J. Phys. Chem. B 2004, 108, 19809-19817
19809
Ornstein-Zernike Random-Walk Approach for Polymers† Johan Skule Høye Department of Physics, Norwegian UniVersity of Science and Technology, N-7491 Trondheim, Norway
George Stell* Department of Chemistry, State UniVersity of New York at Stony Brook, Stony Brook, New York 11794
Chi-Lun Lee Department of Polymer Science and Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003 ReceiVed: June 16, 2004
A new theory of the thermodynamics and structure of polymer-chain fluids is developed. It is based on relating, in a novel way, polymer-chain structure with random-walk structure in the context of Ornstein-Zernike equations. It incorporates much of the interaction-site description of correlation used in reference-system interaction site model (RISM) theory but introduces several simplifying approximations in connecting the RISM equation to random-walk theory that departs in a significant way from the extension of RISM used in polymer RISM (PRISM) theory. Both continuum-space and lattice-gas versions of the theory are summarized, and the extension of the theory to a self-consistent Ornstein-Zernike approach (SCOZA) for polymers is given.
1. Introduction The striking similarity between representative polymer-chain configurations in a solution of such chains and typical randomwalk trajectories has long been used as a theoretical basis for descriptions of polymer-chain solutions.1 More recently, the fact that the integral equation that describes the relation between jump probability and return expectancy in a random walk is of Ornstein-Zernike (OZ) form has also been noted and used to solve a particular set of “inverse” random-walk problems in which the return expectancy is prescribed and the jump probability that yields such an expectancy is sought.2 One can prescribe a walk with a degree of self-avoidance using this observation. The theory we develop here builds on features of this earlier work, in that it exploits the analogy between polymer configurations and random-walk trajectories as well as the fact that a random walk can be described by an equation of OZ form. It goes beyond that earlier work by incorporating the OZ equation in the reference-interaction-site-model (RISM) form that was developed to approximately describe the intermolecular and intramolecular correlations in a molecular fluid.3,4 The reference interaction site model (RISM) has been useful for evaluating correlation functions for molecules that do not possess spherical symmetry.3,4 The molecules are modeled as consisting of spherical atoms that are fused together and described in terms of orientationally averaged correlations between atomic centers; such correlations are spherically symmetric. The RISM approach, extended to describe and evaluate the properties of flexible-chain polymers, is designated as polymer RISM (PRISM).5,6 When compared with Monte Carlo simulations, good agreement has been obtained for the †
Part of the special issue “Frank H. Stillinger Festschrift”. * Author to whom correspondence should be addressed. E-mail address:
[email protected].
atom-atom correlations.7,8 However, the complexity of the equations that must be solved numerically through extended evaluation robs the PRISM results of transparency. A purpose of the present work is to formulate a new simplified approach that is based on relating RISM structure with random-walk structure in a new way. The starting point is to note that polymers in their simplest version can be modeled as chains or rings of stuck-together hard homonuclear beads or spheres. In our modeling, spheres in different polymer chains may also have additional pair interactions beyond their hard cores. Also, all positions in a polymer chain will be considered equivalent, as is exactly the case if the polymers form closed rings; for linear chains, it represents an approximation. Thus, the resulting system is regarded as having a single basic constituentsthe bead or sphere representing the monomeric unitsas is the case in most earlier approaches. In such an approach, however, the spheres do not act as a simple hard-sphere gas, because they are grouped or bound together in polymer units. For us it will be convenient to regard polymers with different lengths and shapes as different species of a polymer mixture. Using RISM theory (for rigid molecules), one should sum over all these polymer species and their intramolecular correlation functions. Because interactions with spheres or sites in other polymers are all considered equivalent, one can formally perform this latter summation by replacing all intramolecular correlations with their average value to be used in the RISM equation. Such use of an average value is consistent with a picture in which the precise shapes of the polymers are not crucial. Thus, the problem reduces to a onecomponent situation: one for which the total pair correlation function for monomers can be written
Γ ˜ (k) )
ω ˜ (k) 1-ω ˜ (k)c˜ (k)
(1)
Here, Γ ˜ (k) ) F + F2h˜ (k), where h(r) is the pair correlation
10.1021/jp0404302 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/03/2004
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function between sites or monomers whose number density is F. Similarly, ω ˜ (k) is the intramolecular pair correlation function along a polymer. The parameter c˜ (k) is the site-site direct correlation function between sites in different polymers. The tilde denotes Fourier transform. With ω ˜ (k) ) F, we are back to the usual problem of a simple hard-sphere fluid (with or without other interactions added). With general ω ˜ (k) values, numerical evaluations are required to solve the resulting RISM problem. However, for certain ω ˜ (k) values that can be suitably used for polymers, the RISM problem simplifies further to make it solvable in analytic form, utilizing Percus-Yevick (PY) and mean-spherical-approximation (MSA) solutions for simple fluids. In these cases, the polymers are not assumed to have one specific length but are characterized by an average length. 2. Intramolecular Correlations First, we note here that a flexible polymer has a close similarity to a random walk. We will use this similarity to describe the intramolecular correlations. Thus, focusing on one site shows that there is a distribution of probability for the location of its neighboring sites. The “jump” probability to go from a site to a neighboring one will be dependent on the details of the polymer model; however, for simplicity, let this function for the moment be a Gaussian,
f(r) )
( )
1 r2 exp - 2 3/2 (2πσ) 2σ
(2)
which has the following Fourier transform:
˜f (k) ) exp
(
)
-2σ2k2 2
(3)
Now, one can go to second and third neighbors, etc., of the original site and ask for their probability distributions. Assuming independent steps in the random walk, the probability distribution of its nth neighbor is obtained by repeated convolutions of f(r), by which its Fourier transform becomes
˜f n(k) ) (f˜(k))n
(4)
The density of particles around the starting site is now the sum of these probabilities, and its Fourier transform is N
F ˜ (k) )
∑
n)0
˜f n(k) )
∞
F ˜ (k) )
1
∑˜f n(k)λn ) 1 - λf˜(k) n)0
(6)
Now, the parameter n may also be considered to be continuous, especially for long polymers. One then can replace the summation of eq 6 by integration, and using eqs 3 and 4, one finds
F ˜ (k) ∝
1 + k2 2
(7)
where 2 is a constant. One sees that eq 6 also is of the same form as eq 7 for small k. Therefore, with ) 0 (i.e., λ ) 1), one notes that this is the Fourier transform of the Coulomb potential by which F(r) ∝ 1/r. Furthermore, the latter is the stationary solution of the diffusion equation for a point source or the solution for a random walk that is not stopped or interrupted. However, with > 0, there is a probability of termination at each step. This again means diffusion with absorption, and the solution of the diffusion equation with an absorption term will be of Yukawa form for a point source, i.e. F(r) ∝ exp(-r)/r. The Yukawa form is also the large-r form of the solution of the OZ equation for an interaction of finite range when approaching the critical point, i.e., f 0. Furthermore, when identifying F(r) with the pair correlation function and f(r) with the direct correlation function, eq 6 has the form of the OZ equation. Thus, we may view the transition probability f(r) as a direct correlation function. This identification is not an exact one, because there will be another random walk starting in the opposite direction, except at the ends of the polymer. However, regardless of the relation between these quantities, we expect the direct correlation function that generates intramolecular correlations to be similar to f(r) and have the same range. Introducing the direct correlation function also facilitates a means of making the random walk self-avoiding, at least on the average. Therefore, in conclusion, the usual OZ equation for a one-component fluid can be used to generate the intramolecular correlation function of homopolymers by making simple assumptions about the corresponding direct correlation function. (This can be extended to the OZ equation for mixtures when there are different sites along the polymers.) 3. Polymers on a Lattice
1 - (f˜(k))N+1 1 - ˜f (k)
With the probability λ (< 1) for the random walk to terminate at any instant, the aforementioned density distribution becomes (as N f ∞)
(5)
where the Nth neighbor is the last one, and we have also included the starting site itself, n ) 0. For a given chain or polymer, one should also make a similar sum in the other direction and, in addition, sum over the various sites as starting sites. The resulting density distribution will be the site-site or intramolecular correlation function of the polymer. Performing the details of such a procedure will be a relatively complex task, so we look for further simplification. This can be obtained by introducing a finite probability for the chain to end at any site. All sites then become equivalent, in the sense that the lengths of the various polymers will be distributed like the time intervals between radioactive decays - in other words, we assume the polymers can be terminated anywhere with equal probability. This equivalence obviates the necessity of the additional summations mentioned previously.
The arguments and derivations given previously also apply to polymers or random walks on a lattice. In the spirit of eq 6, the intramolecular pair correlation function ω(r) can be written in terms of a direct correlation function cω(r) as
ω ˜ (k) )
F 1 - Fc˜ ω(k)
(8)
On a lattice, the cω(r) can, for instance, be limited to nearest neighbors, but it can also be a more general and more longranged function of inverse range γ. To obtain ω(r), one can consider the low-density limit in which the correlations with other polymers are not present. In the case of a lattice, we can write
c˜ ω(k) ) c0 + c1˜f (k)
(9)
where ˜f(k) can be associated with a given transition probability,
Ornstein-Zernike Random-Walk Approach for Polymers
J. Phys. Chem. B, Vol. 108, No. 51, 2004 19811
as done previously, whereas c0 and c1 are determined via the hard-core condition (i.e., self-avoiding walk) and compressibility in the ideal-gas limit F f 0. Therefore, we have (with lattice cells of unit volume) the conditions
ω(0) ) F
(10a)
ω ˜ (0) ) nF
(10b)
where n is the average length or number of monomer units in a polymer. Equation 8 can now be written as
ω ˜ (k) )
(
)(
F 1 1 - Fc0 1 - z0˜f (k)
)
(11a)
interaction, whereas β is the usual inverse temperature. If φ(r) ) 0 is considered, it follows that only c0 can change as density increases. Therefore, for the lattice case, at higher density, eq 13 is modified by the core condition to
F + F2h(0) )
F P(z) ) F - F2 1 - Fc0
or
1 - Fc0 )
P(z) 1-F
Fc1 ) z0P(z0) ) (11b)
For the core condition, we need the integral
P(z) )
1 (2π)d
∫ 1 -dkzf˜(k)
(12)
where d is dimensionality. Setting z ) z0 for F f 0, we then get
ω(0) )
F P(z ) ) F 1 - Fc0 0
(13a)
(17b)
Inserting both eq 13 (z ) z0) and eq 17 in eq 11b for z, we obtain
with
Fc1 z0 ) z ) 1 - Fc0
zP(z) 1-F
(18)
This equation then determines z as a function of F, because z0 is determined by eq 14 for a given average polymer length n. Thus, the solution of eq 18 yields the sought pair correlation function for monomers by which the equation of state can be determined via the compressibility relation. We then have
1 - Fc0 - Fc1˜f (0) 1 2 1 ∂(βp) ) ) ) 2 F ∂F F F(1 - F) F + F h˜ (0) (19a) with
2 ) (1 - z)P(z)
or
1 - Fc0 ) P(z0)
(13b)
Compared to that of a gas of monomers, the ideal gas pressure of the polymers is reduced by a factor 1/n, so eqs 10 and 11 mean that, for F f 0 (f˜(0) ) 1),
F 1 ) ) (1 - Fc0)(1 - z0) ) 20 ω ˜ (0) n
(14a)
20 ) (1 - z0)P(z0)
(14b)
where
When put into eq 11, the solution of eq 14 for z0 yields the intramolecular correlation function when used with eqs 12 and 13. Extension to higher densities turns out to be rather straightforward. If, for instance, one has a reference system where only hard-core interactions are present, only the parameter c0 will change, because the transition (or jump) probability c1˜f(k) will stay fixed in the absence of other interactions when the RISM approximation is used. The RISM correlation function h(r) between any pair of monomers follows from eqs 1 and 8:
F + F2h˜ (k) )
F 1 - F(c˜ ω(k) + c˜ (k))
(15)
by which the sum c˜ ω + c˜ will be the direct correlation function of the resulting monomer fluid. Now, in the RISM, as in the MSA, the c(r) is given by
c˜ (r) ) -βφ(r)
(16)
for interactions outside hard cores. Here, φ(r) is the pair
(17a)
(19b)
where P(z) is given by eq 12. The variable p denotes pressure. 4. Flory Polymers Here, we want to make contact with the well-known FloryHuggins theory for polymers on a lattice.9 We find that it can be regarded as a mean-field theory in the sense that it can be obtained by taking the limit γ f 0 in the aforementioned formalism; i.e., the “jump” probability becomes infinitely longranged. We have P(z) ) 1 + O(γ3) (in three dimensions). Thus, P(z) f 1 as γ f 0, and the solution of eqs 14 and 18 yields
z0 ) 1 -
1 n
(20a)
and
(
z ) (1 - F)z0 ) (1 - F) 1 -
1 n
)
(20b)
to be inserted in eq 19 to obtain
∂(βp) 1 - z 1 1 ) ) - 1∂F 1-F 1-F n
(
)
(21)
or, after integration,
(
βp ) -ln(1 - F) - 1 -
1 F n
)
(22)
This is precisely the pressure of Flory-Huggins polymer theory. In view of the limit γ f 0, this equation also has immediate physical interpretation as a mean-field theory. The first term is the pressure of hard-core monomers. However, in the present case, the monomers are not free but, instead, are
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glued together into polymers. Thus, the ideal-gas pressure of free monomers is subtracted while the corresponding ideal-gas pressure of polymers with density F/n is added. Furthermore, this result can be understood by noting that, for γ f 0, monomers bound into polymers can still have all the configurations of free (nonoverlapping) monomers. Thus, a polymer can be obtained by grouping together any choice of (an average of) n monomers from such configurations. (We note that with this construction of a polymer, neighbors along the polymer chain need not be neighbors on the lattice.)
2 ) (1 - z)P(z) ) )
[ (
βp ) -ln(1 - F) + ln 1 - 1 -
To get a better idea how the version of the reference-system interaction site model (RISM) that we have constructed for polymers works, and also to obtain some estimate of its quantitative accuracy, we will consider polymer rods in one dimension, which also can be treated exactly for comparison. For rod polymers, one can assume that the jump probability is limited to nearest neighbors. This means
˜f (k) ) cos k
(23)
and the intramolecular correlation function eq 11 becomes (along with eq 13)
1 F P(z0) 1 - z0˜f (k)
(24)
From eq 12, we find, after integration (-π < k < π),
P(z) )
1
(25)
x1 - z2
To determine z0, eq 14 is used (F f 0) to give
1 ) (1 - z0)P(z0) ) n
x
1 - z0 1 + z0
(26)
or
z0 )
n2 - 1 n2 + 1
(27)
Next, we evaluate for arbitrary density and use eq 18 to get
zP(z) ) (1 - F)z0P(z0) z
x1 - z2
) (1 - F)
z0
x1 - z02
)x
(28a)
with
n2 - 1 2n
x ) (1 - F)
(28b)
Therefore, after solving for z, we have
z)
x
x1 + x2
and inserting in eq 19, we find
x
x1 + x2 - x 1 ) 2 x1 + x + x x1 + x2 + x
(29)
(30)
This can be compared to the exact result, which, for a pressure p, is
5. Polymers Rods in One Dimension
ω ˜ (k) )
x11 -+ zz
1 F n
)]
(31a)
or
1 - (1/n) ∂(βp) 1 ) ∂p 1 - F 1 - [1 - (1/n)F] )
1 n(1 - F)[1 - (n - 1)/n]F
(31b)
To assess the deviation in our theory from the exact result, one can thus compare the denominator of eq 30 with n[1 - (n - 1)F/n] ) 1 + 2nx/(n + 1). For F ) 0 and F ) 1, i.e., for x ) (n2 - 1)/(2n) and x ) 0, these expressions are equal, but at values between these bounds, they deviate somewhat. By differentiation, one finds that (x1+x2 + x)/[1 + 2nx/(n + 1)] has a minimum at x ) n/(n + 1) - (n + 1)/(4n), and the minimum value is [n/(n + 1) + (n + 1)/(4n)]-1, which, for n ) 2, is equal to 24/25 and approaches 4/5 as n f ∞, whereas F ) 1 - 2nx/(n2 - 1) f 1 (i.e., a very narrow minimum at F ≈ 1). In the limit n f ∞, this ratio becomes 1 for all F (except for the point where the minimum value 4/5 is located). Therefore, the exact solution is obtained for continuum rods. Although we will not do it here, eq 19 can also be integrated explicitly to obtain βp with 2 given by eq 30. For this integrated quantity, the relative error will be smaller than for 2, because integration will smooth out deviations. As a comment, we note here that the description of the polymer mixture with polymers of different lengths as a onecomponent fluid consisting of monomers with a fixed length distribution cannot be quite correct. For a monomer to be anywhere at thermal equilibrium, the length distribution should change, according to equations of chemical equilibrium (which, for low density at least, are well defined). Accordingly, the average length will increase with density, which again means that the compressibility of the gas increases. This is reflected in ω ˜ (0). The parameter ω ˜ (0) of eq 10b is merely an effective one (for the polymer mixture), whereas the true one (for the one-component monomer fluid) will be larger, because longer chains are weighted more than short chains, because there are more monomers in the former, and these monomers are correlated with all other monomers in the same chain. However, after integration to get the pressure p, this error or deviation is removed, at least for low density. The physical reason for this is that p is determined by the length distribution itself (for a given density), not by whether it changes or not with density. By evaluation for a mixture with fixed chains, one finds that ∂(βp)/∂F ) 1/〈q〉, and for monomers with “chemical equilibrium”, one similarly finds ∂(βp)/∂F ) 〈q〉/〈q2〉 at low density, where q is the length of the chains. This approximation can be studied in more detail for the previously mentioned polymer rods, and we note that eq 24 is in accordance with the exact one (eq 33 below) if the lengths of the rods are distributed such that
Ornstein-Zernike Random-Walk Approach for Polymers
p(q) ) (1 - λ)λq-1
(for q ) 1, 2, 3,‚‚‚)
J. Phys. Chem. B, Vol. 108, No. 51, 2004 19813
(32)
where p(q) is the probability for a rod or polymer to be of length q (∑qp(q) ) 1). (Similarly, (1 - λ)2qλq-1 will be the probability for a site or monomer to be in a rod of length q.) The (average) site-site correlation function for these rods, when evaluated, will be
ω ˜ e(k) ) F
(
1 - λ2 1 + λ2 - 2λ cos k
)
(33)
From eq 32, we find, for average values,
1 1-λ
(34a)
1+λ (1 - λ)2
(34b)
〈q〉 ) and
〈q2〉 )
or, taken together with eqs 35 and 36,
z0P(z0) + Fβ )
zP(z) 1-F
With z0 determined by eq 14, this relation determines z, which again gives the equation of state via the compressibility relation when inserted in eq 19. However, for perturbing interactions, the energy route generally is more accurate. (But the reference system still must be evaluated via eq 19, unless input from simulations or some other input is used.) We then need the internal energy, which is also needed in SCOZA to enforce thermodynamic self-consistency between the energy and compressibility routes. This we obtain from the pair correlation function, which still can be written in the form of eq 15, but with new c0 and c1:
F + F2h˜ (k) )
ω ˜ (k) 1-ω ˜ (k)c˜ (k)
Thus, w˜ e(0) ) F〈q2〉/〈q〉, whereas in eqs 10b and 14a, the effective value w˜ e(0) ) F〈q〉 is used.
)
F 1 - F(c0 + c1˜f (k))
6. Self-Consistent Ornstein-Zernike Approach (SCOZA) for RISM Polymers
)
F(1 - F) 1 P(z) 1 - zf˜(k)
Our self-consistent Ornstein-Zernike approach (SCOZA) is based on the requirement that the thermodynamics obtained from evaluating the internal energy as the ensemble average of the Hamiltonian be identical to the thermodynamics obtained from an evaluation of the isothermal compressibility via fluctuation theory.10-13 In RISM, as in the MSA, the contribution outside the hard core is given by eq 16. Now, if this interaction is proportional to, or equal to, f(r) the situation becomes especially simple. On a lattice with both nearest-neighbor interaction and nearest-neighbor jump probability, this will be the case. The pair correlation function will have the form of eq 15. Now, with ˜f(k) proportional to φ˜ (k), the c˜ (k) can be absorbed into c˜ ω(k) by merely adjusting the coefficients c0 and c1 in eq 9. With the MSA form (eq 16), the redefinition of coefficients c0 and c1 means
c1 ) c1r + β
(35)
when the interaction is normalized, such that -φ˜ (0) ) ˜f(0) ) 1. Here, the parameter c1r is c1 when β ) 0, i.e., with eq 18,
Fc1r ) z0P(z0)
-Fu1 )
P(z) 1-F
(38)
Therefore, eq 18 for the relationship between c1 and z remains unchanged:
Fc1 )
zP(z) 1-F
∫(F + F2h˜ (k))f˜(k) dk
(39)
˜f (k)
)
F(1 - F) 1 2P(z) (2π)3
)
F(1 - F)(P(z) - 1) 2zP(z)
∫1 - zf˜(k) dk (42)
where P(z) is given in eq 12. Now, part of the energy in eq 42 is merely a “self-energy” from particles bound together in the same polymer chain. It can be subtracted, but this subtraction is inconsequential, because it is only a constant contribution from each polymer and, thus, can have no other influence on the equation of state. When evaluated, this self-energy us is given by
-Fus ) )
1 1 2 (2π)3
∫ω˜ (k)f˜(k) dk ˜f (k) dk
1 F˜ ∫2P(z ) (2π)3 1 - z ˜f (k) 0
)
F(P(z0) - 1) 2z0P(z0)
Furthermore, c0 and z are related, as in eq 17:
1 - Fc0 )
1 1 2 (2π)3
(36)
(37)
(41)
Thus, the energy per particle (u1) that is due to correlations is
where, as observed previously, z0 is determined by eq 14. The parameter z will be defined as observed previously, i.e., eq 11b is kept:
Fc1 z) 1 - Fc0
(40)
0
∝F
(43)
using eqs 11-14 with z ) z0. Note that, with n ) 1, the aforementioned expressions are those of the standard lattice gas or the Ising model, where eq 40 with eq 39 represents the MSA. For n ) 1, we have z0 ) 0 and P(z0) ) 1. With the aforementioned expressions, the corresponding SCOZA problem is easily defined. All expressions, except the MSA condition in eq 40, are kept. The latter is replaced by thermodynamic self-consistency, which is given by
∂2 ∂ 1 ∂βp ) 2[F(u0 + u1)] ∂β F ∂F ∂F
(
)
(44)
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where u0 is the mean field internal energy:
1 -Fu0 ) F2 2
1
1
∑r f(r) ) 2F2˜f (0) ) 2F2
(45)
then obtained if the transition or jump probability has a Yukawa form outside the hard core, whereas the hard-core condition on the correlation function dictates its value inside. Thus, we have
cω(r) ) K
Inserting eqs 19, 42, and 45 into eq 44, we obtain the SCOZA partial differential equation:
{
∂2 ∂2 ) -F(1 - F) 1 + 2[F(1 - F)F(z)] ∂β ∂F
}
(46a)
exp[-z(r-1)] r
(for r > 1)
Solving the OZ equation with a core condition, one obtains the inverse compressibility as
∂(βp) ) 1 - Fc˜ ω(0) ) a ∂F
with
F(z) )
P(z) - 1 2zP(z)
(46b)
(47)
where γr characterizes the inverse range of interaction. One easily sees that eq 46 is identical to the SCOZA equation for the Ising model, despite the large difference between polymers and monomers. Mathematically, this large difference lies in the boundary conditions for β ) 0, the reference system. For the Ising model (n ) 1), we have z0 ) 0, P(z0), and thus ) 1 for β ) 0. With n > 1, eq 14 yields z0 (>0), eq 18 yields z ) zr for β ) 0, and eq 19 thus yields the parameter 2 ( 1). One can especially note that the second virial coefficient of eq 64 is an order-of-magnitude smaller than that in eq 65. Expanding eq 64 for small ξ, one sees that the term linear in ξ has a coefficient proportional to 1/xn, whereas the corresponding term of the “mean-field” Flory-type equation is on the order of 1. This 1/xn dependence seems to reflect the volume of interaction for a pair of random-walk-type polymers whose diameter is proportional to xn and, thus, has a volume proportional to n3/2. With a monomer density F, the number density of polymers is F/n. Thus, the contribution to the pressure from the second virial coefficient is proportional to
(xn)3
Thus,
(64)
(Note that, in the latter two equations, the p again means pressure.) Compared to simulations for polymers of fixed length,15-17 this very simple and explicit result yields a pressure that is somewhat below the simulated one. The difference increases rapidly with increasing n and persists over a wide density range (see Figures 1 and 2). This difference seems to be largely due to the fact that the full self-avoiding character of polymer chains is not taken into account in our theory leading to eq 64. The self-avoidance makes the polymers less flexible and larger in end-to-end distance and pressure for a given n. The result from eq 64 can be also compared with the “Flory” theory for the continuum case that we have developed earlier.13 This theory is obtained in the mean-field limit γ f 0, which leads to eq 22 in the lattice case. In the continuum case, one similarly finds in the mean-field limit, i.e., z f 0,
(
by which eqs I2.35 and I2.38 yields
)
() ( )
F2 1 2 ) F n xn
(66)
in accordance with eq 64. (On the third virial-coefficient level, the same argument will give [(xn)3]2(F/n)3 ) F3, also in accordance with eq 64. However, this argument will not yield accurate results for still higher virial coefficients, which, in our theory, are on the order of 1, as observed for the third virial coefficient.) From these considerations, one would expect that eq 64 would be adequate for very small values of n, and the comparison of our eq 64 with the simulation results and the Flory-type result of eq 65 suggests that we can obtain an improved theory using a suitably determined nonzero value of z that will be dependent on both n and density, rather than the z f 0 limit that yields eq 65. By making z nonzero, one is describing an effective or renormalized random walk in which the effective step length becomes longer, thus expanding the polymer size and pressure for a given number of steps. Figures 1 and 2, which show a comparison of our eq 64 with simulation results and with the analytic theory of Honnell and Hall,18 enable us to confirm that, for an appropriately chosen value of z(n), we can obtain good agreement with the best estimates of the compressibility factor. For example, in Figure 3, one observes that, for n ) 8, letting z ) 10-20 gives a result
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Figure 1. Compressibility factor (Z) versus reduced density (ξ) for n ) 2, 4, and 8 (from bottom to top). Crosses are simulation data from ref 15.
Figure 2. Compressibility factor (Z) versus reduced density (ξ) for n ) 16, 32, and 51 (from bottom to top). Crosses are simulation data from ref 15 (n ) 16), ref 17 (n ) 32), and ref 16 (n ) 51).
for the compressibility factor that is very similar to the quite accurate theoretical result of Honnell and Hall18 (which is slightly lower than the estimated result obtained from the simulation results of Chang and Sandler15 for high densities and vice versa for medium densities). One of the goals of our future work is to find a simple recipe that results in a z(n,ξ) expression that is capable of yielding accurate compressibility factors over a wide range of values for n and ξ. Acknowledgment. C.-L. Lee acknowledges the support of the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy. The contribution of G.S. to this work has been supported in part by a grant from the National Science Foundation (under Award No. 29963). Appendix. Check of Equation of State Relation eq 58, which led to the equation of state shown in eq 64, can be given an extra check. In the limit z f ∞, the
direct correlation function (eq 50) will become a δ-function at the hard-core surface. This δ-function will reappear in the correlation function h(r), i.e., for r = 1 (K ∝ z f ∞),
g(r) ) h(r) + 1 f K
exp[- (r - 1)z] r
(A.1)
This, together with the definition given in eq I2.4 for V, thus gives
V ) 24ξKez
2
∫1∞exp(-zr)g(r)r dr f 12ξKz
(A.2)
Solution of the OZ equation, on the other hand, gives the relation presented in eq I2.37, such that
V)K
(σ1--τyy ) f zK(1y - 1)
(A.3)
Now, y f 1; therefore, eq 57 must be expanded further to obtain
Ornstein-Zernike Random-Walk Approach for Polymers
J. Phys. Chem. B, Vol. 108, No. 51, 2004 19817
Figure 3. Compressibility factor for various z values with a chain length of n ) 8 (solid curves), with z ) 0, 5, 10, 20, and ∞ (from top to bottom). Dashed curve is plotted from the analytic approximation derived by Honnell and Hall,18 where crosses are simulation data.15
the final result for V. We then need the expansion
xp2 + 2pzxA + z2p ) zxp 1 + 1xA + ‚‚‚
(
z
when utilizing eq 58. Finally, inserting this into eq A.3, one obtains eq A.2, as we wanted to show.
)
(A.4)
Inserted in eqs 54-57, this changes these equations to
1 1 U0 ) z(xp - xA) 1 + (xp + 2xA) + ‚‚‚ 2 2
[
1 1 2xpΓ ) z2xp(xp - xA) 1 + xA + ‚‚‚ 2 2
[
U1 1 1 4 ) - z 1 + (xp - xA) - + ‚‚‚ U0 2 2 z
[
]
]
]
4 U1 1 2 )- 1+ y z z U0
(
)
1 ) 1 + (xp - xA) + ‚‚‚ 2 )1+
12ξK + ‚‚‚ z2
(A.5)
References and Notes (1) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and Row: New York, 1971. (2) Cummings, P. T.; Stell, G. J. Stat. Phys. 1983, 33, 709. (3) Chandler, D.; Andersen, H. C. J. Chem. Phys. 1972, 57, 1930. (4) Chandler, D. In Studies in Statistical Mechanics; Montroll, E. W., Lebowitz, J., Eds.; North-Holland: Amsterdam, 1982; Vol. III. (5) Schweizer, K. S.; Curro, J. G. Phys. ReV. Lett. 1987, 58, 246. (6) Schweizer, K. S.; Curro, J. G. AdV. Polym. Sci. 1994, 116, 319. (7) Yethiraj, A.; Hall, C. K. J. Chem. Phys.1992, 96, 797. (8) Yethiraj, A.; Curro, J. G.; Schweizer, K. S. J. Chem. Phys. 1993, 98, 1635. (9) (a) Flory, P. J. J. Chem. Phys. 1941, 9, 660. (b) Huggins, M. L. J. Chem. Phys. 1941, 9, 440. (10) Dickman, R.; Stell, G. Phys. ReV. Lett. 1996, 77, 996. (11) Pini, D.; Stell, G.; Dickman, R. Phys. ReV. E 1998, 57, 2862. (12) Høye, J. S.; Borge, A. J. Chem. Phys. 1998, 108, 4516. (13) Høye, J. S.; Lee, C.-L.; Stell, G. Mol. Simulat. 2003, 29, 727. (14) Høye, J. S.; Stell, G. Mol. Phys. 1976, 32, 195. (15) Chang, J.; Sandler, S. I. Chem. Eng. Sci. 1994, 49, 2777. (16) Escobedo, F. A.; de Pablo, J. J. J. Chem. Phys. 1995, 102, 2636. (17) Escobedo, F. A.; de Pablo, J. J. J. Chem. Phys. 1995, 103, 1946. (18) Honnell, K. G.; Hall, C. K. J. Chem. Phys. 1989, 90, 1841.
