Compressible Regular Solution Model (CRS)

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The CRS model is a free energy expression for the mixing of weakly interacting polymer blends that accounts for their compressibility, using as input parameters pure component properties only. The CRS has been capable to predict, at least qualitatively, the phase behavior of more than 30 polymer pairs, capturing upper critical solution temperature (UCTS) and lower critical solution temperature (LCST) behaviors.¹ The phase behavior of ternary polymer mixtures can also be capture by the extension of the CRS model to multicomponent systems. Other components, such as solvents and small molecules like plasticizers, can also been incorporated in this model.²

A full derivation of the CRS equation, explaining the corresponding assumptions and limitations, is presented in reference 1. The final equation for two components, A and B, is:

where the variables are:

ÝÝÝÝÝÝ reduced density (density/hard core density),

ν_i ÝÝÝÝÝÝÝ hard-core molar volume,

δ_i,0 and δ_iÝ solubility parameters at 0 K and temperature T,

N_i ÝÝÝÝÝÝ number of repeat units per chain

f_i ÝÝÝÝÝÝÝ volume fraction

K ÝÝÝÝÝÝÝ Boltzman constant

Input parameters for the calculation of the free energy can be obtained experimentally or can be estimated by group contribution methods.

The CRS model captures the components free volume changes with temperature by introducingÝ a reduced density. The reduced density, , is nothing more than the ratio of the density at certain thermodynamic conditions r(T,P) over an extrapolated value of density at 0 Kelvin, or hard-core density, r*, which accounts for the presence of free volume in the system:

The energy associated to each repeat unit is represented by the well-known solubility parameter theory (also know as cohesive energy density). For the components cross interaction, the CRS model invoke the standard Berthelot mixing rule, such that the solubility parameter of the mixed state is a geometric average of the component values as described by:

The effect of density changes on the solubility parameter, are assumed to follow:

where is the reduced density at a certain temperature and Ýis the reduced density at 298 K. is the cohesive energy density at 298 K, temperature at which is often found in literature of estimated by group contribution methods.

As for any free energy of mixing expression,Ý phase diagrams of polymer mixtures can be calculated with the CRS model using standard thermodynamic relationships for spinodal and binodal equilibrium curves:

Spinodal

Binodal

where

Ý ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝ Ý

References

1. Ruzette, A.-V. G. & Mayes, A. M. ìA simple free energy model for weakly interacting polymer blendsî.Ý Macromolecules 34, 1894-1907 (2001).

2. Gonzalez-Leon, J.A. & Mayes, A.M. ìPhase behavior prediction of ternary polymer mixturesî. Macromolecules 36, 2508-2515 (2003).