We study the Casimir torque arising from the quantum electromagnetic fluctuations due to the interaction of two interfaces in a system formed by a dense array of metallic nanorods embedded in dielectric fluids. It is demonstrated that as a consequence of the ultrahigh density of photonic states in the nanowire array it is possible to channel the quantum fluctuations, and thereby boost the Casimir torque by several orders of magnitude as compared to other known systems (e.g., birefringent parallel plates).
©2013 Optical Society of America
In the last few years, the Casimir effect [1–3] has received considerable attention, mainly because of its importance in micro- and nano-electromechanical systems (MEMS and NEMS) [4–7]. The Casimir interactions result from the confinement of the quantum-mechanical zero-point fluctuations of the electromagnetic fields and may cause the permanent adhesion of nearby surface elements in MEMS/NEMS (a phenomenon known as ‘stiction’, or static friction [4,5]), severely affecting the device reliability. On the other hand, the Casimir interaction phenomena may also open new and promising directions in the field of micro and nanomechanics , potentially allowing for the design of, for example, Casimir oscillators , nanoscale racks and pinions , and Casimir ratchets .
The Casimir effect was first studied by H. Casimir in 1948. In his pioneering work , an attractive force between two electrically neutral metallic plates in a vacuum was predicted, as a result of the zero-point energy of the electromagnetic field. This theory was some years later generalized by E. Lifshitz to the case of dielectric plates , and subsequently extended to the case of having a third dielectric in between the plates . A further generalization of the Casimir theory included the possibility of considering anisotropy in the plates, and was first investigated in the nonretarded regime (small separation distances) , and later extended to arbitrary distances . In these works [12,13], it was also shown that in systems formed by two parallel birefringent plates (with in-plane optical anisotropy) a mechanical torque may arise that forces the rotation of the plates towards the position that minimizes the zero-point energy, corresponding to the alignment of the optical axes of the plates. This Casimir torque was further investigated in Refs [14,15], and some experiments aiming to observe and measure the torque were suggested [14–16]. The Casimir torque was also studied in a system formed by two parallel corrugated plates  when the corrugation directions are not aligned.
Recently, the Casimir-Lifshitz interaction was also investigated in structured materials (metamaterials) [18–23]. In particular, in [24,25] it was demonstrated that the Casimir interaction in systems comprising nanowire materials may be characterized by ultralong-range forces, in contrast to the short-range forces inherent to systems with isotropic backgrounds. Nanowire materials support quasi-transverse electromagnetic (q-TEM) photonic states, whose dispersion contours are hyperbolic . The enhanced Casimir forces in nanowire media are a consequence of the ultra-large density of photonic states in such materials in the low frequency limit, which promotes the quantum induced interactions between the two plates in the Casimir problem, resulting in an ultralong-range force. In this work we further study the Casimir interaction in nanowire material environments, proving that the Casimir torque in this kind of systems is qualitatively and quantitatively very different from the usual torques in either birefringent parallel plates [14,15] or corrugated metallic parallel plates .
2. Zero-point energy in nanowire materials
The geometry of the structure under study is illustrated in Fig. 1. It consists of a dense array of parallel metallic nanowires arranged in a square lattice with period a. The nanowires have radius and are oriented along the direction (forming an angle with the z-direction) (Fig. 1), where and are the unit vectors along the coordinate axes. The nanowires are embedded in three dielectric fluids with relative permittivities , , and , respectively.
Next, we describe the effective medium model used in this work to calculate the zero-point interaction energy of the system (Fig. 1) and discuss its validity.
2.1 Effective medium model
In general, the uniaxial wire medium is characterized by strong spatial dispersion and its electromagnetic response is described by a nonlocal dielectric function that depends on the component of the wave vector parallel to the wires ( is the wave vector) [26–29]. Within this framework, the wire medium is modeled by the permittivity tensor , where is the permittivity of the host material, and , being the wave number in the dielectric host, is the volume fraction of the metal, a is the lattice constant, is the complex permittivity of the metallic wires, and is a structural parameter with the physical meaning of plasma wave number such that . In this work, we are interested in the case where the metallic wires are densely packed (limit with fixed, being the lattice constant and the length of the wires), such that the relevant physics in the Casimir problem is determined by the frequency range , i.e. by frequencies much smaller than the effective plasma frequency of the composite medium. In such a scenario, reduces to the simple formula28,29], such that the wave propagation in the uniaxial wire medium can be described solely in terms of a transverse electric (TE) mode (ordinary wave) and of a q-TEM mode (extraordinary wave). Note that for noble metals is approximately real (in case of low loss) and negative, because has the same property. Hence, since the transverse permittivity (along directions perpendicular to the wires direction) is , the uniaxial wire medium behaves as an hyperbolic medium.
It can be proven (e.g. using the argument principle [22,30,31]) that, at zero absolute temperature, the regular part of the zero-point interaction energy per unity of the cross-sectional area of the system under study (Fig. 1) can be written as22].
The characteristic function can be readily obtained asFig. 2), and are the propagation matrices for the forward (waves travelling along the direction) and backward (waves travelling along the direction) waves, respectively, which propagate in medium 2 (see Fig. 2). The reflection matrices are defined asEq. (1) and depending on the region. Note that the optical axis of the effective media is tilted with the respect to the direction normal to the interfaces (z-direction). In this work, it is assumed that the metal is the same in the three regions, such that follows the Drude dispersion model , where is the high-frequency permittivity, is the metal plasma oscillation frequency, and is the collision frequency.
On the other hand, the propagation matrices are as followsEq. (1) with . In the particular case where the plasma oscillation frequency tends to infinity, , the metal behaves as a perfect electric conductor (PEC) and , so that the effective medium is characterized by “extreme anisotropy”. In such a scenario, the propagation constants for the extraordinary waves are simply given by
2.2 On the validity of the effective medium model
To assess the validity of the effective medium model in the calculation of the Casimir-Lifshitz forces – particularly the fact that spatial dispersion is neglected - next we compare the results obtained based on the hyperbolic medium approximation [Eq. (1)] with the more general theory of Ref  that takes into account the nonlocal dispersive behavior of the nanowire materials [26,27]. In these calculations, we only consider the case so that the wires are perpendicular to the interfaces.
In Ref , it was shown that the Casimir force in nanowire environments is an ultralong-range force that decays as , which contrasts markedly with the characteristic decay of in usual systems with isotropic backgrounds. This anomalously high Casimir interaction stems from the fact that arrays of nanowires support q-TEM waves that allow channeling the quantum oscillations of the electromagnetic field, boosting in this way the intensity of the Casimir force at large distances. It was demonstrated in  that the contribution to the Casimir force of the other electromagnetic modes (TM and TE modes) that propagate in the wire medium may be negligible (especially for large distances), since these modes contribute only to short-range forces that decay quickly with the distance. Hence, based on Ref , the Casimir force (or more rigorously, the contribution of the q-TEM modes to the Casimir force) per unit of area in systems formed by arbitrary nondispersive magnetodielectrics embedded in nanowire materials is given by:Fig. 3], the Casimir force is simply
In Fig. 3, we compare the normalized Casimir force as a function of the distance calculated using the hyperbolic medium model [, where is the zero-point energy obtained from Eq. (2)] (discrete symbols) with the results based on the nonlocal model of Ref  (solid lines). There is an excellent agreement between the two theories, even when the effect of loss and dispersion in the metal nanowires is considered, supporting in this manner the validity of the simplified effective medium model in the context of the Casimir problem. Note, however, that such an agreement can be obtained only by restricting the integration in Eq. (2) to the first Brillouin zone, which in effect takes into account the wire medium granularity and recovers the dependence of the Casimir force with distance in such systems. It is also clear that the effect of metallic loss and dispersion is quite mild, since the magnitude of the Casimir force for Ag nanowires is only slightly lower than that for PEC nanowires.
The results of Fig. 3 confirm that the contribution of the TE mode to the Casimir force is indeed residual. Indeed, in our calculation (discrete symbols of Fig. 3), both the contributions of the q-TEM modes (extraordinary wave) and TE modes (ordinary wave) are taken into account. Finally, it is important to point out that the Casimir force is attractive for the configuration I (Fig. 3(i)), whereas in the other scenario the Casimir force is repulsive (Fig. 3(ii)).
3. Casimir torque in nanowire materials
In this section, we calculate the Casimir interaction torque acting on the metallic nanowires in the system illustrated in Fig. 1 formed by two interacting interfaces separated by a distance . One important point is that for this type of systems the Casimir torque may be nonzero even when the distance between the two interfaces approaches infinity. Indeed, it should be clear that even for a single interface the zero-point energy stored in the system may depend on the angle that defines the orientation of the nanowires, and therefore, one should expect a remnant Casimir torque when . This property is specific of our system, and is not found in other geometries studied hitherto [14,15,17]. For example, due to the isotropy of the vacuum the zero-point energy of a single birefringent plate standing alone in free-space is independent of its relative orientation, and thus the Casimir induced torque for a pair of birefringent plates vanishes when .
Obviously, because of the electromagnetic coupling between the two interfaces, the torque will depend on , such that the total torque can be written as , being the remnant torque associated with the first interface (between the dielectrics and ), the remnant torque associated with the second interface (between the dielectrics and ), and the interaction torque resulting from the coupling between the two interfaces. In this work, we present a detailed analysis of the calculation of the interaction torque, and derive a rough analytical estimation for the single-interface Casimir torques ( and ).
3.1 Theoretical analysis
The Casimir torque per unit of area can be expressed in terms of the zero-point energy of the system as Eq. (2). However, because our background (region 2) is not isotropic, here we need to proceed carefully. Indeed, it should be noted thatFig. 4(a)) and 2-3 material interfaces, and and are the corresponding zero-point energies, which are independent of d.
The interaction torque can be readily evaluated with the help of Eq. (2). In the next two subsections we derive an analytical estimation for and discuss how the single-interface torques and can be determined.
3.2 Analytical expression for the interaction torque
To obtain a closed analytical formula for the Casimir interaction torque induced in the nanowire material (see Fig. 1), next we generalize the theory of  and compute the contribution of the q-TEM photonic modes to the Casimir interaction energy when the angle is in the range . To do this, we consider the case wherein the nanowire materials are characterized by extreme anisotropy so that in the relevant frequency range. In such a situation the reflection coefficient for a q-TEM wave incident upon an interface of two nanowire materials (let us say, propagating from region 2 to region 1) with different dielectric hosts is nearly independent of (), and is approximately equal to , being the wave impedance in the pertinent dielectric host. Therefore the zero-point energy in the nanowire configuration sketched in Fig. 1 can be written as follows:Eq. (8)] and and are the reflection coefficients at the two interfaces. Making the change of variables , the Casimir interaction energy at zero temperature becomes13–15].
3.3 Single-interface torque
In order to determine ( is determined in the same manner), we consider the scenario of Fig. 4(b), which represents a twin-interface configuration wherein and the distance between the two interfaces is . The same analysis that led to Eq. (13) shows that for the scenario of Fig. 4(b) one can write:Fig. 4(b) must vanish, because in such a situation the nanowires are embedded in a uniform background. Thus, we can write or equivalently:Fig. 4(b). Notice that the limit needs to be taken in the calculation. This is challenging for several reasons.
Indeed, this torque is determined by the near-field interactions (at the interface) of the metallic wires and the host materials, and thus the contribution of higher-order modes (not described by the effective medium model) is likely to be important. In the context of the Casimir problem the use of the effective medium model is accurate only when .
Nevertheless, one may estimate based on the continuous medium approximation, but even in such a case things are not plain. In fact, unless one adopts a realistic dispersive model for the dielectrics and metal so that their electric response vanishes for , the zero-point energy typically diverges in the limit .
Considering these limitations, next we obtain an analytical formula for based on the continuous material approximation, under the hypothesis that all the material responses cease at , beyond which the contribution of higher-order modes should be included and the effective medium theory breaks down. It is evident that in general this limit may depend on the wire tilting angle . It is reasonable to assume on physical grounds that . Proceeding as in Subsection 3.2, we find that:Eq. (18) we obtain the desired result:
It is interesting to look at the problem from a different perspective, and suppose that the nanowires are held fixed, whereas the dielectric hosts (e.g. two immiscible fluids such as water and olive oil) are free to be rotated. In this scenario, the Casimir torque induced in the system may force the dielectric fluids to move in such a manner that the interface moves toward the position of stable equilibrium where . Perturbations in the alignment of the interfaces with respect to the orientation of the nanowires may provide a “signature” of the Casimir torque that may permit detecting it experimentally.
3.4 Numerical results and discussion
Next, we present a numerical study of the Casimir interaction torque (), i.e. of the perturbation of the total torque resulting from the electromagnetic coupling of the two interfaces, for several illustrative cases.
We have calculated the Casimir interaction torque induced in the nanowire system sketched in Fig. 1 for different angles and for a fixed distance , using both the analytical model [Eq. (16)] and the hyperbolic medium model [, where is the zero-point energy obtained from Eq. (2)]. The results are depicted in Fig. 5(i).
First, it is important to highlight the very good agreement between the results obtained with the two different calculation methods. Similar to the analytical model (black dashed curve), the numerical results (green (light gray) solid curve) also predict that the Casimir interaction torque varies as . Moreover, the maximum magnitude of the Casimir interaction torque occurs at . On the other hand, the Casimir interaction torque is null when the nanowires are parallel () or perpendicular () to the interface planes, similar to the single-interface torques. For the case of twin-material interfaces () the sign of the interaction torque is opposite to that of the single-interface torques, consistent with the fact that if the gap closes the total torque should vanish. Therefore, in this case, the interaction of the two interfaces leads to a depression of the Casimir torque.
The results of Fig. 5(i) also prove that the contribution of the TE mode to the Casimir interaction can be neglected, since the result predicted by our numerical calculation (which takes into account the contribution of the TE mode) is quantitatively very similar to the one obtained from the analytical model (which only takes into consideration the contribution of the q-TEM mode).
In the previous example, it was assumed that the nanowires are PECs. To assess the effect of metallic loss and dispersion, we have also calculated the Casimir interaction torque in the nanowire system of Fig. 1 for Ag nanorods (Fig. 5(ii)) [Ag is described by a Drude dispersion model ]. As can be seen, the behavior of the Casimir interaction torque with the angle for Ag nanowires is qualitatively similar to that of PEC nanowires, but the system with Ag nanowires generates a slightly weaker Casimir interaction torque.
We have also calculated the magnitude of the Casimir interaction torque at as a function of the distance (Fig. 6). We show the results of the Casimir torque for both PEC nanowires [black dashed curve] and Ag nanowires [blue solid curve]. Again, the dependence of the Casimir interaction torque on metallic loss and dispersion is relatively weak, since the difference between the PEC and Ag nanowires results is modest.
In addition, we also show in Fig. 6 the magnitude of the Casimir torque in the system discussed in Ref  formed by a two parallel birefringent plates (calcite and barium titanate) separated by a vacuum. It is seen from Fig. 6 that the Casimir interaction torque induced in the nanowire configuration illustrated in Fig. 1 is several orders of magnitude larger than the torque generated in the birefringent plates system . In particular, the difference between the magnitudes of the torques is increasingly pronounced as the distance is increased. In fact, this result was expected because for the configuration of Ref  the Casimir torque decays approximately as , whereas it was found in Subsection 3.2 that in the proposed nanowire system the Casimir interaction torque decays as (see Eqs. (15-16)). It is important not to forget that the total torque also depends on and which typically are nonzero in case of the nanowire system, but vanish in the system of Ref .
In this work, we have studied analytically and numerically the Casimir interaction torque induced in a system formed by a dense array of metallic nanowires embedded in three dielectric fluids (modeled as hyperbolic media). It was demonstrated that the perturbations in the torque resulting from the interaction of the two interfaces is several orders of magnitude larger than the torque generated in other structures reported in the literature (e.g., birefringent parallel plates, or even corrugated metallic parallel plates). In particular, it was found that in the nanowire structures the Casimir interaction torque decays as , in contrast to the decay that characterizes the usual setups comprising two parallel plates. The reason is that nanowire hyperbolic materials have a large density of photonic modes in the low-frequency limit, which promotes the quantum induced interactions. We have also estimated a contribution to the Casimir torque resulting from the near-field interactions of the nanowires passing through a single dielectric interface. Generally, this contribution to the torque cannot be obtained within the framework of a continuous medium model, as it depends on the microstructure of the metamaterial in the vicinity of the interface.
This work is supported in part by Fundação para Ciência e a Tecnologia under project PTDC/EEATEL/100245/2008.
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