Why Charged Molecules Move Across a Temperature Gradient: The Role of Electric Fields

Methods to move solvated molecules are rare. Apart from electric fields, only thermal gradients are effective enough to move molecules inside a fluid. This effect is termed thermophoresis, and the underlying mechanisms are still poorly understood. Nevertheless, it is successfully used to quantify biomolecule binding in complex liquids. Here we show experiments that reveal that thermophoresis in water is dominated by two electric fields, both established by the salt ions of the solution. A local field around the molecule drives molecules along an energy gradient, whereas a global field moves the molecules by a combined thermoelectrophoresis mechanism known as the Seebeck effect. Both mechanisms combined predict the thermophoresis of DNA and RNA polymers for a wide range of experimental parameters. For example, we correctly predict a complex, nonlinear size transition, a salt-species-dependent offset, a maximum of thermophoresis over temperature, and the dependence of thermophoresis on the molecule concentration.

Figure 1

Local and global electric fields move molecules along a temperature gradient. (a) Around a charged molecule, dissolved ions form a shielding capacitor with Debye length ${\lambda }_{DH}$. The energy stored in the capacitor decreases in the cold and leads to a positive Soret coefficient $S_T^{CM}$. For molecules with radius $R$ smaller than the Debye length ${\lambda }_{DH}$, the radial capacitor can be approximated as a point charge; for larger molecules, it can be approximated as a plate capacitor. The result is a nonlinear size transition depending on ${\lambda }_{DH}/R$. (b) The differential Soret coefficients of ions in solution, here ${\mathrm{K}}^{+}$ and ${\mathrm{Cl}}^{-}$, create a global electric field. The resulting electrophoresis cannot be readily distinguished from thermophoresis. This Seebeck effect results in an ion-species-dependent offset $S_T^{EL}$ that is independent of the Debye length for the used experimental conditions.

Figure 2

Nonlinear size transition of capacitive thermophoresis. (a) The Soret coefficient $S_T$ is measured for single-stranded DNA with lengths of 2, 5, 10, 22, 50, and 80 bases and plotted against Debye length ${\lambda }_{DH}$ at $15°\mathrm{C}$. The radius R is measured from diffusion; the effective charge describes the amplitude, and a constant offset $S_T({\lambda }_{DH}=0)=S_T^{EL}+S_T^{NI}+1/T$ is determined. (b) After rescaling the data according to Eq. (2), the measurements fall onto a single master curve and confirm in detail the size transition of the capacitor model. Broken lines denote the limiting cases for ${\lambda }_{DH}\ll R$ and ${\lambda }_{DH}\gg R$. (c) The effective charge per base fitted from the capacitor model decreases with increasing length. The number of bases is used as a measure of molecule length; thus, only half of the bases of the double stranded species is counted. It matches the effective charge known from electrophoresis shown as a solid line [32]. (d) Thermophoresis measurements using divalent salt ions equally follow the same capacitor model.

Figure 3

Seebeck contribution and dependence on concentration and temperature. (a) The Seebeck contribution $S_T^{EL}$ is extracted from salt-species-dependent measurements (Supplemental Material S5 [26]) by extrapolating to ${\lambda }_{DH}=0$, subtracting $1/T$, and removing the nonionic, molecule-specific contribution $S_T^{NI}$ according to Eq. (1). The theoretical Seebeck contribution [Eq. (3)] matches the experimental $S_T^{EL}$ for positively charged rhodamine 6G and negatively charged 2mer, 22mer, and 80mer ssDNA, with small deviations of lithium salts for 22mer and 80mer. (b) The DNA concentration dependence of thermophoresis matches the prediction based on the Seebeck effect. (c) After subtracting $S_T^{CM}$, $S_T^{EL}$, and $1/T$ from the measurements, the remaining nonionic contribution $S_T^{NI}$ matches the empirical Eq. (4) proposed by Piazza [35]. Its magnitude $S_T^{\infty }$ scales linearly with DNA length (inset). (d) Ionic thermophoresis decreases with temperature [Eq. (2)] but increases with the nonionic contribution [Eq. (4)]. Their combination directly explains the nontrivial maximum of thermophoresis at intermediate temperatures.