Theoretical evaluation ofKp in size-exclusion chromatography from a thermodynamic viewpoint

Abstract

The Flory‐Huggins formalism developed for a ternary solvent(1)‐polymer(2)‐polymer(3) system has been used to evaluate the binary and ternary interaction functions. The raw data are the two‐phase equilibrium compositions of the three components determined by liquid chromatography. These interaction functions have been used to evaluate theoretically the distribution coefficient of solute‐gel interactions in size exclusion chromatography (SEC), Kp. For a system solvent(1)‐polymer(2)‐gel matrix(3), a new expression for Kp has been obtained from a thermodynamic point of view. The proposed equation is a function of the fraction of gel matrix in the ternary phase, of the molar volumes of solvent and polymer and of the binary and ternary interaction functions. For the systems studied, values of Kp higher and lower than unity have been obtained at different hydrodynamic volumes which evidences the existence of secondary mechanisms. These calculations lead to a better understanding of the separation mechanism in chromatographic experiments.

Thermodynamic Interpretation of the SEC Behavior of Polymers in a Polystyrene Gel Matrix

“…where subscripts 1, 2 and 3 refer to the different components of the ternary mixture, and the prime and double prime refer to the two coexisting phases. The expressions for the change in the chemical potentials of the solvent, Dl 1 = (qDG/qn 1 ) n 2 ;n 3 ;p;T and of the two polymers, with mean polymerization degrees x 2 and x 3 , Dl 2;x 2 = (qDG/ qn 2 ) n 1 ;n 3 ;p;T and Dl 3;x 3 = (qDG/qn 3 ) n 1 ;n 2 ;p;T , have been previously reported [17,18] with v 1 = n 1 /(n 1 + n 2 x 2 + n 3 x 3 ), v i = n i x i /(n 1 + n 2 x 2 + n 3 x 3 ) (i = 2, 3) and n i being the number of moles of component i in the ternary mixture. Therefore, from the equilibrium condition (Equation (2)), we obtain the following unknowns: (g 12 )9, (g 13 )9, (g 23 )9, (g T )9, (g 12 )99 (g 13 )99 (g 23 )99 (g T )99, as well as their derivatives: (dg 12 /dv 2 )9, (dg 13 /dv 3 )9, (dg 23 /dv 3 )9, (qg T /qv 3 )9, (dg 12 /dv 2 )99, (dg 13 /dv 3 )99, (dg 23 /dv 3 )99 and (qg T /qv 3 )99.…”

“…In some others, however, there is an initial decrease of K p with a subsequent increase. An increase of K p with v 3 would be expected [18] since an increase of the gel volume fraction in the ternary phase indicates more interactions between segments of the polymeric solute 2 and the gel 3, and consequently a greater value of K p . This trend could indicate that the experimental (K p , v 3 ) values should be in the upward curve.…”

“…With the aim of determining the interaction functions and in the framework of the Flory-Huggins theory, [17,18] the phase separation of a ternary polymer system can be described by the Gibbs free energy of mixing by mol of occupied lattice sites, DG, as a function of concentrationdependent interaction functions:…”

“…This approach using the equilibrium composition of five tie-lines gives 15 equations that allow to determine 15 unknowns. To eliminate the one remaining unknown, the best approach [18] consists in assuming that either g 12 or g 13 functions change linearly with polymer composition. Consequently, either (dg 12 / dv 2 ) or (dg 13 /dv 3 ) are constant at any polymer concentration and do not change for the two conjugate phases of a given tie-line neither between the different tie-lines of a given run:…”

“…However, if enthalpic effects, due to interactions between the polymeric solutes and the pore walls take place, the distribution coefficient of real SEC is K D = K SEC K p , and the retention volume is as follows: [21] V e = V 0 + K SEC K p V p (7) with K p a 1, if solute-gel interactions are repulsive (i. e. electrostatic type) or K p A 1 if they are attractive (reversible adsorption of polymer onto the matrix packing). In this regard, recently, we have derived an expression to evaluate K p from a thermodynamic starting point, that at v 20 e 0 (the usual condition in SEC experiments) can be written as: [18]…”