A model of thermophoresis of colloidal proteins in water using non-Fickian diffusion currents

Abstract

In 1928, Chapman generalised Einstein's theory of diffusion for non-uniform fluids to show the presence of a non-Fickian diffusion current, which he considered important in thermodiffusion (Ludwig-Soret effect). In 1941, Kiyosi Itô proposed the formal methods of stochastic calculus in the presence of spatially dependent diffusion, yielding the same non-Fickian diffusion current as shown by Chapman. The phenomenon of thermodiffusion and thermophoresis happens in the presence of a temperature gradient, which makes diffusion space-dependent. The role of solvation forces in thermophoresis will only be clearer once that of diffusion is understood properly. In this paper, we investigate the importance of Chapman's non-Fickian diffusion current on the thermophoretic motion of colloidal particles in water (with weak salt concentration). We show that all the general features of variations of the Soret coefficient $S_T$ with temperature can be captured using Chapman's non-Fickian diffusion current. We compare our theoretical results with experimental plots of the Soret coefficients for three polypeptides in aqueous solution: Lysozyme, BLGA, and Poly-L-Lysine, and find a strong match. We emphasise that, in addition to the yet-to-be-understood details of solvation forces, Chapman's non-Fickian diffusion current is an indispensable element that needs to be taken into account for a complete understanding of thermophoresis and thermodiffusion.

Figures (6)

• Figure 1: Water properties as a function of temperature. Variation of (a) density $\rho_{w}$ of water, (b) its coefficient of thermal expansion $\alpha_{w}$, and (c) $\frac{d}{dT}\rho_{w}T$ as a function of temperature is shown.
• Figure 2: Soret coefficient comparison between experiment data, empirical form, and model predictions shown for Lysozyme, BLGA, and Poly-L-Lysine. The experimental data shown have been extracted from Fig.1 of the reference iacopini2006macromolecular using an online data-digitization tool (PlotDigitizer).
• Figure 3: The figure (a) shows the variation of viscosity and (b) that of $D_0(T)$ with temperature, where (c) shows the variation of the temperature gradient of the logarithm of $D_0(T)$ with temperature. The protein particle radius taken is 1.7 nm iacopini2003thermophoresispiazza2004thermophoresis.
• Figure 4: Competition between Fickian, non-Fickian diffusion currents and solvation force-driven drift shown for Lysozyme, BLGA, and Poly-L-Lysine.
• Figure 5: Figure shows variation of (a) concentration (normalised), (b) derivative vs temperature plot obtained from our model for thermophoresis of Lysozyme, BLGA, Poly-l-lysine, respectively. In (c), we display the Comparison of normalised saturation concentration (normalised) with temperature obtained from equilibrium experiment and the model, shown for chicken egg-white lysozyme in the main body figureforsythe1999tetragonalcacioppo1991solubilityiacopini2003thermophoresiscarpineti2004metastabilitypiazza2004thermophoresis. The upper-right inset (a semi-log plot) shows the variation in saturated concentration (un-normalised) with temperature.