Anisotropies in thermal Casimir interactions: ellipsoidal colloids trapped at a fluid interface
We study the effective interaction between two ellipsoidal particles at the interface of two fluid phases which are mediated by thermal fluctuations of the interface. In this system the restriction of the long–ranged interface fluctuations by particles gives rise to fluctuation–induced forces which are equivalent to interactions of Casimir type and which are anisotropic in the interface plane. Since the position and the orientation of the colloids with respect to the interface normal may also fluctuate, this system is an example for the Casimir effect with fluctuating boundary conditions. In the approach taken here, the Casimir interaction is rewritten as the interaction between fluctuating multipole moments of an auxiliary charge density–like field defined on the area enclosed by the contact lines. These fluctuations are coupled to fluctuations of multipole moments of the contact line position (due to the possible position and orientational fluctuations of the colloids). We obtain explicit expressions for the behavior of the Casimir interaction at large distances for arbitrary ellipsoid aspect ratios. If colloid fluctuations are suppressed, the Casimir interaction at large distances is isotropic, attractive and long ranged (double–logarithmic in the distance). If, however, colloid fluctuations are included, the Casimir interaction at large distances changes to a power law in the inverse distance and becomes anisotropic. The leading power is 4 if only vertical fluctuations of the colloid center are allowed, and it becomes 8 if also orientational fluctuations are included.
When a fluctuating medium with long–ranged, power–law correlations is confined between a set of boundaries, forces with likewise long–ranged character are induced between the boundaries. There are different possible sources of such fluctuations in a medium: in a quantum–mechanical system it is the zero point energy of the vacuum (or ground state), and in a classical system it is the finite temperature which causes order parameter fluctuations kargol_modrev ().
This kind of force was discovered theoretically by Casimir in 1948 for the case of two parallel, conductive and uncharged plates immersed in vacuum which he attributed to zero–point fluctuations of the electromagnetic field casimir (). For a review on recent progress and the status of experimental verification of this quantum mechanical Casimir effect, see Ref. Mil04 (). A classical equivalent of the Casimir force observable between objects immersed in a fluid in the vicinity of its critical point was predicted 30 years later degennes (). The fluctuations of the order parameter field near the critical point are long–ranged, and thus they give rise to a Casimir–like, fluctuation-induced force. This effect has recently been observed in an experiment probing the force on colloidal particles which have been immersed in a near–critical binary mixture in the vicinity of a wall Gam08 (). Another classical variant of the Casimir interaction is found between particles (colloids) that are trapped at membranes goulian (); rods () or at the interface of two fluid phases cw (). In this two–dimensional latter instance, thermally excited height fluctuations of the interface which have long–ranged nature are disturbed by the presence of colloids. The energy spectrum of the fluid interface on a coarse–grained level is very well described by an effective capillary wave Hamiltonian which governs both the equilibrium interface configuration and the thermal fluctuations around this equilibrium. Since capillary waves are the Goldstone modes of the broken translational symmetry pertaining to a free interface, their correlations decay logarithmically in the absence of gravity and the corresponding fluctuation–induced forces are a manifestation of the Casimir effect for a Gaussian theory in two dimensions. Compared to manifestations of the Casimir effect in a three-dimensional bulk medium, a new phenomenon arises here: The boundary of the fluctuating interface, which is represented by the contact line on the colloid surface, is itself mobile due to position fluctuations of the colloids and finite surface tensions of the colloid–liquid interfaces and thus the Casimir force receives another contribution due to these fluctuating boundaries. This effect has been noticed first for colloidal rods trapped at membranes and films rods (). For a system composed of two spherical colloids trapped at a fluid interface, various realizations of these fluctuating boundary conditions are possible and it has been shown martin1 (); martin2 (); martin3 () that the fluctuation-induced force sensitively depends on the type of boundary conditions, with the asymptotics of the force ranging from to .
In recent work, the general form of the fluctuation-induced interactions between a finite number of compact objects of arbitrary shapes and separations has been calculated for a fluctuating medium of scalar Gaussian type jaffe1 () and an electromagnetic medium jaffe2 () with fixed boundary conditions on the surface of the objects. This has been achieved by viewing the Casimir interaction as resulting from fluctuations of source distributions (of the fluctuating field) on the surfaces of the objects which are decomposed in terms of multipoles. Then by a functional integral over the effective action of multipoles, the resulting interaction has been found. In such a way, the effect of anisotropic object shape on the Casimir force appears to be tractable. However, the objects’ shape and the boundary conditions enter the effective action by its scattering matrix which is a nontrivial object already for simple shapes jaffe1 (). Studies with explicit calculations for objects other than spheres, cylinders and walls have partially been focused on the effect of wall corrugations Bue04 (); Gie08 (), but also sharp edges Gie06 () and rectangular bodies Mil08 () have been investigated. Also, in a recent work, the quantum Casimir interaction between two ellipsoidal particles as well as an ellipsoidal particle and a plane has been studied emig09 (). In the present work, we investigate the fluctuation–induced interactions between two ellipsoidal colloids that are trapped at the interface of two fluid phases with special emphasis on the effect of anisotropy. Ellipsoids (spheroids) of varying aspect ration allow a smooth interpolation between spheres martin1 () and rods rods (). The ellipsoidal colloids are assumed to be of Janus type, therefore the interface contact line is always pinned to the colloids surface, nevertheless the vertical position of colloids and their orientation may fluctuate, giving rise to the already mentioned feature of fluctuating boundaries. For the calculation of the Casimir force, we employ techniques which have been introduced in previous work martin1 (); martin2 (); rods () and which partially can also be interpreted in terms of the scattering matrix ansatz of Refs. jaffe1 (); jaffe2 () such that our results are a concrete example of the general theory for the effects of anisotropic object shape. As stressed before, however, at an interface we always have the influence of the fluctuating boundary conditions. In order to study the effect of the mobile boundaries in our work, we have divided our investigations into two main parts; an interface fluctuation part and a second part where the colloid fluctuations are included. In the interface fluctuation part, the positions of the colloids and thus the contact lines are fixed and therefore the problem reduces to the “usual" Casimir problem with Dirichlet boundary conditions. The Casimir interaction is determined by multipoles of an auxiliary field on the (1d) contact lines which is similar to the source field in the language of Ref. jaffe1 (). In the second part, we include that the colloid position may fluctuate in all possible ways (height and tilts) which turns out to lead to Casimir interactions determined by multipoles of an auxiliary field defined on the 2d domain enclosed by the contact lines. The fluctuating boundaries result in certain restrictions on these multipoles. (Incidentally, we note that for small fluctuations amplitudes, an exact separation of interface and colloid fluctuation in the partition sum is possible martin1 (); martin2 (). This route, however, appears to be very difficult to follow for other than spherical colloids and is not taken here.)
The paper is structured as follows. In Sec. II, we introduce the effective Hamiltonian of the model system and the partition function which is defined by functional integrals over colloid and interface fluctuations. The implementation of the different boundary conditions is discussed. In Secs. III and III.1, we determine the fluctuation–induced force for the interface fluctuation part in the long and short distance regimes analytically and for intermediate distances numerically, respectively . In Sec. IV, we include the colloid fluctuations to the problem and discuss the modified long–distance asymptotics of the Casimir interaction. Sec. V contains a brief summary. Some technical details of the calculations have been relegated to Apps. A–D.
The investigated system consists of two nano- or microscopic, uncharged spheroidal colloids with principal axes , , (), which are trapped at the interface of two fluid phases I and II. The effective interaction between the colloids is mediated by thermal height fluctuations of the (sharp) interface. Without fluctuations, the equilibrium interface is flat and is set to be at . The corresponding equilibrium position of the colloids is assumed to be symmetrical with respect to , such that at the contact line the contact angle is . The elliptic cross-section of the ellipsoids with the equilibrium interface is denoted by which is an ellipse with major and minor axes , respectively. may also be expressed in confocal elliptic coordinates by the elliptic radius , see App. A for the coordinate definitions. The equilibrium interface at without the two elliptic holes cut out by the colloids is termed the reference meniscus . Deviations from this planar reference meniscus are considered to be small, without overhangs and bubbles, therefore the Monge representation is employed to describe the interface position. The colloids are of Janus type, thus the contact line is always pinned to their surface. The total Hamiltonian of the system which is used for calculating the free energy costs of thermal fluctuations around the flat interface is determined by the change in interfacial energy of the interface I/II:
Where expresses the energy needed for creating the additional meniscus area associated with the height fluctuations. In Eq. (1), is the meniscus area projected onto the plane (where the reference interface is located) and is the meniscus in the reference configuration mentioned above. The first line of Eq. (1) constitutes the drumhead model which is well-known in the renormalization group analysis of interface problems, but is also used for the description of elastic surfaces (c.f. Ref. Jas86 ()). In the second line we have applied a small gradient expansion which is valid for slopes and which provides the long wavelength description of the interface fluctuations we are interested in. The small gradient expansion entails that
is the change in projected meniscus area with respect to the reference configuration. We rewrite this change in projected meniscus area in terms of the interface position at the reference contact line ellipses . corresponds (in second order approximation) to the contact line of the colloid with fluctuating center position and fluctuating orientation. The contact line which is a function of the elliptic angle only (see App. A for the definition of elliptic coordinates) is expanded as
and we refer to the coefficients and as boundary multipole moments below. The desired expression of in terms of boundary multipole moments proceeds as discussed in Ref. Oet05 () and allows us to identify it as a sum over boundary Hamiltonians for each colloid (see also App. B):
of the capillary wave Hamiltonian which describes the energy differences associated with the additional interfacial area over the reference configuration and the boundary Hamiltonians which can be viewed as the energy cost due to fluctuations of the contact line (and which in turn are caused by colloid height and tilt fluctuations). As is well-known, the Hamiltonian is plagued with both a short-wavelength and a long-wavelength divergence which, however, can be treated by physical cutoffs. The natural short-wavelength cut-off is set by the molecular length-scale of the fluid at which the capillary wave model ceases to remain valid. The long wavelength divergence is reminiscent to the fact that the capillary waves are Goldstone modes. Of course, in real systems the gravitational field provides a natural damping for capillary waves. Accounting also for the costs in gravitational energy associated with the interface height fluctuations, therefore, introduces a long wavelength cutoff and leads to an additional term (“mass term”) in the capillary wave Hamiltonian,
Here the capillary length is given by , where is the mass density in phase and is the gravitational constant. Usually, in simple liquids, is in the range of millimeters and, therefore, is by far the longest length scale in the system. In fact, here it plays the role of a long wavelength cutoff of the capillary wave Hamiltonian , and we will discuss our results in the limit and . However, as we will see below, care is required when taking the limit (corresponding to ), since logarithmic divergencies appear Saf94 (). Another common way to introduce a long-wavelength cut-off is the finite size of any real system. As discussed in Ref. Oet05 (), the precise way of incorporating the long-wavelength cut-off is unimportant for the effects on the colloidal length scale. As an example, in both approaches the width of the interface related to the capillary wave is logarithmically divergent, .
Via the integration domain of , the total Hamiltonian of the system, Eq. (5), implicitly depends on the geometric configuration. This leads to a free energy which depends on the distance between the colloid centers and the orientation angles and of their major axes with respect to the distance vector joining the colloid centers (see Fig. 1). The free energy is related to the partition function of the system by
The partition function is obtained by a functional integral over all possible interface configurations and boundary configurations ; the relation between interface and boundary configurations is included by -function constraints,
Here is a normalization factor such that
and ensures a proper regularization of
the functional integral.
Via the -functions the interface field is
coupled to the contact line height and therefore,
the boundary Hamiltonians
have a crucial influence on the resulting effective
interaction between the colloids.
The kind of possible contact line fluctuations is solely determined by the colloid fluctuations since the contact line is pinned. These fluctuations are vertical fluctuation of the colloids on the axis normal to the equilibrium interface (height) and orientational fluctuations around that axis (tilts). In order to incorporate various boundary counditions into the solutions, we categorize them into three cases:
colloids are fixed in the reference configuration, thus there are no integrations over the boundary terms.
colloid heights fluctuate freely without tilting, thus the boundary monopoles must be included in the integration measure so that .
unconstrained height and tilt fluctuations. Up to second order in the tilts this corresponds to the inclusion of boundary dipoles in the integration measure, thus .
Case (A) corresponds to the “standard" Casimir effect in 2d with Dirichlet boundary conditions . We call this the interface fluctuation part and it will be treated in Sec. III. Note that a short summary of this part has already been given in Ref. meandmartin (). The inclusion of the colloid height and tilt fluctuations in (B) and (C) is given in Sec. IV.
Iii Interface fluctuation part
The partition function for fixed contact lines is given by
The disapearance of the interface fluctuations at the colloids boundaries is included by the Dirac delta function. In this section, analytical expressions for the fluctuation induced force in the intermediate asymptotic regime are calculated. We express the -functions in Eq. (9) by their integral representation via auxiliary fields defined on the reference contact lines . This enables us to integrate out the field leading to
where is the infinitesimal line segment on the circles . After this integration, the normalization factor is changed, , such that still holds. In Eq. (10) we introduced the Greens function of the operator which is given by where is the modified Bessel function of the second kind. In the range and , we can use the asymptotic form of the for small arguments, such that . Here, is the Euler-Mascheroni constant exponentiated. We introduce auxiliary multipole moments as the Fourier-transforms of the auxiliary fields on the reference contact line ,
where is the elliptic angle pertaining to a coordinate system centered around each colloid , respectively, such that the –axis in this colloid–specific coordinate system joins the two foci of . Furthermore, is the scale factor in elliptic coordinates (see App. A). The lengthy calculation leading to the multipole (Fourier) decomposition for the Greens function (for general orientations and of the ellipsoids) is given in App. C. The final results is collected in Eq. (48) and Eqs. (C)–(64). Using this, the double integral in the exponent of Eq. (10) can be written as a double sum over the auxiliary multipole moments (Fourier components), consisting of a self-energy part when and reside on one ellipse and when the points and reside on different ellipses, respectively. The functional integral over the auxiliary fields becomes a product of integrals over their multipole moments, , and the resulting partition function then reads
where the vectors with and contain the auxiliary multipole moments of colloid . The coupling matrix which contains the Fourier modes of the Greens function has a block structure. The self energy submatrix which describes the coupling between auxiliary moments of the same colloid are diagonal, and its form can be determined from definition (11) and Eq. (48).
where . The offdiagonal blocks characterise the interaction between the multipole moments residing on different colloids. It is convenient to split the matrix into a block structure describing the interaction of cosine and sine multipoles:
From Eq. (12) we find that the fluctuation part of the free energy reads
where . The factors , given in Eqs. (63) and (64), contain the dependence on the orientation angles and of the ellipsoids (see Fig. 1). As can be seen from above, the interaction coefficients between multipoles of order and take the form of a series in , starting at . (For spherical colloids, this multipole interaction coefficient only contains the order martin2 ().) In principle, the matrix is infinite dimensional and is divergent and its regularisation is provided by the normalization factor . The explicit series for the elements of allows for a systematic expansion of the logarithm in Eq. (19) in powers of ,
where the coefficients depend on the logarithms and , as well as the angles and . The number of auxiliary multipoles included in the calculation of the asymptotic form of in Eq. (20) is determined by the desired order in . Inclusion of multipoles up to order leads to an asymptotics correct up to . In the limit the free energy expansion coefficients in Eq. (20) up to fourth order are111Note that a factor of is missing in the expression for in Eq. (29) of Ref. meandmartin ().
The double–logarithmic divergence in in the leading coefficent is a reflection of the fact that the interface itself becomes ill–defined for due to the capillary waves. For the Casimir force itself, however, we find a finite value for all in the limit . Anisotropies in the Casimir interaction appear here first in the subleading term . Their angular dependence stems from the monopole–dipole interaction of the auxiliary field, and the attraction is maximal if both ellipses are aligned tip–to–tip.
In the opposite limit of small surface–to–surface distance , (where is the distance of the closest approach between ellipses) the fluctuation force can be calculated by using the Derjaguin (or proximity) approximation Der34 (). It consists in replacing the local force density on the contact lines by the result for the fluctuation force per length between two parallel lines with a separation distance and integrating over the two opposite contact lines to obtain the total effective force between the colloids. The Casimir force density between two parallel surfaces was calculated in Ref. Li91 () in a general approach for arbitrary dimensions. Applied to two dimensions we obtain the force line density . Integrating this density over the opposing contact lines yields meandmartin ()
Here, and are the curvature radii of the two ellipses at the end points of the distance vector of the closest approach. It is seen that the fluctuation force diverges as upon contact of the ellipsoids ().
iii.1 Intermediate distances: Numerical calculation
For intermediate distances the fluctuation induced force has to be calculated numerically. This can be done in principle by including a number of multipoles in the numeric evaluation of the determinant in Eq. (19), see Ref. jaffe1 (). In order to avoid the algebraic evaluation of the multipole coefficients of , it is possible to apply a method which was introduced in Ref. Bue04 (). The starting point is Eq. (10) for the partition function . Introducing a mesh with points , , on the reference contact line converts the double integral in the exponent to a double sum. Thus the functional integrals over the auxiliary fields are replaced by ordinary Gaussian integrals over the , . In the exponent, the are coupled by a matrix with elements . performing the Gaussian integrals and disregarding divergent and -independent terms immediately leads to for the fluctuation free energy. Here, . It contains the self energy contributions and is needed for the regularization of the free energy. Deriving with respect to , the Casimir force can be written as
The advantage of the direct calculation of the force is that Eq. (23) does not contain any divergent parts which would require regularization, thus easing the numerical treatment considerably. The determinant is computed by using a standard LU decomposition press02 (). We find good convergence of the numerical routine. The convergence can be sped up by distributing more points in the regions where the ellipses face each other. We note that computing the force by the multipole series seems to be more efficient jaffe1 (); this can partially be compensated by the point distribution on the ellipses. In Fig. 3a (ellipse aspect ratio 2) and 3b (aspect ratio 6) we compare the analytical results of Eqs. (20) and the Derjaguin approximation (Eq. (22) with the numerical results. As it is shown the analytical expressions show very good agreement with the numerical data points for both long- and short range behavior and almost cover the whole distance regime. At large distances , the leading term of the free energy expansion in Eqs. (20) mainly determines the behavior of the Casimir interaction because of its long-ranged nature, hence the orientation dependence of the subleading terms can be neglected. In order to demonstrate the anisotropy of the Casimir interaction, we show results for a fixed, intermediate distance between ellipsoid centers and varying orientation of the second ellipsoid, see Fig. 3c (aspect ratio 2, ) and 3d (aspect ratio 6, ). The orientation of the first ellipse was fixed to three values, , and . As can be seen, the fluctuation–induced interaction is maximally attractive for (tip–to–tip configuration). When deviates from zero then the resulting force reduces. This behavior holds for both aspect ratios 2 and 6.
Iv Inclusion of colloid fluctuations
In general, the inclusion of colloid height and tilt fluctuations into the partition function (Eq. (8)) can be realized by an approach used in Refs. martin1 (); martin2 (). In this approach, the partition function is split into a product of a colloid fluctuation part and the interface fluctuation part. The latter contains only the contribution of the fluctuating interface, with Dirichlet boundary conditions on the colloid surface (see previous section). In the colloid fluctuation part, the fluctuations of colloid heights and tilts are weighted by a Boltzmann factor which contains the energy of the mean field solution (Euler–Lagrange equation) to the capillary problem with the boundary conditions set by the fluctuating contact line. This decomposition is possible due to the fact that capillary wave Hamiltonian is Gaussian in the field . In principle, it is possible to use this method also for the special case of ellipsoidal colloids considered here. However, finding the mean field solution in such a geometry for arbitrary contact lines is rather cumbersome. To bypass this difficulty, we employed a trick adapted from Ref. rods () in which effective forces between rods on fluctuating membranes and films have been investigated. We extend the fluctuating interface height field which enters the functional integral for to the interior of the ellipses . Thus the measure of the functional integral for is extended by and the integration domain in the capillary wave Hamiltonian is enlarged to encompass the whole . On the colloid surfaces, the interface height field is given by the three phase contact line, . We extend continuously to the interior of the circles . Such a continuation is not unique. However, the partition function remains unchanged (up to a constant factor), if the energy cost of such a continuation is zero (as it is physically required since the interface is pinned to the ellipsoid surface). This has to be insured by appropriate counterterms martin1 (); martin2 (). We choose the continuation:
where and are the elliptic coordinates with respect to ellipse . The specific choice above is convenient for the further calculations since in . Extending the integration domain of the capillary wave Hamiltonian in Eq. (5), generates an additional energy contribution which has to be subtracted from the extended capillary wave Hamiltonian . Therefore, the total Hamiltonian reads:
The correction Hamiltonian is calculated in App. B, and we recall the boundary Hamiltonian:
In Eq. (26) we have already omitted the contributions from the gravitational term in which are of order .
As in the previous section the partition function is written as a functional integral over all possible configurations of the interface position and the boundary lines, expressed by ,
where the product over the -functions enforces the pinning of the interface at the positions of the colloids. In contrast to Eq. (8), this product extends over all instead of , only. The -functions can again be expressed by auxiliary fields , now defined on the two–dimensional elliptical domains as opposed to the auxiliary fields of Sec. III which are defined on the one–dimensional ellipses :
In contrast to the evaluation of the fluctuation term in Sec. III, there will be constraints on the lowest multipoles which contribute to . To see this we note that the Hamiltonian does not depend on the boundary monopole moments and the dipole moments (through a cancellation between and ), and the only dependence of on these moments is through the constraint function . Recalling the definition of the integration measure for the two boundary conditions (B) and (C) and performing the integration over (B) and and (C), we immediately find
Having noticed these constraints on the auxiliary fields, we proceed by integrating over the field in Eq. (28):
where – as in Eq. (10) – is the Greens function of the capillary wave Hamiltonian. A somewhat longer calculation shows that can be split into into an interaction part (coupling the auxiliary multipole moments , and for different colloid labels ), a self–energy part (depending on , and for each value of separately) and a remainder (the sum of boundary and correction Hamiltonian):
The interaction part
turns out to be a bilinear form in the auxiliary multipole moments; this is shown using the already used multipole expansion of the Greens function (valid for ) which is presented in App. C in more detail. This bilinear form reads
where the submatrices , and have already been encountered in the calculation of the fluctuation part and are given by Eqs. (15)–(18). The self–energy part (different from the corresponding one in the calculation of the fluctuation part) is evaluated in App. D, with the result
where the vectors – in contrast to in Sec. III – contain all involved auxiliary and boundary multipole moments. The elements of the matrix describe the coupling of these multipole moments, where the self-energy block couples multipoles defined on the same ellipses . Thus the diagonal part of the self energy matrix can be read off Eq. (26) and Eq. (35) while the off–diagonal part is determined by the term in Eq. (32). The elements of the interaction matrix are determined by the interaction energy in Eqs. (33) and (34) and couple the auxiliary multipole moments of different colloids. All matrix elements representing couplings of other multipoles are zero. Similar as in Eq. (12), the exponent in Eq. (36) is a bilinear form, however, here combined for all types, boundary multipole moments , and auxiliary multipoles , . The computation of the partition function amounts to the calculation of . Again this is found as a series expansion in , and we may define a similar expansion for the free energy as before in Eq. (20):
The leading coefficients in case (B) (inclusion of fluctuations in the ellipsoids’ vertical positions) are given by: