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A11102616108

Towards A Theory for the

Orientation Dependent Packing

Entropy of Inhomogeneous

Polymer Systems

Paul van der Schoot

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Polymers Division

Gaithersburg, MD 20899

October 1986

Final Report

Issued December 1986

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U.S. DEPARTMENT OF COMMERCE

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RESEARCH

INFORMATION CENTER

TOWARDS A THEORY FOR THE

ORIENTATION DEPENDENT PACKING

ENTROPY OF INHOMOGENEOUS

POLYMER SYSTEMS

Paul van der Schoot

U. S DEPARTMENT OF COMMERCE

National Bureau of Standards

Polymers Division

Gaithersburg, MD 20899

October 1986

Final Report

Issued December 1986

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
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1
Towards a Theory for the Orientation

Dependent Packing Entropy of

Inhomogeneous Polymer Systems

By

Paul van der Schoot

Polymers Division

Polymer Building, 224/B210

National Bureau of Standards

Gaithersburg, MD 20899

SUMMARY

This report can be thought of as consisting of three parts. In the first

part a review is given of Di Marzio's site fraction concept to calculate the

configurational entropy of polymers in a homogeneous system. The second part

is concerned with Monte Carlo calculations, performed to check the validity

of this concept. Results are presented of the packing of rigid, rodlike

polymer chains on a square lattice, in the spirit of earlier work by

McCrackin. Using a slightly adapted algorithm we also present results of

calculations done on a refined lattice. It is shown that (a) the concept

gives quite satisfactory results in the volume fraction range of 0.05 (the

lowest volume fraction tested) to approximately 0.6, but (b) that one should

be suspicious of the theory's validity at volume fractions approaching 1.0;

it is furthermore shown that (c) the theory works better for small rods than

for larger ones. This last point is partly ascribed to a Markovian

approximation used in the theory. The third and last part of this report

deals with the application of the site fraction treatment to the calculation

of the configurational entropy of a polymeric system having a density

gradient in one direction.

2
CHAPTER 1

INTRODUCTION

Over the past many years various theories have been presented that model

inhomogeneous polymer systems, like polymers adsorbing on a plane wall

(1,2,3) or polymer-polymer interfaces (4,5). One of these theories, the so

called Scheut jens-Fleer (3) theory of polymer adsorption, is particularly

promising. There are several reasons why one may say so. The theory does

not, to give an example, assume the chains to be of infinite length, like so

many other theories do. It also not only enables us to obtain detailed

structural information of the solution near the adsorbing wall, but also

provides us with quite extensive thermodynamic information. Furthermore,

the theory is proven to be very flexible in the sense that extensions to

other polymeric systems, like bilayer membranes (6) and semi-crystalline

bulk polymer (7), appear to be relatively straightforward.

The Scheut jens-Fleer theory (SF-theory) is a lattice theory, and in

essence a sophisticated "hybrid" of Flory's theory on polymer solutions (8),

and Di Marzio's (and Rubin's) theory of polymer between plates (1). Without

going to deeply into the subject, this "hybrid" combines Flory's volume

fraction concept to calculate entropy and energetics of the polymer system,

and Di Marzio's matrix method to carry out the step weighted random walk

representing the configuration of a polymer molecule.

Using Flory's volume fraction concept to calculate the entropy of polymer

chains may give a reasonable approximation in the case of a non-oriented,

serai-dilute system, but is less appropriate for oriented, more concentrated

systems. It is the goal of this study to investigate the possibilities of

introducing the more accurate orientation dependent site fraction concept,

3
developed by Di Harzio (9) , to describe systems having a gradient in the

polymer density. Such systems include adsorbing polymers, bilayer membranes

etc. To achieve this goal, we will first describe Di Marzio's method for

the case of rigid molecules (chapter 2) . We shall then check the site

fraction concept by means of Monte Carlo calculations (chapter 3) , and

finally concern ourselves with the problem of deriving an expression for

the configurational entropy of a polymer solution near a wall, or more

correctly, of a polymer solution which has a density gradient in one of the

directions (chapter 4) . Unfortunately, we are at the moment only able to

give a general formalism describing how to solve the problem.

4
CHAPTER 2

THE ENTROPY OF CHAIN MOLECULES

2.1 INTRODUCTION

The (configurational) entropy of polymer molecules, be it in bulk polymer,

in polymer solutions or in more complicated systems like interphases,

consists of three contributions. First of all, there is the contribution

from the many "shapes" (conformations) of the molecules. Second, the

position of the centers of mass of the molecules also contribute to the

entropy of the system. Note that this contribution depends on how we define

the conformation of the polymer chain, i.e. whether a different spatial

position of a certain "shape" of a molecule be counted a different

conformation. Last, but not least, we have to account for the entropy that

arises from the competition of the molecules for space. This contribution to

the (configurational) entropy of a polymer system accounts for the effect

that volume restrictions have on the average "shape" of the chains due to

spatial interference.

Thus, we can write the configurational entropy S of a polymer system is

(1) S=S (conf ) +S (cm) +S (int)

where S(conf) is the conformational part of the entropy, S(cm) the part due

to the freedom of the centers of mass of the molecules and S(int) that part

that arises from the interference of the chains. Obviously, both S(conf)

and S(cm) contribute positively to the entropy, whereas S(int) contributes

negatively to the entropy of the system. That is to say that S(conf) i 0,

S(cm) > 0 and S(int) < 0 (10). One can easily see that S(int) ^ 0 because

5
interference of chains reduces the freedom of chains with respect to

unhindered chains.

The entropy due to the conformations and centers of mass of the

(unhindered) polym.er molecules can be determined with relative ease,

depending on how accurate the calculation has to be. It is predominantly

the interference part of the entropy that tends to give headaches to all

those who are involved in the statistical mechanics of chain molecules.

To illustrate the importance of the interference of the molecules to the

total entropy, we shall at first restrict ourselves to stiff, rodlike

molecules. By doing so we can concentrate on the competition of the

molecules for space, and the cooperative effect it has on the overall

orientation (and by definition on the conformation) of the chains. The

rigid rod problem has in itself importance because of the fact that rigid

rods are used to model liquid crystals of asymmetric molecules (see for

instance (11) and (12)). We end the chapter by applying the general ideas

developed for the rigid rods to polymers of arbitrary shape in a homogeneous

system.

2.2 .RIGID RODS

We want to calculate the entropy of a system consisting of stiff, rodlike

polymer molecules, (rigid rods). There are several approaches to this

problem, each with their own advantages and disadvantages.

The first approach, those of the continuum models, is the most realistic.

It allows both chains and their constituent segments to occupy all positions

of a certain volume in space (and phase space) . All within the physical

limits determined by for instance excluded volume and, in the case of

flexible polymers, bond angles of course. One need not say that these

6
models tend to be rather complicated, and in many cases almost impossibly

complicated. Despite the fact that one can model the physical reality with

the highest possible accuracy, at least in theory, usually the analysis

becomes too complicated, and one has to take recourse to mathematically

involved approximations or restrict the validity of the analysis to special,

very limited cases. To avoid this, "coarse grained" methods have been

developed.

The lattice treatment, the second way of approaching the problem we

mentioned, is such a "coarse grained" method. The lattice theories represent

the space to be filled partially or completely by polymer molecules by a

lattice. The lattice sites are then thought to be occupied by polymer

segments and, say, solvent molecules or "holes" (free volume) . This is done

in such a way that each site is either filled by a segment or a solvent

molecule (or empty in the case of "holes"). Figure (2.1) visualizes this

concept. One can easily understand that the achieved discretization of the

space to be occupied by the molecules of our system reduces the set of

configurations from an uncountable, in principle infinite set to at least a

countable set. This simplifies the problem considerably, be it at the cost

of accuracy.

We shall restrict ourselves to the lattice approach, obviously because of

the convenience of the concept, but also because of the fact that at least

qualitative behavior of the molecules have been predicted reasonably well

for many different polymeric systems. Examples can be given form the field

of polymer solutions, bulk polymer, amorphous and semi-crystalline

interphases of copolymers and so forth. In the next section we shall deal

with the lattice theory of rigid rods as proposed by Di Marzio (9)

7
2.1.1 A Lattice Approach

Consider a simple cubic lattice of N lattice sites. On this lattice we

place Nr rigid rods each r segments long in the directions of the three

orthogonal base vectors. Our population of rods consists consequently of

three species; one for every direction. Thus

(2) Nr = I N(i)

i = 1

where N(i) > 0 the number of rods pointing in the i direction (i=l,2,3). The

rods are placed in such a way that each segment occupies only one lattice

site and that all the sites are at the most singly occupied. Thus, each rod

occupies r consecutive lattice sites in a certain direction (fig. (2.2)).

Obviously, N=Nr*r+No, if N(0)>0 is the number of unoccupied sites ("holes").

To calculate the entropy of such a system, we have to find out how many ways

we can place the population (N(i)} of rods on the lattice.

It turns out to be handy to calculate the number of ways we can place a

single rod in the i direction, given the presence of a num.ber of rods on the

lattice. Let us assume that there are j (1) , j (2) and j(3) rods on the

lattice, where j (Ic) (lc=i,2,3) is the number of rods on the lattice in the

kth direction. If we want to )cnow the number of ways to place the

(j(i)+l)th molecule we must ]<now how many of the remaining

N-r* ( j (1) +j (2) +j (3)

empty sites have r-1 contiguous sites in the i

direction that are empty. Let us define r {k) to be the position on the

cubic lattice of the Ic-th segment of a chain we intend to put on the

lattice. This site has to be empty because we do not allow for double

occupancy of lattice sites. Since we are dealing with rigid, rodli)ce

polymers we know that

8
(i=2, . . . ,r)

(3)

'r (i) - r (i-1) I = b.e (k)

where b is the bond length, e(i) the i-th base vector and k a constant for

all i. The probability that, having chosen an empty site for the first

segment, there are (r-1) contiguous empty sites in the i direction can now

be written as

(4)

p(n r(i) ' r(D) = n p(r(i) fn' r(k))

1 = 2 ~ 1=2 ~ k=l“

where we have used standard probability theory notations and Bayes’ theorem

(17). This equation is hard to solve for r>l let alone for r>3 (16). We

can however use a Markovian approximation, where we forget about long range

correlations along the chain, and write

(5)

p(Fl r(i) ’r(l))- n''p(r(i) i r(i-l))- (p(r(2) r(l))^

^

1 = 2 ~ ~ 1 = 2 ~ ~ ~ “

Physically equation (5) means that we start at an empty site, calculate the

probability that (r-1) steps in the i direction are successful, i.e. that

they each end at a empty site, and assume that the step probability is

independent of the ranking number of the segment we want to place.

The step probability can be calculated from the number of neighbors to

segments that are potentially empty and the number of potentially empty

neighbors to holes in the direction anti-parallel to the i direction (the

(-i) direction) , that is to say anti-parallel to the direction of the i-th

9
base vector (9)

The ratio of these numbers is equal to the ratio of the

conditional probabilities of respectively finding a neighbor to a segment in

the (-i) direction and finding a neighbor to a hole in the (-i) direction.

We write for this ratio of probabilities the following expression:

(6) ((l-r)*j(i) + r * ^ j(k))/(N-r * Z 3 (k) ) k — X k — X

We can easily derive (6) by keeping in mind that when stepping in the i

direction one can "see" all r segments from each rod in the perpendicular

(j=i) directions, but just 1 segment from each rod in the (j=i) direction:

the other r-1 segments are shielded by the first segment. Because the sum

of the conditional probabilities equals 1, we find for the probability that

a step in the i direction ends on an empty site, given the presence of j (i)

(i=l,2,3) rods,

(7) p(i;j(l), j(2), j(3)) = (N-r* | j (k) ) / (N+ (1-r) (i)

)

k= 1

Obviously, to place the (j(i)+l)th rod, we have to make (r-1) steps,

starting at an empty site. Thus, placing the (j(i)+l)th rod can be done in

:8) (N-r* ^j(k))* (p(i;j(l), j (2) , j(3))

k = 1

•-1

different ways.

What we are going to do now is first place N(l) rods, one by one, and

10
calculate the number of ways we can do that. Then, we put the N(2)

molecules on the lattice, again one by one, and calculate the number of ways

we can do that given the presence of the N(l) molecules in direction 1.

Finally, we place N(3) rods of direction 3 and determine the number of ways

to put them on the lattice, given the presence of the N(l) and N(2) rods of

respectively type 1 and 2.

For placing the N(l) rods in the 1-direction, we get

where the factor N(l)! accounts for the indistinguisability of the rods in

the 1 direction. Next we put the N(2) rods in the 2 direction on the

lattice. This is possible in

N(l)-1 r-1

(9) n (N-r * j (1) )* (p(l;j (1) , 0, 0)

3 ( 1)=0

/ N(l)

N(2)-l r-1

(10) A (N-r*N(l)-r*j (2))*(p(2;N(l) , j(2),0))

j ( 2)=0

/N(2)

ways. Finally, we place the rods in the 3 direction, giving

N(3)-l r-1

(11) A (N-r*N(l)-r*j(2))*(p(3;N(l) ,N(2) ,j(3))

j (3)=0

/N(3)

for the number of ways to pack. The product of (6)

(7) and (3) gives the
number of ways g ( {N (i) 1

, N (0) ) we can pack Nr=N (1) +N(2) +N (3) rigid molecules

on a square lattice. Using the approximation

(12) - y!/(y-x) !

for x<<y, (13) one can show that

g({N(i)l, N(0)) = a(N-

r-1) *N (i) ) ! / {N (0) ! (N ! )

nN(i)!!

i = 1 i 1

Note that we have implicitly taken care of the degeneracy of the rods in the

+i and the -i direction (i=l,2,3). One can futhermore prove that the result

(equation (13)) is not dependent on the order in which we lay down the rods

on the lattice, to calculate g ( {N (i) i ,N (0) )

Herzfeld for instance obtains

the same result by first placing Nr monomers on the lattice and then

subsequently "extending" N(l) monomers in the 1 direction, N(2) in the 2

direction and N(3) rods in the 3 direction (13).

The entropy of a system of lN(i)} rigid rods on a cubic lattice of N sites

can be calculated from (9) and is equal to

(14)

S/k =j2n (g(lN(i)l, N(o)))

where k is Bolzmann's constant. For large enough N, using the Stirling

approximation, this can be written into the following form

(15)

S/k-r-Nr = Vr ^l^((l-v(i) + v (i) /r ) in (1-v (i) +v (i) /r)

i = 1 - (v (i) /r) inv (i) /r) ) -v (o) inv/ (o)

giving the entropy per segment, where V (i) =N (i) *r/N the volume fraction of

12
rods of species i, V{0)=N(0)/N the volume fraction solvent or "holes" and V,-

the overall volume fraction of rods. Choosing an appropriate reference

state we can calculate AS=S-S(ref), being the entropy of our system with

respect to this reference state (see below)

2.2.2. Results of Calculations

As we said before, rigid rods on a lattice can be used as a model for

liquid crystals, for which the constituent molecules are known to be

asymmetric. Now, liquid crystals have shown to give transitions from the

disordered, isotropic state to more ordered states like the nematic state,

where on the average the molecules point their longest axis in a certain

preferred direction, or the cholesteric state where (to put it simply) the

unique direction is perpendicular to the longest axis of the molecules, (see

footnote at the end of this section) (for instance (14)). In terms of rigid

rods on a lattice ( and neglecting fluctuations) , one can expect transitions

from a system where N (1) =N ( 2

) =N ( 3 ) (the isotropic phase) to a system where

N(l) >N(2) ,N(3) (the nematic phase) or where N(1)=0 and N(2)=N(3) (the quasi

cholesteric phase) . Permutations of the three directions in the definitions

of the different phases do not change the definitions, of course.

We shall now show that these transitions can at least partly be due to th'.

packing effects of asymmetric molecules. In other words, we shall show that

a system can have a lower free energy dF=dU-TdS in a ordered state than in a

disordered, even when dU=0. To do this we use equation (15) to calculate

the free energy dF=-TdS=-T* ( S-S (ref ) ) , and we define the isotropic state to

be the reference state. The orientation of the molecules is defined by the

order parameter S(i), the fraction of molecules pointing in the i direction.

Of course, S (1) +S (2) +S ( 3) =1 . S(i) is not to be confused with the entropy S.

13
Figure (2.3) shows the free energy per segment as a function of the

fraction rods of length 10 in the 1 direction [S(l)] for various volume

fractions Vr . The 2 and 3 direction are thought to be symmetric with

respect to one another, i.e. S{2)=S(3). S(l)^ corresponds to the quasi

cholesteric phase, S(l)=l/3 to the isotropic phase and S(l)^l to the nematic

phase. We see that for a dilute system, Vr low, the lowest free energy is

obtained for S(l)=l/3, namely dF=0 (remember that the reference is the

isotropic state). This is of course to be expected. For higher values of Vr,

one can clearly see that the free energy for both the nematic and the

cholesteric states becomes lower than that of the corresponding isotropic

state. This means that for these rigid molecules by increasing the density

a change in the average orientation can be observed, from isotropic to

nematic (stable) or cholesteric (metastable). See also figure (2.4) and

(2.5), where we have plotted for the three cases S(1)=0, S(l)=l/3 and S(l)=l

the free energy per segment as a function of the volume fraction Vr rods.

Figure (2.5), determined from figure (2.4), suggests the most probable curve

for the free energy as a function of the volume fraction. We observe a

transition from isotropic to nematic, at least for this molecule, at a

volume fraction of Vr=0.4. To reach the cholesteric phase a energy term

would be needed to assist, i.e. to lower the free energy curve (9). We have

also plotted the free energy for S(1)=0, S(l)=l/3 and S(l)=l as a function

of the length of the molecule. Figure (2.5) gives the result for a closely

paclced system, Vr=0.99, and figure (2.7) for a less densely paclced system,

Vr=0.6. We can see that even for a closely packed system, there is a

minimum chain length before nematic packing becomes more favorable

(approximately 4) , and we start seeing long range orientational order

Ik
(cf.l3). For lower volume fractions this minimum chain length increases to

higher values.

Footnote . A true cholesteric state is a little more complicated. One can

view it as stacked two-dimensional nematic phases, where the direction of

the molecules in each 2-D nematic phase is shifted a little with respect to

the direction of the molecules of the lower neighbor. The essence of this

phase is the absence of the molecules who have their longest axis in

direction perpendicular to the planes in which the 2-D nematic phases lie.

The cholesteric state we describe with the model is simpler than a true

cholesteric state, only the just described basic feature of not having

molecules pointing toward a certain direction is modelled. We shall

therefore refer to this simplified cholesteric state with "quasi cholesteric

state"

.

2.2.3 Limitations of the Lattice Approach

We learned from figure (2.3) that at high enough a volume fraction the

state with the lowest free energy would be the nematic state, for which

S(l)^l. This means that virtually all rods point in the preferred direction

of the liquid crystal. We define the orientational order parameter

? (1) = (3*<cos (a (i) ) >-l) /2 , where <.> indicates an ensemble average over all

molecules with angle a(i) between the long axis of the molecule and the

preferred direction. In a cubic lattice, with only three orientations, this

expression reads p (1) = ( 3*S (1) -1) /2 . We see that whenever S(l) equals 1,

P(l) also equals 1. Experiments usually provide us with values of

approximately P(l)<0.6 for the nematic state (14). The discrepancy between

experimental and theoretical results is of course due to the fact that in a

cubic lattice we only allow the rods to be pointing in the 3 orthogonal

15
directions. Another factor is likely to be the fact that we confine the

rods to the lattice sites, i.e. the discretization of real space. Both

adequacies can be corrected, at least in an approximate way. Let us start

with allowing the rods to have more orientations on the lattice than the

those of the three orthogonal base vectors. This can be done by placing the

first segment of a rod on the lattice in the ordinary way, and then placing

the remaining consecutive segments in a given direction, where these

segments need not occupy a single lattice site. Figure (2.6) shows this.

Say we allow for n different rod angles (with respect to the 1 direction)

,

and which we shall refer to as angles i=l,2,...,n. The excluded volume a

rod experiences from another rod is now a function of the angle between the

orientations of both rods. This means that p(i; j (1) , j (2) , . . . , j (n)

)

is not

only a function of the number of rods, but also of the angles between the

rods (recall equation (4) , j (i) now denotes the number of rods at an angle

i) . We know from previous results that two rigid rods i and j at an angle

e(i,j)=0 experience the mole fraction type of interference, while two rods

at an angle of e(i,j)=n/2 interfere in the volume fraction type of way (i.e.

all r segments are "felt", see section 2.2.1). For rods i and j at an angle

0le(i,j)<r/2 we would expect a behavior that lies between the mole fraction

and volume fraction type of behavior. Imagine that we place a rod of length

r on a lattice that already contains a number of parallel rods, also of

length r. We place this new rod at an angle e with respect to the already

present rods. The interference effect these rods have on the

configurational entropy of our "new" rod is thought to be equal to a system

where a rod is placed perpendicularly to a field a parallel rods of a

projected length. The projection is perpendicular to direction of the newly

l6
placed rod, and equal to approximately 1 + (r- 1 ) *sin (e) if one takes into

account the finite thickness of the rigid molecules (9). See figure (2.9).

This means that

(N-r*! j (k)

(16) p(i;j(l),...,j(n)) =

5 {N-r* T j(k)+^(l+(r-l)*sin(e(i,k)))*j(k) 1 (r= 1 k = 1

Note that if we restrict ourselves to the original three perpendicular

orientations we obtain equation (4)

Eliminating the lattice as far as confinement of segments to the discrete

lattice sites is concerned, is possible by refining the lattice. In order to

do that a generalization of Di Marzio's treatment of the packing of "thick"

rods, i.e. rods R(x) sites long and R(y) sites wide in the case of a two

dimensional lattice, is necessary. Such a generalization has been proposed

by Herzfeld (13). The connection between the discrete lattice and a

continuous "lattice" becomes clear in the following. We start with a (for

reasons of simplicity) square lattice of N sites, on which we have placed a

number of rods R(x) sites long and R(y) sites wide. Having a generalized

theory we can now calculate the packing entropy S=k*ln (g ( I N ( i ) 1 , N (0) ) ) of

this system. We then replace the lattice by a refined lattice, where each

site is subdivided in 1^ sites. The number of lattice sites increases from N

to N* 12 , whereas the length of the rods in units of lattice sites increases

from R(x) to R(x)*l. The width of the rods increases of course also from

R(y) to R(y)*l. Having done so, the expression for the entropy becomes a

function of 1: S=k*ln (g ( i N ( i) 1

, N (0) , 1) ) . If we let 1 go to infinity the

17
"lattice" becomes continuous and we got rid of the discrete nature of the

lattice. Herzfeld has done this and analyzed the effect the discretization

has on the properties of the system.

In the next chapter we will test the Herzfeld formulae by comparing the

results with Honte Carlo calculations, to get an idea how well the

generalization of the site fraction treatment works.

2.3 RIGID CHAINS OF ARBITRARY SHAPE

The lattice treatment we used to deter.mine the entropy due to orientation

dependent packing of linear, rodlike molecules can be extended for the case

of polymer molecules of arbitrary shape. Central in the described method is

the expectancy p(i) that a lattice site is empty, given that an adjacent

site in the (-i) direction is potentially empty. We calculated this

expectancy by observing the following. Say we step form an empty site in

the i direction to an adjacent site. The step is considered to be

successful whenever this site is empty. In other words, if the original

site from which we are stepping adjoins a hole in the (-i) direction, then

the step is successful. The number of times that this happens is

proportional to N(0), the number of such neighbors to holes (equal in all

directions) . If we step from an empty site in the i direction to a

neighboring site, and this site happens to be occupied, than the step is

considered to be unsuccessful. The probability that this happens is

proportional to the number of (potentially vacant) sites adjoining polymer

chains in the (-i) direction B(-i). This means that p (i) =N (i) / (N(i) +B (-i)

)

since the sum of both probabilities equals 1. B(-i) equals the number of

sites adjacent to polymer segments minus the number of bonds in the i

direction, because this is the number of sites not accessible to new polymer

18
segments. See figure (2.10).

2.3.1. Cylindrically Symmetric Molecules

We shall restrict ourselves in this section to chains which are

cylindrically symmetric along one of the axis of the configuration. Let 1',

2' and 3' be the three local base vectors of the polymer configuration, and

let 1, 2 and 3 be the base vectors of the cubic lattice. Any polymer

molecule on the lattice occupies r lattice sites, being the degree of

polymerization, and each lattice site can be occupied by at most 1 segment

(on the average). We assume that a fraction a(i) of the bonds of the

molecules point in the i=-3 ' , -2 '

, -1 ' , 1

' , 2 ' , 3

' directions of the local base,

where negative values indicate directions anti-parallel to the respective

base vectors. To calculate the entropy due to the packing of a number of

these chains on the cubic lattice, i.e. the center of mass part and the

interference part of the entropy, we only need to take into account the

fractions a (x) :=a (1 ' ) +a (-1 ' ) , a (y )

: =a ( 2

' ) +a (-2 ' ) and a (z) : =a ( 3

' ) +a (-3)

because of symmetry considerations. Cylindrical symmetry implies that, say,

a(x)=a(y) where we define the z-axis to be the main axis of the

configuration. Since the local z-axis can point in either three directions

of the lattice base, we can distinguish between three possible orientations

of the molecules (provided that they're not spherical). Let N(i) be the

number of molecules of which the local z-axis points in the i direction. Wh

calculate the packing entropy of the distribution lN(i)l of chains in a

similar way as we did in the case of the linear molecules, namely by adding

one molecule at the time to the lattice.

Let us for starters assume that there are j (k) (k=l,2,3) molecules of type

k on our cubic lattice of N sites. We can place the (j(k)+l)th chain of

19
type k in at most

3 (17) N(0) = N-r-a (k)

k=l

different ways, being the number of empty lattice sites. We have to correct

this number for the fact that not all of these sites can harbour a first

segment of a chain because of interfering segments of other polymer

molecules elsewhere along this chain. In order to find the correction

factor we will carry out an analogue of the step by step procedure followed

in the rigid rod case, i.e. we will take a(x)*(r-l) steps in the x

direction, a(y)*(r-l) in the y direction and a(z)*(r-l) steps in the z

direction and for each step determine the probability that the step is

successful

.

The conditional probability p(i; j (1) , j (2) , j (3)

)

that a step in the i

direction is successful, given the presence of j (k) (k=l,2,3) molecules,

equals

(18) p(i;j(l), j(2),j(3)) = N(o)/(N(o)+r-^(k)-(r-l)-(a(z)-j(i)-a(x)*j(i)+^a(x)*j(k)})

k=l k=l

where N(0) = (N-r*S(k))

k=l

This means that we can place the (j(i)+l)th chain in

20
(a (z) 5k . i + (l-5k ,i)a(x))*(r-l)

(19) (N-r • S (k))-n p(k;j(l),j(2),j(3))

k=l k=l

(I) (ID

different ways, where 5k , i is the Kronecker delta. (I) denotes the

uncorrected center of mass part and (II) the interference correction term

If we first place the N(l) type of chains on the lattice, then we can do

that in

( 20 ) N(l)-1 a(z)-(r-l) a(x)-(r-l) a(x)-(r-l)

A (N-r- j (1) ) -pd; j (1) ,o,o) • p(2; j (1) ,o,o) -p(3; j (1) ,o,o) /N(l)

J(l)= 0

different ways. Next we put the N(2) type of molecules on the lattice,

given the presence of the N(l) type of polymers, which is possible in

N(2)-l a(x)-(r-l)

(21) A(N-r-N(l)-r- j (2) ) •p(l;N(l) , j (2)

,

0 )

j ( 2) =0

a (z) • (r-1)

p(2;N(l) , j (2) ,0)

a(x) • (r-1)

p(3;N(l) , j (2) ,0) /N(2)!

ways. Finally, we lay down the N(3) type of chains, possible in

21
N(3)-l

(22) n{N-r-N(l)-rN{2)-r- j (3) )*p(l;N(l) ,N(2) , j (3)

)

j (3) =0

a(x) . (r-1) *

•p(2;N(l) ,N(2) ,j

(3)^^^^

•p(3;N(l) ,N(2) , j (3))^^^^ ' N(3) !

ways. The product of these three expressions gives us the total number of

ways to pack {N(i)l molecules of length r on the lattice, g ( {N (i) } ,N(0) )

.

We can write by approximation

(23) g( lN(i) } ,N(o) ) nN(k) ! • (N!)2 . N(o) ) k

(N-(r-l) • (a(z) • N(l) + a(x) • N(2) + a(x) • N(3)))! *

(N-(r-l) • (a(x) * N(l)+a(z) • N(2) + a(x) • N(3)))! *

(N-(r-l) • (a(v) • N(l) + a(x) *N(2) + a(z) -N(3)))!

Note that this equation is invariant to permutation of the base vectors, a

necessary condition. Note furthermore that the expression reduces to the

expression derived by Di Marzio for the case N(2)=N(3)=0 (10).

The total entropy of the presented system equals

S=k*ln (g ( {N ( i) }

, N (0) ) ) +Nr*k*ln (g ' ( { a (i) 1

, r ) ) where Nr=N(l) +N(2) +N(3) . The

second term represents the configurational entropy of the individual

chains, and equals

22
(24) g' (!a(i) } ,r) = (r-1)

n(a (i) ^ (r-1) ! n(a (-i) ^ Tr^l) ) ! i i

We refer to the paper by Di Marzio for model calculations, in a application

to the theory of rubber elasticity (10)

23
CHAPTER 3

MONTE CARLO SIMULATION OF THE PACKING OF RODS

In chapter 2 we derived equations for the number of ways to pack oriented

rodlike polymer molecules on a cubic lattice. Using the same general

principles we were also able to derive analogous equations for arbitrarily

shaped chain molecules. The equations only give (mean field)

approximations, so we cannot expect them to be accurate under all regimes.

To find out how accurate they really are, and under what circumstances major

deviations start to occur, we performed Monte Carlo simulations of the

random packing of (oriented) rods on a square lattice. We had to restrict

ourselves to the 2 dimensional case to keep the required computing time

within reasonable bounds. We shall first repeat some of the calculations

carried out by McCrackin, comprising the simulation of packing of "thin"

rods, i.e. of rods which are r lattice sites long and 1 lattice site wide

(15) . We will also present results of calculations done on systems

consisting of "thick" rods on a square lattice, i.e. of rods R(x) sites long

and R(y) sites wide. The results of the calculations will be compared with

the results of 2 dimensional versions of the analytical formulae for both

thin rods and thick rods.

3.1 THIN RODS

3.1.1. The Algorithm

The algorithm we used to generate N(l) "horizontal" and N(2) "vertical"

rods of r segments in such a way on a square lattice of N sites that they

are nonoverlapping has been proposed earlier by McCrackin, but we shall

nevertheless give a brief description of it.

The general idea of the algorithm is as follows. First we choose randomly

2k
an empty, accessible site to place the first segment of the rod (what this

means will become clear below) . We then consider this site and r-1 sites to

the right to be occupied by segments. This is repeated for each of the

horizontal rods. Once the horizontal rods have been laid down on the

lattice we do exactly the same for the vertical rods, where we choose

randomly an accessible site and use this site plus the r-1 consecutive

sites below to harbor the segments of the rod. An infinitely large system

is simulated by using periodic boundaries.

During the placing of the rods we keep record of how many accessible sites

are left each time we put a "new" rod on the lattice. This enables us to

calculate (an estimate of) the number of ways to pack N(l) horizontal and

N(2) vertical rods. How to do this is easily explained. First we generate

a table which gives all empty sites which have r-1 consecutive empty sites

to the right. It is clear that when the lattice is empty, all N sites are

"accessible" for the placing of a first segment of a new rod. Let G(i) be

the number of sites the i-th rod to be placed has available, equal to the

number of entries in the just mentioned table. For the first rod we have

G{1)=N. Once we've chosen a site to lay down the first segment of the first

rod, we remove this site and successive r-1 sites to the right from this

table. We have thus removed all sites from the table occupied by segments.

But we also have to remove r-1 successive sites to the left of our randomly

chosen site, since rods starting at one of these sites imply double

occupancy of sites. G(2) is the number of remaining sites. From these G(2)

sites we choose randomly a site to place the first segment of the second

horizontal rod we are going to put on the lattice. We remove this site and

r-1 successive sites the right from our table, these sites are occupied by

25
segments of the second rod. We also eliminate a maximum number of r-1 sites

to the left, as far as they have not been removed already. The number of

accessible sites is reduced to G(3). We repeat this process till all N(l)

are placed on the lattice. Now we have to update the table with accessible

sites for the vertical rods. This is done using the following scheme: (1)

regenerate the table for the case of an empty lattice, (2) remove all sites

occupied by segments of horizontal rods and (3) remove all sites that are at

most r-1 sites above occupied sites, as far as not already removed. This

gives the first vertical rod G(N(1)+1) lattice sites to start at. Once

placed, sites are removed from the table in a similar way as we did for the

horizontal rods. This gives G(N(l)+2). We repeat this process until all

N(2) vertical rods are put on the lattice.

The product of all G(i)'s, denoted by G, is an estimate of the number of

configurations. Repeating the described process n times gives us an average

estimate <g>. Thus, if

26
M(l)+N(2)

(25) G = A G(i)

i=l

is an estimate for the number of configurations, then

1 n

(26) <g> = * ^ g(i)

n i=l

is the average of the estimates, where g{i) is the estimate G of the i-th

trial and n is the number of trials. McCrackin showed that this is an

unbiased estimate of the number of ways to pack rods, given the distribution

over horizontal and vertical rods.

Having determined the degeneracy of packing of rods on a square lattice we

can also calculate the entropy of the system, using the equation

(27) S = k * J2n(<g>/N(l) !N(2) !

where the factorials correct double counting due to indistinguishability of

respectively the horizontal and the vertical rods. In the next sections we

shall compare the calculated results with the two dimensional version of the

analytical result (13) , reading

2 2 (28) g ( (N(i) } ,N(o) ) = n (N- (r-1 )

• N ( i) ) i /N (o)

!

N ( i) ! i=l 1=1

27
3.1.2 Isotropic Trimers

In this section we shall concentrate on the packing of trimers on a square

lattice of 15x15 sites, where we chose the order parameter

S (1) :=N (1) / (N (1) +N(2) ) to be equal to 0.5 (isotropic system). This is to

illustrate some of the general characteristics of the calculations and to

give a first impression of how well Monte Carlo results match the analytical

results

.

There are three (rather obvious) points we should keep in mind when

engaging in Monte Carlo simulations. First of all, the number of "trials"

must be large enough to sample a sufficiently large part of phase space in

order to obtain a reasonable estimate of the entropy. In other words we

want to repeat the process described in the last section sufficiently often

to obtain a small a standard deviation. Remember that the standard deviation

decreases with the inverse of the square root of the number of samples.

Secondly, we have to keep in mind that, in spite of periodic boundaries,

edge effects may infuence the results our calculations. Thus, the lattice

has to be large enough that edge effects are negligible. The last point is

that we have to optimize both lattice size and number of iterations, and

still be able to do the calculations within a reasonable amount of computer

time. Because of this, and because of the fact that we made use of a

minicomputer (a DEC VAX-11730) , we were limited in the extent to which we

could study the system.

Now doing the calculations for relatively dense systems may cause serious

difficulties, because of the above mentioned points. A major problem is

that many times when we try to lay down the rods randomly (given the

distribution) we will find that we cannot do so because there is not enough

28
space. This can happen even when N>r*Nr because of steric interference of

the already placed rods. These "configurations" do count in the averaging

(with a g(i)=0), but reduce the number of positive contributions to <g>

This means that in a series of runs we may not be able to generate

sufficient positive (g{i)>0) contributions and thereby not sample large

enough a part of phase space to get a reasonable estimate of <g>. We shall

come back to this subject below.

Figure (3.1) shows the average number of configurations of trimers on a

square lattice as a function of the number of samples taken, i.e. as a

function of the number of runs. The curve is typical for volume fraction

less than approximately 0.6, and shows that <g> levels off reasonably well

in 10^ runs (the standard deviation of the sampling is roughly 0.02%) . At

higher volume fractions however, problems do arise. Figure (3.2) gives <g>

as function of the number of runs for trimers on the same square lattice,

but now for a volume fraction of 0.6667. We see that the levelling off is

rather poor, there are still considerable fluctuations and jumps in the

curve (the standard deviation is about 7.9%). That we may expect jumps to

occur at higher volume fractions is easily seen when considering the

following. Since we place first the horizontal rods, it may very well be

that at times the first number of rods are placed in a relatively dense

fashion, i.e. close to each other. This leaves as a consequence a

relatively large number of sites accessible for the remaining rods, larger

at least than when the first rods would have been distributed more evenly

over the lattice. This means that at such occasions we would get a large

contribution to <g>. Because the probability of events like this is

expected to be rather low, we see them as "sudden" jumps in the <g> vs. the

29
number of runs curve. This effect is more pronounced in dense systems than

in diluted systems. In order to obtain a more acceptable value for the

standard deviation we have to do more runs. One should however remember that

the standard deviation (s.d.) of the sampling decreases with the inverse

square root of the number of samples. This means that in order to get a s.d.

of 0.8% we have to increase the number of runs from 10® to lO"^ . This,

however, leads to unacceptably long computing times (remember that all

calculations are carried out on a minicomputer)

.

To illustrate the decrease of the s.d. with an increase of the number of

runs, we redid the calculation for 10® runs, which should decrease the s.d.

with a factor of the square root of ten. The result of this calculation is

shown in figure (3.3). We see that the average levels off reasonably well,

but the s.d. is still rather high (3.8%). Note that the average of this

calculation is well within two times the s.d. of the previous calculation.

That calculations for dense systems are indeed difficult is illustrated in

figure (3.4), where we've plotted the fraction of samples in which we were

not able to place all rods as a function of the volume fraction. We see

that for volume fractions higher than approximately 0.75 the fraction of

such rejects begins to come very close to a 100%, making the results of

calculations inaccurate.

Comparison of the theoretical predictions by Di Marzio and the Monte Carlo

calculations is presented in figure (3.5), for the case of trimers on a

square, 15x15 sites lattice. We have also included a calculation based on a

Flory type of approach (volume fraction approach as opposed to the site

fraction approach) . The agreement between Monte Carlo and the theoretical

30
prediction by the Di Marzio treatment seems to be quite good. The Flory

type of treatment is less accurate as one can see, except at low volume

fractions

Figure (3.8) gives the relative difference (in percent) of the Monte Carlo

calculations and the theoretical results, defined as 100* (Monte Carlo -

Theory) / (Monte Carlo). The curve shows that agreement is very good for

volume fractions up to approximately Vr=0.6, but appears to diverge for

denser systems.

Note that the s.d.'s are very small indeed. This is partly due to the

fact that we are not showing raw data: remember equation (27)

Figure (3.7) shows for three arbitrary cases that although one would thinlc

that a 30x30 lattice is huge compared to the size of the trimer, edge

effects are not altogether negligible. Figure (3.5) is obviously an upper

bound. Although Figure (3.5) is an upperbound, the general trend is

preserved using larger lattices, i.e. that the relative error increases with

Vr .

3.1.3 More Extensive Calculations

This section deals with more extensive calculations to checlt the validity

of the analytical results. We will present results for a variety of systems

and lattice sizes. The variation in lattice size is necessary, as stated

before, to see wether results of calculations of relatively small lattices

carry over to larger systems (edge effects!). Calculations involving longer

rods are necessary to check whether the Markovian approximation breaks down

with increasing chain length. A more complete account of the simulation of

the packing of thin rods can be found in McCrackin's paper.

The first figure (3.7) gives a comparison between theory and Monte Carlo

31
"experiment" for both a isotropic and a (nearly) nematic system of trimers.

Results are given for two lattice sizes. Agreement between the theory and

the simulations is remarkable, especially when one concentrates on the

larger lattices. The observation that the theory may give erroneous results

for volume fractions coming close to 1.0 is however not contradicted (see

also McCrackin's paper). The theory gives better results for more oriented

systems than for "isotropic" ones (see also Fig. (3.9)). Figure (3.9) gives

the relative "errors" of the theoretical predictions as a function of the

volume fraction of hexamers. We have chosen for three different

orientational states, namely "nematic" (S(l)=l/6), isotropic (S(l)=0.5) and

something in between (S(l)=0.25). Although we have only checked volume

fractions less or equal than 0.48, we already see that the relative error is

much larger than in the case of the trimers (2 to 3 times as large)

Figure (3.10) shows the influence of the lattice size on the results, for

hexamers in a system where S(l)=0.5

3.2 THICK RODS

3.2.1 Herzfeld's Solution for the Thick Rod Problem

In the previous chapter we referred to the thick rod problem as a way to

bridge, at least in part, a lattice approach and a continuum approach. To

calculate the entropy of packing of thick rods on a cubic lattice one can

make use of a generalization of the Di Marzio theory of thin rods, as

described in section (2.2). We shall not give a detailed account of this

generalization, due to Herzfeld (13) , but instead just point out the

strategy used to tackle the problem. We restrict ourselves here to the two

dimensional case, for reasons of convenience. In the following section

Herzfeld's predictions and Monte Carlo calculations are compared.

32
Consider a square lattice of N sites. We want to calculate the entropy of

a number of rigid rods placed on this lattice, where each of the rods occupy

R(x)*R(y) sites. Here R(x) denotes the length of the rods, in units of

lattice sites, and R{y) the width. Each lattice site is to be occupied by

at most one segment. Say we want N(l) rigid rods to point their longest

axis in the 1 direction and N(2) rods in the 2 direction. In order to

calculate the number of ways we can place the rods on the lattice we start

putting on the lattice N(l)+N{2) single segments. One can easily calculate

the number of ways to do that. Once we've done that, we have to correct

this number for the excluded volume effect of the thick rods we're going to

build around these single segments, in a fashion similar to the way we

solved the thin rod problem and the flexible chain problem. That is to say

that we make use of a Markovian approximation, where the probability that we

can place the rod is replaced by a product of step probabilities. These step

probabilities are the conditional probabilities that a site neighbors an

empty site in a certain direction.

First we "add" consecutive segments in the, say, 1 direction, segment by

segment (see figure (3.11)). This means that we have to add R(x)-1 segments

on N(l) and R{y)-1 on N(2) of the placed single segments. Since we are

adding in the 1 direction, and there are consequently only rods pointing in

the 1 direction, the obstruction the adding of segments may encounter is

strictly of the mole fraction type. That means that we only see one segment

of the obstructing rods, the knowledge from which we can determine the s’:'.p

probability in the 1 direction. Once this process is finished, we add to

the original single segments consecutive segments in the 2 direction. Thus,

we place R(y)-1 segments for each N(l)-type rod and R(x)-1 segments for each

33
N(2)-type rod. Note that we are now dealing with L-shaped molecules. When

adding the consecutive segments in the 2 direction, we may find obstruction

from N (1) *R (x) +N (2) *R (y) segments (we do not see the others). Finally we

fill in the "L"'s. Now only obstructing corner segments can prevent us from

entirely filling in the L's, as one can easily see. Using this scheme, one

arrives at the conclusion that

(29) g(!N(i) } ,N(o))= (N(o)+N(l)+N(2) ) ! N(l) !N(2) !NlN(o) ! k

(N-(R(x)-l)*N(l)-(R(y)-l)*N(2)) 1 (N- (R (x) -1) *N (2) - (R (y) -1) *N (1) )

(N-(R(x)+R(y)-2) • fN (1) +N(2) )

where N (0) =N-R (x) *R (y) * (N (1) +N (2) ) the number of remaining empty sites.

Note that the equation is symmetrical with respect to the interchange of the

base vectors, a necessary condition. Note furthermore that for R(x)=r and

R(y)=l equation (29) reduced to equation (28).

Since the probability of finding R(x)*R(y) empty sites in a certain

direction is replaced by a Markovian approximation, losing correlations

between non-neighboring segments, we have to expect that the theoretical

predictions may be less accurate than for the linear polymers.

3.2.2 Results of Calculations

Since extension of the Monte Carlo algorithm for thin rods to an algorithm

that is able to handle thick rods is straightforward, we will not waist any

paper giving account of it. Instead, results of Monte Carlo calculations

are presented involving the placing of rods of dimensions 3x2 3x3 and 6x3

(in units of lattice sites) .
Figure (3.12) shows the entropy of "rods" 3 sites long and 3 sites wide as

a function of the volume fraction. Agreement between theory and experiment

appears to be remarkably good. However, studying the relative differences

(fig. (3.12)) shows that it is not as good as for the thin trimers, as one

should expect, and that "divergence" occurs at a much lower volume fraction.

The shape of the curve strongly suggests that a correction term of quadratic

nature could improve the results considerably. Edge effects are quite

large, but do not seem to affect the shape of the curve: it just shifts to

more negative values (see also fig. (3.14).

Similar, but less pronounced, behavior can be found for rods of 3x2 sites

(fig. (3.15)). The influence of the size of the rods on the results can be

evaluated with help of figures (3.16) and (3.17). We see that the agreement

between theory and M.C. "experiment" is indeed a function of the rod size.

The relatively poor results for the thick rods indicate that one should be

very careful extrapolating to "continuous" lattices (13)

3.3 SUMMARY AND DISCUSSION

We have shown that the site fraction concept gives quite satisfactory

results, at least for rigid, rodlike molecules in the volume fraction range

of approximately 0.05^Vr^0.6. We have not investigated volume fractions of

less than 0.05.

Agreement between theory and simulation for the case of "thin" trimers is

better than 0.5% for the nematic phase and better than 1.0% for the

isotropic phase. This is true for volume fractions less than approximately

0.67. Results for larger "thin" rods seem to be not as good, but are still

quite impressive. One finds that in the case of hexaraers agreement is

better than 2% for the nematic phase and better than 5% (possibly even

35
better than 3%) for the

0.05<Vr<0.5.

isotropic phase in the volume fraction range of

Calculations indicate that the relative difference between theory and

Monte Carlo simulations increases rapidly with the volume fraction, casting

doubt upon the validity

value of 1.0.

of the theory at volume fractions approaching the

The generalization of the site fraction concept for refined lattices,

where rigid rods have a thickness of more than one lattice site, appears to

be less accurate as one

greater than 0.5.

would like to see, especially at volume fractions

Rods three sites long and two sites wide show a relative discrepancy

between theory and "experiment" of less than 2% , while square rods of 3x3

sites exhibit a relative deviation of less than 4% (Vr<0.6). However, these

discrepancies seem to increase roughly quadratically with the volume

fraction, giving values of respectively ca. 4% and ca. 9% for a volume

fraction close to 0.8. Thick rods of six sites length and three sites width

also exhibit a similar quadratic behavior of the error as a function of the

volume fraction (error less than 8% for Vr^O.6 and ca. 16% as Vr->0.8).

Let it be said that the calculations are by no means sacred, i.e. that it is

not possible to exclude the possibility of having introduced systematic

errors in the calculations. We believe that errors are indeed introduced in

the high volume fraction range, because edge effects are more likely to be

felt at high volume fractions. However, in spite of this we think that

there are enough indications (also from trends at lower volume fractions)

that support the statement that one should be skeptical about the theory's

validity at high densities (Vr->1.0). Obviously, this statement does not

36
apply to densities less than approximately 0.6, where theory and simulation

agree very well indeed.

Improvement of Di Marzio's site fraction concept is possible by allowing

for longer memory of the segments along the chain, i.e. instead of using a

Markovian approximation (giving two-segment correlations) including

three-segment correlations in the calculation of the interference part of

the entropy (see appendix A). This will, however, probably not improve the

situation for closely packed systems. As is pointed out elsewhere (11),

mean field approximations do not take proper account of long range

correlations between rods (like in the case of multi-colored cycles) . It is

argued that (especially for longer rods) endsegment volume exclusion is

neglected to a certain extent.

37
CHAPTER 4

POLYMER SYSTEMS HAVING A DENSITY GRADIENT

In the next few sections we will give account of an attempt to describe

the entropy of a polymer solution having a density gradient in one

direction. It is a generalization of the site fraction concept we

encountered in the previous chapters. The first section shall deal with the

derivation of a general equation for the interference and center of mass

part of the configurational entropy. Equations for the "Markovian" step

probabilities in terms of site fractions are presented. The second section

describes the relationship between the volume fraction step probabilities

(zeroth order) and the site fraction step probabilities (first order). 4.1

Configurational Entropy and Step Probabilities

Consider a simple cubic lattice of which the layers are numbered

i=l , 2 , . . .m, . . . . On the lattice we place polymer molecules and solvent

molecules (or "holes") in the usual way. The polymers occupy r sites each.

We assume there to be a gradient in the direction perpendicular to the

lattice layers (the +z and -z direction) but not in the directions parallel

to these layers (the x,y,-x and -y directions). The x,y and z directions

represent the directions of the orthogenal base vectors, and -x,-y and -z

antiparallel to the orthogonal base vectors.

The configuration of a single polymer chain can be characterized by giving

the layer numbers in which we can find the consecutive chain segments (3)

An alternative way to describe a configuration is to give the number of

bonds a chain has in each direction as a function of the lattice layer

number. Thus, one can characterize a chain configuration c by giving

38
a n(+ 2 ,i)c, being the number of bonds a chain of configuration c has

starting in layer i and ending in layer i+1 (defined to be in

the +z direction)

b n(-z,i)c, ditto for bonds starting in layer i and ending in layer i-1 (-

z direction)

c n(±x,i)c, and n(±y,i)c, ditto for bonds laying in the plane of the

lattice layer (begin and end in layer i)

Obviously, in our definitions of n(a,i)c (a=±x,±y,±z) we have implied a

numbering of the chain segments in order to determine the direction of a

bond. Expressions derived making use of this concept should be insensitive

to inversion of the chain numbering. One has furthermore to note that

because of assumed homogeneity of the density in the (x,y) plane, we cannot

distinguish between bonds in the ±x and ±y directions. We assume therefore

that n(±x,i)c = n(±y,i)c.

As stated before, we shall in this section concern ourselves with the

packing part of the configurational entropy. Let us assume that there are

j (k) chains of configuration k=l

,

2 , . . ., c , . .

already on the lattice. We are

now going to calculate the number of ways we can place a (j(c)+l)th molecule

of configuration c. How this can help us in determining the packing entropy

of an amount of N(k)^j(k) k=l

,

2 ,..., c ,... polymer chains on a lattice will

become clear below.

The number of ways we can place the (j(c)+l)th chain assuming that there

are j (k) polymer chains on the lattice can formally be written as

(30) No(j(l),j(2),...j(c),..;i(c))*f (j(l),j(2),...j(c),..)c

39
where No ( j (1) , . . . ;i (c) ) equals the number of remaining empty sites on layer

i(c), the layer where conformation c has it's first segment; f(j(l),...)c is

a factor that accounts for the interference the (j(c)+l)th chain experiences

from the already present chains. No ( j (1) , . . . ; i) equals the total number of

sites in a layer i, say L, minus the number of segments in that layer. The

factor f(j(l),...)c is determined using a Markovian approximation similar to

the one we used in the case of the rigid rods or the rigid chains of

arbitrary shape. Define the following conditional probabilities:

a P+z ( j (1) , j (2) . . . , j (c) , . . . ; i)being the probability that a site in layer

i+1 is empty given that the site it neighbors in layer i is potentially

empty;

b P-z ( j (1) , j (2) , . . . , j (c) . . .

; i)this is the probability that a site in

layer i-1 is empty given that the site it neighbors in layer i is

potentially empty;

c Px ( j (2) , j (2) , . .

. j (c) , . . . ; i) , py ( j (1) , j (2) , . . . j (c) , . . ; i) , ditto for two

sites in layer i; we do not distinguish between +x and -x, +y and -y because

of assumed symmetry in the lattice layers. For clarity we retain the x and

y notation, but obviously px=py.

By approximation we may write

31) f(j(l),...j(c)...)c = n(P.z(j(l),..;i)^^'^^'^^" .P-z(j(l)...;iy^

Px (j (1)

.^)n(x,i)c

Py (J1 ) .^)n(y,i)c

We can make use of (30) and (31) to determine the number of ways to place an

ensemble of iN(k)! (k=l,2,...) polymer molecules on the lattice. A way to

4o
do that is first to place all the chains of configuration 1 on an empty

lattice giving

(32) W(l) = (j (1 ) ,0 ,o;i(l)).f(j(l),o,...,o)i

j ( 1 ) = 0

for the number of ways to do that. Subsequently placing M{2) molecules of

configuration 2 is possible in

(33) ¥(2) = '"n’''No (N(l) , j (2) , 0 , o ; i ( 2 ) ) . f (N ( 1 ) , j ( 2 ) , o ;o )2

j ( 2 ) = 0

different ways. To pack N(k) chains of configuration k on a lattice on

which are already placed N(l) ,N(2) , . . . ,N(k-l) molecules gives

(34) W(k) (N(l) ,N(k-l) J (k) ,0 o;i(k))*

j ( k ) = 0

f(N(l),...,N(k-l),j(k),o,...,o)k

This means that the packing part of the configurational entropy of an

ensemble of lN(k)l molecules on a lattice having L sites in each layer

equals

(35) Sp = k* In (g(!N(k) 1 ,L)/rN(k) !)

k

where

(36) g( lN(k) 1 ;L) = ^rW(k)

Our concern is now to determine expressions for No ( j ( 1 ),...; i ) and

f ( j (1) ,

. . . )c , i.e., for No and the P's.

hi
can be ieternined by

segments, (bl direction determines lere tne second segment is

:e first segment (c) tnereio! :ountin: a__ oc:

:n a ‘ J ^ —

in that layer (not comniing tie

(37: No (j(l),j(2),...:(c),...;i: =

:>) . (n (^z, i-l^k-n ;x, 1} k^n (y, i)k-n ;-z, i-l ; k~ 3 i . i ( k >

)

Kronecxer :lta 5i . i

.

accou: tor - ^ r = r V r 3

tion k.

'^e '^1 oresent ez^oressions tor tne

_c i _ 1 — ) V e

es

,

^s tne " ^ l'’ :t finding

probability of finding an empty site in layer i-l to a empty site in layer

1, a conditional probability. Since tie site in layer 1 is thought to be

empty not all segments in layer i-^-l are able to "obstruct our path", namel

all those segments connected to bonds coming from cr going to layer 1.

Using this insight one sees immediately that

^2
(38) p+z (j(l),j(2),...;i)

No(j(l),i(2) ;i+l)

No ( j (1) , j (2) , . . . ; i+1)

j

(k) . { n (x, i+1) k+n (y , i + 1) k +

+n(-z,i+2)k-n(-z,i + l)u+5i , i < k > 1

or that

(39) ptz (j(l),j(2),...;i) =

(j (1) ,j (2) , . . .;i + l)

L - ^j(k).(n( + z,i) (-Z , i + l)

j^) k

Similarly

(40) p-z (j (1) ,j (2) , . . .;i) =

N

o

L

(j(l),j(2)..;i-l)

-Ij(K).(n(-z,i) +n (+z , i-1)

k

^ k

)

A step in the plane of a latice layer is a step in a homogeneous system,

something we have dealt with before. This provides us with

43
(41) px ( j (1) , j (2) f • * . ;i)

( j (1) , j (2) . . . ; i-1)

N^(j (1) , j (2) , . . i)+Ij (k) . (n( + z,i-l) ^+n (-Z, i+1) ^+n (y, i)

k

or

(42) Px (j (1) , j (2) , . . . ; i)

N^(j (1) , j (2) ;i)

L p(k) .n (x, i)

We already know that py (

j

(1) ,

j

(2) , . . . ; i) = px ( j (1) ,

j

(2) , .

.

i) .

4.2 Connection with the Volume Fraction Approach

We derived earlier that a step in the +z direction from an empty site in

layer i to a neighboring site in layer i+1 has a probability

(43)

ptz(i(l),j(2),..;i) = No (j(l),j(2)...;i + l) /L-^ j (k) . (n( + z,i)k+n(-z,i+l)k

of being successful given the presence of j (i) (i=l,...) molescules of

conformation i. If we devide numerator and denominator by L, thenC'fS)

becomes the limit sum of a harmonic series

44
(j(l)/j(2)... ; i) =N^ ( j (l),j(2),..;i + D*

( 44 )

L

00 *I I j (k) . (n (+z , i) +n (-Z , i + 1) , )

L

because

I J(k) . (n (+z, i) (-Z , i+1) <L, at least

k

for volume fractions smaller than one. For not too pathological ases one

may even assume that

Ij (k) . (n( + zs,i)k+n(-z,i+l)k) < <L

k

holds, true of course for dilute systems, but probably also for more

concentrated systems. Given this to be true, one can truncate the summation

neglecting second and higher order terms:

(45) P+z(j(l),..;i)~(No(j(l),..;i + l)/L)*ll+p(k)-(n(+z,i)k+n(-z,i + l)iJ/LI

Note that this equation consists of two parts:

(I) A classical Flory type of volume fraction term, giving the lowest order

of approximation for the step probability;

(II) A correction term accounting for the cooperative ordering of bonds.

Similarly,

(46) P-z (j (1) , . . .;i) = (No (j (1) , . . ;i-l) /L*) *ll+p (k) (n ( -z , i ) +n ( + z ; i-l ) ) 'L ‘

^5
( 47 )

Px ( j (1) , j (2) . .

; i) (No ( j (1) , . .

; i) /L) * ! l+I j (k) .n (x, i) k /L

1+6
Appendix A Higher Order Approximations

Consider a simple cubic lattice on which are placed rigid rods of length

(conform section (2.2.1). The probability that one can find (r-1)

consecutive empty sites next to an empty site reads

r r k-1

(AI) p(rir(k) r (1) =

-p (r (k) lilr (i)

)

k=2 k=2 1=1

We approximated (AI) by

r r (AID p(nr(k) r(D) - v p (r (i) r (i-1) ) - (p(r(2) rd)))''"'

k=2 1=2

this being a Markovian approximation. We might however want to include

"three-segment correlations" in our analysis, i.e.

r r (AIII) p (Dr (k) r (1) ) -p (r (2) 'r (1) ) *

-T) (r (k) r (k-2) • r (k-1 ) ) - k=2 k=3

-p(r (2) r (1) )*(p(r (3) r(l) -r (2) ) )^~^

It is not easy to determine p(r(3) r(l)-r(2)), but let use see how far we

get using some simple notions from probability theory.

One can easily show that

(AIV) p(r (3) 'r (1) -r(2) ) = p(r(3)-r(2) r(D)

p(r(2) r(D)
where p(r(2) r(D) is a known quantity. We further know that

p(r(3)-r(2) ’r(l)) obeys the equation

(AV) p(r(3)-r(2) r (1) ) +P (r (3) -flir r (1) ) ) +

+pTFTT)-r(2) 'r(l) )+p(7TT) TTT) r(l))=l,

the horizontal bars indicating occupancy of a site. Let B( ) and B( ) be

the number of (potentially empty) neighbors to rods pointing in respectively

the same and a perpendicular direction to the direction of the three sites

we are investigating. We define further more B( +)=B()+B('),

being the sum of both. Using the same "Ansatz" as in Di Marzio's thin rod

theory we find for the last term in (AV)

(AVI) p(F(3) -F(2) r (1) ) =

3( ) ^ B( ) B( + ) N(o) +B( + ) N(o) + B( + I) N(o) + B( + ' |)

where N(o) is the number of empty sites. Equation (AVI) can be derived

making use of the following observations:

a) if an empty site happens to neighbor a rod in the direction

the rod is pointing, both consecutive sites in this

direction are occupied by segments;

b) if an empty site happens to neighbor a rod in the direction

perpendicular to the direction the rod is pointing in, then

we know that if neighbor is occupied but we do not know

whether the next to neighbor is occupied. We have to

48
correct for that.

Using similar arguments we get for the second term of equation (AV)

(AVID p(r (3) -r (2) 'r(l) ) =

B( ) N(o)

N(o) + B{ +|;) N(o) + B( + )

Note that (AVI and (AVII) obey the condition

(A VIII) pT7TT)17T) 'r(D) + p(r(3)TTI)) 'r(l)=pTrT7) r(l))

furthermore that for example

(A IX) p(r(3)-r(2) r(l))=p(r(2) r(l))-p(r(2) r(D) indicating that we have

gone further than the two-segment correlations of section (2.2.1).

The third term of (AV) is not as easy to determine as the second and

fourth ter.m. However, we do not expect p(r(3).r(2) r(D) and

p(r(3).r(2) r(D) to differ much and some physical insight may help us to

find a relationship between the two. Say we have a row of three consec ;t i"c

sites on the cubic lattice, and say that the first site is empty, the sec -nd

one occupied by a segment and the third again empty. The second si'^e ra.;

only be occupied by a segment from rods perpendicular to our three sites.

Let us have a loolc again at three consecutive sites but now the first two

are empty and the third occupied. The third site can be occupied ei"her by

segments from rods perpendicular to the three sites or by a (single; Scgmc;.*:

from rods parallel to the three sites. This means that except for a

correction term for the just mentioned parallel rods, both cases should h*

49
equally probable. This implies that

(AX) p(r(3).r(2) r (1) ) =p (r (3) .r (2) r(l)).(l-B( ) /3 ( + )).

This is good physics. To show that it is indeed good physics assume that

the three sites be parallel to a "field" of nematic rigid rods. This means

that B( + ) = B ( ) , or that p(r(3).r(2) r(l))=0 which is of course to be

expected. If the three sites be perpendicular to this nematic field of

rigid rods, then B( ) = 0 and p(r(3).rT2T r (1) )=p(r (3) .r (2) r(l)). This is

obviously also true.

Using (AX) we find

(AXI) p(r(3).FT2T r(D) = N(o).B( + )/(N(o)+B( + ))^

mar)cedly enough a Markovian type of expression.

We now have enough information to determine p(r(3).r(2) r(l)), namely

equations (AV) , (AVI)AVI!) and (AXI)

This leads to

(AXII) p(r(3).r(2) r(l)) = N (o) ^ / (N (o) +B ( + ))^ again an expression one

would also obtain in the Markovian approximation.

This means that (a) our analysis is too naive, or (b) long range

correlations along the chain are indeed taken care of at least to a second

order approximation.

50
(1) E. A. Di i-Iarzio and R. J. Rubin, J. Chem. Phys., 4313

(1971)

(2) R. J. Roe, J. Chem. Phys., 60, 4192 (1974)

(3) J. M. H. :!. Schetjens and G. J. Fleer, J. Chem., i3, 1619

(1979)

(4) E. Helfand, J. Chem. Phys. 63, 2192 (1975)

(5) E. Helfand, Macromolecules, 9, 307 (1976)

(6) F. A. M. Leermakers, et al., Biophys. Chem., 1^, 353 (1983)

(7) F. A. M. Leermakers, et al.. Polymer, 25_, 1577 (1984)

(8) P. J. Flory, "Principles of Polymer Chemistry", Cornell

University Press: Ithaca, N.Y. (1953)

(9) E. A. Di Marzio, J. Chem. Phys, 3^, 658 (1961)

(10) E. A. Di Marzio, J. Chem. Phys. 36_, 1563 (1962)

(11) J. Stecki, J. Chem. Phys., 62, 948 (1977)

(12) D. E. Matire et. al., J. Chem. Phys., 64, 1456 (1976)

(13) J. Herzfeld, J. Chem. Phys., 1S_, 4185 (1982)

(14) S. Chandrasekhar, "Liquid Crystals", Cambridge University Press:

Cambridge, N.Y. (1980)

(15) F. L. McCrackin, J. Chem. Phys. 69_, 5419 (1978)

(16) A. R. Miller, "The Theory of Polymer Solutions" Clarendar Press,

(1948)

(17) H. Jeffreys, "Theory of Probability" Oxford Univ. Press (1961)

0>:f 0

51
( 2 . 1 )

Fig. (2.2)

Fig. (2.3)

Fig. (2.4)

Fig. (2.5)

Flexible polymers on a square lattice.

Rigid, rodlike molecules on a square lattice; 1

and 2 indicate the directions of the orthogonal

base '-ectors.

The free energy dF as a function of the

orientational state of a 10-mer on a cubic

lattice. The verical axis represents the free

energy per )c.T, per chain segment, the

horizontal axis the fraction of rods S(l)

pointing in the 1 direction. Different curves

represent different volume fractions: (1) Vr =

0.01, (2) Vr = 0.1, (3) Vr = 0.2, (4) Vr = 0.4,

(5) Vr = 0.6, (6) Vr = 0.8, (7) V,- = 0.99. The

reference state is the "isotropic" state (S(l)

0.5). Athermal system.

The free energy per k.T, per poly.m.er segment as

a function of the volume fraction 10-mers on a

cubic lattice. The reference state is the

"isotropic" state. Three curves are indicated:

(1) S(l) = 0.5 ("isotropic" state), (2) S(l) =

1.0 (nematic state) and (3) Sfl) = 0.0 ("quasi

cholesteric" state) . Athermal system.

Most probable curve for the free energy as a

function of the volume fraction 10-mers on a

cubic lattice (see also fig. (2.4)). A

transition for the "isotropic" phase 1 (S(l) =

52
0.5) to the nematic phase 2 (S(l) = 1.0) can be

seen. In order to get a transition to the

"quasi cholesteric" phase 3 we need an energy

term to assist.

Fig. (2.6)

Fig. (2.7)

Fig. (2.8)

Fig. (2.9)

Fig. (2.10)

Fig. (3.1)

Free energy as a function of the chainlength r

for rods on a cubic lattice at a volume fraction

of Vr = 0.99. The dashed curve represents the

"isotropic" state (S(l) = 0.5); the reference

state is chosen to be the isotropic one.

Results for S(l) = 1.0 (nemetic system) and S(l)

= 0.0 (quasi cholesteric system) are indicated.

Identicial to Fig. (2.6), but now for Vr = 0.6.

Rigid rods on a lattice: the classical way to

pack the rods (left) and the way to pack when we

want to include more than 2 directions (right)

Calculation of the excluded volume two rods

experience when at angle e.

When stepping in the i direction, only A, 3 and

C can obstruct our path, not D. D is shielded

by B. Similarly, when stepping in the j

direction only A and D can obstruct our path,

not B and E.

The average number of configurations <g> as a

function of the number of runs (i.e. the number

of iterations) for trimers on a square lattice

of N=225 sites. There are 10 trimers on the

53
lattice (Nr = 10) , half the rods point in the 1

Fig. (3.2)

Fig. (3.3)

Fig. (3.4)

Fig. (3.5)

Fig. (3.6)

Fig. (3.7)

direction and half in the other direction (S(l)

= 0.5). The resulting volume fraction rods is

Vr = 0.1333.

As Fig. (3.1), but now for Vr = 0.6667.

As fig. (3.2), but now for a larger number of

iterations

The fraction of unsuccessful trials as a

function of the volume fraction trimers (S(l) =

0.5, N =225). A trial is unsuccessful whenever

not all rods intended could be placed.

Entropy per lattice site as a function of the

volume fraction. System: trimers on a square

lattice of N=225 sites (S(l) = 0.5); (1) Monte

Carlo results, (2) theoretical prediction using

the site fraction method, (3) theoretical

prediction using "Flory" type of volume fraction

approach

Relative difference ("deviation") between Monte

Carlo (MC) calculation and site fraction

prediction of the entropy, as a function of the

volume fraction. Error bars are indicated.

Identical system as in fig. (3.5).

Influence of the lattice size on the results:

The vertical axis represents the relative error

(between MC and theoretical prediction of the
entropy) , the horizontal axis the inverse

lattice size 1/N. Results for three volume

fractions Vr are shown (R=3, S(l) = 0.5).

Fig. (3.8) Comparison of different orientational states.

The relative error as a function of volume

fraction is given for two orientational states:

S(l) = 0.1 and S(l) = 0.5, on two different

lattice sizes (N=225 and N=900)

,

Fig. (3,9) Identical to Fig. (3.0), but now for hexamers

and three orientational states: S(l) = 0.1667,

S(l) = 0.25 and S(l) = 0.5.

Fig. (3.10) Lattice size influence on the relative

difference between theory and MC for hexamers

(S(l) = 0.5). Three volume fractions are

indicated

,

Fig. (3.11) The Herzfeld method to do the lattice statistics

for thicJc rods. Explanation of the method can

be found in the text.

Fig. (3.12) Configurational entropy of squares of size 3x3

sites as a function of the volume fraction.

Open circles give the Y.C results, the curve

gives the theoretical prediction due to

Herzfeld. The lattice size is N=225.

Fig. (3.13) The relative error as a function of the volume

fraction. Squares of 3x3 sites on three

different lattices: N=225, N=900 and M=2025.

55
Fig. (3.14)

Fig. (3.15)

Fig. (3.16)

Fig. (3.17)

The relative difference between MC and

theoretical prediction as a function of the

inverse lattice size. Squares of 3x3 sites;

three volume fractions are indicated: Vr =

0.12, Vr =0.38 and Vr = 0.60.

The relative difference between MC calculation

and theoretical predictions of the entropy of

rods of dimensions 3x2 on a square lattice as a

function of the volume fraction. Results for

two orientational states and lattice sizes are

given

As fig. (3.15) for rods of size 6x3, results for

three lattice sites N are given.

Lattice size dependence of the relative

difference between theory and "experiment", for

rods 6 sites long and 3 sites wide. Nematic

state (S(l)=1.0). Results for three volume

fractions Vr are given.

56
Acknowledgements

I wish to express my gratitude to a number of people. First of all there is

Ed Di Marzic. Apart from proposing the subjects studied in this report, he

spent a huge amount of time discussing them with me. He has proven to be a

great teacher, and has made my stay at NBS more than worthwhile. Then there

is Frank McCrackin, who provided the basic algorithm for the Monte Carlo

simulations and was of invaluable help when I programmed this algorithm.

Last, but not least, I wish to thank Leslie Smith and Gerard Fleer. Without

their help my visit to NBS would have been impossible.

57

2 . /

z.z
S

F/(k*T*r*Nr) i

F/(k*T*r*Nr)
2.S
I

i.?

(g)/1.0E+23

run/ 100

run/ 100
J.3

run/1000

0.5 0.7 0.9

volume fraction Vr

rel.

difference

(%)

^el.

difference

(%)

R=3

S(l)=0.5

Vr=0.5333

AVr=0.2667

• Vr=0.1333

N=225 S(l)=0.1

N=900 S(l)=0.1

o N=225 S(l)=0.5

• N=900 S(l)=0.5
rel.

difference

(%)

reL

difference

(%)

oN=225 S(l)=0.1667

N=800 S(l)=0.1667

aN=225 S(1)=0.25

*N=900 S(l)=0.25

oN=225 S(l)=0.5

• N=900 S(l)=0.5

4 - t

3

-

i

R=6

S(l)=0.5

Vr=0.48

AVr=0.32

• Vr=0.16

0.000 0.002 0.004

l/N

J./o
W)

i./z
volume fraction Vr

• N=225

aN=900

N=2025

1/N

R(x)=3

R(y)=3

^Vr=0.12

Vr=0.38

• Vr=0.60
rel.

difference

(%)

rel.

difference

(%)

A S(l)=0.5 N=225

• S(l)=1.0 N=225

X S(l)=1.0 N=900

J./5

0.0 0.4 0.8

volume fraction Vr

• S(l)=1.0 N=324

S(l)=1.0 N=1296

o S(l)=1.0 N=2916

X S(l)=0.5 N=324

/‘i- ^
1/N

R(x)=6

H(y)=3

• Vr=0.1111

Vr=0.3333

A Vr=0.6111
Ok

Oi
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instruction s)

1. PUBLICATION OR

REPORT NO.

86-3466

2. Performing Organ. Report No 3. Publication Date

NOVEMBER 1986

4. TITLE AND SUBTITLE

Towards a Theory for the Orientation Dependent Packing Entropy of Inhomogeneous

Polymer Systems

5. AUTHOR(S)

Paul van der Schoot

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s) 7. Contract/Grant No,

national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State. ZIP)

10. SUPPLEMENTARY NOTES

I

I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT ( A 200-word or less factual summary of most significant information. I f document includes a significant

bi bliography or literature survey, mention it here) This report can be thought of as consisting of three parts. In the first part a

review is given of Di Marzio’s site fraction concept to calculate the configurational

entropy of polymers in a hcmc^eneous syston. The second part is concerned with

Monte Carlo calculations, performed to check the validity of this concept. Results

are presented of the packing of rigid, rodlike polymer chains on a square lattice,

in the spirit of earlier work by McCrackin. Using a slightly adapted algorithm we

also present results of calculations done on a refined lattice. It is shown that (a)

the concept gives quite satisfactory results in the volume fraction range of 0.05

(the lowest volume fraction tested) to approximately 0.6, but (b) that one should

be suspicious of the theory's validity at volume fractions approaching 1.0; it is

furthermore shown that (c) the theory works better for small rods than for larger one;.

This last point is partly ascirbed to a Markovian approximation used in the theoi^^.

The third and last part of this report deals with the application of the site fraction

treatment to the calculation of the configurational entropy of a poljTneric system

having a density gradient in one direction.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semi colon s) packing entropy, orientation dependence, spacially dependent oritentation effect,

liquid crystals. , surface induced orientation.

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