Polymer statistics

Physica A 176 (1991) 514-533 North-Holland

POLYMER STATISTICS I. PHASESEPARATION IN POLYMER SOLUTIONS: A NEW STATISTICAL TREATMENT

F. AGUILERA-GRANJA hstitctro de Fisica ” Matwl Snndovnl V~llartca”, Universidad San Lks Porosi, S. L. P. 7NN0, Mexico

Auhotna

de San Lttis Potosi.

Ryoichi KIKUCHI Dcparttnen~ of Marerials Science CA 90024. USA

and Engineering,

University of California,

Los Angeles,

Received 2 1 December 1990 Revised manuscript received 21 March 1991

A new statistical mechanics treatment of polymer systems is presented, built on the lattice model as used by Flory and Huggins, but using the pair approximation of the cluster variation method (CVM). The present paper is the first of a series, and shows how the new method compares with the most common theories in the polymer ficld. Phase separation diagrams arc qualitatively similar to those of Flory. A noticcabie differcncc is that our result for x, deviates from the value i proposed by Flory for any coordination number z. WC also calculate the osmotic pressure for the polymer solution using our approximation and some comparison with Flory (F), Flory-Huggins (FH), Bawcndi-Freed (BF) theories and the Monte Carlo (MC) simulations are done for the a thermal case. The qualitative agreement between our results and the FH results and the MC results supports the reliability of the present treatment.

1. Introduction When a solution containing polymers is cooled, the solution may separate into two phases of different polymer concentrations. Phase diagrams of this kind were determined experimentally [l-4], and the phenomen’jn was theoretically challenging. Flory [5] and Huggins [6] independently developed a statistical mechanical theory of polymer solutions including the phase separation. Their theory has been used extensively in polymer sciences and is regarded as one of the basic theories in the field. There has been a variety of attempts to improve Flory’s theory. In one, Flory’s mean field theory is improved such that it recovers the exact result for a one-dimensional lattice [7]. In others, the multicomponent (:haraLi~r of poly03%43’71/91/$03.SO @

1991 - Elsevicr Science Publishers B.V. (North-Hollmd)

F. Aguilera-Grana,

R. Kikuchi I Polymer statistics I

515

mer solutions is taken into account in order to improve agreement between theory and experiments [8-lo]. Bawendi et al. used the lattice spin field theories with an expansion in l/z, z being the lattice coordination number, in order to find corrections for finite z [ll]. In contrast to all these theories which generalize Flory’s theory so that the original Flory formulation comes as a limit or as a special case, there are a few attempts to improve the statistical counting of the classical Flory theory. Probably the most elaborate improvement of this kind is the work by Kurata et al. [ 121, which is an application to the polymer problems of the cluster variation method (CVM) [ 131 originally designed for alloys and the Ising model. The present paper applies the CVM using a pair as the basic cluster to study the equilibrium structure of polymer solutions. The pair approximation has a distinct advantage “over” the point approximation upon which Flory’s and related theories depend. The pair approximation we formulate in here is somewhat different from Kurata’s pair approximation [ 121. and is of the kind used by one of the authors [14] in working out the polygon statistics of the quantum liquid He problem, which supplemented Feynman’s path integral formulation [15] and successfully calculated the A-shaped specific heat. In the present paper we treat only one-component neutral polymers, for the sake of simplicity, altt,r;gh the model can be easily extended to other cases like polyelectroiytes and block copolymers without difficulty. We show the reliability of the new approach, by comparing our results with Flory (F). Flory-Huggins (FH), Bawendi-Freed (BF) theories and Monte Carlo (MC) simulations. The emphasis of the paper is in paving the road for forthcoming applications. In subsequent papers we apply the pair approximation of the CVM to polymers adsorbed on a solid surface. Polymers may be neutral or

polyelectrolytes, or can be a variety of copolymers. One of the distinct advantages of the pair approximation of the CVM is that simulations can be constructed on the basis of the numerical output of the method [ 161. This simulation is different from, and much faster than, the Monte Carlo method. Papers are to come on the CVM simulations for polymers in a bulk solution, and also for those adsorbed on solid surfaces. As an incidental comment, we may point out that Flory’s model is still being worked out, as is done in the present paper, for various purposes as we see in refs. [11, 17-191.

2. The model and the metho

In order to make it possible to apply the established lattice statistics, j.,ve assume an underlying

entropy expression of crystal lattice in which polymers and

F. Aguilera-Grarlja, R. Kikuchi I Polymer statistics I

516

solvent molecules are placed. This lattice structure requires that our polymer is

made of segments, and each segment occupies a lattice point. Since the lattice is hypothetical, we assume a simple lattice structure in the present treatment for the sake of simplicity. The coordination numbel z is the only parameter we need in specifying the lattice. In the present treatment, we do not include polyelectrolytes or copolymers, and thus a polymer is made of one kind of segment. For the purpose of defining the length of a polymer, we distinguish the segment on an end of a polymer (an “end” segment) and an “internal” segment. Thus, at each lattice site we find either an end segment (i = I), or an internal segment (i = 2) or a solvent molecule (i = 3). The special feature which makes the polymer problem different from, for example, the Ising mode1 is that not only the chemical species i on a lattice site but the direction of the chemical bonding which connects two segments of a polymer is considered explicitly. An end segment is bonded to an adjacent segment in one of the z (= W,) directions, and an internal segment is bonded to its two neighboring segments in one of z(z - 1)/2 (=w,) ways. The solvent molecules (i = 3) are not bonded. These numbers are listed in table I. Including the bonding directions, there are W, different “end segments” and w2 different “internal segments”. Therefore, our mathematical problem is to find the equilibrium distribution of W, + W, + 1 different species over the lattice points, with the requirement that chemical bondings be consistently distributed. The inclusion of bonding directions in the definition of species is the same procedure as was used in the polygon statistics of ref. [ 141. The difference between our treatment and Kurata’s is that in the latter, chemical bonds are distributed over the lattice, but the bonding directions are not included in defining “species”. After the model is thus defined, our approach is to write the free energy F in terms of the state variables, which describe the state of the system, and then to minimize F to find the equilibrium state. The difference between ours and previous work lies in the choice of the state variables and the way of writing the entropy expression. The pair approximation of the CVM uses two sets of state variables: one is the set of probability variables for a configuration of a Table I Statistical weight factors for the single site probabilities defined in our model as a function of the coordination number 2.

i

W,

Comment

1 2 3

z Z(Z - 1)/2 1

end segment internal segment solwit. molecule

F. Aguilera-Granja,

R. Kikuchi I Polymer statistics I

517

point, xi, and the other is for a pair of nearest-neighboring points, yii. The energy E and the entropy S of a system are written as functions of these variables. In deriving a phase diagram, it is more convenient to use the grand potential 0 rather than the Helmholtz free energy F. The former is defined as ,n({xi),

{Yij}) = E({xi),

{Yijl) - Ts({xil~ {Yijl) - NC &Pi 3 i

w

where T is the absolute temperature, lui is the chemical potential and pi is the density of the ith species. The number of lattice points in a system is written as N throughout this paper. Note that the definition of F does not contain the last term in (1). The equilibrium state is derived by minimizing fl with respect to {vii} keeping {E.ci},rather than the composition {pi} c fixed. The advantage of using fl rather than F is twofold: (i) when two phases coexist, the chemical potentials in the two phases are equal. We can determine the coexisting two phases more accurately and more conveniently by plotting 0 against ti rather than drawing a common tangent in the F vs. p diagram. (ii) The special iteration method we use, called the natural iteration method (NIM) [20], was proved to converge when 64 is minimized, keeping pi fixed. The NIM is a very stable iteration method, but it does not necessarily converge when F is minimized, keeping p fixed. Flory’s theory and their modifications used only the point variables, xi. Both &rata’s treatment and ours use the pair variables; Kurata also presented treatments using clusters larger than a pair. In the following sections 3 and 4 we present mathematical details.

3. Variables 3.1. Definibzs We let X, denote the probability of finding, on a certain lattice point, an end segment with its bonding pointing toward one of the w, possible directions. This is equivaient to saying that the number of such end segments in a system is Nx,, where N is the total number of lattice points in a system. For an internal segment, we similariy define x2 with the weight factor w,. The probability of finding a solvent molecule is written as x,, with wR = 1 since it has no chemical bonding direction. We next consider a pair of nearest-neighboring lattice points. When the segments on thes c two points are adjoining segments of a polymer, we call them the “connected” pair, and write the probabilities of finding the configura-

for these configurations are counted tion (i - j) as y!y’. The weight factors I+$‘..~ from the allowed directions of chemical bonds and arc listed in table II. The probability variables yj,!” are similarly defined for a pair of species i and j which are not directly bonded inside a polymer (“unconnected” pair). Their weight factors wl”:i are listed in table III. Inspect&n of tables II and III suggests that the weight factors in them can be written in a condensed form when we define the “semi-weights”, WY’.. and wit: as in table IV. Then w(il)as =

w(il)

w(il)

except

?‘:r/ y:i .V:i w’“!,= #)y-iWy*i (h) ?‘;I/ .

(2 a)

wjt:\, = 0 ,

UJ)

l

We also see the relations from which can write U’iin table

I

as

Table II Statistical weight factors for the connecting pairs (w’f:,) defined in our model as a function of the coordination number z.

Table III Statistical weight factors for non-connectins function of the coordination number z. i\ 1

2 3

j

1

2

(z - I)? (i - l)‘(i

(z-

pairs (w:‘::,) dcfincd in our model as a

1)

3

(z - 2)/2

-

- 1)l)i(z (2 - 2)!2 2)-/d ;zc_ - l)(i - 2)/2

(f - 1) (z - l)(z I

Table IV Semi-weight factors as a function of the coordination number 2.

- 2)/2

3.2. Reduction relrtiom As is the standard procedure in the CVM formulation. we USCgeometrical relations to write cluster probability variables as linear combinations of larger cluster variables. Thus s’s are written in terms of y’s as

when the bonding

is within the pair (thus cscluding

i = 3). and as

when the pair is not bonded. 3.3. Corzstraint rehtions When we choose the pair variables as the basis of formulation. variables are subject to several constraints. . 3.3. I. Norrmzliztrtiotz Since the s’s and ~3 arc probability c W$; =

vatk~blcs. they arc normalized

thcsc

to unity:

1 .

(3

i

which can also be written as linear combinations of Fs when WC SC (-la) ;md (4b). When we minimize the grand potential in the nest section. we write the nr-~e multiptier h in the terms _ normalization constraint using a Lab.G..P

Note that the expression in the small parenthesis as derived from (5). (3) and (2).

is for the normalization

of _V*S

3.3.2. Comisterlcy cow 0-h t The single site probabilities can be written in different w;rys as we see in (-la) and (4b). Since the system is isotropic. the two A-, exprcssioas in (-1) NC‘ equivalent. In the minimization procedure, this consistency constrirint can bc written as

V? where (yi (i =

1 0:’

2) are the Lagrange

multipliers.

F. Aptilertl-Grmja,

520

R. Kikuchi

I Polymer

statistics 1

Polymer length In minimizing the grand potential, we specify the average length of polymers in a system. The average length is the total number of segments, N(w,x, + w+), divided by half the number of end segments, Nw,x, 12. Using a Lagrange multiplier y, the length constraint can be written as 3.33.

C,

z

2 Cam,‘!!”with cii = WiW:h~(RSi, - Si,) + WjW~t’Ti)(R~~~ - Sjz) 9 . 9

y

ij

(8)

where Sii is Kronecker’s delta and the constant R is related to the average length L as L=2(R+

is)

1).

Note in (8) that the subscript internal segment.

i =

1 is for an end segment and 2 is for an

3.3.4. Composition constraint The concentration of polymers is fixed using the chemical potential terms:

where p, and pu, are the chemical potentials for the polymer and the solvent, respectively. When there are no vacancies in the system, the two $s are not independent, but their linear combination is a constant. Thus we may choose without loss of generality p, + pJ = 0 and call p = p, = --CA.; simply the “chemical potential” in the rest of the paper. In analogy with sections 3.3.1, 2, 3, the chemical potential can be interpreted as a Lagrange multiplier for the composition.

4. Energy, entropy and grand potential

4.1. Gc?wd potential (~$2) In view of experiments that show that phase separation occurs in the polymer solution, we assume that polymer segments and solvent molecules repel each other when they sit next to each other (as the nearest neighbors) in the lattice. We define the energy parameters as follows: Eij=

J

0

for a polymer-solvent pair, when the pair is between particles of the same type

,

(19

F. Aguilcra-Crarrja,

R. Kihcki

1 Polymer statistics I

521

where J > 0. When there are no vacant sites assumed in the lattice, Ed,are the only energy parameters we need. The energy for the total sy,;em is written as a sum of the nearest-neighbor energies for the entire system as

(12) Note that this energy expression is exact based on the given model and the

variables. Different from the energy, the entropy is written only approximately in terms of the pair variables. The CVM formula is [ 13) S = kN( if - 1 + 2 (2 - l)wiCF(Xi) i

where k is the Boltzmann constant and 3’(u) represents the function u In u - u. A reader who is familiar with the quasi-chemical approximation [ 211 will recognize the coefficient in this expression. The advantage of the CVM is that the entropy expression can be improved systematically when we choose a larger cluster as the basis of the formulation. and the CVM expression is the most efficient (for practical purposes) for the chosen cluster. When the particle density is a variable parameter and is not required to be fixed, as was pointed out in section 2. it is convenient to minimize the grand potential G defined in (1) while keeping the chemical potential p fixed, rather than the Helmholtz free energy. fi is written explicitly as i-2 = E - TS - Ccl + CA + C,, + C, .

Note that the constraint 4.2. Minimization

(14)

terms C of (6), (7), (S), and (10) are included.

of l2

Tho equiiibrium state of the system is found as a minimum of the grand potential J2 for given values of the interaction energies. the temperature T. the length parameter R, and particularly the chemical potential value. Differentiations with respect to yiy’ and to yi,!” lead to the standard form of the CVM basic equations: Y = AXUMAr

,

where each factor represents

(1 5,’

the following.

522

(i) The variable for the basic cluster (which is the pair in the present case) is on the Icft-hand side. Namely, Y in (15) is actually y:,!” or yj,? (ii) A is the normalization factor: A = cxp(2hlzkT).

uw

(iii) X is the product of probabilities for the two species forming the pair sisi, raised to the power (z - 1) /z, which is characteristic of the pair approximation of the CVM and rcflccts the gcomctry of the lattice: x = (_&$$(“ ‘- ’ 1’: ,

( 1w

(iv) The Boltzmann cncrgy factor is given by U, which contains &ii. It appears only in yj:’ bccausc WC do not write the intra-polymer bonding energy explicitly in yii”. The cxprcssions of U for y;;” and ~71)“are

u

(;I)

=

1 and

U, ,,, = cxp( - ~lkT)

.

( 164

(v) The M factor contains the chemical potentials

p,.:

M = cxp

( 16d)

This cxprcssion is the same for (a) and (13). (vi) The factor A takes care of the constraints

A(“’

ai -+&;’

=

A

(I>)

\’ .I

(vii) The I’ factor is for the length constraint cxprcssion

P” =

which is written

in the (b)

in the following form: 1

and

r(“’

-CXP(-

3)

l

(Sf)

\ .ii

where the c;, arc dctincd in (8). The variables yii” and yj:’ arc found from thcsc equations by using the NIM technique. The iteration processes go as follows. (i) WC start with a set of guessed values of the point variables, x’;, and the Lagrange variables cy, and y. Usually WC may start with CPI, = y = 0.

(ii) We fix the values of xi and determine (Y~and y so that the corresponding constraint equations are satisfied. We call this step the “minor” iteration. The equations to determine the values of cy, are (4a) = (4b) for i = I 7 2, and we choose cyJ= 0 without loss of generality. The length equation for y is C, = 0. These three equations are written in such a form that the input values for cyi and y are used on the right-hand sides and the output comes in on the left. l!t is our experience that this iteration always converges. (iii) After (Y,and y are determined, we use them in the Y expressions in ( 15) and determine A from the normalization condition. (iv) We substitute the values for yii thus dctcrmined in the reduction relations (4) and obtain the next set of input values of si. The iteration step from 1 to 4 is called the “major” iteration. (v) The major iteration steps are repeated until the process convcrgcs. As the convergence criterion, we may use the sum of absolute values of differences between the input and the output s’s. 4.3. Phase-separation diagram The equilibrium state when several phases coexist is determined from the common tangent on the free energy density curves. However, in this paper this calculation is carried out and described in an easier way using the grand potential and chemical potential. When two phases coexist. the equilibrium conditions for the coexistence of the two phases is expressed as the simultaneous equality of the grand potential and the chemical potentials. The procedure is illustrated in fig. 1. Suppose we equilibrium state A for the chemical potential p(A). in the next step we

_______ __ _ -~__ 1

__-_____ Chemicai Fig. 1. Schematic diayam vs. p diagram.

________.____-potential

p

to show how the cocsisting phase arc numcricall~ dckrmind

in the f1

524

F. Agm’kw~-Grartjn,

R. Kikuchi

I Polymer

statistics I

slightly increase the p value and move towards B. As this procedure is repeated, the equilibrium point moves from A to B, and approaches to C. The points on the B-C segment are metastable but the CVM can calculate them because the grand potential becomes a local minimum at these points. After the spinodal point C is reached, the state to which the CVM converges jumps from C to the stable equilibrium point D. This discontinuous jump signals the existence of the lower branch B-D-E which belongs to the second phase, and can be calculated by decreasing p towards B. At B the equilibrium state is that of the two coexisting phases having the same values of 0 and p but different compositions. The unstable branch C-H-G cannot be calculated by the CVM because the points on it are not a local minimum.

5. Phase diagrams

In fig. 2 we present our results for the phase separation diagram based on the simple cubic lattice for different polymer lengths (molecular weights). Let us call the critical point where the phase separation curve reaches its maximum. Therefore, the critical concentration, p, and the critical temperature I’, are defined for this point. The temperature scale is normalized by T,. We can see that pc is shifting to the lower polymer concentration as the polymer becomes longer. pc approaching zero for the infinite length. This behavior is in agreement with Flory’s calculations [2,5]. It is clear that the asymmetry in the phase

Z=6

2 0.5 ----

L=12

0.5 Concentration

Fig. 2. Phase separation diagrams calculated by the present CVM theory based on the simple cubic lattice (z = 6) and for different polymer lengths L.

F. Aguilera-Grarlja, R. Kikuchi I Polymer statistics 1

Ob

0

Fig. 3. Coordination and (b) 1002.

0.5

number (2) dependence

525

I.0 Concenttotion of the phase separation diagram. for lengths (a) 102

diagrams is due to the large difference in the molecular size of the two components, polymers and solvent molecules, and is therefore an entropy effect. The dependence of pc on the coordination number is shown in fig. 3 for i = 6 (SC) and 12 (fee). We can see that as z becomes larger. pc shifts to the low polymer concentration. The shift of pc is smaller when the polymer becomes longer, as is seen in fig. 3b. This weak dependence is the reason why Flory’s curves [S] show at least qualitatively correct behavior although his theory does not depend on z explicitly. The z dependence was first taken into account by Huggins [6]. Fig. 4 compares experiments ]2] on polyisobutylene in diisobutyl ketone with the theories. The experimental lengths of the polymers, 160 and 2008, were estimated by Flory as the ratio of the molar volume of a polymer to that of a solvent molecule. Both his theory and our CVM use the same lengths in (a) and (b). The general trends of the curves are the same. Using the experimental T,- values of 293 K in (a) and 319 K in (b). we can estimate the theoretical interaction constant. When we assume z = 6, the J values are 28 K k in Flory’s theory and 35 K k in the CVM, for both lengths. If we use z = 12, both values decrease between 40% and 50%. When we compare our theory with other experiments [4] with i;olystyrene in methycyclohexane of L = 990 in fig. 5, the pair interaction turns out to be the same as that for fig. 4, J =SScKk based on z=6. AXrough the exact agreement may be a coincidence l

l

8.0,

I

-1

I

(a) -

T C,Exp = 292.5

9.0-

I

(b)

CVM

K

.

I

I

-w-w

Flory M

‘.O-

L = 2008

0.4

Concentration

A mean-field theory like Flory’s has the advantage of simplicity, and even the possibility to dcrivc an analytical cxprcssion for the critical temperature T, as is given in ref. [ !I]:

(17) For an infinitely long polymer. this cxprcssion goes to x,(x) = zJIkT, = i, where T, is callcc! the 0 tempcraturc. According to Flory’s theory, at 0 the second virial cot, Cent vanishes and therefore the effect of the excluded volume disappears, making the polymer behavior a random walk. In our pair treatment. howevc r, it is not possible to obtain an analytical expression of T, or xC. However, n*Jmerical evaluation of .y, as a function of the t is possible, as we will discuss in Ihc next section.

F. Agdera-Granja,

R. Kihchi

J Polymer statistics 1

527

L= IO00

Z= 6 Tc, Exp = 322.71

K

CVM

0.1

0.2

0.3

0.4

Concentration Fig. 5. Experiments (dots) on polystyrene the CVM theory for L = 1000.

in methycyclohexane

of length L = 990 compared

with

6. Osmotic pressure Using the pair approximation of the CVM, we can derive the equation of state for the polymer solution. We calculate the osmotic pressure II from the Helmholtz free energy F as I]=.--

-t3F ( dV )

T.particlch

.

(is)

where V is the volume of the system. and the derivation is done kee the number of particles fixed. The CVM free energy expression leads to s=(z--l)lnx,-

$lnr$‘, ._

(19)

where x’? and y$’ are the solvent concentration and the probability of finding a solvent-solvent pair, respectively, and u,, is the unit volume per cell in the lattice structure. As was mentioned in the previous section, in Flory’s theory the 0 temperature plays an important role. This temperature has the following three characteristics: (i) at @, polymer configurations obey the random waik ruie; (ii j 6 corresponds to the critical temperature of the phase separation diagram for infinitely long polymers; and (iii) the second virial coefficient vanishes at 8. In the following, we use the third property to derive our value of 0. To do that let us define a function g(p, z, L, T) in the following way:

g(p. z, L,

T)

=

!f$ .

528

F. Aguilera-Gralzja, R. Kikuchi I Polymer statistics I

This function tends to unity for the low density limit, in which case the polymers behave like the ideal gas [22]. The function g(p, z, L, T) can be expanded in powers of p as g( p, z, L, 7’) = 1+ B,(z, L, T) P + B,(t, L, 7’) P’ +

l

.

l

3

(21)

where Bi(Z, L, T) are the virial coefficients. The function g( p, z = 6, L = 10, T) for different temperatures is shown in fig. 6. Fig. 6a is for kTIJ = 5 or T < %“M3 w is for kTIJ = 10 or T > BcVM, and (c) is for T = 0. The corresponding function due to Flory is shown for comparison. We can de-

3.0 2.0.

CVM Flory

0.0 0.0

0.2

_______s

0.4

0.6

0.8

I

1.0

Concentration Fig. 6. The function g( p, z, L, T) for different temperature T = 0, where 0 is the 0 temperature Z= 6.

values: (a) T < Q, (b) T > 0 and (c) in the respective theories. For polymer length L = 10 and

F. Aguilera-Granja,

R. Kikuchi 1 Polymer statistics I

529

termine 0 as the temperature at which the curve in fig. 6 starts out horizontally when p = 0, as is illustrated in (c) for the CVM calculation and Flory’s theory. Fig. 7 plots this temperature in the form of xc = zJlk@ for different z values. As for the length L dependence, our numerical results show that xc does not depend on L, in agreement with Flory’s result [22]. However, fig. 7 shows how xc deviates from the value i proposed by Flory for any t. The values of xc we obtained for different lattices are: 0.77 for a square lattice, 0.675 for SC,0.61 for bee and 0.57 for fee. From these results it is possible to conclude that Flory’s theory is safe to use at large values of z. When polymers are in contact with a surface, the effective z is interpreted as small, and thus Flory’s theory is less adequate and we have to take correlations into account more carefully. Before we finish this section, we would like to comment that the comparison of the xc calculated using the CVM pair approximation for z = 4, with the one calculated by Saleur using the transfer matrix method [23] for the square lattice (x, = 0.7), shows a qualitatively good agreement. On the other hand, the approximation level of our calculations and of those works in the literature using the Bethe approximation [24-261 is basically the same, although there are some differences in the details. For more information about the CVM pair approximation and its relationship with the Bethe approximation, the reader may consult refs. [ 13,27-291.

-_-

6

6

IO

12

I6

I

I8

Coordinatim Number Fig. 7. xc calculated by the CVM for different coordination aid for the eye.

numbers. The continuous curve

is an

530

F. Aguilera-Granja, R. Kikuchi I Polymer statistics 1

7. AthermaPcase In this last section we compare our calculation in the athermal case with the most common theories used in the polymers field. WC compare our calculation with Flory (F) [S], Flory-Huggins (FH) [ii], Monte Carlo simulation (MC) [ 17,181 and the 1lz expansion of Bawendi-Freed (BF) [ 11,191. In fig. 8, we compare our g( p, z, L, T) for a square lattice (z = 4) in the athermal case for different polymer lengths with others in the literature [17]. The results in (a) and (b) show that the g for short polymers is very similar in FH, MC (circles and triangles respectively) and CVM, and the g calculated with the plain Flory is the one that deviates more from the other calculations. As the polymer length increases as shown in (c) the FH and the CVM results deviate from the MC calculations (diamonds) and once again the plain Flory is the one that shows the largest deviation. The comparison of the g function for the F, FH, BF, MC and CVM in the athermal case for the simple cubic lattice and polymers lengths of 5, 10 and 20 are shown in the series of table Va, b and c, respectively. From the series of table V, we can conclude that our statistical formulation is much better than the plain Flory and just a little bit better than the FH if we regard the MC as the “exact results”, and that the BF results to second order in 1lz are in between the CVM results and the MC.

L....I*...J

0.0

0.5

10

Concentration

Fig. 8. The g function for a square lattice (i = 4) in the athermal case as a function of the polymer concentration for different polymer lengths, L = 5, L = 10 and L = 20, respectively in (a). (b) and (c). The FH and the CVM calculations are closer to the MC results than the plain F theory. The circles. the triangles and the diamonds correspond to the MC simulations.

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F. Aguilera-Graltja, R. Kikuchi I Polymer statistics 1

8. Summary In this paper we have proposed a new theoretical approach to the calculation of the statistical mechanics of polymers in a solution. It uses the pair approximation of the cluster variation method (CVM). Differing from usual applications of the CVM pair method, we have to include directions of chemical bonds in the definition of species. We calculated the phase separation diagrams of polymers in solution, and found that the diagrams are qualitatively similar to those of Flory. A noticeable difference is that our results for xc deviates from the value i proposed by Flory for any coordination number z. Our results show that Flory’s approximation for xc = 4 is good only at high coordination number. The qualitative agreement between our results and FH and MC simulations for the athermal case gave us the confidence t\, apply the technique to other features of polymer statistics, as will be reported in the papers to follow.

Acknowledgements The authors cordially acknowledged to Dr. Ilhan A. Aksay and also technical discussions with the members of the Advanced Materials Technology Program of the University of Washington, in particular with Dr. Wan Young Shih. One of us (FAG) also acknowledges to Dr. F. Mejia-Lira for a critical reading of the manuscript and to CONACyT for the economical support.

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F. Aguilera-Granja,

R. Kikuchi

I Polymer statistics I

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