^{1,2}, Ali Naji

^{3,a)}, David S. Dean

^{4}, Ron R. Horgan

^{3}and Rudolf Podgornik

^{2,5}

### Abstract

We investigate the effect of monopolar charge disorder on the classical fluctuation-induced interactions between randomly charged net-neutral dielectric slabs and discuss various generalizations of recent results [A. Naji *et al.*, Phys. Rev. Lett.104, 060601 (2010)] to highly inhomogeneous dielectric systems with and without statistical disorder correlations. We shall focus on the specific case of two generally dissimilar plane-parallel slabs, which interact across vacuum or an arbitrary intervening dielectric medium. Monopolar charge disorder is considered to be present on the bounding surfaces and/or in the bulk of the slabs, may be in general quenched or annealed and may possess a finite lateral correlation length reflecting possible “patchiness” of the random charge distribution. In the case of quenched disorder, the bulk disorder is shown to give rise to an additive long-range contribution to the total force, which decays as the inverse distance between the slabs and may be attractive or repulsive depending on the dielectric constants of the slabs. By contrast, the force induced by annealed disorder in general combines with the underlying van der Waals forces in a nonadditive fashion, and the net force decays as an inverse cube law at large separations. We show, however, that in the case of two dissimilar slabs, the net effect due to the interplay between the disorder-induced and the pure van der Waals interactions can lead to a variety of unusual nonmonotonic interaction profiles between the dielectric slabs. In particular, when the intervening medium has a larger dielectric constant than the two slabs, we find that the net interaction can become repulsive and exhibit a potential barrier, while the underlying van der Waals force is attractive. On the contrary, when the intervening medium has a dielectric constant between that of the two slabs, the net interaction can become attractive and exhibit a free energy minimum, while the pure van der Waals force is repulsive. Therefore, the charge disorder, if present, can drastically alter the effective interaction between net-neutral objects.

D.S.D. acknowledges the support from the Institut Universitaire de France. R.P. acknowledges support from ARRS through the program P1-0055 and the Research Project J1-0908. A.N. is supported by a Newton International Fellowship from the Royal Society, the Royal Academy of Engineering, and the British Academy. J.S. acknowledges generous support by J. Stefan Institute (Ljubljana) provided for a visit to the Institute.

I. INTRODUCTION

II. MODEL

III. FORMALISM

A. Correlated quenched disorder

B. Correlated annealed disorder

IV. SYMMETRIC CASE OF TWO IDENTICAL SLABS

A. Quenched disorder-induced interactions

B. Annealed disorder-induced interactions

V. ASYMMETRIC CASE OF TWO DISSIMILAR SLABS

A. Interaction between a disordered and a disorder free slab

B. Nonmonotonic interaction between dissimilar slabs with

C. Nonmonotonic interaction between dissimilar slabs with

VI. DISCUSSION

### Key Topics

- Annealing
- 67.0
- Dielectrics
- 40.0
- Van der Waals forces
- 38.0
- Surface charge
- 28.0
- Dielectric constant
- 27.0

## Figures

We consider two semi-infinite net-neutral slabs (half-spaces) of dielectric constant and interacting across a medium of dielectric constant . The monopolar charge disorder (shown schematically by small light and dark patches) is distributed as random patches of finite typical size (correlation length) in a layered structure in the bulk of the slabs and on the two bounding surfaces at . It may be either quenched or annealed.

We consider two semi-infinite net-neutral slabs (half-spaces) of dielectric constant and interacting across a medium of dielectric constant . The monopolar charge disorder (shown schematically by small light and dark patches) is distributed as random patches of finite typical size (correlation length) in a layered structure in the bulk of the slabs and on the two bounding surfaces at . It may be either quenched or annealed.

(a) Magnitude of the rescaled total force, [Eq. (25)], between two identical net-neutral dielectric slabs in *vacuum* bearing *quenched* monopolar charge disorder as a function of the rescaled distance, . The results are plotted here for fixed , (no surface disorder) and bulk disorder variance , and varying disorder correlation lengths , 200, , , and (from top to bottom). Inset shows the ratio of the total force to the pure zero-frequency vdW force (34) between the slabs in the absence of charge disorder for the same range of . (b) Same as (a) but here we fix , , and and vary the bulk disorder variance in the range , , , , and (from top to bottom). Solid curve is the pure vdW force (34). (c) Same as (b) but here we fix , , and and vary the dielectric constant of slabs as (top dotted curve), 10, 40, and 100 (bottom dot-dashed curve). All graphs are plotted in log-log scale.

(a) Magnitude of the rescaled total force, [Eq. (25)], between two identical net-neutral dielectric slabs in *vacuum* bearing *quenched* monopolar charge disorder as a function of the rescaled distance, . The results are plotted here for fixed , (no surface disorder) and bulk disorder variance , and varying disorder correlation lengths , 200, , , and (from top to bottom). Inset shows the ratio of the total force to the pure zero-frequency vdW force (34) between the slabs in the absence of charge disorder for the same range of . (b) Same as (a) but here we fix , , and and vary the bulk disorder variance in the range , , , , and (from top to bottom). Solid curve is the pure vdW force (34). (c) Same as (b) but here we fix , , and and vary the dielectric constant of slabs as (top dotted curve), 10, 40, and 100 (bottom dot-dashed curve). All graphs are plotted in log-log scale.

(a) Ratio of the total force (30) to the zero-frequency vdW force (34) between two identical net-neutral dielectric slabs in vacuum bearing *annealed* monopolar charge disorder as a function of the rescaled distance, . The results are plotted here for fixed , (no surface disorder), , and varying disorder correlation length , 200, , and (from top to bottom). Annealed curves are bounded by the perfect conductor result, Eq. (37) (top solid line), for large disorder and the vdW result, Eq. (34) (bottom solid line), for no disorder. (b) Same as (a) but here we have fixed , , , and bulk disorder variance varied in the range , , , , and (from top to bottom). (c) Same as (a) but here we fix , , and and vary the dielectric constant of the slabs as , 10, 40, and 100 (from top). Inset shows a closer view of the curves for and 100 (from top). Top dotted lines correspond to Eq. (37). All graphs are plotted in log-log scale.

(a) Ratio of the total force (30) to the zero-frequency vdW force (34) between two identical net-neutral dielectric slabs in vacuum bearing *annealed* monopolar charge disorder as a function of the rescaled distance, . The results are plotted here for fixed , (no surface disorder), , and varying disorder correlation length , 200, , and (from top to bottom). Annealed curves are bounded by the perfect conductor result, Eq. (37) (top solid line), for large disorder and the vdW result, Eq. (34) (bottom solid line), for no disorder. (b) Same as (a) but here we have fixed , , , and bulk disorder variance varied in the range , , , , and (from top to bottom). (c) Same as (a) but here we fix , , and and vary the dielectric constant of the slabs as , 10, 40, and 100 (from top). Inset shows a closer view of the curves for and 100 (from top). Top dotted lines correspond to Eq. (37). All graphs are plotted in log-log scale.

(a) Same as Fig. 3(a) but here we fix , , , and and vary the dielectric constant of the intervening medium as , 10, 20, and 40 (from bottom to top). (b) Same as (a) but here we fix the dielectric constant of the slab as and vary that of the intervening medium as , 40, 60, and 80 (from top to bottom). Top dotted lines correspond to Eq. (37).

(a) Same as Fig. 3(a) but here we fix , , , and and vary the dielectric constant of the intervening medium as , 10, 20, and 40 (from bottom to top). (b) Same as (a) but here we fix the dielectric constant of the slab as and vary that of the intervening medium as , 40, 60, and 80 (from top to bottom). Top dotted lines correspond to Eq. (37).

The rescaled magnitude of the (attractive) force, Eq. (30), between two identical net-neutral dielectric slabs bearing annealed monopolar charge disorder in a *dielectrically homogeneous* system as a function of the rescaled distance, . Here we plot the resulting force for , 10, 40, and 100 (from top to bottom) and for uncorrelated bulk disorder in both slabs with , , and . Top solid line shows the universal limiting expression (37).

The rescaled magnitude of the (attractive) force, Eq. (30), between two identical net-neutral dielectric slabs bearing annealed monopolar charge disorder in a *dielectrically homogeneous* system as a function of the rescaled distance, . Here we plot the resulting force for , 10, 40, and 100 (from top to bottom) and for uncorrelated bulk disorder in both slabs with , , and . Top solid line shows the universal limiting expression (37).

Same as Fig. 3(a) but for two dissimilar dielectric slabs in vacuum with one slab being disorder free and the other slab containing annealed charge disorder of variance , . Here, we fix and vary the disorder correlation length as , 200, , and (from top to bottom). The results in this case are bounded by the limiting values given by Eqs. (34) and (42) [solid lines labeled by and , respectively]; see Eq. (41). The top solid line [labeled by ] is from Eq. (37).

Same as Fig. 3(a) but for two dissimilar dielectric slabs in vacuum with one slab being disorder free and the other slab containing annealed charge disorder of variance , . Here, we fix and vary the disorder correlation length as , 200, , and (from top to bottom). The results in this case are bounded by the limiting values given by Eqs. (34) and (42) [solid lines labeled by and , respectively]; see Eq. (41). The top solid line [labeled by ] is from Eq. (37).

(a) The rescaled total force, [Eq. (25)], between two dissimilar net-neutral dielectric slabs interacting across a medium of dielectric constant varying in the range , 15, 25, and 40 (dashed curves from bottom). Here, we have fixed the dielectric constant of the slabs as and , which contain uncorrelated quenched disorder of fixed disorder variances and . Inset shows a closer view of the region around the minimum. (b) Same as (a) but plotted for correlated quenched disorder. Here, we fix and vary the correlation length as , 20, 100, 500, and , which is taken to be equal in both slabs . Inset again shows a closer view of the region around the minimum. (c) Same as (a) but for (uncorrelated) annealed disorder obtained from Eq. (30) for , 20, and 40. Inset shows the ratio of the total force to the pure zero-frequency vdW force (18) in the absence of charge disorder for a wider range of separations. The horizontal dotted lines show the limiting expression (37).

(a) The rescaled total force, [Eq. (25)], between two dissimilar net-neutral dielectric slabs interacting across a medium of dielectric constant varying in the range , 15, 25, and 40 (dashed curves from bottom). Here, we have fixed the dielectric constant of the slabs as and , which contain uncorrelated quenched disorder of fixed disorder variances and . Inset shows a closer view of the region around the minimum. (b) Same as (a) but plotted for correlated quenched disorder. Here, we fix and vary the correlation length as , 20, 100, 500, and , which is taken to be equal in both slabs . Inset again shows a closer view of the region around the minimum. (c) Same as (a) but for (uncorrelated) annealed disorder obtained from Eq. (30) for , 20, and 40. Inset shows the ratio of the total force to the pure zero-frequency vdW force (18) in the absence of charge disorder for a wider range of separations. The horizontal dotted lines show the limiting expression (37).

(a) The rescaled total force, [Eq. (25)], between two dissimilar net-neutral dielectric slabs interacting across a medium of dielectric constant varying in the range , 40, 60, and 100 (dashed curves from bottom). Here, we have fixed the dielectric constant of the slabs as and , which contain uncorrelated quenched disorder of fixed disorder variances and . Inset shows a closer view of the region around the maximum. (b) Same as (a) but for annealed disorder obtained from Eq. (30). Inset shows the ratio of the total force to the vdW force (18) for a wider range of separations; here the horizontal dotted lines correspond to Eq. (37). (c) Same as (a) but for one slab being disorder free and the other slab containing annealed charge disorder of variances and (dashed curves from top correspond to , 40, 60, and 100). Inset shows the ratio for a wider range of separations; the horizontal dotted lines correspond to Eq. (42) where should be replaced by

(a) The rescaled total force, [Eq. (25)], between two dissimilar net-neutral dielectric slabs interacting across a medium of dielectric constant varying in the range , 40, 60, and 100 (dashed curves from bottom). Here, we have fixed the dielectric constant of the slabs as and , which contain uncorrelated quenched disorder of fixed disorder variances and . Inset shows a closer view of the region around the maximum. (b) Same as (a) but for annealed disorder obtained from Eq. (30). Inset shows the ratio of the total force to the vdW force (18) for a wider range of separations; here the horizontal dotted lines correspond to Eq. (37). (c) Same as (a) but for one slab being disorder free and the other slab containing annealed charge disorder of variances and (dashed curves from top correspond to , 40, 60, and 100). Inset shows the ratio for a wider range of separations; the horizontal dotted lines correspond to Eq. (42) where should be replaced by

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