Phonon thermophoresis of crystalline nanoparticles in liquids.

Our nonequilibrium thermodynamic model of thermodiffusion in molecular liquid systems is used to examine the role of thermal phonons in the thermophoresis of liquid suspensions of crystalline nanoparticles, which tend to have high thermal conductivity. The Soret coefficient used to characterize stationary thermodiffusion is related to differences in entropy between a particle and the body of liquid that it displaces. Calculated phonon Soret coefficients for graphite and diamond nanoparticles in three polar solvents are used to establish parameters where the phonon mechanism is expected to dominate particle thermophoresis compared to slip-flow caused by forces induced in the surface layer by the temperature gradient. Because the active mode of thermal conductivity in crystals varies with particle size, phonon thermophoresis is expected to dominate within a specific range of particle size, which varies with the properties of the particle and suspending liquid. For graphite and diamond particles in polar solvents the model estimates a size range of around 10-100 nm. Finally, thermophoretic particle accumulation is ultimately limited by the increasing concentration of particles having high thermal conductivity because the zone of particle concentration decreases the local temperature gradient that drives thermophoresis. The respective nonperturbing concentration is evaluated as the function of the size of a given material.

Nonequilibrium thermodynamic model of thermoelectricity and thermodiffusion in semiconductors.

We present a self-consistent model rooted in nonequilibrium thermodynamics for defining concentration gradients in the electron/hole pairs and electric-field gradients in an intrinsic semiconductor created upon exposure to a temperature gradient. The model relies on the equation for entropy production expressed through phenomenological equations for the electron/hole flux, with the imposed condition that the resulting concentration profiles of the electrons and holes are identical. The chemical potentials of electrons, holes, and parent atoms of the lattice, which are contained in the flux equations, are calculated on the basis of the temperature-dependent equilibrium dissociation reaction: lattice atom ↔ electron + hole. Electron/hole concentration profiles resulting from the temperature gradient, along with the associated gradient in the electric field, are expressed through equilibrium microscopic parameters of the semiconductor, which include the effective masses of electrons and holes, the energy gap width, and the Debye temperature. The resulting expressions contain neither kinetic nor fitting parameters, and predict values in reasonable (order-of-magnitude) agreement with empirical data. Finally, the model predicts a measurable additional thermodiffusion-based Seebeck effect when the temperature difference is on the order of several tens of degrees across a nonisothermal semiconductor working as a power supply under conditions of optimal power transport.

Nonequilibrium thermodynamic material transport equations for thermodiffusion of isotopes and isomers in multicomponent systems.

We extend our nonequilibrium thermodynamic model of thermodiffusion in binary systems to multi-component mixtures. The fundamental parameter is the difference in molecular entropy of the components, which can be obtained in one of three ways; (i) derived as temperature derivatives of the respective equilibrium chemical potentials at constant pressure using equilibrium statistical mechanics; (ii) obtained in the literature from computer simulations; or (iii) obtained as empirical values in the literature. The model is used to relate thermodiffusion in multicomponent mixtures of related isomers or isotopes to isomer/isotope effects in binary mixtures that are commonly enumerated in one of two ways: (i) as a difference in the Soret coefficients measured on two binary mixtures, each containing one of two related isomers/isotope in a common solvent; or (ii) as this difference from two binary mixtures, each consisting of a common solute dissolved in one of the two related isomers/isotopes as the solvent. The model is used to estimate the concentration profiles established in neat multicomponent mixtures of hexane isomers, and for the prediction and optimization of separating isotopes of cyclohexane in various organic solvents by thermodifusion.

Statistical Mechanic and Phenomenological Approaches to Isomeric Effects in Thermodiffusion.

We present a model that explains variance in the thermodiffusion of hydrocarbon isomers in binary liquid mixtures. The model relies on material transport equations for binary nonisothermal liquid systems that were derived through a nonequilibrium thermodynamic approach in a previous work, coupled with one of two methods: (i) use of equilibrium chemical potentials for each component under conditions of constant pressure, derived using statistical mechanics or (ii) use of the temperature derivative of chemical potential expressed phenomenologically as molecular entropy. The model is evaluated using Soret coefficients (ST) measured in binary solutions of heptane isomers in benzene. The statistical mechanic approach yields moderately acceptable agreement with experimental data. The phenomenological approach, which relies on both measured and calculated values of molecular entropy from the literature, yields values of ST centered around the experimental data, with the scatter likely due to poor precision in the measured or calculated values of entropy. For the latter case, we identify several methods for calculating entropy that yield good agreement with experimental data.

Internal Degrees of Freedom as a Basis for Isotopic Effects in Thermodiffusion.

We present a model that relates isotope effects in thermodiffusion to changes in internal degrees of freedom associated with rotational and vibrational motion. The model uses general material transport equations for binary non-isothermal liquid systems, derived using non-equilibrium thermodynamics in our previous work. The equilibrium chemical potentials of the components at constant pressure are derived using statistical mechanics. In evaluating the model, we use experimental data on changes in the Soret coefficient of various hydrocarbons in perprotonated and perdeuterated cyclohexane. We also compare predictions of the model with experimental data on the Soret coefficient in isotopic mixtures. In all cases, the model is consistent with experimental data and computations.

Thermophoretic Random Walks and Enhancement of Diffusion.

The contribution of the stochastic thermodiffusion to the diffusion enhancement is studied. The thermodiffusion of particles suspended in a liquid may hold place when the spontaneous endo- or exothermal nanoscale events similar to elementary acts of enzymatic reactions occur as the random series in the space and time. In these events, the energy can be emitted or absorbed at nanoscale during few to hundreds of picoseconds. It may cause local spontaneous temperature spikes spreading quickly in the space and decaying with time. The random local temperature spikes create local transient temperature gradients, where thermodiffusion of the molecules and particles holds place as well as the change in the physical properties of the suspending medium due to heating. These thermodiffusion random walks may appear as the enhanced usual Stokes-Einstein diffusion when the energy absorption/generation is high enough. The evaluated relative contribution of the mentioned effect to the molecular mobility is shown to be in agreement with experimental data on enzymatic reactions from the literature.

Coupled Heat and Mass Transport in Liquid Films that Contain Thermophoretically Active Particles.

We consider the spontaneous redistribution of thermophoretically active particles suspended in a thin film of liquid, made to absorb energy and transmit it in the form of heat to the surrounding medium. When the opposing boundaries to the thin dimension are maintained at a constant temperature, a nonuniform temperature profile is formed across the film because of differential heat dissipation, which is maximized at the boundaries. Thermophobic particles move and concentrate at the opposing (cooler) boundaries, whereas thermophilic particles concentrate within a layer midway between the boundaries. The Dufour-like effect results in a synergism between concentration and temperature profiles that enhances the temperature gradient. Increases in particle concentration up to 10-fold can be achieved rapidly for non-Janus particles at room temperature using low energy input to maintain transverse temperature differences of a few degrees Kelvin. This level of concentration is much higher than that predicted for Janus particles, where mass diffusion coefficients are larger and less dependent on temperature. Finally, in contrast to Janus particles, the system is expected to remain stable with both positive and negative thermophoresis.

Thermoosomosis in microfluidic devices containing a temperature gradient normal to the channel walls.

We analyze a microfluidic pump from the literature that utilizes a flat channel with boundary walls at different temperatures and tilted elongated pillars within in order to construct an adequate theory for designing devices in which the temperature gradient between channel walls is transformed into a longitudinal temperature gradient along the channel length. The action of the device is based on thermoosmosis in the secondary longitudinal temperature gradient associated with the specific geometry of the device, which can be described using physicochemical hydrodynamics without invoking the concept of thermophoretic force. We also describe a rotating drive device based on the same principle and design.

Comment on "Soret motion in non-ionic binary molecular mixtures" [J. Chem. Phys. 135, 054102 (2011)].

The material transport equations derived in the article by Leroyer and Würger [J. Chem. Phys. 135, 054102 (2011)] do not adequately provide a description of material transport in liquid binary systems. An alternate approach based on non-equilibrium thermodynamics is presented.

Molecular thermodiffusion (thermophoresis) in liquid mixtures.

Thermodiffusion (thermophoresis) in liquid mixtures is theoretically examined using a hydrodynamic approach. Thermodiffusion is related to the local temperature-induced pressure gradient in the liquid layer surrounding the selected molecule and to the secondary macroscopic pressure gradient established in the system. The local pressure gradient is produced by excess pressure due to the asymmetry of interactions with surrounding molecules in a nonuniform temperature field. The secondary pressure gradient is considered an independent parameter related to the concentration gradient formed by volume forces, calculated from the generalized equations for mass transfer. Values of Soret coefficients for mixtures of toluene and -hexane are calculated using parameters in the literature. When the molecules are assumed to be similar in shape, the calculated Soret coefficients are lower than the empirical values found in the literature. However, by introducing an asymmetry parameter, which is calculated from independent measurements of component diffusion in the literature, very good agreement is obtained.

Symmetric diffusion equations, barodiffusion, and cross-diffusion in concentrated liquid mixtures.

In models of diffusion in multicomponent mixtures, the current practice is to derive equations for an isobaric system. The equations are nonsymmetric in relation to the components of the mixture, and the concentration of solvent is assumed to be governed by the conservation of mass instead of its own corresponding diffusion equation. For concentrated mixtures, the solvent component is selected arbitrarily, which makes interpretation of the experimental data dependent on the choice of the interpreter. In this work, we derive a symmetric system of equations, made possible by the introduction of a spontaneously produced secondary pressure gradient. The effect of that pressure gradient is barodiffusion (barophoresis), defined by the force expressed as the secondary pressure gradient multiplied by the molecular volume. The model also considers the cross-diffusion (diffusiophoresis) that results from the hydrodynamic stresses associated with the local concentration-induced pressure gradient in liquid layers surrounding individual molecules. The resulting system of diffusion equations, which contains the secondary pressure gradient and component concentrations, is applied to a binary (nonionic) mixture of benzene and 1,2-dichloroethane. The steady-state system is placed in a uniform force field, and the effect of the secondary pressure gradient on the field-induced migration is discussed. Fluctuation dynamics in a system with no external force field is also discussed. The numerical results predict the establishment of lower concentration gradients compared to standard theory. Also, the predicted concentration dependence in the effective diffusion coefficient measured by dynamic light scattering is different compared to standard theory. Finally, experiments are proposed to further evaluate differences between the new model and the standard approach.

Correction for particle-wall interactions in the separation of colloids by flow field-flow fractionation.

In the characterization of materials by field-flow fractionation (FFF), the experienced analyst understands the importance of incorporating additives in the carrier liquid that minimize or eliminate interactions between the analyte and accumulation wall, particularly in aqueous systems. However, as FFF is applied to more difficult samples, such as those with high surface energies, it is increasingly difficult to find additives that completely eliminate particle-wall interactions. Furthermore, the analyst may wish to use specific conditions that preserve the high surface energy of particles, to study their interaction with other materials through their behavior in the FFF channel. With this in mind, Williams and co-workers developed a model that quantifies the effect of particle-wall interactions in FFF using an empirically determined interaction parameter. In this work, the model is evaluated for the application of flow FFF in carrier liquids of low ionic strength, where particle-wall interactions are magnified. The retention of particles ranging in size from 64 to 1000 nm is measured using a wide range of field strengths and retention levels. The model is found to be generally valid over the entire range, except for minor discrepancies at lower levels of retention. Although retention levels are dramatically affected by particle-wall interactions, the point of steric inversion (500 nm), where the size-based elution order reverses, is not affected. When particle-wall interactions are not accounted for, they lead to a bias in particle sizes calculated from standard retention theory of up to 70%. The model can also be used to refine the measurement of channel thickness, which is important for the accurate conversion of retention parameters to particle sizes. In this work, for example, errors in channel thickness led to systematic errors on the order of 10% in particle diameter.