On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results.

Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. In the QED case, the medium is usually a vacuum and the massless excitations are photons, while in the CC and HF cases the medium is usually a critical or correlated fluid and the fluctuations of the order parameter are the cause of the force between the macroscopic or mesoscopic bodies immersed in it. We discuss the importance of the presented results for nanotechnology, especially for devising and assembling micro- or nano-scale systems. Several important problems for nanotechnology following from the currently available experimental findings are spelled out, and possible strategies for overcoming them are sketched. Regarding the example of HF, we explicitly demonstrate that when a given integral quantity characterizing the fluid is conserved, it has an essential influence on the behavior of the corresponding fluctuation-induced force.

Exact expressions for the partition function of the one-dimensional Ising model in the fixed-M ensemble.

We obtain exact closed-form expressions for the partition function of the one-dimensional Ising model in the fixed-M ensemble, for three commonly used boundary conditions: periodic, antiperiodic, and Dirichlet. These expressions allow for the determination of fluctuation-induced forces in the canonical ensemble, which we term Helmholtz forces. The thermodynamic expressions and the calculations flowing from them should provide insights into the nature and behavior of fluctuation-induced forces in interesting and as-yet unexplored regimes.

Behavior of the van der Waals force between a plate and a single-walled carbon nanotube under uniform hydrostatic pressure: a theoretical study.

We study the behaviour of the non-retarded van der Waals force between a planar substrate and a single-walled carbon nanotube, assuming that the system is immersed in a liquid medium which exerts hydrostatic pressure on the tube's surface, thereby altering its cross section profile. The shape of the latter is described as a continual structure characterized by its symmetry index n. Two principle mutual positions of the tube with respect to the substrate are studied: when one keeps constant the minimal separation between the surfaces of the interacting objects; when the distance from the tube's axis to the substrates bounding surface is fixed. Within these conditions, using the technique of the surface integration approach, we derive in integral form the expressions which give the dependance of the commented force on the applied pressure.

Sign change in the net force in sphere-plate and sphere-sphere systems immersed in nonpolar critical fluid due to the interplay between the critical Casimir and dispersion van der Waals forces.

We study systems in which both long-ranged van der Waals and critical Casimir interactions are present. The latter arise as an effective force between bodies when immersed in a near-critical medium, say a nonpolar one-component fluid or a binary liquid mixture. They are due to the fact that the presence of the bodies modifies the order parameter profile of the medium between them as well as the spectrum of its allowed fluctuations. We study the interplay between these forces, as well as the total force (TF) between a spherical colloid particle and a thick planar slab and between two spherical colloid particles. We do that using general scaling arguments and mean-field-type calculations utilizing the Derjaguin and the surface integration approaches. They both are based on data of the forces between two parallel slabs separated at a distance L from each other, confining the fluctuating fluid medium characterized by its temperature T and chemical potential µ. The surfaces of the colloid particles and the slab are coated by thin layers exerting strong preference to the liquid phase of the fluid, or one of the components of the mixture, modeled by strong adsorbing local surface potentials, ensuring the so-called (+,+) boundary conditions. On the other hand, the core region of the slab and the particles influence the fluid by long-ranged competing dispersion potentials. We demonstrate that for a suitable set of colloids-fluid, slab-fluid, and fluid-fluid coupling parameters, the competition between the effects due to the coatings and the core regions of the objects involved result, when one changes T, µ, or L, in sign change of the Casimir force (CF) and the TF acting between the colloid and the slab, as well as between the colloids. This can be used for governing the behavior of objects, say colloidal particles, at small distances, say in colloid suspensions for preventing flocculation. It can also provide a strategy for solving problems with handling, feeding, trapping, and fixing of microparts in nanotechnology. Data for specific substances in support of the experimental feasibility of the theoretically predicted behavior of the CF and TF have been also presented.

Manipulation and amplification of the Casimir force through surface fields using helicity.

We present both exact and numerical results for the behavior of the Casimir force in O(n) systems with a finite extension L in one direction when the system is subjected to surface fields that induce helicity in the order parameter. We show that for such systems, the Casimir force in certain temperature ranges is of the order of L^{-2}, both above and below the critical temperature, T_{c}, of the bulk system. An example of such a system would be one with chemically modulated bounding surfaces, in which the modulation couples directly to the system's order parameter. We demonstrate that, depending on the parameters of the system, the Casimir force can be either attractive or repulsive. The exact calculations presented are for the one-dimensional XY and Heisenberg models under twisted boundary conditions resulting from finite surface fields that differ in direction by a specified angle, and the three-dimensional Gaussian model with surface fields in the form of plane waves that are shifted in phase with respect to each other. Additionally, we present exact and numerical results for the mean-field version of the three-dimensional O(2) model with finite surface fields on the bounding surfaces. We find that all significant results are consistent with the expectations of finite-size scaling.

Critical and near-critical phase behavior and interplay between the thermodynamic Casimir and van der Waals forces in a confined nonpolar fluid medium with competing surface and substrate potentials.

We study, using general scaling arguments and mean-field type calculations, the behavior of the critical Casimir force and its interplay with the van der Waals force acting between two parallel slabs separated at a distance L from each other, confining some fluctuating fluid medium, say a nonpolar one-component fluid or a binary liquid mixture. The surfaces of the slabs are coated by thin layers exerting strong preference to the liquid phase of the fluid, or one of the components of the mixture, modeled by strong adsorbing local surface potentials ensuring the so-called (+,+) boundary conditions. The slabs, on the other hand, influence the fluid by long-range competing dispersion potentials, which represent irrelevant interactions in renormalization-group sense. Under such conditions, one usually expects attractive Casimir force governed by universal scaling function, pertinent to the extraordinary surface universality class of Ising type systems, to which the dispersion potentials provide only corrections to scaling. We demonstrate, however, that below a given threshold thickness of the system L(crit) for a suitable set of slabs-fluid and fluid-fluid coupling parameters the competition between the effects due to the coatings and the slabs can result in sign change of the Casimir force acting between the surfaces confining the fluid when one changes the temperature T, the chemical potential of the fluid µ, or L. The last implies that by choosing specific materials for the slabs, coatings, and the fluid for L≲L(crit) one can realize repulsive Casimir force with nonuniversal behavior which, upon increasing L, gradually turns into an attractive one described by a universal scaling function, depending only on the relevant scaling fields related to the temperature and the excess chemical potential, for L≫L(crit). We present arguments and relevant data for specific substances in support of the experimental feasibility of the predicted behavior of the force. It can be of interest, e.g., for designing nanodevices and for governing behavior of objects, say colloidal particles, at small distances. We formulate the corresponding criterion for determination of L(crit). The universality is regained for L≫L(crit). We also show that for systems with L≲L(crit), the capillary condensation phase diagram suffers modifications which one does not observe in systems with purely short-ranged interactions.

Reply to "Comment on 'Casimir force in the O(n→∞) model with free boundary conditions' ".

The preceding Comment raises a few points concerning our paper [Phys. Rev. E 89, 042116 (2014)]. In this Reply we stress that although Diehl et al. [Europhys. Lett. 100, 10004 (2012) and Phys. Rev. E 89, 062123 (2014)] use three different models to study the Casimir force for the O(n→∞) model with free boundary conditions we study a single model over the entire range of temperatures from above the bulk critical temperature T(c) to absolute temperatures down to T=0. The use of a single model renders more transparent the crossover from effects dominated by critical fluctuations in the vicinity of the bulk transition temperature to effects controlled by Goldstone modes at low temperatures. Contrary to the assertion in the Comment, we make no claim for the superiority of our model over any of those considered by Diehl et al. [Europhys. Lett. 100, 10004 (2012) and Phys. Rev. E 89, 062123 (2014)]. We also present additional evidence supporting our conclusion in Dantchev et al. [Phys. Rev. E 89, 042116 (2014)] that the temperature range in which our low-temperature analytical expansion for the Casimir force increases as L grows and remains accurate for values of the ratio T/T(c) that become closer and closer to unity, whereas T remains well outside of the critical region.

Casimir force in the O(n→∞) model with free boundary conditions.

We present results for the temperature behavior of the Casimir force for a system with a film geometry with thickness L subject to free boundary conditions and described by the n→∞ limit of the O(n) model. These results extend over all temperatures, including the critical regime near the bulk critical temperature Tc, where the critical fluctuations determine the behavior of the force, and temperatures well below it, where its behavior is dictated by the Goldstone mode contributions. The temperature behavior when the absolute temperature, T, is a finite distance below Tc, up to a logarithmic-in-L proximity of the bulk critical temperature, is obtained both analytically and numerically; the critical behavior follows from numerics. The results resemble-but do not duplicate-the experimental curve behavior for the force obtained for He4 films.

Surface integration approach: a new technique for evaluating geometry dependent forces between objects of various geometry and a plate.

We present a new approach, which can be considered as a generalization of the Derjaguin approximation, that provides exact means to determine the force acting between a three-dimensional body of any shape and a half-space mutually interacting via pairwise potentials. Using it, in the cases of the Lennard-Jones, standard and the retarded (Casimir) van der Waals interactions we derive exact expressions for the forces between a half-space or a slab of finite thickness and an ellipsoid in a general orientation, which in the simplest case reduces to a sphere, a tilted fully elliptic torus, and a body obtained via rotation of a single loop generalized Cassini oval, a particular example of which mimics the shape of a red blood cell. The results are obtained for the case when the object is separated from the plane via a non-polar continuous medium that can be gas, liquid or vacuum. Specific examples of biological objects of various shapes interacting with a plate like substrates are also considered.

Casimir force in the rotor model with twisted boundary conditions.

We investigate the three-dimensional lattice XY model with nearest neighbor interaction. The vector order parameter of this system lies on the vertices of a cubic lattice, which is embedded in a system with a film geometry. The orientations of the vectors are fixed at the two opposite sides of the film. The angle between the vectors at the two boundaries is α where 0≤α≤π. We make use of the mean field approximation to study the mean length and orientation of the vector order parameter throughout the film--and the Casimir force it generates--as a function of the temperature T, the angle α, and the thickness L of the system. Among the results of that calculation are a Casimir force that depends in a continuous way on both the parameter α and the temperature and that can be attractive or repulsive. In particular, by varying α and/or T one controls both the sign and the magnitude of the Casimir force in a reversible way. Furthermore, for the case α=π, we discover an additional phase transition occurring only in the finite system associated with the variation of the orientations of the vectors.

Finite-size effects in presence of gravity: the behavior of the susceptibility in 3He and 4He films near the liquid-vapor critical point.

We study critical-point finite-size effects on the behavior of susceptibility of a film placed in the Earth's gravitational field. The fluid-fluid and substrate-fluid interactions are characterized by van der Waals type power-law tails, and the boundary conditions are consistent with bounding surfaces that strongly prefer the liquid phase of the system. Specific predictions are made with respect to the behavior of 3He and 4He films in the vicinity of their respective liquid-gas critical points. We find that for all film thicknesses of current experimental interest the combination of van der Waals interactions and gravity leads to substantial deviations from the behavior predicted by models in which all interatomic forces are very short ranged and gravity is absent. In the case of a completely short-ranged system exact mean-field analytical expressions are derived, within the continuum approach, for the behavior of both the local and the total susceptibilities.

Casimir force in O(n) systems with a diffuse interface.

We study the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry infinity;{d-1}xL , where 2infinity limit of O(n) models with antiperiodic boundary conditions applied along the finite dimension L of the film. We observe that the Casimir amplitude Delta_{Casimir}(dmid R:J_{ perpendicular},J_{ parallel}) of the anisotropic d -dimensional system is related to that of the isotropic system Delta_{Casimir}(d) via Delta_{Casimir}(dmid R:J_{ perpendicular},J_{ parallel})=(J_{ perpendicular}J_{ parallel});{(d-1)2}Delta_{Casimir}(d) . For d=3 we derive the exact Casimir amplitude Delta_{Casimir}(3,mid R:J_{ perpendicular},J_{ parallel})=[Cl_{2}(pi3)3-zeta(3)(6pi)](J_{ perpendicular}J_{ parallel}) , as well as the exact scaling functions of the Casimir force and of the helicity modulus Upsilon(T,L) . We obtain that beta_{c}Upsilon(T_{c},L)=(2pi;{2})[Cl_{2}(pi3)3+7zeta(3)(30pi)](J_{ perpendicular}J_{ parallel})L;{-1} , where T_{c} is the critical temperature of the bulk system. We find that the contributions in the excess free energy due to the existence of a diffuse interface result in a repulsive Casimir force in the whole temperature region.

Interplay of critical Casimir and dispersion forces.

Using general scaling arguments combined with mean-field theory we investigate the critical (T approximately Tc) and off-critical (T not equal Tc) behavior of the Casimir forces in fluid films of thickness L governed by dispersion forces and exposed to long-ranged substrate potentials which are taken to be equal on both sides of the film. We study the resulting effective force acting on the confining substrates as a function of T and of the chemical potential mu. We find that the total force is attractive both below and above Tc. If, however, the direct substrate-substrate contribution is subtracted, the force is repulsive everywhere except near the bulk critical point (Tc, mu(c)), where critical density fluctuations arise, or except at low temperatures and (L/a)(beta(Delta)(mu))=O(1), with Delta(mu)=mu-mu(c)<0 and a the characteristic distance between the molecules of the fluid, i.e., in the capillary condensation regime. While near the critical point the maximal amplitude of the attractive force if of order of L(-d) in the capillary condensation regime the force is much stronger with maximal amplitude decaying as L(-1). In the latter regime we observe that the long-ranged tails of the fluid-fluid and the substrate-fluid interactions further increase that amplitude in comparison with systems with short-range interactions only. Although in the critical region the system under consideration asymptotically belongs to the Ising universality class with short-ranged forces, we find deviations from the standard finite-size scaling for xi(ln)(xi/xi0(+/-)) >>L even for xi, L>>xi0(+/-), where xi[t=(T-Tc)/Tc-->+/-0,Delta(mu)=0]=xi0(+/-)/t/-nu, is the bulk correlation length. In this regime the dominant finite-size contributions to the free energy and to the force stem from the long-ranged algebraically decaying tails of the interactions; they are not exponentially small in L, as it is the case there in systems governed by purely short-ranged interactions, but exhibit a power law decay in L. Essential deviations from the standard finite-size scaling behavior are observed also within the finite-size critical region L/xi=O(1) for films with thicknesses L less than or approximately equal Lcrit, where Lcrit=xi0(+/-)(16/s/)nu/beta, with nu and beta as the standard bulk critical exponents and with s=O(1) as the dimensionless parameter that characterizes the relative strength of the long-ranged tail of the substrate fluid over the fluid-fluid interaction. We present the modified finite-size scaling pertinent for such a case and analyze in detail the finite-size behavior in this region. The standard finite-size scaling behavior is recovered only for L>>Lcrit.

Finite-size effects on the behavior of the susceptibility in van der Waals films bounded by strongly absorbing substrates.

We study critical point finite-size effects in the case of the susceptibility of a film in which interactions are characterized by a van der Waals-type power law tail. The geometry is appropriate to a slablike system with two bounding surfaces. Boundary conditions are consistent with surfaces that both prefer the same phase in the low temperature, or broken symmetry, state. We take into account both interactions within the system and interactions between the constituents of the system and the material surrounding it. Specific predictions are made with respect to the behavior of 3He and 4He films in the vicinity of their respective liquid-vapor critical points.

Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: general considerations and exact spherical-model results.

We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L --> infinity and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx-(d+sigma) as x --> infinity, with 2

Critical Casimir force and its fluctuations in lattice spin models: exact and Monte Carlo results.

We present general arguments and construct a stress tensor operator for finite lattice spin models. The average value of this operator gives the Casimir force of the system close to the bulk critical temperature T(c). We verify our arguments via exact results for the force in the two-dimensional Ising model, d -dimensional Gaussian, and mean spherical model with 2 = k(b) T(c) (d-1)Delta/ (L/a)(d), where L is the distance between the plates and Delta is the (universal) Casimir amplitude.

Universality of the thermodynamic Casimir effect.

Recently a nonuniversal character of the leading spatial behavior of the thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys. Rev. E 66, 016102 (2002)]. We reconsider the arguments leading to this observation and show that there is no such leading nonuniversal term in the systems with short-ranged interactions if one treats properly the effects generated by a sharp momentum cutoff in the Fourier transform of the interaction potential. We also conclude that lattice and continuum models then produce results in mutual agreement independent of the cutoff scheme, contrary to the aforementioned report. All results are consistent with the universal character of the Casimir force in the systems with short-ranged interactions. The effects due to dispersion forces are discussed for the systems with periodic or realistic boundary conditions. In contrast to the systems with short-ranged interactions, for L/xi>>1, one observes leading finite-size contributions governed by power laws in L due to the subleading long-ranged character of the interaction, where L is the finite system size and xi is the correlation length.