Sample-to-sample fluctuations of electrostatic forces generated by quenched charge disorder

It has been recently shown that randomly charged surfaces can exhibit-long range electrostatic interactions even when they are net neutral. These forces depend on the specific realization of charge disorder and thus exhibit sample-to-sample fluctuations about their mean value. We analyze the fluctuations of these forces in the parallel slab configuration and in the sphere-plane geometry via the proximity force approximation. The fluctuations of the normal forces, which have a finite mean value, are computed exactly. Surprisingly, we also show that fluctuations in lateral forces are present, despite the fact that they have a zero mean, and that they have the same scaling behavior as the normal force fluctuations. The measurement of these lateral force fluctuations could help us to characterize the effects of charge disorder in experimental systems, leading to estimates of their magnitudes that are complementary to those given by normal force measurements.

Figure 1

A schematic top view of a spherical atomic force microscopy (AFM) tip (right: side view) with disordered charge distribution above a planar substrate with similar charge distribution. Three different realizations of the experiment, i.e., three different lateral positions of the tip above the substrate, are shown corresponding to three different samples of force data. Each sample would show a different measurement of the normal as well as lateral force with a sample-to-sample variance calculated in the main text.

Figure 2

A schematic view of the bounding surfaces of two parallel semi-infinite slabs separated by a distance $l$. The surfaces are net neutral but carry a finite amount of quenched charge disorder. The slab ($S_2$) whose surface is at $z=0$ has a dielectric constant ${\epsilon }_2$, and the slab ($S_1$) whose surface is at $z=l$ has a dielectic constant ${\epsilon }_1$. We denote by ${\epsilon }_m$ the dielectric constant of the intervening material. In general, we assume that the charge distribution on slab $S_1$ is restricted to a finite area $A$; we then take the limit of large $A$.

Figure 3

Contour plot of the rescaled lateral force fluctuations, $f({\epsilon }_1/{\epsilon }_m,{\epsilon }_2/{\epsilon }_m)$ [Eqs. (22) and (23)], between two parallel slabs carrying quenched charge disorder as a function of ${\epsilon }_1/{\epsilon }_m$ and ${\epsilon }_2/{\epsilon }_m$ shown here on a $\log {}_{10}-\log {}_{10}$ scale.

Figure 4

Contour plot of the rescaled normal mean force, $G({\epsilon }_1/{\epsilon }_m,{\epsilon }_2/{\epsilon }_m,g_{2s}/g_{1s})$ [Eqs. (46) and (47)], between two parallel slabs carrying quenched charge disorder with $g_{2s}=g_{1s}$ as a function of ${\epsilon }_1/{\epsilon }_m$ and ${\epsilon }_2/{\epsilon }_m$ shown here on a $\log {}_{10}$–$\log {}_{10}$ scale.

Figure 5

Contour plots of (a) $\log {}_{10}{D}_{11}$, the contribution to the force fluctuations due to the self interactions, i.e., of the charges on surface 1 with their images, and (b) $\log {}_{10}{D}_{21}$, the contribution to the force fluctuations due to the interaction between the charges on surface 1 with those on surface 2 as a function of ${\epsilon }_1/{\epsilon }_m$ and ${\epsilon }_2/{\epsilon }_m$. (c) The rescaled normal force fluctuations $\log {}_{10}L({\epsilon }_1/{\epsilon }_m,{\epsilon }_2/{\epsilon }_m,g_{2s}/g_{s1})$ for $g_{1s}=g_{2s}$. All plots are shown in a $\log {}_{10}$–$\log {}_{10}$ scale for ${\epsilon }_1/{\epsilon }_m$ and ${\epsilon }_2/{\epsilon }_m$.