One-dimensional harmonic chain model of vibration-mode matching in solid-liquid interfacial thermal transport

Understanding the atomistic mechanism of interfacial thermal transport at solid-liquid interfaces is a key challenge in thermal management at the nanoscale. A recent molecular-dynamics study demonstrated that interfacial thermal resistance (ITR) at the interface between a solid and a surfactant solution can be minimized by adjusting the molecular mass of the surfactant. In the present study, we explain the mechanism of this ITR minimization in view of vibration-mode matching using a one-dimensional (1D) harmonic chain model of a solid-liquid interface having an interfacial adsorption layer of surfactant molecules. The equation of motion for the 1D chain is described by a classical Langevin equation and is analytically solved by the nonequilibrium Green's function (NEGF) method. The resultant ITR is expressed in a form of vibrational matching, and its relationship to the overlap of the vibrational density of states is also discussed. The analysis leads to a conclusion that the damping coefficient η in the Langevin equation should be a finite and sufficiently large value to represent the rapid damping of vibration modes at solid-liquid interfaces. This conclusion provides a clue to seamlessly extend the conventional NEGF-phonon transmission picture of solid-solid interfacial thermal transport, which assumes η to be infinitesimal, to solid-liquid interfaces.

Figure 1

Interface between the solid and surfactant solution in the reference system of Ref. [19] (top), and its one-dimensional harmonic chain (bottom) model, consisting of $n_{\mathrm{h}}$ h-particles, one a-particle, and $n_{\mathrm{c}}$ c-particles.

Figure 2

Interfacial thermal resistance of the 1D chain model is plotted as a function of the scaled a-particle mass ${\chi }_{\mathrm{a}}=m_{\mathrm{a}}/m_{\mathrm{c}}$. Parts (a), (b), and (c) show the results for the systems with different chain lengths $n_{\mathrm{tot}}=3$, 5, and 101, respectively. The red lines are the analytic solutions by the NEGF theory, while MD results are shown by the black circles with error bars.

Figure 3

Chain length dependence of the location of the ITR minimum calculated by the NEGF theory. The left axis is the scaled a-particle mass that gives the minimum ITR. The bottom axis is the chain length in terms of $n_{\mathrm{tot}}=n_{\mathrm{h}}+n_{\mathrm{a}}+n_{\mathrm{c}}$ under the conditions $n_{\mathrm{a}}=1$ and $n_{\mathrm{h}}=n_{\mathrm{c}}$.

Figure 4

VDOS profiles normalized by the number of particles. Parts (a), (b), and (c) are the results for the h-, a-, and c-particles, respectively, in the case of $n_{\mathrm{tot}}=3$ and ${\chi }_{\mathrm{as}}=0.2$, while (d), (e), and (f) are those for $n_{\mathrm{tot}}=101$ and ${\chi }_{\mathrm{a}}=0.3$. The red lines show the analytic solution by the NEGF theory, while the black circles show the MD results.

Figure 5

Overlap between the partial transmission functions ${\alpha }_{\mathrm{hc}}$ (black dotted lines) and ${\alpha }_{\mathrm{a}}$ (red solid lines) for the case of $n_{\mathrm{tot}}=3$. Parts (a), (b), and (c) compare the curves for different values of the scaled a-particle mass ${\chi }_{\mathrm{a}}$. The value of the interfacial thermal resistance for each case is also included as R.

Figure 6

Contour of the minimum location of ITR-${\chi }_{\mathrm{a}}$ curve, ${\chi }_{\mathrm{a},min}$, for the chain with $n_{\mathrm{tot}}=101$. A calculation point is denoted by a filled circle or a cross depending on whether the ITR-${\chi }_{\mathrm{a}}$ curve is downward convex with a single minimum or not, respectively. The calculation points that well reproduce the results of the reference system with ${\chi }_{\mathrm{a},min}=0.2–0.3$ are emphasized by red.