Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams

Abstract

The Edmond−Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B 11 , B 12 , B 22 ) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B 11 , B 12 , B 22 ). Since the Edmond−Ogston model is equivalent to the Flory− Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory−Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B 12 .

Addition to “Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams”

“…In addition, eq is an expression for S c that explicitly shows that it satisfies the transformation ( B 11 , B 12 , B 22 , S c ) → ( B 22 , B 12 , B 11 , 1/ S c ). This property was already clear from eqs 20–21 in ref but was not obvious from the explicit solutions as presented in eqs 22–23 in ref .…”

“…It was recently noted that eq 20 in ref , which expresses the stability requirement in the critical point where -S c is the slope of the tangent of the binodal and spinodal in the critical point and ( B 11 , B 12 , B 22 ) are the virial coefficients for the mixture can be rewritten as As a result, by using eqs 24 and 25 in ref , it is found that Here ( c 1, c , c 2, c ) are the coordinates of the critical point in molar concentration units. This is a useful relation in the case where the critical point is known.…”

“…From eq 29 in ref , it follows that all triplets of virial coefficients ( B 11 * , B 12 * , B 22 * ) that correspond to a certain critical point ( c 1,c , c 2,c ) can be written in vector notation as Here, -S c is the slope of the tangent of the binodal and spinodal in the critical point; λ 1 can be any real number provided . Equation has the interesting consequence that when combined with a rearranged version of equations originally provided by Edmond and Ogston (eqs 26–27 in ref )­ it allows one to rewrite eq in a form that solely contains the location of the critical point ( c 1,c , c 2,c ): This expression gives all possible values for the virial coefficients ( B 11 * , B 12 * , B 22 * ) leading to the same critical point ( c 1,c , c 2,c ). Here λ 2 > 1/(2­( c 1,c 2/3 c 2,c 1/3 + c 1,c 1/3 c 2,c 2/3 )) to satisfy the phase separation criterion .…”

“…Here λ 2 > 1/(2­( c 1,c 2/3 c 2,c 1/3 + c 1,c 1/3 c 2,c 2/3 )) to satisfy the phase separation criterion . The advantage of this method is that no prior knowledge of the virial coefficients is needed, in contrast to eq 29 in ref .…”

“…Regarding our previous publication entitled “Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams”, we wish to add a short discussion on the following topics: (1) a practical expression for the slope of the tangent to binodal and spinodal in the critical point, (2) an extension on how to find the virial coefficients, when the critical point is known, (3) a demonstration that eq 37 from ref is always met by the solution of the coexistence equations, and (4) an extension on how to find the virial coefficients when the composition of two coexisting phases is known.…”