTE CARLO SIMULATION OF POLYMER CHAINS ON A LATTICE



Here we present an overview of lattice Monte Carlo techniques that have been

used for the simulation of polymers. Our main focus will be on the challenges and

practical considerations in our lattice simulations of polymers. A comprehensive review

of all early methods of lattice simulations of polymers was done by Binder and Kremer.87

Review has also been made of the more recent advances in Monte Carlo (and molecular

dynamics) simulation of polymers.88 5

Why are lattices still used to model polymers even now that we have such

comparatively powerful computers at our disposal? One reason is that the number of

important configurations that a polymer can take, even on a simple lattice is

overwhelming even for a modern computer. To attempt to incorporate more detail into

the model requires a sacrifice in the range of systems one can study. The classical

example of this is the scaling in dilute solution of the radius of gyration with chain

length. While atomic details of chain conformations depend on the model used, the

overall chain size is one of many universal features of polymer conformations. It would

be wasted effort to attempt to obtain the Flory exponent, v = 3/5, from a continuum

model. For the long chain lengths necessary to obtain the exponent accurately, the effect

of the lattice vanishes. Such long chain lengths are not well suited to "weighty"

continuum approaches. Therefore the most exact Monte Carlo calculations of universal









exponents have come from lattice studies.21 There are many other problems in which

coarse-grained information is desired and the advantage of these methods is evident.



Enumeration, an Exact Theoretical Treatment for Model Oligomers



In many ways, Monte Carlo simulation is a substitute for the full enumeration of

all possible configurations of a system. On a simple cubic lattice, a 100-step random walk

can trace 6100 1078 conformations. While the fraction of these that are self avoiding is

extremely small (10-26 %), the number of self avoiding walks is nevertheless staggering.

For the same size chain a self-avoiding walk (SAW) has ~1050 conformations. Current

high-speed computers can only generate of order 109 conformations,4 thus a modern

computer cannot even begin to enumerate the statistics for polymer problems of modern

interest. Enumerations are not simulations, they are exact statistical mechanics

computations (for lattice models).

Enumerations seek to generate each term in the summations representative of

thermodynamic quantities. For such a property, A, in the canonical ensemble, its standard

configurational average is



A(rN)exp[-P(f(rN)]
(A)NV = rN (49)
rN









where the summations range over the entire set of states possible for N-components on a

lattice of size V. Actually the denominator in Equation 49 is often enough to calculate

properties of interest.

Consider the exact enumeration of a SAW of only two steps. On the simple cubic

lattice with coordination number z = 6 and step length a, there are

S= z(z -1)= 65 = 30 self-avoiding walks that are possible once the first bead has been

placed. One-fifth of these are straight conformations with an end-to-end distance of 2a,

while the other four-fifths are bent conformations with an end-to-end distance of 212a.

Therefore the average end-to-end distance is -2a + 2a = 1.55a .

The number of self-avoiding walks grows quite rapidly. For long chains this

number has the following scaling.2



Q,, = cz"N (50)



where y (=1/6 in 3 dimensions) is a universal exponent called the susceptibility, c is a

constant and z ( < z )is the effective coordination number roughly equal to 4.5 for the

simple cubic lattice. Enumeration studies are usually limited to chains with N 20.87

As the number of steps, N, increases, the fraction of all random walks that are

self-avoiding is vanishingly small. This is evident from the limit of the ratio










Q =A c /N16 (51)
Q RW Z



which approaches zero for large N.



Monte Carlo Methods


Static Monte Carlo



Monte Carlo integration. The first Monte Carlo methods were used for relatively

simple numerical integration, or convolution, as it is called. Instead of using techniques

like Euler's method to evaluate the integrand at a sequence of points within the range of

integration, in Monte Carlo integration the integrand is evaluated at a large number of

random points. This proves to be useful for quickly varying functions.

The advent of statistical mechanics made it necessary to improve the methods of

multi-dimensional integration. To compute integrals or summations like Equation 49, a

microstate, r", must be generated at random. The attempt must be abandoned if, in the

SAW example, there is a multiply occupied lattice site. Even for a single chain, though

not quite as severe as Equation 51 (because only non-reversal random walks (NRRW) are

considered), the number of successful attempts is discouragingly low. Thus little or no

progress was possible in more concentrated systems.

Importance sampling. Improvements were made on the Monte Carlo method that

sample the integrand in a range biased towards the most important contributions to the

integral. "Importance sampling" must be corrected with the appropriate weighting factor









for the non-uniform distribution from which it samples. The improvement of the

sampling distribution is the focus of ongoing development in Monte Carlo algorithms.

Of many proposed static Monte Carlo methods for polymer chains (e.g., simple

sampling, dimerization, enrichment), the most successful is the biased sampling proposed

by Rosenbluth and Rosenbluth89 in 1955. In this method, a chain is generated one step at

a time. The chain avoids overlaps during its growth by selecting from only the

unoccupied neighboring sites. They were able to induce, from a simple example, the form

of the bias that this selection procedure imposes.

Consider the two-dimensional construction of a 4-step walk. There are exactly

100 possible 4-step SAW's starting from the origin. Consider the following two defined

by coordinates (0,0) (1,0) (1,1) (0,1) (0,2) and (0,0) (1,0) (1,1) (2,1) (2,2), shown in

Figure 8. These should occur with equal probability since we are not assigning any

interaction energies. However in this scheme, move A occurs with a probability

(1/4)(1/3)2(1/2) while move B occurs with probability (1/4)(1/3)3. The properties of A

need to be weighted by the factor (2/3) in order to correct for the bias.

A B






Figure 8. Two of the 100 conformations available to a 4-step self-avoiding walk and a
two dimensional square lattice.


To generalize, the weighting function, WN, required to correct any given N-step

walk is defined by the recursive relation (W1 = 1):









W+1 = (n /5) W,. (52)



where n is the number of empty neighboring sites at the mth stage of the walk. This

weighting function is used for thermodynamic averages, instead of Equation 49 we have

(for the case of zero energy of interaction)




C. W,(,-)A(r-")
(A)N =im (53)




where the summations range over the sample states generated (and therefore it is an

approximation). This algorithm generally works well for single chains smaller than 100

segments. Its power is considerably extended when substituted into a dynamic Monte

Carlo scheme


Dynamic Monte Carlo



The dynamic Metropolis Monte Carlo method was developed by Metropolis et

al.90 in 1953 and has been applied to everything from sub-atomic physics to colloidal

suspensions. In one sense the method is a leap forward in selection of a biased sample.

This is because it samples from a Markov chain of states. In other words, each trial state

is generated from the previous state of the system. This allows incremental changes to

occur in large systems rather than abandoning most of the trial attempts as in the static

Monte Carlo methods.









The method allows any set of "dynamic" moves to be used, provided that they

follow two important criteria: ergodicity and semi-detailed balance. A move is ergodic as

long as every configuration of the system can, in principle, be reached after a finite

number of moves. Non-ergodic algorithms can be alternated with ergodic ones which

makes the set of moves as a whole ergodic.

The semi-detailed balance condition requires that the frequency of moves any

state, i, should be equal to the frequency of moves out of that state into other states.

Normally this condition is hard to formalize and the condition of detailed balance is used

instead (a more restrictive criterion than semi-detailed balance). According to this, the

frequency of forward and reverse exchange between a pair of states, i and j, should be

identical. The condition is then subject to the transition probability, H,,,between two

states by a given move. Let P, be the Boltzmann weight, exp(-1A(r,)), of state i. The

detailed balance and ergodic conditions ensure that the number of configurations of i that

are generated will be proportional to B,. Thus the detailed balance condition is



Pn,, = Pn, ,. (54)



When a trial, j, move is fully generated, it can be subjected to an acceptance condition. It

will be accepted with a probability A,. When the move probabilities are symmetric

between all pairs the acceptance test is the only constituent of lH,. In this case the best

solution to Equation 54 is to use the following acceptance rule.


A, = min [ 1, B / B, ]


(55)











Therefore a Metropolis scheme is first proposed and the appropriate acceptance rule is

derived from Equation 55. This forces the sampling distribution to be the Boltzmann

distribution. The consequence of this is that the Boltzmann weight drops out of the

thermodynamic averages of Equation 49 yielding instead




(A) = lim (56)



One of the earliest algorithms designed for lattice polymers uses local rotations of

one or two segments along the backbone of the chain called kink-jump91 and crankshaft

moves."7 This is currently an excellent method to use for melt-like concentrations. Also

its correspondence with Rouse dynamics makes it attractive for kinetics studies.

A second method known as reputation was proposed by Wall and Mandel.92 In this

the last segment of a chain is removed and added to the other end. It has a faster

relaxation time than kink-jump methods but is not appropriate for dynamic studies.

A method that is good for very dilute systems is the pivot algorithm.87 It can be

used with very long chains to calculate the Flory exponent to high precision. In this

method a random site along the chain is tagged as the pivot point and one side of the

chain is rotated through a random angle.

The bond fluctuation method is also a well-used lattice algorithm for polymers.93

This uses a more complex lattice scheme with a coordination number larger than 100 in

order to capture some of the angular and stretching flexibility of polymers. It has been

used in a number of dynamic studies.94 Each move consists of a random segment









displacement (with a small maximum) and therefore resembles a Brownian dynamics

simulation.


Configurational Bias Monte Carlo



The method we use is a combination of Rosenbluth sampling with the Metropolis

method called configurational bias.95 It has a higher efficiency in simulations of long,

multiple chain systems than most other algorithms. This method will be discussed at

length in the next chapter as well as some extensions to it. The algorithm is as follows:

1. Randomly select a chain and a unit of that chain.

2. Erase all segments between that unit and the end of the chain.

3. Calculate the Rosenbluth weight (see below) Wold of the erased portion of chain.

4. Regrow the chain to its full length by randomly choosing directions that are

unoccupied.

5. Calculate the Rosenbluth weight of this new portion of chain Wnew.

6. Accept or reject this new portion of chain with probability Wnew Wold (or 1).

7. Go to step 1.

Where the Rosenbluth weight is designed to ensure the proper importance

sampling. It is calculated from the number of contacts with the surface and the solvent

that the section of chain has. For example, a segment touching the surface that has two

solvent contacts will contribute a factor: exp[-(Xs+2X)/kT].

This method is an improvement over those which use local moves such as

reputation, bond rotation, etc. Large sections of chain are moved at once, which decreases









level of "freezing" that occurs with the local moves. This allows the simulation of very

long chains.

We will discuss in Chapter 5, a significant advantage gained by a small

modification to the above algorithm. In step 1 we impose a maximum on the number of

segments that can be erased. This is useful for the simulation of long chains or dense

systems, since the probability of being able to regrow a chain that was cut in the middle

becomes negligible and is therefore a waste of time. In fact, the success ratio for

regrowths of a given size decays exponentially with the increasing size of the regrowth.

Also, instead of regrowing the chain from the original spot, it is regrown from the

opposite end from where the cut takes place. That way, the segments in the middle of

long chains will eventually have the opportunity to move unlike the standard method, in

which the interior segments of long chains are frozen.



Simulation of Bulk Properties



Bulk properties typically have faster relaxation times and are simpler for other

reasons as well. An excellent check of a simulation method is to test it against some of

the well known properties of bulk solutions. Critical exponents can be calculated such as

the Flory exponent (v) or the "susceptibility" exponent y. Here we will examine some

simulations that capture the correct scaling of the radius of gyration and end-to-end

distance.

Table II lists simulations performed on a single chain in good solvent. The size of

the simulation box must be several times the expected radius of gyration. Preferably 4-6

since fluctuations are of order of the rms radius of gyration. That poses a challenge on










the computer's memory for large chains. The Rg and Rend given for N=1-3 were

calculated by hand merely to show the limiting behavior at short chain lengths. Lattice

effects are obviously very important for short chains but appear only in the prefactor of

the scaling law at large chain lengths. The scaling laws Rg=0.41N.594 and Rend-0.99N0594

were found for the large chains (N=200-1000).




Table II. Properties calculated from the simulation of an isolated chain in an athermal
solvent.


N R g R end
1 0 0
2 0.5 1
3 0.6966 1.5492
5 0.9862 2.2760
10 1.5393 3.6579 0.6545
20 2.3739 5.7096 0.3569
50 4.1586 10.0402 0.1660
100 6.3050 15.1859 0.0952
200 9.5639 22.9356 0.0546
400 14.4466 34.6935 0.0317
1000 24.86975 59.59039 0.0155


The plot of this was shown in the Chapter 1 against corresponding simulations in

a near 0-solvent.. For the near 0-solvent the fitted power laws were Rg=0.57N.485 and

Rend=1.34N0.481. The power law develops even at chain lengths as low as 20 and 50, and

is accurate to less than a percent above 100.

Not only do simulations like this give confidence in the technique, but they

provide valuable information. The radius of gyration and end-to-end distance are

important length scales in all polymer problems and, as we will see, help to interpret

them. The overlap volume fraction (see Chapter 2) is also listed in the table and is a









critical parameter where many property transitions occur in polymer solutions. It is

N
calculated directly from the radius of gyration and chain length from 0* =
47R3 /3

Also, the prefactors of the power laws gives us the approximate correspondence

that the lattice size has to the Kuhn length of a polymer. For the 0-solvent an exact

correspondence of 1K = 1.48a is found by comparing the power law for the end-to-end

distance with the freely jointed chain. The end-to-end distance in a good solvent has a

prefactor of the same order of magnitude but slightly lower indicating that it describes

slightly smaller length scales.

To our knowledge simulations attempting to obtain the scaling of isolated chains

for N>1000 have not been done. While we did not use sophisticated extrapolation

procedures, our answer tends more toward the value 0.592 predicted from critical states

theory.11 At one time it was a great challenge to calculate the critical exponents with

simulations. Researchers extrapolated enumeration and Monte Carlo calculations to find

exponents for all dimensions and lattice types.21 Naturally it was found that only the

dimensionality of the lattice matters. The same critical exponents were obtained from all

3 dimensional lattices such as the simple cubic, diamond, and face centered cubic lattices.



Simulation of Adsorption



We have had success in the simulation of chains as long as 10,000 segments (see

the snapshot in Chapter 2). Most simulations reported for similar systems involve chains

only 50-100 segments in length. With simulation of multiple chains in equilibrium with

an adsorptive surface we are limited (for several reasons) to 1000-2000 segment chains.









Fortunately to obtain macroscopic properties of a polymer solution we do not

have to simulate 1023 (1 mole) chains. Simulations of bulk solution are done using a

small number (typically 10 100) of chains in a box with the usual periodic boundary

conditions that allow the chains to leave one side of the box and come in on the opposite

side. The box should be several times the average size of the chains (Rg). A box that is at

least 4 Rg wide will ensure, even in highly fluctuating systems, that the chains do not

interact with themselves through the periodic boundaries. As can be deduced from Table

II and the simulations described in Chapter 7, the smallest simulations boxes we used

were 3 4 Rg in the more concentrated systems (radius of gyration is smaller for these

systems) for N=200 and 1000 while the more typical case was 5 6 Rg.

We study the case of equilibrium between a bulk solution and adsorption onto a

flat surface. This is arranged by having periodic boundaries in the x and y directions and

2 impenetrable walls at z = 0 and L. L should be large enough so that the region in the

middle (z = L /2) is effectively a bulk solution. The symmetry of this situation is more

preferable to simulation with only one surface. There it would not be easy to create bulk

conditions even with a reflective boundary.

Equilibrium conditions. The basic criterion for equilibrium in bulk simulations is

that the simulation time be much longer than the relaxation time of the radius of gyration.

For simulation of adsorption, equilibrium conditions are harder to define or reach. The

characteristic relaxation times of surface conformations are typically longer than bulk

conformations.96 The main criterion we use however is that there be sufficient exchange

of chains between the bulk and the surface. This criterion becomes demanding for high

chain length in dilute solution since it literally requires 100's of kT to "rip" a chain from









the surface. At some separation distances there is more than one datum. The lower ones

are for simulations where the compressing surface is repulsive. A regular simulation can

never escape from a well of 100 kT. We have had difficulty in simulating chains of 200

or 300 segments using the traditional methods. When such chains have 50 or more

segments coupled to the surface, they are frozen to the local moves that traditional

methods are based on. It is the configurational bias methodology that can "tunnel"

through such steep activation barriers.

Solvent conditions. Athermal conditions can be ensured simply by setting the

monomer-monomer interaction energy, e, to zero. It is the simplest situation to study in

simulations and also it is well within the range of good solvent conditions (the most

relevant experimentally). Simulations are more naturally suited to study the effect of

excluded volume than theories. Recall the radius of gyration and end-to-end distances in

plotted in Chapter 2 which were obtained from simulations in athermal conditions. It was

much easier to find the Flory exponent than the exponent for a random walk.

That leads us to the question of 0-solvent conditions. A 0-solvent is defined as

the solvent that has a Flory-Huggins X= 1/2 which can be achieved experimentally by

varying the temperature of the solvent. For large or infinite length chains, this condition

results in Gaussian statistics. Thus in practice we can approximate 0-solvent condition by

varying the interaction energy between segments, e, so that the radius of gyration follows

the scaling law of an ideal chain, N1 2. Our best estimate comes from simulations for e =

0.25, 0.27 and 0.28 kT, yielding the following scaling laws at high chain length: Rend =

1.06N.53, 1.16NV.53, and 1.34N053 respectively. For these fits, the chain lengths 200, 400,









1000 were used for 0.25 and 0.27 and 400, 1000, 2000 for 0.28. A 2nd order polynomial

interpolation yields the 0-solvent condition: e=0.274 with Rend = 1.22NV500.

There could be subtle differences between a 0-solvent and a true random walk.

While e = 0.274 produces the square root law in the bulk solution, the prefactor may be a

function of concentration. As an example of the consequences, it is known that the

random walks confined in a slit or pore, will not expand laterally under compression.2

However, there is some evidence from simulations that finite length chains in a 0-solvent

do expand laterally.97

Markov matching: estimating the size of the lattice spacing a. In general an nth

order Markov process can be "matched" with a zeroth order Markov process. Alluded in

Chapter 2, this "Markov matching" (first proposed for polymers by Kuhn9) can give us

important information about the size of the lattice unit the chains on the lattice in

comparison to real polymer chains. Thus the quantities a / lc and M / N will be of

interest, where the former is the size ratio of the lattice to monomer unit and the latter is

the ratio of molecular weight to number of units in a lattice chain. In advance we know a

/ lc will be larger than unity because real polymers have more stiffness than lattice chains.

That is, polymer chains are a higher order Markov process.

This exercise consists of two parts: i) Matching the simulation chain to the freely

jointed chain using the correlations we just found for Rend in a 0-solvent; and ii) Matching

the chemical characteristics of real polymer to the freely jointed chain using either

experimental observations or statistical mechanics incorporating the chemical specifics of

various polymer types. In both cases, the matching proceeds as follows. Given N units

of an nth order Markov chain each of length 1, we match them to the 0th order chain (freely









jointed) which will have a smaller number NK units and a larger length per unit IK. The

conditions that the contour length L and the rms end-to-end distance Rend do not change

must be met.



L =NK=IN
(57)
= 1 /1/2 = clN1/2
Rend LKK 1 2



which leads to lK/I = c2 where c is the constant found either experimentally or in

simulation.

For case i) / is the lattice spacing a and we have found for isolated chains in a

0=solvent that c=1.22. This yields for the number of lattice segments in a Kuhn segment

lK/a = 1.5. This seems like a very reasonable answer since one would expect directional

memory of links to disappear somewhere between 1 and 2 links in a random walk on a

lattice. Bitsanis and ten Brinke98 have found for the same simulation method but in a

melt (which is also ideal) that the ratio is 1.1.

For case ii) we can use experimental measurements of Rg which have been

collated in the book by Fleer et al.4 Another option is to use calculations of what Flory

calls C_ = c2 which can be found in his second book.12 We will pursue the former now.

For a polymer with molecular weight M, where the molecular weight of each monomer is

m, we can use N = M / m in Equation 57. The length I = lc (chemical segment length),

will be obtained from the planar zig-zag conformation of polymers, using tetrahedral

valence angles of 109.50. The covalent bond length of carbon-carbon atoms is taken as

0.134 nm. Thus for a monomer that adds two carbons to the backbone of the polymer, lc

= 2(0.134) sin(109.5 / 2) = 0.2 nm.










I I
C C


C C





Figure 9: Length between carbons along the chain backbone.



Table III shows the results of these calculations for three types of polymer. In

general, the calculations by Flory for C are roughly the same as the experimental results

for k / lc. An improvement on the valence angle we chose could improve the agreement.

There are in these cases, between two and eight chemical segments in one our lattice

segments, and the level of detail of our lattice is approximately 1 nm. A 105 molecular

weight chain of polystyrene would correspond roughly to a 100 200 segment lattice

chain, while the same molecular weight version of polyethylene-oxide would be only 10

20 segment lattice chain.




Table III. Comparison of the simple cubic lattice to real polymer chains. Kuhn length, IK,
of the simple cubic lattice is IK = 1.49a.

R / Ml/2
Polymer/ Monomer c t R,/ Ik /c a a M/N M M
solvent weight (nm) (nm) N=100 N=200
PS / CH 104 10.2 0.0288 10.81 7.26 1.6 755.0 75501 151001
PEO / aq. 44 4 0.0343 2.88 1.94 0.6 85.2 8519 17039
PMMA/ (var.) 100 6.9 0.0261 8.53 5.73 1.3 573.3 57330 114660
PE/decanol 28 0.0435 6.64 4.46 1.0 124.9 12485 24970
PM/(var.) 14 6.7 3.32 2.23 0.5 31.2 3121 6243
PDMS / toluene 74 6 0.025 5.79 3.89 0.9 288.0 28803 57607

t C = 0 / NI2 calculated from rotational isomeric state model by Flory.8
tExperimental measurements collated by Fleer et al.4









Initial conditions. Before the attraction at the surfaces was turned on, the chains

were randomly grown. During the growth the chains also were subjected to local

relaxation moves (i.e., reputation, kink jump, rotation, and crankshaft) as outlined by

previously.98 These local relaxation moves have been shown to be effective for

simultaneous growth and equilibration, which is important for chains longer than 30 50

segments. The objective here is only to generate an initial configuration while the main

part of the system relaxation and collection of statistics are done with configurational

bias.

Polymer induced forces. There are two general methods99' 100 used to calculate

the equilibrium forces between a many body system and an object from a dynamic Monte

Carlo simulation on a lattice. We will discuss the latter, which is more accurate. Force is

related to the gradient of a potential energy field as f = -V Therefore, in a canonical

ensemble, the force required to maintain two flat surfaces at a separation H is related to

the derivative of the Helmholtz free energy, F, as



f(H) = F -F(H+a/2)-F(H-a/l2) -kTn Q+
dH a a Q_



We know that the canonical partition function, Q = exp(-U, /kT), can only be

obtained from simulation of systems for which the set of distinguishable configurations is

fully enumerable (note: there is a procedure that can estimate the partition function from

static Monte Carlo89). However, there is no fundamental reason why the ratio of the

partition functions of two similar systems cannot be obtained.









In practice it is not difficult to calculate the partition function ratio for two

systems of equal size, only differing in their potential fields. This means we cannot

actually move the wall from H + a / 2 to H a / 2. What is equivalent the approach

towards an infinite repulsive potential V in the layer next to the compressing wall. Thus

we wish to calculate the limit: Q(H; V = 0)/ Q(H; V- ).

Bennet101 developed a procedure for optimizing the accuracy in the calculation of

such a ratio (for an arbitrary system) through the identification of an adjustable weighting

function. Jimenez and Rajagopalan100 used this to obtain an implicit expression for the

change in free energy, AF, between two lattice systems with difference in potential at the

upper wall of AV



kmax pb a (AF-kAV)/kT
k (AF-kA )kT =0. (59)
k=0 + e- /



All that is required to solve Equation 59 for the change in free energy between

two systems is the probability distribution of surface contacts for each system: Pk. This is

just the probability that there are k segments at the compressing wall. Equation 59 is

valid for any change in potential energy AV If this change is infinite, the equations

reduce to


AF/kT = InPa


(60)









and all that is required is one simulation for V = 0 to find the probability of zero contacts.

In practice, if the probability of zero contacts is very low (i.e., strong force dense

systems) then it cannot be obtained accurately. In that case one incrementally changes

the repulsive potential V and calculate AF at each step from Equation 59. Eventually, for

some highly repulsive potential, the probability of zero contacts can be found and thus

the final term from Equation 60 is added.

We summarize calculation of force at a distance H as follows:

1. Simulate the system with a certain potential at the upper wall.

2. Simulate the system again with a slightly higher potential. The change in potential

must be small so that the new distribution overlaps with the previous one allowing the

solution of Equation 59 for the incremental change in free energy.

3. Repeat 2 until the repulsive potential is high enough that the probability of zero

contacts can be calculated with accuracy. The final increment in free energy comes

from Equation 60.

The simulation of two physisorbed layers is a little more difficult since one must

start from a potential at the upper wall that is attractive, and increment gradually in the

repulsive direction until extrapolation to infinity is possible. This is only stage one

however since we have only calculated the free energy difference between a system with

an adsorptive upper wall at z = H + a / 2 and a system with a non-adsorptive upper wall

located at z = H a / 2. In stage 2 we need to find the free energy difference of re-

adsorbing the chains. This is done by incrementally decreasing the potential, starting

from zero, and continuing until the original adsorption strength is obtained. For dual

physisorbed layers, the energies from stage 1 and 2 are large number with opposite signs






81


which is the reason for poor accuracy for the calculation of weak forces ( < 0.001 kT/ a3)

when the separation is large. The number of simulations this requires altogether can be

as many as 15 or 20. Nevertheless, we have had some success in formulating this

technique to obtain unique information on bridging forces.102














CHAPTER 5
REPTATING CONFIGURATIONAL BIAS MONTE CARLO
FOR LONG CHAIN MOLECULES AND DENSE SYSTEMS



Introduction



The exponential attrition of configurational bias for long chains is reduced to

quadratic by a simple modification of the basic move. Each trial move begins instead on

the side of the remaining sub chain opposite to the cut location. This type of move is

akin to a large-scale reputation and in the limit of a one-segment cut is reputation. The

extension is proven to require the same Rosenbluth weighting scheme as the original

algorithm. Several examples are used to demonstrate the improved performance of the

new method. A single chain is analyzed in isolated and more concentrated environments.

Also, some characteristics of the more demanding problem of polymer physisorption are

elucidated. Finally, a further extension to the method that forces repeated selection from

one end of the chain only is considered. The method while having the advantage of

completely eliminating attrition is rigorously incorrect. However it may be teleologically

correct in many situations.

Development of algorithms for sampling statistics of long chain molecules has

been a challenge since the first days of Monte Carlo. One of the most successful models

because of its simplicity is a self-avoiding walk on a cubic lattice. While this model

suffers in its description of microscopic properties it has been used to analyze large-scale









aspects of many important problems. It corresponds to a coarse-grained polymer chain in

an athermal solvent. Sometimes nearest-neighbor contact interactions are used as well.

This problem will be the subject of much of this chapter, although most of our results can

be applied to off-lattice models as well.

Many of the early methods for polymer chains used static Monte Carlo variants in

which each microstate of a system is generated independently. The first method used for

polymer was called "simple sampling." Because Boltzmann statistics require non-

overlapping and overlapping configurations to be generated with equal probability the

method fails for all but exceptionally short chains. The success rate for chain creation

faces a dramatic exponential attrition with increasing chain length.

"Enrichment" and "biased sampling" were among the more successful attempts to

overcome this problem. However their effect is only to "postpone" the inevitable

attrition. Biased sampling, developed by Rosenbluth and Rosenbluth,89 is the basis of

configurational bias. In contrast to simple sampling, biased sampling allows a choice

only from unoccupied neighboring sites for the sequential placement of trial chain

segments. These early lattice methods are reviewed at length by Kremer and Binder.87

Ultimately complex polymer systems yield their secrets only to dynamic (i.e.,

Metropolis) Monte Carlo.103 "Pivot," "kink jump" and reputation algorithms have all had

some degree of success in simulating longer chains and denser systems and can even be

combined. As opposed to these "traditional" Metropolis methods, configurational bias is

a more "forward-looking" algorithm. It determines each trial configuration with respect

to the system's current configuration in a way that hopefully produces a microstate with

frequency more proportional to its weight in the Boltzmann distribution of microstates.









Developed originally for a canonical ensemble on a lattice104' 95 it has been extended to

off-lattice,105, 106 grand canonical ensemble and Gibbs ensemble systems.107 These

techniques have been reviewed in the book by Frenkel and Smit,5 as well as a

generalization the configurational bias method to all (i.e., not necessarily polymeric)

molecular systems.

Several advances have appeared since then in CBMC. Algorithms which form

trial moves by regrowth of internal instead of end segments have received much

attention.5, 108, 109, 110, 111 The first implementation was on a lattice by Dijkstra et al.108 and

uses a priori knowledge of the closure probability from the properties of a random walk.

This class of algorithms have the potential to generalize CBMC to polymer systems of

many topologies. Applications exist towards simulation of long linear chains, branched

chains, long grafted chains and ring chains to name several problems that are presently

difficult because of their low concentration of endpoints. However the growth of a sub

chain between two fixed points forces a sacrifice of the method's simplicity. Another

advanced extension that overcomes some of CBMC's inherent limitations is the recoil

growth method.112 This is useful in dense, long-chain systems since each growth in the

trial move uses a "retractable feeler" to look several steps forward as compared to the

one-step pre-cognizance of conventional CBMC.

CBMC has been implemented in many phase equilibrium studies.113 Therefore

serious attempts to improve its efficiency for large systems with complicated interaction

potentials have been made.114' 115, 116, 117 Interactions between long chains and interfaces

have been largely ignored by CBMC and require specialized methods.









The organization of this chapter is as follows. In the next section two extensions

to CBMC are proposed and analyzed theoretically. These make long chain simulations

easier through very simple means. The following section contains several comparisons

between conventional and new CBMC in terms of accuracy of results and performance.

This is succeeded by a section summarizing our main findings.



Proposal and Detailed Balance of Configurational Bias Extensions


End-Switching Configurational Bias



In this communication, certain issues are addressed regarding modifications of the

original configuration bias Monte Carlo (CBMC) algorithm proposed by Siepmann and

Frenkel.95 We analyze two modifications that are aimed at relaxing more effectively the

interior segments of long chains in dense systems. In general these methods involve the

removal and random regrowth of pieces of the chains in the system. In contrast to the

original CBMC algorithm (Figure 10a) we propose to grow the trial sub chain opposite

the end from which a sub chain is cut, as shown in Figure 10b. The advantage of this

method is in the relaxation of the interior segments of the chain. Reptation dynamics tells

us that the rate of backbone renewal will increase quadratically with the number of

segments in the chain instead of exponentially. That is to say, the average number of

successful end-switching CB moves that produce a complete renewal of the chains

backbone is proportional to the square of the number of subchains, -(N/Nc)2 N2,

which amounts to a much milder attrition.









For all Monte Carlo methods that generate ensemble snapshots sequentially, the

primary criterion is that the average transition rate between any two configurations must

be equal in the forward and reverse directions. This condition is known as "detailed

balance." A microstate (or configuration), I say, is defined exclusively by a set of

n(N +1) position vectors, where n is the number of chains and N is the number of links

in each chain. For simplicity, lets denote any such set by the notation rf The number

density, PI, of such configurations in the equilibrium ensemble is proportional to the

Boltzmann factor, exp[-p&D(r,")]. As usual, P and ( refer to the inverse thermal energy

and the potential energy respectively. In what follows, we will consider only athermal

systems in which the Boltzmann factor is either 0 if overlaps are present in the system, or

1.0 if no overlaps exist. We can consider the transition rates between arbitrary states I and

J as the product of the number density of each state and the probability, H, of reaching J

from I, or vice versa, via the simulation algorithm. Therefore detailed balance is written

as



PT1,, = PHn,,. (61)



This transition probability can be separated into two factors due to the probability, TI1, of

generating trial state J from state I and the acceptance / rejection rule, AI1 for this trial

move. Normally Monte Carlo methods are devised by proposing a set of moves and thus

fixing TIZ, and then finding an acceptance rule that is consistent with Equation 54.

It is useful to consider the example of two 4-mer configurations shown in Figure

11. For simplicity we use two-dimensional chains inscribed on a square lattice.









Additionally, we will only incorporate removal / regrowth moves of Nc = 2 links at a

time (normally Nc is picked at random). What is the probability, TI1, for for generating

trial move J from I? First a chain in the system is selected at random, and then one end of

the marked chain is selected at random. Thus a factor (1/2)(1/2) is contributed towards

TIj in the example since there are two chains (and of course, two chain ends). At each

stage of the trial insertion, J, a link is added randomly to one of the z available directions.

Thus TIj = (1/2)(1/2)(1/3)(1/3) for the forward move and for the reverse move, Tj, =

(1/2)(1/2)(1/2)(1/2). Therefore the acceptance rule must compensate for the fact that

moves from J to I in this case are more likely. Specifically, A1I / Ajj, is required to be

[zz 2]J /[Z z2] = (2- 2)/(3- 3) 4 / 9.

A generalization for the acceptance rule to athermal lattice systems of any size is

suggested by the above example. An arbitrary configuration J is generated from I with a

probability,




Tj const (62)
n2[Z1 Z2ZN1 W,




where Wj is the Rosenbluth weight generated by the trial portion of the chain as it is

grown in configuration J. That is to say, Wj = W(Nc ) where W(i +1) = (zI / 5)W(i) and

the factor of 5 corresponds to the number of non-backfolding directions on a cubic lattice

(for a square 2D lattice it would be 3). Therefore we find the acceptance rule, in the

absence of any energetic interactions, must satisfy,









Tjl WJ A,
(63)
TI, W, A,



and the best choice for the acceptance rule is therefore



A,j = min[1, W, / W,] (64)



This is exactly the expression used in standard CBMC (without end-switching) and thus

constitutes a proof that they are the same in terms of weighting functions.

We should point out that in these algorithms, the weighting functions a priori

have Nc factors in them, not N factors. That the N Nc factors representing the uncut

portion of the chain, if included into the weighting functions, would cancel out in the

standard method but would not this end-switching method, is purely coincidental.


Persistent End-Switching Configurational Bias



One might entertain the notion that by continually cutting segments from the same

side of each chain in a system. An advantage would be gained in the relaxation of

exceptionally large chains since a chain renewal would be achieved in exactly (N/Nc)

moves. It is tempting to raise the objection that, due to the fact that immediate reversal of

a move is impossible, the scheme automatically violates the law of detailed balance.

However in reality the principle of detailed balance makes no such implication. This

misconception might be due to the popular usage of the term "microscopic reversibility"

when the term "detailed balance" is more a realistic description. One example in previous









literature of a scheme that is "non-reversible" is the persistent reputation algorithm118

(however they switch directions when the chain meets a dead end). As stated above,

detailed balance is only a sufficient condition. The necessary and sufficient condition is

sometimes referred to as "semi-detailed balance" which is less restrictive but harder to

prove.105 This just states that the exchange between two states I and J must be equal.

The difference between detailed balance and semi-detailed balance can be non trivial

(and several examples exist to demonstrate this).

It is best to clarify the persistent end-switching CBMC proposal by the use of an

example. Figure 12 shows two possible configurations of a system of tetramers, again on

a two-dimensional lattice. The trial move algorithm is to remove links 1 and 2 and add

them randomly to link 4. If such a move is accepted, the asterisk (denoting head of chain)

is updated. Any state I in the ensemble then is either of two kinds as regards asterisk

position, which we will call sub states i and i'. For property calculations we do not need

to distinguish between these two sub states, but there certainly is a distinction in terms of

the transition probabilities generated by the algorithm of choice. In Figure 12 the

probability, T,, is (1/2)(1/3)(1/3) and the probability T7,,, is (1/2)(1/2)(1/2). All of the

other combinations, i.e.,, T7,,, Tj,,, Tj,,, T, are zero since these transformations cannot

occur. For instance, i cannot be reached from j' by our proposed move. The only

possibilities are a transition from i toj or a transition from' to i'.

For an arbitrary athermal lattice system it is necessary to recognize that the

requirement of detailed balance still holds in the form of Equation 54. A detailed balance

on sub states is not required! If it were, then the departure of this algorithm from the

detailed balance condition would be of the most extreme kind. All that is required is that









the "flow rate" between states is balanced in the reverse and forward directions. We have

shown that the flow rate in the forward direction consists only of i toj transitions. Similar

is the reverse direction flow rate, which completes the detailed balance expression,

P'ni, = PnH (65)

The obvious question is whether the number of sub states are equal in any given

state, i.e., does P, = P,= PI? If this were true then in the absence of energetic

interactions all configurations are equally likely. Consequently P, = P and an expression

such as Equation 63 would define the appropriate acceptance rule, and the same weights

as in Equation 55 would produce a correct Monte Carlo algorithm. However this

assumption is unfounded. The condition can only be enforced either by a detailed or

semi-detailed balance on sub states; clearly it violates the former condition. In what

follows we show that the persistent end-switching scheme must even be in violation of

the semi-detailed balance condition. This is accomplished by picking illustrative

examples for which P, # P, rather than by a general proof.

Let us attempt to find an example microstate for which the algorithm a priori will

not generate equal numbers of chains with the asterisk located on either end. Again we

shall use only a two-dimensional square lattice to simplify matters. A first attempt is to

consider a fully extended dimer normally adjacent to a surface (or a line). This is shown

in Figure 13a with the asterisk located at the far end and in Figure 13b with the asterisk at

the near end of the chain. Clearly there are 3 possible configurations from which the

former can be generated and only 2 for the latter, and therefore the second configuration

should be 2/3 as likely as the first.









The second example, depicted in Figure 14, also shows that sub states are non

ergodic. The same dimer is now up against an object (either an irregular surface or

another chain) that traps its last segment. Therefore the configuration j with asterisk in

the trap cannot be reached by a single segment move. The opposite configuration j' can

be reached from 3 other possible configurations. If the trap is a fixed object than it is an

example of a non ergodic configuration and there will be zero such states. Even if the

trap is not fixed, i.e., it is another chain, the microstates I and J do not balance.



Tests of the Proposed Algorithms



The methods will first be assessed by the simulation of an isolated chain. The

accuracy of the end-to-end distance of the chain and the associated scaling exponent will

be examined, and in a more sensitive test, the segmental pair correlations and radial

density gradient will be analyzed. This is followed up with the analysis of two further

systems. To illustrate the effect of system density on the algorithms, a semidilute and

concentrated bulk solution will be analyzed. Finally, a problem which inspired much of

this work will be outlined. This is the physisorption of polymers onto a flat surface in

equilibrium with a bulk reservoir of polymer solution. To facilitate discussion the

abbreviations: CBO, CB1 and CB2 will often be used in reference to the standard, random

end-switching and persistent end-switching CBMC methods.




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