As mentioned in the abstract, to find the heat transmitted to the second medium due to the Van der Waals interaction between the surfaces, we take the continuum limit where phonons can be modelled as elastic waves. To relate the expressions of the Van der Waals force found in the abstract to the stress in the medium (taken to be isotropic), we look at the displacements in the medium in the presence of elastic waves. While it is possible to have three polarisation modes – one longitudinal and two transverse (polarised in the horizontal and vertical planes, also referred to as SH and SV waves, respectively), since the transverse mode polarised in the horizontal plane (SH mode) does not result in out-of-plane deformation it does not contribute to phonon transmission, and we only consider the longitudinal and SV modes henceforth. Consider first an incident planar longitudinal wave of unit amplitude in medium 1, as shown in Figure 1b. The equation of the incident wave is of the form ${u}_{0}={a}_{0}{\text{e}}^{-\text{i}{k}_{lx}x}{\text{e}}^{-\text{i}{k}_{y}y}$ where **a**_{0} is a unit vector in the direction of propagation of the incident wave and can thus be expressed in terms of the incident angle *θ*_{l} as **a**_{0}=−cos*θ*_{l}**x**−sin*θ*_{l}**y**, and *k*_{lx} and *k*_{y} are the components of the wave vector perpendicular and parallel to the surface, respectively. The displacement vector of the two reflected waves can be written as ${u}_{I}{}^{\mathrm{(}1\mathrm{)}}={R}_{l}{a}_{I}{}^{\mathrm{(}1\mathrm{)}}{\text{e}}^{+\text{i}{k}_{lx}x}{\text{e}}^{-\text{i}{k}_{y}y}$ and ${u}_{t}{}^{\mathrm{(}1\mathrm{)}}={R}_{t}\mathrm{(}z\times {a}_{t}{}^{\mathrm{(}1\mathrm{)}}\mathrm{)}{\text{e}}^{+\text{i}{k}_{lx}x}{\text{e}}^{-\text{i}{k}_{y}y},$ where *R*_{l} (*R*_{t}) is the reflection coefficient of the longitudinal (transverse) component, and ${a}_{l}{}^{\mathrm{(}1\mathrm{)}}$ $\mathrm{(}{a}_{t}{}^{\mathrm{(}1\mathrm{)}}\mathrm{)}$ is a unit vector in the direction of propagation of the reflected longitudinal (transverse) wave. Thus the components of displacement in medium 1 along the *x* and *y* directions, ${u}_{x}^{\mathrm{(}1\mathrm{)}}$ and ${u}_{y}^{\mathrm{(}1\mathrm{)}},$ are given by

$${u}_{x}^{\mathrm{(}1\mathrm{)}}=\mathrm{[}-\mathrm{cos}{\theta}_{l}{\text{e}}^{-\text{i}{k}_{lx}x}+{R}_{l}\mathrm{cos}{\theta}_{l}{\text{e}}^{\text{i}{k}_{lx}x}+{R}_{t}\mathrm{sin}{\theta}_{t}{\text{e}}^{\text{i}{k}_{tx}x}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}$$(8)

$${u}_{y}^{\mathrm{(}1\mathrm{)}}=\mathrm{[}\mathrm{(}-\mathrm{sin}{\theta}_{l}\mathrm{)}{\text{e}}^{-\text{i}{k}_{lx}x}+{R}_{l}\mathrm{(}-\mathrm{sin}{\theta}_{l}\mathrm{)}{\text{e}}^{\text{i}{k}_{lx}x}+{R}_{t}\mathrm{cos}{\theta}_{t}{\text{e}}^{\text{i}{k}_{tx}x}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}.$$(9)

Since the parallel component of the wave vector has to be conserved across the interface, the angles *θ*_{l} and *θ*_{t} are related by sin*θ*_{l}c_{t}=sin*θ*_{t}c_{l}. The stresses ${\sigma}_{xx}^{\mathrm{(}1\mathrm{)}}$ and ${\sigma}_{xy}^{\mathrm{(}1\mathrm{)}}$ in medium 1 can be obtained from the normal and shear strains, ${u}_{xx}^{\mathrm{(}1\mathrm{)}},$ ${u}_{yy}^{\mathrm{(}1\mathrm{)}},$ and ${u}_{xy}^{\mathrm{(}1\mathrm{)}}$ [16], as

$${\sigma}_{xx}^{\mathrm{(}1\mathrm{)}}=2\rho {c}_{t}^{2}{u}_{xx}^{\mathrm{(}1\mathrm{)}}+\rho \mathrm{(}{c}_{l}^{2}-2{c}_{t}^{2}\mathrm{)}{u}_{ll}^{\mathrm{(}1\mathrm{)}}$$(10)

$${\sigma}_{xy}^{\mathrm{(}1\mathrm{)}}=2\rho {c}_{t}^{2}{u}_{xy}^{\mathrm{(}1\mathrm{)}}$$(11)

where *u*_{ll}=*u*_{xx}+*u*_{yy}, *ρ* is the density, and *c*_{l} and *c*_{t} are the longitudinal and transverse velocities of sound in the isotropic material. This gives the stress at the surface (*x*=0), which should be equal to the Van der Waals stress derived in Section 2, to be

$$\begin{array}{l}{\sigma}_{xx}^{\mathrm{(}1\mathrm{)}}=\mathrm{[}\text{i}\rho {c}_{t}^{2}\mathrm{(}\mathrm{cos}2{\theta}_{l}{k}_{l}+{R}_{l}\mathrm{cos}2{\theta}_{l}{k}_{l}+{R}_{t}\mathrm{sin}2{\theta}_{t}{k}_{t}\mathrm{)}\\ \text{}+\text{i}\rho \mathrm{(}{c}_{l}^{2}-{c}_{t}^{2}\mathrm{)}{k}_{l}\mathrm{(}1+{R}_{l}\mathrm{)}]{\text{e}}^{-\text{i}{k}_{y}y}\end{array}$$(12)

$${\sigma}_{xy}^{\mathrm{(}1\mathrm{)}}=\text{i}\rho {c}_{t}^{2}[\mathrm{(}1-{R}_{l}\mathrm{)}{k}_{l}\mathrm{sin}2{\theta}_{l}+{R}_{t}{k}_{t}\mathrm{cos}2{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}.$$(13)

In the second medium, for ease of analysis we initially assume the material to be the same as in medium 1, before extending to the case of different materials. The angles *ϕ*_{l} and *ϕ*_{t} in Figure 2 are thus equivalent to the angles *θ*_{l} and *θ*_{t}. The displacement vector in the second medium due to the presence of transmitted longitudinal and transverse waves can be written as ${u}^{\mathrm{(}2\mathrm{)}}={T}_{l}{a}_{l}^{\mathrm{(}2\mathrm{)}}{\text{e}}^{-\text{i}{k}_{lx}x}{\text{e}}^{-\text{i}{k}_{ly}y}+{T}_{t}\mathrm{(}z\times {a}_{t}^{\mathrm{(}2\mathrm{)}}\mathrm{)}{\text{e}}^{-\text{i}{k}_{lx}x}{\text{e}}^{-\text{i}{k}_{y}y}$ giving us the components of the displacement vector along the *x* and *y* axes to be

$${u}_{x}^{\mathrm{(}2\mathrm{)}}=\mathrm{[}{T}_{l}\mathrm{(}-\mathrm{cos}{\theta}_{l}\mathrm{)}{\text{e}}^{-\text{i}{k}_{lx}x}+{T}_{t}\mathrm{sin}{\theta}_{t}{\text{e}}^{-\text{i}{k}_{tx}x}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}$$(14)

$${u}_{y}^{\mathrm{(}2\mathrm{)}}=\mathrm{[}{T}_{l}\mathrm{(}-\mathrm{sin}{\theta}_{l}\mathrm{)}{\text{e}}^{-\text{i}{k}_{lx}x}+{T}_{t}\mathrm{(}-\mathrm{cos}{\theta}_{t}^{\mathrm{(}2\mathrm{)}}\mathrm{)}{\text{e}}^{-\text{i}{k}_{tx}x}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}.$$(15)

From (6), (8), and (14) we get the normal force acting on the surface (*x*=0) of medium 1, *F*_{1}, to be

$$\begin{array}{l}{F}_{1}=-\frac{\text{H}}{4\pi}{K}_{2}\mathrm{(}{k}_{\left|\right|}d\mathrm{)}\frac{{k}_{\left|\right|}^{2}}{{d}^{2}}\mathrm{[}-{T}_{l}\mathrm{cos}{\theta}_{l}+{T}_{t}\mathrm{sin}{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}\\ \text{}+\frac{\text{H}}{2\pi {d}^{4}}\mathrm{[}-\mathrm{cos}{\theta}_{l}+{R}_{l}\mathrm{cos}{\theta}_{l}+{R}_{t}\mathrm{sin}{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}\end{array}$$(16)

and the expression for the normal force acting on the surface (*x*=0) of medium 2, *F*_{2}, is

$$\begin{array}{l}{F}_{2}=-\frac{\text{H}}{4\pi}{K}_{2}\mathrm{(}{k}_{\left|\right|}d\mathrm{)}\frac{{k}_{\left|\right|}^{2}}{{d}^{2}}\mathrm{[}-\mathrm{cos}{\theta}_{l}+{R}_{l}\mathrm{cos}{\theta}_{l}+{R}_{t}\mathrm{sin}{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}\\ \text{}+\frac{\text{H}}{2\pi {d}^{4}}\mathrm{[}-{T}_{l}\mathrm{cos}{\theta}_{l}+{T}_{t}\mathrm{sin}{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}.\end{array}$$(17)

From (14) and (15) expressions for the compressive and shear stress at the surface of the second medium can be arrived as

$${\sigma}_{xx}^{\mathrm{(}2\mathrm{)}}=\mathrm{[}\rho {c}_{t}^{2}{T}_{l}\text{i}{k}_{l}\mathrm{cos}2{\theta}_{l}-\text{i}\rho {c}_{t}^{2}{k}_{t}{R}_{t}\mathrm{sin}2{\theta}_{t}+\rho \mathrm{(}{c}_{l}^{2}-{c}_{t}^{2}\mathrm{)}{T}_{l}\text{i}{k}_{l}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}$$(18)

$${\sigma}_{xy}^{\mathrm{(}2\mathrm{)}}=\rho {c}_{t}^{2}\mathrm{[}{T}_{l}\text{i}{k}_{l}\mathrm{sin}2{\theta}_{l}+{T}_{t}\text{i}{k}_{t}\mathrm{cos}2{\theta}_{t}\mathrm{]}{\text{e}}^{-\text{i}{k}_{y}y}.$$(19)

Equating the stresses at the surface to the Van der Waals force gives us the four boundary conditions

$${\sigma}_{xy}^{\mathrm{(}1\mathrm{)}}=0$$(20)

$${\sigma}_{xx}^{\mathrm{(}1\mathrm{)}}={F}_{1}$$(21)

$${\sigma}_{xy}^{\mathrm{(}2\mathrm{)}}=0$$(22)

$${\sigma}_{xx}^{\mathrm{(}2\mathrm{)}}={F}_{2}$$(23)

where the expressions for stresses ${\sigma}_{xx}^{\mathrm{(}1\mathrm{)}},$ ${\sigma}_{xy}^{\mathrm{(}1\mathrm{)}},$ ${\sigma}_{xx}^{\mathrm{(}2\mathrm{)}},$ ${\sigma}_{xy}^{\mathrm{(}2\mathrm{)}},$ and the forces *F*_{1} and *F*_{2} in terms of the reflection and transmission coefficients are substituted from (12), (13), (18), (19), (16), and (17), respectively. The four equations can be solved analytically to obtain the transmission coefficients. Analytical expressions for the transmission coefficients are

$${T}_{l}^{\mathrm{(}l\mathrm{)}}=\frac{-2{Q}_{1}{Q}_{4}}{\mathrm{(}{Q}_{1}+{Q}_{2}+{Q}_{3}\mathrm{)}\mathrm{(}{Q}_{1}+{Q}_{2}-{Q}_{3}\mathrm{)}+{Q}_{4}^{2}\mathrm{)}}\mathrm{;}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{T}_{t}^{\mathrm{(}l\mathrm{)}}=-{T}_{l}^{\mathrm{(}l\mathrm{)}}\frac{{k}_{l}\mathrm{sin}2{\theta}_{l}}{{k}_{t}\mathrm{cos}2{\theta}_{t}}$$(24)

where the superscript has been included to indicate the polarisation of the incident wave (longitudinal, in this case). A similar analysis for an incident transverse wave gives the transmission coefficients:

$${T}_{l}^{\mathrm{(}t\mathrm{)}}=\frac{\mathrm{(}{Q}_{5}-{Q}_{2}\mathrm{)}{Q}_{4}}{\mathrm{(}{Q}_{1}+{Q}_{2}+{Q}_{3}\mathrm{)}\mathrm{(}{Q}_{1}-{Q}_{3}\mathrm{)}+{Q}_{4}^{2}}\mathrm{;}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{T}_{t}^{\mathrm{(}t\mathrm{)}}=-{T}_{l}^{\mathrm{(}t\mathrm{)}}\frac{{k}_{l}\mathrm{sin}2{\theta}_{l}}{{k}_{t}\mathrm{cos}2{\theta}_{t}}$$(25)

where

$${Q}_{1}=\text{i}\rho {k}_{l}{c}_{t}^{2}\mathrm{cos}2{\theta}_{l}+\text{i}\rho {k}_{l}\mathrm{(}{c}_{l}^{2}-{c}_{t}^{2}\mathrm{)}$$(26)

$${Q}_{2}=\text{i}\rho {k}_{l}{c}_{t}^{2}\frac{\mathrm{sin}2{\theta}_{t}\mathrm{sin}2{\theta}_{l}}{\mathrm{cos}2{\theta}_{t}}$$(27)

$${Q}_{3}=-\frac{\text{H}}{4\pi}{K}_{2}\mathrm{(}{k}_{l}\mathrm{sin}{\theta}_{l}d\mathrm{)}\frac{{k}_{l}^{2}{\mathrm{sin}}^{2}{\theta}_{l}}{{d}^{2}}\mathrm{(}\mathrm{cos}{\theta}_{l}+\frac{{k}_{l}\mathrm{sin}2{\theta}_{l}\mathrm{sin}{\theta}_{t}}{{k}_{t}\mathrm{cos}2{\theta}_{t}}\mathrm{)}$$(28)

$${Q}_{4}=\frac{\text{H}}{2\pi {d}^{4}}\mathrm{(}\mathrm{cos}{\theta}_{l}+\frac{{k}_{l}\mathrm{sin}2{\theta}_{l}\mathrm{sin}{\theta}_{t}}{{k}_{t}\mathrm{cos}2{\theta}_{t}}\mathrm{)}$$(29)

$${Q}_{5}=2\text{i}{k}_{t}\rho {c}_{t}^{2}\mathrm{sin}2{\theta}_{t}.$$(30)

These expressions are equivalent to that derived in [10]. This analysis can be extended to the case when you have different materials across the vacuum gap. The corresponding transmission coefficients are then given by

$${T}_{l}^{\mathrm{(}l\mathrm{)}}=\frac{-2{Q}_{4}{Q}_{1}}{\mathrm{(}{S}_{1}+{S}_{2}+{S}_{3}\mathrm{)}\mathrm{(}{Q}_{1}+{Q}_{2}-{Q}_{3}\mathrm{)}+{S}_{4}{Q}_{4}}\mathrm{;}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{T}_{t}^{\mathrm{(}l\mathrm{)}}=-{T}_{l}^{\mathrm{(}l\mathrm{)}}\frac{{{c}^{\prime}}_{t}\mathrm{sin}2{\varphi}_{l}}{{{c}^{\prime}}_{l}\mathrm{cos}2{\varphi}_{t}}$$(31)

$${T}_{l}^{\mathrm{(}t\mathrm{)}}=\frac{\mathrm{(}{Q}_{5}-{Q}_{2}\mathrm{)}{Q}_{4}}{\mathrm{(}{S}_{1}+{S}_{2}+{S}_{3}\mathrm{)}\mathrm{(}{Q}_{1}-{Q}_{3}\mathrm{)}+{S}_{4}{Q}_{4}}\mathrm{;}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{T}_{t}^{\mathrm{(}t\mathrm{)}}=-{T}_{l}^{\mathrm{(}t\mathrm{)}}\frac{{{c}^{\prime}}_{t}\mathrm{sin}2{\varphi}_{l}}{{{c}^{\prime}}_{l}\mathrm{cos}2{\varphi}_{t}}$$(32)

where the additional factors *S*_{1}, *S*_{2}, *S*_{3}, and *S*_{4} are given by

$${S}_{1}=\text{i}{\rho}^{\prime}{{k}^{\prime}}_{l}{{c}^{\prime}}_{t}^{2}\mathrm{cos}2{\varphi}_{l}+\text{i}{\rho}^{\prime}{{k}^{\prime}}_{l}\mathrm{(}{{c}^{\prime}}_{l}^{2}-{{c}^{\prime}}_{t}^{2}\mathrm{)}$$(33)

$${S}_{2}=\text{i}{\rho}^{\prime}{{k}^{\prime}}_{l}{{c}^{\prime}}_{t}^{2}\frac{\mathrm{sin}2{\varphi}_{t}\mathrm{sin}2{\varphi}_{l}}{\mathrm{cos}2{\varphi}_{t}}$$(34)

$${S}_{3}=-\frac{{\text{H}}_{\text{eff}}}{4\pi}{K}_{2}\mathrm{(}{{k}^{\prime}}_{l}\mathrm{sin}{\varphi}_{l}d\mathrm{)}\frac{{{k}^{\prime}}_{l}^{2}{\mathrm{sin}}^{2}{\varphi}_{l}}{{d}^{2}}\mathrm{(}\mathrm{cos}{\varphi}_{l}+\frac{{{k}^{\prime}}_{l}\mathrm{sin}2{\varphi}_{l}\mathrm{sin}{\varphi}_{t}}{{{k}^{\prime}}_{t}\mathrm{cos}2{\varphi}_{t}}\mathrm{)}$$(35)

$${S}_{4}=\frac{{\text{H}}_{\text{eff}}}{2\pi {d}^{4}}\mathrm{(}\mathrm{cos}{\varphi}_{l}+\frac{{{k}^{\prime}}_{l}\mathrm{sin}2{\varphi}_{l}\mathrm{sin}{\varphi}_{t}}{{{k}^{\prime}}_{t}\mathrm{cos}2{\varphi}_{t}}\mathrm{)}.$$(36)

Here, ${\rho}^{\prime}\mathrm{,}\text{\hspace{0.17em}}{{c}^{\prime}}_{t},$ and ${{c}^{\prime}}_{l}$ are the density, transverse velocity of sound, and longitudinal velocity of sound in medium 2, respectively. H_{eff} is the effective Hamaker’s constant for the two objects made of different materials, an estimation for which can be obtained from the Lifshitz formula [13]:

$${H}_{\text{eff}}=\frac{3}{2}{k}_{B}T\mathrm{(}\frac{{\epsilon}_{1}-1}{{\epsilon}_{1}+1}\mathrm{)}\mathrm{(}\frac{{\epsilon}_{2}-1}{{\epsilon}_{2}+1}\mathrm{)}+\frac{3h}{4\pi}{\displaystyle {\int}_{0}^{\infty}\mathrm{(}\frac{{\epsilon}_{1}\mathrm{(}\text{i}\nu \mathrm{)}-1}{{\epsilon}_{1}\mathrm{(}\text{i}\nu \mathrm{)}+1}\mathrm{)}\mathrm{(}\frac{{\epsilon}_{2}\mathrm{(}\text{i}\nu \mathrm{)}-1}{{\epsilon}_{2}\mathrm{(}\text{i}\nu \mathrm{)}+1}\mathrm{)}\text{d}\nu}$$(37)

where *ε*_{1} and *ε*_{2} are the static dielectric constants of medium 1 and medium 2 and *ε*(i*ν*) denotes the values of the dielectric functions at imaginary frequencies.

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