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We discuss theoretical predictions for the thermal Casimir force and compare them with available experimental data. Special attention is paid to the recent claim of the observation of that effect, as predicted by the Drude model approach. We show that this claim is in contradiction with a number of experiments reported so far. We suggest that the experimental errors, as reported in support of the observation of the thermal Casimir force, are significantly underestimated. Furthermore, the experimental data at separations above 3 μm are shown to be in agreement not with the Drude model approach, as is claimed, but with the plasma model. The seeming agreement of the data with the Drude model at separations below 3 μm is explained by the use of an inadequate formulation of the proximity force approximation.

We discuss the possibility of determining the properties and quality of spherical surfaces used in precise experiments with the help of capacitance measurements. The results of this kind measurements for the lens-plane and sphere-plane, Au coated surfaces are compared with theoretical predictions from various models of perfect and broken sphericity. It is shown that capacitance measurements are incapable of discriminating between models of perfect and modified centimeter-size spherical surfaces in an experiment demonstrating the anomalous scaling law for the electric force. Claims to the contrary in the recent literature are explained by the use of improper comparison. The data from capacitance measurements in an experiment measuring the Casimir force by means of a micromechanical torsional oscillator employing micrometer-size spheres are shown to be in excellent agreement with theoretical predictions using the model of a perfect spherical surface.

We comment on progress in measurements of the Casimir force and discuss what is the actual reliability of different experiments. In this connection a more rigorous approach to the usage of such concepts as accuracy, precision, and measure of agreement between experiment and theory, is presented. We demonstrate that all measurements of the Casimir force employing spherical lenses with centimeter-size curvature radii are fundamentally flawed due to the presence of bubbles and pits on their surfaces. The commonly used formulation of the proximity force approximation is shown to be inapplicable for centimeter-size lenses. New expressions for the Casimir force are derived taking into account surface imperfections. Uncontrollable deviations of the Casimir force from the values predicted using the assumption of perfect sphericity vary by a few tens of percent within the separation region from 1 to 3 μm. This makes impractical further use of centimeter-size lenses in experiments on measuring the Casimir force.

Additional information is provided on the effect of the significant (up to 35%) reduction in the magnitude of the Casimir force between an Au-coated sphere and an indium tin oxide film which was observed after UV treatment of the latter. A striking feature of this effect is that the reduction is not accompanied with a corresponding variation of the dielectric permittivity, as confirmed by direct ellipsometry measurements. The measurement data are compared with computations using the Lifshitz theory. It is shown that the data for the untreated sample are in a very good agreement with theory taking into account the free charge carriers in the indium tin oxide. The data for the UV-treated sample exclude the theoretical results obtained with account of free charge carriers. These data are found to be in a very good agreement with theory disregarding the free charge carriers in an indium tin oxide film. A possible theoretical explanation of our observations as a result of phase transition of indium tin oxide from metallic to dielectric state is discussed in comparison with other related experiments.

Measurements of the Casimir force are used to obtain stronger constraints on the parameters of hypothetical interactions predicted in different unification schemes beyond the Standard Model. We review new strong constraints on the Yukawa-type interactions derived during the last two years from recent experiments on measuring the lateral Casimir force, Casimir force in configurations with corrugated boundaries and the Casimir-Polder force. Specifically, from measurements of the lateral Casimir force compared with the exact theory the strengthening of constraints up to a factor of 24 millions was achieved. We also discuss further possibilities to strengthen constraints on the Yukawa interactions from the Casimir effect.

The gradient of the Casimir force between carefully cleaned Au surfaces of a sphere and a plate is measured using a dynamic atomic force microscope in the frequency modulation regime in high vacuum. The electrostatic calibration of the setup did not reveal any effect of patches or surface contaminants. The experimental data for the force gradient are found to be consistent with theory using the plasma model approach over the entire measurement range. The Drude model approach is excluded by the data at separations from 235 to 400 nm at a 67% confidence level.

We report development of a high-sensitivity torsion balance to measure the thermal Casimir force. Special emphasis is placed on experimental investigations of a possible surface electric force originating from surface patch potentials that have been recently noticed by several experimental groups. By gaining a proper understanding of the actual contribution of the surface electric force in real materials, we aim to undertake precision force measurements to resolve the Casimir force at finite temperature in real metals, as well as in other semiconducting materials, such as graphene.

We present analytical and simple expressions to determine the free energy, internal energy, entropy, as well as the pressure acting at the interface of a perfectly conducting rectangular Casimir piston. We show that infrared divergencies linear in temperature become cancelled within the piston configuration, and show a continuous behavior consistent with intuitive expectations.

Using nonstandard recursion relations for Fresnel coefficients involving successive stacks of layers, we extend the Lifshitz formula to configurations with an inhomogeneous, n-layered, medium separating two planar objects. The force on each object is the sum of a Lifshitz like force and a force arising from the inhomogeneity of the medium. The theory correctly reproduces very recently obtained results for the Casimir force/energy in some simple systems of this kind. As a by product, we obtain a formula for the force on an (unspecified) stack of layers between two planar objects which generalizes our previous result for the force on a slab in a planar cavity.