Effects of dielectric disorder on van der Waals interactions in slab geometries

We analyze the effects of disorder on the thermal Casimir interaction for the case of two semi-infinite planar slabs across an intervening homogeneous unstructured dielectric. The semi-infinite bounding layers are assumed to be composed of plane-parallel layers of random dielectric materials. We show that the effective thermal Casimir interaction at long distances is self-averaging and can be written in the same form as the one between nonrandom media but with the effective dielectric tensor of the corresponding random media. On the contrary, the behavior at short distances becomes random, and thus sample dependent, dominated by the local values of the dielectric constants proximal to each other across the central homogeneous unstructured dielectric layer. We extend these results to the regime of intermediate slab separations by using perturbation theory for weak disorder as well as by extensive numerical simulations for a number of systems where the dielectric function has a log-normal distribution. Numerical simulation completely corroborates all the main features of the disorder dependent thermal Casimir interaction deduced analytically.

Figure 1

(Color online) A schematic presentation of the model. Two finite slabs with disordered plane-parallel dielectric layers interacting across a dielectrically homogeneous slab of thickness $\ell$. $z$ axis is perpendicular to the plane of the slabs.

Figure 2

Numerical and theoretical plots of the distribution $p\left(A,k,y\right)$ for the telegraph model defined in Sec. 4 for ${ϵ}_A=5.0$, ${ϵ}_B=10.0$, $\omega =1.0$, and for $k=0.1,0.5,1.0$. The agreement is good and the minor discrepancy for $k=1.0$ is a symptom of the discrete nature of the numerical procedure. The value $k=0.5$ corresponds to the transition where $p\left(A,k,y\right)$ becomes nonzero at $y={ϵ}_A$. In general, for $k<1/2\omega$ we have $p\left(A,k,y={ϵ}_A\right)=0$ and for $k\ge 1/2\omega$ $p\left(A,k,y={ϵ}_A\right)>0$. This behavior has a strong effect on the Hamaker coefficient.

Figure 3

Numerical and theoretical plots of the distribution $p\left(B,k,y\right)$ for the telegraph model defined in Sec. 4 for ${ϵ}_A=5.0$, ${ϵ}_B=10.0$, $\omega =1.0$, and for $k=0.1,0.5,1.0$. The agreement is good and the minor discrepancy for $k=1.0$ is a symptom of the discrete nature of the numerical procedure. The value $k=0.5$ corresponds to the transition where $p\left(B,k,y\right)$ becomes nonzero at $y={ϵ}_B$. In general, for $k<1/2\omega$ we have $p\left(B,k,y={ϵ}_B\right)=0$ and for $k\ge 1/2\omega$ $p\left(B,k,y={ϵ}_B\right)>0$. This behavior has a strong effect on the Hamaker coefficient.

Figure 4

The effective Hamaker coefficient $H\left(l\right)$ defined in Eq. (93) for the telegraph model defined in Sec. 4 for ${ϵ}_A=1.0,{ϵ}_B=10.0,\omega =1.0$. The distribution function is $p\left(k,y\right)=p\left(A,k,y\right)+p\left(B,k,y\right)$ where $p\left(A/B,k,y\right)$ are given in Eqs. (81, 82). Note that the curve is strongly dependent on separation $l$ for $l\sim 1.0$ which corresponds to the transition in the behavior of $p\left(k,y\right)$ at $k=0.5\equiv 1/2\omega$. We would expect in the general case that this strong dependence occurs for $l\le 1/2\omega$. The error bars on the points are within the symbol size. The asymptotic values for $l\to 0$ and $l\to \infty$ are also shown. In this case they are $H\left(l\right)/k_{B}T\to 3.688{10}^{-3}l\to 0,H\left(l\right)/k_{B}T\to 5.666{10}^{-3}l\to \infty$.

Figure 5

Probability density for $\alpha$ for $k=1,0.1,0.01$ for $ϵ\left(z\right)=\text{exp}\left(\psi \left(z\right)\right)$ where $\psi$ is the OU Gaussian process with correlation function given by Eq. (69). The distribution $p\left(y,k\right)$ becomes sharply peaked around $y=1$ which is as predicted for small $k$, ${ϵ}^{\ast }$ given by Eq. (31)

Figure 6

The filled circles are the probability density for $\alpha$ for $k=10$ for $ϵ\left(z\right)=\text{exp}\left[\psi \left(z\right)\right]$ here $\psi$ is the OU Gaussian process with correlation function given by Eq. (69) with $\omega =1$. The solid line is the probability density function $q\left(y\right)$ given in Eq. (90) for the same distribution of $ϵ$. The two distributions are already very close for $k=10$ in agreement with the prediction of Eq. (52).

Figure 7

Probability density for $\alpha$ for $k=1,0.1,0.01$ for $ϵ\left(z\right)=\text{exp}\left[\psi \left(z\right)\right]$ where $\psi$ is the Gaussian process with correlation function given by Eq. (91). As for the OU process the distribution $p\left(y,k\right)$ becomes sharply peaked around $y=1$ which is as predicted for small $k$, and ${ϵ}^{\ast }$ is given by Eq. (31)

Figure 8

The filled circles are the probability density for $\alpha$ for $k=100$ for $ϵ\left(z\right)=\text{exp}\left[\psi \left(z\right)\right]$ here $\psi$ is the Gaussian process with correlation function given by Eq. (91) with $\omega =1$. The solid line is the probability density function $q\left(y\right)$ given in Eq. (90) for the same distribution of $ϵ$. The two distributions are already very close for $k=10$ in agreement with the prediction of Eq. (52).

Figure 9

The effective Hamaker coefficient $H\left(l\right)$ defined in Eq. (93) for the distribution defined by Eqs. (55, 91) for ${ϵ}_g=1.0,1.414$. In both cases the curves are asymptotic to the Hamaker constant for a homogeneous system with dielectric constant ${ϵ}^{\ast }=\sqrt{⟨ϵ⟩/⟨1/ϵ⟩}$. Notably, ${ϵ}^{\ast }=1$ for ${ϵ}_g=1.0$ and we see that $H\left(l\right)$ is asymptotic to zero from above showing that the force is attractive for all $l$ even in this case. As is also seen, the curves are asymptotic either above or below depending on the value of ${ϵ}_g$ and is not necessarily monotonic.

Figure 10

The effective Hamaker coefficient $⟨H\left(l\right)⟩$ defined in Eq. (93) for the distribution defined by Eqs. (55, 91) for ${ϵ}_g=2.0$, 10.0. In both cases the curves are asymptotic to the Hamaker constant for a homogeneous system with dielectric constant ${ϵ}^{\ast }=\sqrt{⟨ϵ⟩/⟨1/ϵ⟩}$.

Figure 11

Examples of the Hamaker coefficient from the ensemble generated by the distribution for $ϵ\left(z\right)$ for ${ϵ}_g=2.0$ plotted versus separation $l$. The average curve of the ensemble is shown in Fig. 10. Note that for small $l$ the curvature of the ensemble curves are not of definite sign

Figure 12

Examples of the Hamaker coefficient from the ensemble generated by the distribution for $ϵ\left(z\right)$ for ${ϵ}_g=10.0$ plotted versus separation $l$. The average curve of the ensemble is shown in Fig. 10. Note that the curvatures of the ensemble curves are not of definite sign.