Modeling electrostatic patch effects in Casimir force measurements

Electrostatic patch potentials give rise to forces between neutral conductors at distances in the micrometer range and must be accounted for in the analysis of Casimir force experiments. In this paper we develop a quasilocal model for describing random potentials on metallic surfaces. In contrast to some previously published results, we find that patches may provide a significant contribution to the measured signal and thus may be a more important systematic effect than was previously anticipated. Additionally, patches may render the experimental data at distances below 1 $\mu$m compatible with theoretical predictions based on the Drude model.

Figure 1

Comparison of the sharp-cutoff and quasilocal patch models described in Secs. II$$A (dashed lines) and II$$B (solid lines), respectively. Plot (a) shows the voltage correlation functions in real space while plot (b) shows the associated spectrum in Fourier space. All plots correspond to the correlation function $C\equiv C_{ii}$ divided by $V_{\mathrm{rms}}^2$. On the lower plot, the sharp-cutoff spectrum discussed in II$$A is multiplied by a factor of 100 in order for it to appear at the scales shown. The parameters used for both models, discussed in Secs. II$$A and II$$B, are taken from [9] but do not correspond to the same average patch size.

Figure 2

The voltage correlation function, $C(\mathbf{x},{\mathbf{x}}^{\prime })$, is constructed by summing over all circular patches which contain both $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$. This is undertaken by integrating over patch centers $\mathbf{y}$ and accounting for the distribution in patch sizes with the distribution $\Pi \left(\ell \right)$.

Figure 3

Comparison of the residual $\delta P^{\mathrm{Drude}}$ between the experimental pressure in [10] and the Drude prediction (points with error bars at $67\%$ confidence taken from Fig. 2 of [10]) with patch pressure $P_{\mathrm{patch}}$ for four different patch models (more details in the main text): 1. The solid curve is the result of the sharp-cutoff model (with assumptions of Sec. II$$A); 2. The dotted curve corresponds to the quasilocal patch correlation model, assuming that the patch sizes are given by the grain sizes and that the rms voltage is given by the variance of the work function over different crystallographic planes; 3. The long-dashed curve is the result of a best-fit of the parameters (${\ell }_{\max }$ and $V_{\mathrm{rms}}$) of the quasilocal patch correlation model; and 4. The short-dashed curve (underneath the long-dashed curve) is a fit of a phenomenological model proposed in [31]. The inset shows the residual signal resulting from subtracting the fit of the quasilocal model (long-dashed curve) from $\delta P^{\mathrm{Drude}}$.

Figure 4

Comparison of the residual $\delta P^{\mathrm{plasma}}$ (points with error bars at $67\%$ confidence taken from Fig. 2 of [10]) with patch pressures given by the sharp-cutoff model and the quasilocal model: 1. The solid curve is the result of the sharp-cutoff model (with assumptions of Sec. II$$A). We find, consistently with the analysis in [9, 10], that the patch pressure from this model gives a negligible contribution to the measured signal; 2. The dotted curve corresponds to the quasilocal patch correlation model adopting the same parameters used in [9, 10] for the patch size and the rms voltage. In distinction from the sharp-cutoff model, we find that the quasilocal model gives a large signal as compared to the residual $\delta P^{\mathrm{plasma}}$.

Figure 5

Comparison of the residual $\delta F^{\mathrm{Drude}}$ between the data of [11] and the computed sphere-plane force with the associated patch pressure $F^{\mathrm{patch}}$. The dots correspond to $\delta F^{\mathrm{Drude}}$, with the errors bars including only the experimental error in the force determination. The solid line is a best fit of the patch force within the quasilocal model of Sec. II$$B. The inset shows the corresponding residual $\delta F^{\mathrm{plasma}}$ with the same convention employed in the main figure.