Structural Relaxation and Thermodynamics of Viscous Aqueous Systems: A Simplified Reappraisal

Abstract

The attainment of true equilibrium conditions is a dynamic process that encompasses a time span. For slow relaxing systems, non-equilibrium steady states can often look like equilibrium states. This is the case of viscoelastic systems, whose properties reflect their thermo-rheological history. After a summary of the seminal woks by Eyring, Adam & Gibbs and Angell, and mention of promising recent approaches that imply updated theoretical and experimental techniques, the paper suggests a simplified approach for aqueous systems, through a modified expression of the chemical potential of water and use of the “dynamic” phase diagram, so far proposed by Slade and Levine. For homogeneous systems (aqueous solutions), an extra term in the expression of the chemical potential accounts for the energy related to the residual strains produced during the thermo-rheological history of the system. This approach allows estimation of the effect of viscosity on the observed freezing point of polymer solutions. For heterogeneous systems (hydrogels, colloidal glasses), changes of the phase boundaries in the phase diagram explain the gel/sol hysteresis and the syneresis process as the result of water exchange between hosting meshes and trapped aqueous solution. Finally, physical hurdles that hinder inter-phase water displacements and/or the access to the headspace of the system can lead to the coexistence of aqueous phases with different a_W within the same heterogeneous system.

Appendices

Appendix

Equivalence of WLF and VTF Equations [12,13,14]

Taking into account that either equation is empirical and aims at the description of the relaxation time at the temperature T, with reference to T_g and T_o, for WLF and VTF, respectively, one may put:

2.1 WLF

$$\tau_{{\text{R}}} \left( T \right) = \tau_{{\text{R}}} \left( {T_{{\text{g}}} } \right)\exp \left[ { - \frac{{C_{1} \left( {T - T_{{\text{g}}} } \right)}}{{C_{2} + \left( {T - T_{{\text{g}}} } \right)}}} \right]$$

(A1)

2.2 VTF

$$\tau_{{\text{R}}} \left( T \right) = \tau_{{\text{R}}} \left( \infty \right){\text{exp}}\left( {\frac{B}{{\left( {T - T_{{\text{o}}} } \right)}}} \right)$$

(A2)

For T = T_g, Eq. 5 leads to

$$\tau_{{\text{R}}} \left( {T_{{\text{g}}} } \right) = \tau_{{\text{R}}} \left( \infty \right)\exp \left( {\frac{B}{{\left( {T_{{\text{g}}} - T_{{\text{o}}} } \right)}}} \right)$$

(A3)

that can be replaced in Eq. A1 which in combination with A2 gives

$$\tau_{{\text{R}}} \left( T \right) = \tau_{{\text{R}}} \left( \infty \right)\exp \left( {\frac{B}{{\left( {T_{{\text{g}}} - T_{{\text{o}}} } \right)}}} \right)\exp \left[ { - \frac{{C_{1} \left( {T - T_{{\text{g}}} } \right)}}{{C_{2} + \left( {T - T_{{\text{g}}} } \right)}}} \right] = \tau_{{\text{R}}} \left( \infty \right)\exp \left( {\frac{B}{{\left( {T - T_{{\text{o}}} } \right)}}} \right)$$

(A4)

The eventual result is as follows:

$$\frac{{C_{2} + \left( {T - T_{{\text{g}}} } \right)}}{{C_{1} }} = \frac{{\left( {T - T_{{\text{o}}} } \right)\left( {T_{{\text{g}}} - T_{{\text{o}}} } \right)}}{B}$$

(A5)

Putting T = T_o, one gets

$$C_{2} = \left( {T_{{\text{g}}} - T_{{\text{o}}} } \right)$$

(A6)

and

$$B = C_{1} C_{2}$$

(A7)

Taking into account that (T – T_o) = (T – T_g) + (T_g –T_o) = C₂ + (T – T_g), Eqs. A6 and A7 state the formal equivalence of the WLF and VTF equations. However, it is important to recall that the WLF equation seems more adequate for polymer solutions at (T_g + 100 K) > T > T_g and therefore does not cover the (T_o, T_g) range, while the VTF equation holds for T > T_o and describes the experimental evidence of many glass-forming systems [14].

Equation 6 reveals that C₁ corresponds to the abrupt change of the order of magnitude of τ_R on crossing the glass transition threshold T_g [15]:

$$\log_{10} \left[ {\frac{{\tau_{{\text{R}}} \left( {T_{{\text{g}}} } \right)}}{{\tau_{{\text{R}}} \left( \infty \right)}}} \right] = \log_{10} \left[ {\frac{{\eta \left( {T_{{\text{g}}} } \right)}}{\eta \left( \infty \right)}} \right] = \frac{{C_{1} }}{2.3}$$

(A8)

which, for 8 orders of magnitude viscosity drop, implies for C₁ a “universal” value around 17 (while C₂ ≈ 50 °C) [15].

When combined with Eqs. 1, A6 and A7, A8 leads to

$$S_{{\text{c}}} = \frac{C}{B}\frac{{T - T_{{\text{o}}} }}{T} = \frac{C}{{C_{1} C_{2} }}\left( {1 - \frac{{T_{{\text{o}}} }}{T}} \right) = \frac{C}{{\left( {T_{{\text{g}}} - T_{{\text{o}}} } \right) \ln \left( {\frac{{\tau_{{\text{R,g}}} }}{{\tau_{{{\text{R}},\infty }} }}} \right)}}$$

(A9)

where T_o is the temperature at which S_c = 0, possibly coincident with the Kauzmann temperature, T_K, where the entropy of the undercooled liquid becomes equal to that of the thermodynamically stable solid phase.