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.