Physica A: Statistical Mechanics and its Applications
Crossover Flory model for phase separation in polymer solutions
Introduction
It has been established that polymer solutions in low-molecular-weight solvents asymptotically close to the critical point of mixing exhibit the same universal behavior as that of simple fluids and fluid mixtures [1], [2], [3], [4], [5]. In other words, polymer solutions belong to the universality class of the 3-dimensional Ising-model criticality like other fluids [6], [7]. However, the approach to asymptotic universal Ising-like behavior is strongly affected by the degree of polymerization. The higher the degree of polymerization the narrower the range of Ising-like behavior appears [8]. Ultimately, the range of universal Ising-like critical behavior disappears at the theta point which is the critical mixing point in the limit of an infinite degree of polymerization. As a result, the actual near-critical properties of polymer solutions exhibit some kind of intermediate (crossover) behavior rather than universal asymptotic behavior. On the mesoscopic (nanoscopic) level such a crossover is caused by a competition between the correlation length of the critical fluctuations and the size of the polymer chain [8]. Consideration of these crossover effects is essential for the interpretation of experimental studies of critical phenomena in polymer solutions.
In this paper we show how critical fluctuations modify the phase separation of a polymer solution as described by the Flory theory. The Flory theory is the simplest model that explains phase separation in polymer solutions [9]. It is essentially a mean-field theory which, like the van der Waals theory for simple fluids, completely ignores fluctuations. The method presented in this paper can be applied, in principle, to any classical equation of state. The resulting crossover model demonstrates how fluctuations change the classical behavior and how “classical” critical points shift to their actual positions.
There is an extensive discussion in the literature concerning the dependence of various physical properties on the degree of polymerization N. Our theory will show that this dependence is strongly affected by crossover effects.
The paper is organized as follows. First, we briefly recall the predictions given by the Flory theory for the critical behavior of polymer solutions. Then, we consider how fluctuations modify the Gibbs energy of mixing in the Flory model. The classical Flory model is universal: when the temperature is reduced by the theta temperature, it does not contain any system-dependent parameters. The Flory model modified by fluctuations contains only one system-dependent parameter: the range of intermolecular forces in a reference system, namely, in the monomer solution to which the Flory theory is reduced at N=1. To specify this parameter we have fitted recent computer simulation data for a lattice polymer model with N=1 [10] to our crossover Flory model with N=1. We then consider the N-dependence of the critical parameters and critical amplitudes. Specifically, we show that in the commonly used range of N≃10–103 the physical properties do not obey asymptotic (N→∞) power laws. Instead, they are characterized by “effective” (crossover) exponents which themselves depend on N.
Section snippets
Classical Flory model
The separation of a polymer solution into two coexisting phases below the theta point has been explained by Flory [9]. In the Flory theory the molar Gibbs energy of mixing of polymer chains and solvent molecules gmix is represented aswhere N is the number of repeated monomer units in a polymer molecule (degree of polymerization), φ the volume fraction of polymer in the solution, ε=Θ/2T the interaction parameter, T the temperature, Θ the critical temperature
Theory of crossover critical behavior
Properties of fluids near the critical points are affected by long-range density or concentration fluctuations [6]. As a result, the properties asymptotically close to the critical point exhibit power laws with universal exponents different from those predicted by classical (mean-field) theory. The position of the critical point is also shifted and appears at a lower temperature than predicted by a classical equation of state [13]. The validity of the asymptotic power laws is, however,
Crossover Flory model
When the transformation (28) is applied to the expansion (27) for , one obtains an equation that incorporates a crossover between critical mean-field behavior given by the expansion (27) and asymptotic Ising-like behavior. As the expansion implies that τ and Δφ should be small, this procedure is valid only in a relatively narrow region around the critical point. However, since in the Flory model the explicit expression for the critical part of the Gibbs energy is known (Eq. (13)), we can
Dependence of critical parameters and critical amplitudes on the degree of polymerization
While the critical exponents in the power laws, given by , , , are universal, the amplitudes in these power laws depend strongly on the degree of polymerization (proportional to molecular weight). It is assumed that for N→∞ these amplitudes themselves satisfy power laws of the form [35]:In addition, the critical concentration and the critical temperature are assumed to depend on N asWe emphasize the asymptotic character of these equations. As N
Conclusion
In this work we have demonstrated how critical fluctuations modify the classical Flory model for polymer solutions. We are well aware of the restrictions imposed by the Flory theory on the parameters of our crossover model. This is why we have not yet tried to describe experimental or computer simulation data for N different from unity: these restrictions are too tight to describe actual data. Moreover, in addition to the critical fluctuation effects considered, one should expect logarithmic
Acknowledgements
We are indebted to A.Z. Panagiotopoulos for providing us with detailed information on the computer simulation data [10]. We have benefitted from fruitful interactions with T.A. Edison and Yu.B. Melnichenko and have appreciated stimulating discussions with B. Widom and M.E. Fisher. The research has been supported by the Division of Chemical Sciences of the Office of Basic Energy Sciences of the U.S. Department of Energy under Grant No. DE-FG02-95ER-14509.
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