Development of a Unifying Framework for Modeling Multi-component Diffusion in Polymer Solutions

Abstract

In this work a unifying framework for modeling multi-component diffusion in mixed solvent polymer solutions is developed by introducing additional restrictions such as the Onsager reciprocal relations (ORR) and the quasi-equilibrium postulate. More specifically, three different multi-component diffusion models, namely the Zielinski and Hanley model, the Dabral et al. theory and the Alsoy–Duda model are revised by using the above restrictions which are based on sound principles of non-equilibrium thermodynamics. Realistic simulations for the solvent(s) evaporation from the water/acetone/cellulose acetate (CA) and formamide/acetone/CA systems were obtained by combining the above multi-component diffusion models with the ORR and the quasi-equilibrium postulate. It is believed that the results of this work could be used to further study diffusion in multi-component systems appearing in coating and membrane formation.

Appendix

The starting point for the derivation of Miller’s equation in the case of a diffusing polymer ternary solution close to equilibrium and in the absence of external forces is the dissipation function Ψ (see Eq. 7):

$$ \varPsi = TS = \sum\limits_{i = 1}^{3} {J_{i}^{ \ne } X_{i}^{{}} } $$

(36)

Obviously, the thermodynamic force $X_{i} = \frac{{\partial \left( {\mu_{i} } \right)_{T,p} }}{\partial z}$ is subject to the Gibbs–Duhem restrictions:

$$ \sum\limits_{i = 1}^{3} {\rho_{i} X_{i} = 0} . $$

(37)

By taking into account that the fluxes are linearly dependent and there is no volume change during the diffusion experiment, the following equation holds true:

$$ \sum\limits_{i = 1}^{3} {V_{i} J_{i}^{ \ne } = 0} $$

(38)

where $V_{i}$ represents the partial specific volume of the i-th substance. By further using Eq. 36–38 the dissipation function can be written as:

$$ \varPsi = J_{1}^{ \ne } Y_{1} + J_{2}^{ \ne } Y_{2} $$

(39)

where

$$ \begin{aligned} Y_{1} & = \left[ {1 + \frac{{\rho_{1} V_{1} }}{{\rho_{3} V_{3} }}} \right]X_{1} + \left[ {\frac{{\rho_{2} V_{2} }}{{\rho_{3} V_{3} }}} \right]X_{2} = \alpha X_{1} + \beta X_{2} \\ Y_{2} & = \left[ {\frac{{\rho_{1} V_{2} }}{{\rho_{3} {\text{V}}_{3} }}} \right]X_{1} + \left[ {1 + \frac{{\rho_{2} {\text{V}}_{2} }}{{\rho_{3} {\text{V}}_{3} }}} \right]X_{2} = \gamma X_{1} + \delta X_{2} \\ \end{aligned} $$

By applying the linearity postulate in the above equations the following law is derived:

$$ J_{1}^{ \ne } = L_{11} Y_{1} + L_{12} Y_{2} ;\,J_{2}^{ \ne } = L_{12} Y_{1} + L_{22} Y_{2} $$

(40)

where the L _ij are the mass conductivity coefficients.

Fick’s law for a ternary diffusing system can be written in terms of the mass concentration ρ _i as:

$$ \begin{aligned} J_{1m} & = - D_{11} \frac{{\partial \rho_{1} }}{\partial z} - D_{12} \frac{{\partial \rho_{2} }}{\partial z} \\ J_{2m} & = - D_{21} \frac{{\partial \rho_{1} }}{\partial z} - D_{22} \frac{{\partial \rho_{2} }}{\partial z} \\ \end{aligned} $$

(41)

By using the chain rule to write the partial derivatives appearing in Fick’s law in terms of chemical potential and by comparing the coefficients with the reported coefficients in Eq. 40 one obtains:

$$ \begin{aligned} - D_{11} & = \left[ {\alpha \mu_{11} + \beta \mu_{21} } \right]L_{11m} + \left[ {\gamma \mu_{11} + \delta \mu_{21} } \right]L_{12m} = aL_{11} + bL_{12} \\ - D_{12} & = \left[ {\alpha \mu_{12} + \beta \mu_{22} } \right]L_{11m} + \left[ {\gamma \mu_{12} + \delta \mu_{22} } \right]L_{12m} = cL_{11} + dL_{12} \\ - D_{21} & = aL_{21} + bL_{22} \\ - D_{22} & = cL_{21} + dL_{22} \\ \end{aligned} $$

(42)

where a, b, c, d denote the terms in the corresponding square brackets and μ _ij stands for $\frac{{\partial \left( {\mu_{i} } \right)_{T,p} }}{{\partial \rho_{j} }}$. Solution of Eq. 42 for L _ij then yields:

$$ \begin{aligned} L_{11} & = \frac{{bD_{12} - dD_{11} }}{ad - bc},\,L_{12} = \frac{{cD_{11} - aD_{12} }}{ad - bc} \\ L_{21} & = \frac{{bD_{22} - dD_{21} }}{ad - bc},\,L_{22} = \frac{{cD_{21} - aD_{22} }}{ad - bc} \\ \end{aligned} $$

(43)

The necessary and sufficient condition that L ₁₂ = L ₂₁ (ORR) is

$$ aD_{12} + bD_{22} = cD_{11} + dD_{21} ,\,ad - bc \ne 0 $$

(44)

Obviously, Eq. 44 satisfies both the ORR and the quasi-equilibrium postulate.