Critical Casimir Force between Inhomogeneous Boundaries
CCritical Casimir Force between Inhomogeneous Boundaries
Jerome Dubail, Raoul Santachiara, and Thorsten Emig
2, 3, 4 IJL, CNRS & Universit´e de Lorraine, Boulevard des Aiguillettes F-54506 Vandœuvre-l`es-Nancy Cedex, France Laboratoire de Physique Th´eorique et Mod`eles Statistiques,CNRS UMR 8626, Bˆat. 100, Universit´e Paris-Sud, 91405 Orsay cedex, France Massachusetts Institute of Technology, MultiScale Materials Science for Energy and Environment,Joint MIT-CNRS Laboratory (UMI 3466), Cambridge, Massachusetts 02139, USA Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA (Dated: October 15, 2018)To study the critical Casimir force between chemically structured boundaries immersed in a binarymixture at its demixing transition, we consider a strip of Ising spins subject to alternating fixed spinboundary conditions. The system exhibits a boundary induced phase transition as function of therelative amount of up and down boundary spins. This transition is associated with a sign change ofthe asymptotic force and a diverging correlation length that sets the scale for the crossover betweendifferent universal force amplitudes. Using conformal field theory and a mapping to Majoranafermions, we obtain the universal scaling function of this crossover, and the force at short distances.
PACS numbers: 11.25.Hf, 05.40.-a, 68.35.Rh
Fluctuation-induced forces are generic to all situationswhere fluctuations of a medium or field are confined byboundaries. Examples include QED Casimir forces [1, 2],van der Walls forces [3], and thermal Casimir forces insoft matter which are most pronounced near a criticalpoint where correlation lengths are large [4, 5]. The inter-action is then referred to as critical Casimir force (CCF).Analogies and differences between these variants of thecommon underlying effect have been reviewed in Ref. [6].Experimentally, CCFs can be observed indirectly inwetting films of critical fluids [7], as has been demon-strated close to the superfluid transition of He [8] andbinary liquid mixtures [9]. More recently, the CCFbetween colloidal particles and a planar substrate hasbeen measured directly in a critical binary liquid mix-ture [10, 11]. Motivated by the possibility that the lipidmixtures composing biological membranes are poised atcriticality [12, 13], it has been also proposed that inhomo-geneities on such membranes are subject to a CCF [14]which provides an example of a 2D realisation.The amplitude of the CCF is in general a universal scal-ing function that is determined by the universality classesof the fluctuating medium [15]. It depends on macro-scopic properties such as the surface distance, shape andboundary conditions of the surfaces but is independent ofmicroscopic details of the system [5]. Controlling the signof fluctuation forces (attractive or repulsive) is importantto a myriad of applications in design and manipulationof micron scale devices. While for QED Casimir forcesa generalized Earnshaw’s theorem rules out the possibil-ity of stable levitation (and consequently force reversals)in most cases [16], the sign of the CCF depends on theboundary conditions at the confinement. For classical bi-nary mixtures, surfaces have a preference for one of thetwo components, corresponding to fixed spin boundaryconditions (+ or − ) in the corresponding Ising universal- L b a b a b a
FIG. 1: Ising strip of width L with alternating fixed spinboundary conditions on one side, with a typical spin con-figuation indicated by the shading. ity class. Depending on whether the conditions are like(++ or −− ) or unlike (+ − or − +) on two surfaces, theCCF between them is attractive or repulsive. So-calledordinary or free spin boundary conditions are difficultto realize experimentally but can emerge due to renor-malization of inhomogeneous conditions as we shall showbelow [17].Motivated by their potential relevance to nano-scaledevices, fluctuation forces in the presence of geomet-rically or chemically structured surfaces have been atthe focus recently. Sign changes of CCFs due to wedgelike surface structures have been reported very recently[18]. Competing boundary conditions can give rise to in-teresting crossover effects with respect to strength andeven sign of the forces. Here we consider such a situ-ation for the Ising universality class in 2D. At critical-ity, this system can be described by conformal field the-ory (CFT) [19, 20], and CCFs are related to the centralcharge of the CFT [21–23], and scaling dimensions ofboundary operators [24].In this Letter, we show that boundary conditions whichalternate periodically between two spin states (see Fig. 1)give rise to a novel phase transition. Associated with thatis a diverging correlation length that sets the scale for a a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y sign change of the CCF on one side of the transition. Weobtain the critical exponents and exact expressions forthe universal scaling function of the force in the criticalregion. Consider the Ising model on an infinitely longstrip of width L , and assume that the system is at itscritical temperature T c so that it is conformally invari-ant. For homogenous, fixed spin boundary conditions γ , γ = ± on the two boundaries, the critical Casimirenergy per unit strip length, F , is determined by CFT.Since L is the only finite length scale, the energy obeysa simple power law. The amplitude is determined onlyby the central charge c = 1 / h γ γ of the so-called boundary condi-tion changing (BCC) operator from γ to γ (see belowfor details on the BCC operator) [24], F = − π (cid:18) − h γ γ (cid:19) L (1)where we measure here and the following energies in unitsof k B T c . For like boundary conditions γ = γ = +or − one has h ++ = h −− = 0 and hence an attrac-tive force F = − d F /dL . For unlike boundary conditions γ (cid:54) = γ one gets h + − = h − + = 1 / h f + = h f − = 1 /
16 which implies arepulsive interaction in Eq. (1). In the following we con-sider a strip with homogeneous + spins on one boundaryand alternating regions of − and + spins of length a and b , respectively, on the other boundary, see Fig. 1.If the temperature is slightly different from T c , thesystem is in the critical region, where the free energydensity can be decomposed into non-singular ( F ns ) andsingular ( F s ) contributions, F ( t, L, τ ) = F ns ( t, L, τ ) + F s ( t, L, τ ) (2)that depend on the reduced temperature t = T /T c − L , and a scaling variable τ = a/b − t and τ , the singular part is not. For homogeneous bound-ary conditions, t is the only relevant scaling variable, andin the critical region the singular part of the free energydensity is given by a universal scaling function ϑ thatdepends only on L/ξ [5, 15] where ξ ( t → ± ) = ξ ± | t | − ν is the bulk correlation length with amplitude ξ ± and ex-ponent ν = 1 for the Ising model. As we shall see below,the same renormalization-group (RG) concepts apply toa novel, boundary induced critical region that we identifyfor inhomogeneous boundary conditions around a = b .To focus on that region, we assume in the following thatthe system is at its bulk critical point, t = 0. For large L (cid:29) a, b the singular part of the free energy density can be expressed in terms of a universal scaling function ofthe new correlation length ξ c ( τ ) = ( a + b ) | τ | − ν c , F s (0 , L, τ ) = 1 L ϑ [ L/ξ c ( τ )] . (3)Below we shall determine ϑ and the exponent ν c .BCC operators have been introduced in CFT to studysystems with discontinuous boundary conditions [24].When inserted on a boundary, these local operators in-terpolate between the different boundary conditions oneither side of the insertion point. They are highest weightstates of weight h and all such states may be realized byan appropriate pair of boundary conditions. For the crit-ical Ising model, the BCC operator that takes the bound-ary condition from + spin to − spin corresponds to thechiral part of the energy operator (cid:15) ( z, ¯ z ). This can be un-derstood easily in the representation of the Ising model interms of a free Majorana fermion field ψ ( z ) out of whichthe energy operator is composed, (cid:15) ( z, ¯ z ) = iψ ( z ) ¯ ψ (¯ z )[25]: The Jordan-Wigner transformation shows that thefermion creation and annihilation operators flip locallythe spin orientation.Now the BCC operators permit us to relate the parti-tion function of the strip with alternating boundary con-ditions to a correlator for the field ψ ( z ) at positions wherethe boundary conditions change. On the upper complexplane, one has (cid:104) ψ ( z ) ψ ( z (cid:48) ) (cid:105) = 1 / ( z − z (cid:48) ) which yields (af-ter a conformal map) for the partition function of thestrip the Pfaffian, Z = Z (cid:104) ψ ( w ) . . . ψ ( w N ) (cid:105) = Z Pf( G ) = Z det / ( G ) , (4)with G = [ (cid:104) ψ ( w i ) ψ ( w j ) (cid:105) ] i,j =1 ,..., N , where we used theWick theorem for fermions, w j are the positions of the2 N BCC operators on the upper edge of the strip, and Z is the partition function of the homogenous systemwith a = 0. Due to the symmetry under translations by a + b , the matrix G is of block Toeplitz form, G ij = g i − j ,with g j = (cid:18) g [ j ( a + b )] g [ j ( a + b ) − a ] g [ j ( a + b ) + a ] g [ j ( a + b )] (cid:19) , (5)where g ( w ) = π/ [2 L sinh( πw/ (2 L )].The free energy density can be expressed in the ther-modynamic limit as F = − π
48 1 L − lim N →∞ N ( a + b ) log det G . (6)The Szeg¨o-Widom (SW) theorem for block Toeplitz ma-trices states that the determinant can be expressedin terms of the matrix valued Fourier series ϕ ( θ ) = (cid:80) ∞ k = −∞ g k e ikθ as [26]lim N →∞ N log det G = 14 π (cid:90) π dθ log det ϕ ( θ ) (7)where det acts now on a 2 × a < b . Thereason for that is a subtle difference between the Toeplitzmatrix G and the corresponding circulant matrix C thatdescribes periodic boundary conditions along the strip.While for a < b the spectra of G and C become equivalentfor N → ∞ , for a > b there exists a pair of eigenvaluesof GC − that tend to zero exponentially for N → ∞ ,yielding an extra contribution δ that is determined bythe decay of the Fourier integral J = 12 π (cid:90) π dθe − ijθ (cid:2) ϕ − ( θ ) (cid:3) ∼ e − jδ for j → ∞ (8)and has to be subtracted from the r.h.s. of Eq. (7) for a > b . Here (cid:2) ϕ − ( θ ) (cid:3) denotes the 11-element of the2 × ϕ − ( θ ). In the following we apply Eqs. (7)and (8) to compute the critical Casimir force in variousscaling limits.When L (cid:28) a, b , the function g ( w ) defined belowEq. (5) can be replaced by g ( w ) = ( π/L ) e − π | w | / (2 L ) which yields the exact determinantdet ϕ ( θ ) = cos θ − cosh( π ( a − b ) / (2 L ))cos θ − cosh( π ( a + b ) / (2 L )) . (9) For a < b , the SW theorem then yields N log det G = − ( πa ) / (2 L ). For a > b , this is also the correct result asit follows from subtracting the correction δ which followsfrom Eq. (8) and J = e − π | a − b | j/ (2 L ) as δ = π ( a − b ) / (2 L ).It follows that the critical Casimir force for L (cid:28) a, b is F = π
48 23 a − ba + b L + . . . (10)It has an analytic amplitude that varies continously with a/b . This result is identical to an addition of the am-plitudes from Eq. (1) for unlike and like boundary con-ditions, weighted by a/ ( a + b ) and b/ ( a + b ), accordingto their occurrence. Hence, additivity holds at short dis-tances. This has been observed also for a 3D Ising modelin the special case of boundaries with alternating stripesof equal width [17].Next, we consider the case L (cid:29) a, b . Using the Abel-Plana summation formula, it can be shown that in thislimit the elements of the matrix ϕ ( θ ) = πL (cid:18) iγ ( θ ) γ ( θ ) − γ ∗ ( θ ) iγ ( θ ) (cid:19) (11)approach γ j ( θ ) = La + b (cid:8) ˆ γ j ( θ, τ ) − i j +1 [tanh( θL/ ( a + b )) + tanh(( θ − π ) L/ ( a + b ))] (cid:9) (12)with ˆ γ ( θ, τ ) = 1 − θ/π andˆ γ ( θ, τ ) = 1 π (cid:20) − τ + 2 τ + 1 + e iθ ( τ + 2) F (cid:18) , τ + 2 , τ + 3 τ + 2 , e iθ (cid:19) − e − iθ τ + 22 τ + 3 F (cid:18) , τ + 3 τ + 2 , τ + 5 τ + 2 , e − iθ (cid:19)(cid:21) (13)where F is a hypergeometric function. For a < b , the SW theorem then yields the free energy density F = − π
48 1 L − π ( a + b ) (cid:90) π log (cid:26) θ, τ ) (cid:20) tanh θLa + b + tanh ( θ − π ) La + b (cid:21)(cid:27) dθ . (14)where we have subtracted a L -independent contributionthat does not change the force, and defined Γ( θ, τ ) =(2ˆ γ + i (ˆ γ − ˆ γ ∗ )) / ( | ˆ γ | − | ˆ γ | ). In the evaluation ofthe integral, the correlation length ξ c ( τ ) defined aboveEq. (3) becomes important. The integrand is exponen-tially localized around θ = 0 , π over a small range( a + b ) /L . Also, it can be shown that Γ has the scalingproperty lim τ → Γ( τ /ζ, τ ) = Γ ( ζ ) = 1 / (1 + π ζ/
32) forany constant ζ . Hence, in the critical region of small τ ,or ξ c ( τ ) (cid:29) a + b , the proper scaling is obtained by setting ζ = ( Lτ ) / ( a + b ) = L/ξ c (up to a numerical coefficient),showing that the exponent ν c = 2. In the integral, Γ( θ, τ )can be replaced by Γ ( ζ ) and one obtains after a simpleintegration the result for the universal scaling function ofEq. (3) when a < b or τ < ϑ − ( ζ ) = 14 π Li (cid:18)
21 + π ζ/ − (cid:19) (15) where Li ( x ) = (cid:80) ∞ k =1 x k /k is a polylogarithm func-tion. Outside the critical region L (cid:29) ξ c , one has ϑ − ( ζ → ∞ ) = − π/
48 so that the force is fully domi-nated by the boundary regions with like spins. On thecontrary, for L (cid:28) ξ c , and hence τ → − , the frustrationbetween almost equal amounts of fixed + and − spins onthe boundaries leads to a renormalization to effectively free boundary conditions with ϑ − ( ζ →
0) = π/
24. For a > b , the correction δ yields an extra contribution ∆ ϑ ( ζ )determined by ∆ ϑ ( ζ ) tan[∆ ϑ ( ζ )] = π ζ (16)so that the scaling function for τ > ϑ + ( ζ ) = ϑ − ( ζ ) + ∆ ϑ ( ζ ). Since ∆ ϑ ( ζ →
0) = 0, the scaling func-tion is continuous around τ = 0. For L (cid:29) ξ c , how-ever, ∆ ϑ ( ζ → ∞ ) = π/ ϑ + ( ζ → ∞ ) = 23 π/ ~~0 attractive repulsive repulsive~~ Fig.3 FIG. 2: Schematic overview of critical Casimir force ampli-tudes as function of the strip width L and the ratio a/b . For L (cid:29) a, b the solid curves represent the diverging correlationlength ξ c . The horizontal dashed line indicates the cut alongwhich the force amplitude is plotted in Fig. 3. Along the redcurve the sign of the force changes whereas the blue curve in-dicates only a change between two universal (repulsive) limits. Our findings can be summarized by the scheme ofFig. 2. It shows the different scaling regimes and the cor-responding asymptotic amplitudes of the Casimir force.At short distance L (cid:28) a, b the amplitude varies continu-ously across the critical point at a = b , with a sign changeat b/a = 23. For L (cid:29) a, b there exist three distinct re-gions: around a = b appears a region where L (cid:28) ξ c wherethe force is repulsive and approaches for asymptotic L theuniversal amplitude for fixed-free spin boundary condi-tions. For a < b , the force changes sign from attractiveto repulsive when L approaches ξ c , corresponding to astable point. For a > b , the force is always repulsive butthe amplitude crosses over from π/
24 to 23 π/
48 underan increase of L beyond ξ c .The dependence of the force F on | a − b | at fixed L (cid:29) a, b (see dashed horizontal line in Fig. 2) is de-termined by F = − ∂ F /∂L = Θ( x s ) L − with a uni-versal scaling function Θ of the scaling variable x s thatis defined on both sides of the critical point by x s =sign( τ )( L/ξ c ) / ∼ a − b . This function is shown in Fig. 3where we used the results for ϑ ± ( L/ξ c ) of Eqs. (15), (16). FIG. 3: Universal scaling function Θ( x s ) for the criticalCasimir force as function of the scaling variable x s ∼ a − b . In the critical region | x s | (cid:28)
1, one has the expansionsΘ( x s ) = π − π x s + . . . for x s < π + π / √ x s + . . . for x s > , (17)whereas for L outside the critical region, | x s | (cid:29) x s ) = − π +
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