THE CRiTiCaL UNIVERSAL FLUCTUATION-INDUCED

III A. Gambassi 1,2 , C. Hertlein 3, L. Helden 3, S. Dietrich 1,2 , and C. Bechinger 1,3 III DOI: 10.1051/ epn /2009301 aT WORk

᭡ The demixing It is well known that nothing comes from nothing. In , however, this is not of water and the solvent always the case. Surprisingly indeed, two electrically neutral conducting metallic plates 2,6-lutidine placed in about half a micrometer apart attract each other, even at zero (seen here in false colours), temperature, due to a which was theoretically predicted in 1948 by the Dutch is Hertlein and physicist [1]. This force originates from the mechanical colleagues' laboratory fluctuations of the electromagnetic field in vacuum - so, apparently, from nothing. for measuring the critical Casimir force. © C. Hertlein et al. n the absence of the plates, these fluctuations are the such material parameters, which might be sources of same everywhere. But if the plates are introduced, a experimental uncertainties, makes its experimental Idistance L apart, the fluctuation modes need to have verification a particularly stringent test. a node at the plate surfaces, so that only waves with e first attempts to verify Casimir's prediction date wavelength λ=nL/ 2 and integer n are permitted in the back to 1958, but a sufficiently accurate measurement of gap between the plates. Outside, instead, there is no F at the micrometer scale has been accomplished first such restriction. As a consequence the total field outside only in 1997 [2] due to the relative weakness of the the gap produces a on the plates which is force: Two plates of A=1mm 2 at a L=1µm higher than the one produced from inside, so the sur - attract each other with a force of 10 -9 N. Nonetheless, faces are pushed together by a force F. e force per this effect is responsible for the stiction (static friction) unita area A of the plates takes the form occurring in currently available microelectromechani - affiliations cal systems (MEMS), in which different device parts 1 2 - Max-Planck- —F =–—π —hc (1) cling together, hampering their functioning [3]. Institut für Metall - A 24 0 L4 forschung, Heisen - Casimir, in collaboration with D. Polder, actually had bergstrasse 3, 70569 Stuttgart, for L large compared to a characteristic already derived Eq.(1) by showing that the interaction Germany microscopic length Lc≈c/ωp. e value of Lc is roughly potential V(R) between two neutral atoms at a distance R, 2 Institut für Theo - retische und An - set by the material-dependent frequency ωp above due to their fluctuating electric dipole moments, is affected gewandte Physik, which the actual metal is no longer conducting by the finite speed of c (retardation). As a result V(R) Pfaffenwaldring 6 57, 70569 Stutt - (Lc≈0.1µm for copper). changes from the London-van der Waals behaviour 1/R gart, Germany e asymptotic expression of this long-ranged force has to 1/R 7 as R increases. Equation (1) actually reflects this 3 2. Physikalisches Institut, Universi - a universal character in that it depends only on the asymptotic behavior for large R. Surprised by the fact that tät Stuttgart, Pfaf - - fenwaldring 57, c and on Planck's constant hbut not on the final result of their lengthy calculation for V(R) was so 70569 Stuttgart, the material properties of the plates which determine simple in the limit R→∞, Casimir (following a suggestion Germany only the actual value of Lc . e absence in Eq. (1) of by ) looked for a simpler derivation focussed on III

18 EPN 40/1 Article available at http://www.europhysicsnews.org or http://dx.doi.org/10.1051/epn/2009301 Casimir EffECt fEaturEs

III vacuum fluctuations. is kind of focus has far-reaching fluctuations in the density are correlated across the so- consequences when carried over to cases in which fluc - called correlation length ξ which increases with a power

tuations of a different occur. Casimir’s line of law for T→Tc. At scales large compared with ξ0 the resul - argument, indeed, does not rely on the fact that the fluc - ting physical behavior of the system turns out to be tuating quantity is actually the electromagnetic field of the largely independent of its microscopic details (giving vacuum and that its fuctuations are of quantum nature. rise to the concept of “universality” ). In a certain sense, a What really matters is the presence of any such a quantity critical point acts as a magnifying glass for the flucta - and a mechanism by which physical boundaries modify tions which occur in any case at the molecular scale. the spectrum of its allowed fluctuations, so that the corres - Critical points are present in the phase diagrams of a ponding total free depends on the distance wide variety of microscopically different systems, ran - between the boundaries. Due to this dependence, as Casi - ging from classical and quantum fluids (for example mir pointed out, the boundaries experience an effective, 4He close to the superfluid transition) to nuclear matter. fluctuation-induced force. ese basic requirements are so In each case the relevant quantity – called "order para - modest that indeed one encounters such kind of fluctua - meter" –, the thermal fluctuations of which are tion-induced forces in a wide range of circumstances, well magnified upon approaching the critical point, has a beyond the context of , ranging different physical nature but plays the same role in from cosmology to statistical physics, making the subject determining the free energy density of the system. really fascinating (see, e.g. , Ref. [4]).

Thermal fluctuations at work: ᭣ FiG. 1: the critical Casimir effect Schematic phase In the following we focus on thermal fluctuations which diagram of the liquid mixture occur in statistical physics, in particular close to continuous of water with phase transitions where their effects are particularly pro - the oily liquid 2,6-lutidine nounced, such that their confinement leads – in analogy to (dimethylpyri - the Casimir force – to the so-called critical Casimir effect. dine, C7H9N). At ambient pres - In statistical physics fluctuations are due to the thermal sure, the relevant of atoms and molecules. erefore they typi - thermodynamic cally occur on a molecular scale ξ . variables are 0 the temperature Consider the case of a liquid which is kept at room tem - T and the mass perature T. If one sets out to measure its density from a fraction x of lutidine in the 3 sample of volume 1 mm there is no chance to detect e effective force resulting from the confinement of mixure. The appreciable fluctuations around its average. In order to the fluctuations of the order parameter is called "critical schematic side view of a vertical observe such fluctuations one has to decrease the Casimir force" [5] and was first discussed in 1978 by test tube filled volume to a few nm 3, corresponding to the typical spa - Michael Fisher and the late Pierre-Gilles de Gennes in with the binary liquid mixture tial linear scale ξ0 at which fluctuation of the density of their study of the behavior of a binary liquid mixture is shown by (a) a fluid occur under normal conditions. made up of two components and confined between two and (b). In (b) All fluctuation effects, including possible Casimir-like plates [6]. (An independent field-theoretical analysis W and L indicate the water- and forces, are relevant at this scale and disappear at larger was carried out by Symanzik [7].) the lutidine- ones. In addition, one cannot expect these forces to be e phase diagram of a binary liquid mixture, for rich phase, respectively. characterized by universal laws such as the one derived example water and lutidine (a clear oily substance), is by Casimir, because the physical phenomena at the depicted in Fig.1: For a fixed lutidine mass fraction x, molecular scale are dominated by the specific micro - the mixture forms a mixed solution at low temperatures scopic details of the fluid. (as indicated by the inset (a)), whereas it separates into ere are instances, however, in which the fluctuations a water-rich (W) and a lutidine-rich (L) phase if the become relevant and detectable even at scales ξ much solution is heated above the transition line (inset (b)).

larger than the molecular ones, i.e. , ξ >> ξ0. At the critical point Tc≅34 °C and xc ≅0.28 6 (CP in Fig.1), is emerges upon approaching a second-order phase the two phases do not form abruptly, like water solidi - transition point (also called critical point , located at spe - fying into ice. Instead, at the critical concentration and

cific values Tc and Pc of temperature and pressure, below the critical temperature, fluctuating form in respectively) which is characterized by the fact that due the mixture which contain more water or more lutidine. to the emerging collective behavior of the molecules, e closer the temperature gets to the critical point, the III

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III larger these fluctuating areas grow and the longer they Due to this weakness its experimental detection is parti - remain intact. Actually, the typical size of these correlated cularly difficult. Indeed, only in 1999 the first indirect -ν fluctuation diverges as ξ≈ξ0I1- T/T cI , where ν≅0.63 cha - experimental evidence was provided by capacitance stu - racterizes the type of critical point. e order parameter dies of wetting films of liquid 4He close to the superfluid for this mixture is the difference between the local transition [9] and by X-ray studies of wetting films of a concentration of water and lutidine compared with binary liquid mixture in 2005 [10]. Close to the corres - their average values and therefore it increasingly fluc - ponding critical points, the thicknesses of these liquid tuates upon approaching the critical point. films change and this phenomenon can be succesfully If two plates at close distance are introduced in the mix - explained by the occurrence of critical Casimir forces. ture, analogously to the quantum Casimir effect, each of In order to measure directly the force acting on an them sets boundary conditions by preferring to be in object, the simplest approach is to attach to it a suitable contact with one of the two components of the binary dynamometer and then read off the result from the mixture. e corresponding boundary conditions are scale. is approach would suggest the use of the canti - referred to as (++) (or (––) ) if the two plates prefer the lever of an , capable to measure same component of the mixture and (+–) (or (–+) ) if forces with pico-Newton resolution (1 pN = 10 -12 N), but they have opposite preferences. this turns out to be still too difficult. e resulting critical Casimir force F on the plates of e measurement of the Casimir force is actually area A, a distance L apart, is theoretically expected to easier if, instead of considering the force acting between be largely independent of the microscopic details of two parallel plates immersed in the fluctuating medium, the fluctuating medium and of the boundary condi - one considers the force acting on an immersed spheri -

tions. Indeed, for ξ,L>> ξ0 it takes the form cal particle of R when it approaches the wall of the container of the mixture. As in the previous cases —F =—kBTΘ(L/ ξ) (2) both the wall and the sphere impose boundary condi - A L3 tions on the order parameter and as a consequence a

where kB is Boltzmann's constant and kBT the thermal critical Casimir force F acts on the sphere. If the dis - energy. In contrast to the quantum Casimir effect, the tance of closest approach z between the sphere and the range of the force F is now set by the correlation length ξ wall is much smaller than the radius R, the potential

of the critical fluctuations, which ⌽c(z) of the critical Casimir force takes the form can be controlled by the tempera - The force we have ture T. e function Θ(u) is —⌽c(z)=—R ϑ(z/ξ) (3) k T z measured can be easily universal in that it depends only on B turned from attractive certain gross features of the critical where the function ϑ can be expressed in terms of Θ [11] to repulsive by suitable behavior of the system and on the and it shares the same qualitative and universal properties. surface treatments kind of boundary conditions With this change in the geometrical setting the experimen - imposed by the plates on the order tal challenge is to determine the force acting on the sphere. parameter, i.e. , (++) or (+–) for a In order for the effects of F to be detectable on a sphere binary liquid mixture of classical fluids. Most of the details floating in the mixture, its magnitude has to be compa - about the molecular structure of the fluid and its micro - rable with the typical forces at play. is suggests the use scopic interactions with the atoms constituting the plates of micrometer-sized polysterene particles, i.e. , colloids . are of no importance in determining F as long as the cor - relation length ξ and the distance L are much larger than a femto-newton dynamometer the microscopic length scales of the system. e sensitivity necessary to directly measure the critical is universality allows one to determine the function Casimir force can be achieved by total internal reflec - Θ(u) by studying a slab of the Ising model with surface tion microscopy (TIRM [12], see Fig.2) which actually fields. For the type of fluid under consideration such a is capable to measure forces with femto-Newton resolu - study has been recently carried out via Monte Carlo tion [13] (1 fN = 10 -15 N). simulations [8]. It turns out that the force F can be made A laser beam is totally reflected at the silica wall of the repulsive by changing the boundary conditions from container of the mixture, so that an evanescent wave

(++) to (+–) . Such a possibility is still debated for the penetrates into the mixture with an intensity Ieν (z)ϰexp(- κz ), quantum Casimir force. A rough estimate of F reveals which decreases exponentially upon increasing the dis - that at room temperature, for two plates with A=1mm² tance z from the wall, with a decay length κ-1 . When the and at a distance L=1µm, the expected force is about 10 -9 N. colloidal particle approaches the wall with surface to III

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III surface distance z, it scatters light off the evanescent wave ᭣ FiG. 2: The femto-Newton and the scattered intensity is proportional to Ieν (z). A sin - dynamometer. gle counter measures the scattered intensity Isc (z), (a) Scheme of -1 the TIRM set-up. from which one infers the distance z=- κ ln[ Isc (z)/ Isc (0)]. A colloidal parti - An optical tweezer provides the lateral force which cle undergoes avoids the dri of the particle out of the objectives of the thermal motion in the evanes - microscope. e colloid is actually so small that it is sen - cent Ieν sitive to the collisions with the molecules of the fluid created by a which cause it floating randomly in the mixture as in the totally reflected blue laser beam celebrated Brownian motion. As a result, the distance z and scatters randomly changes in time and this is actually detected by light, whose intensity Isc (z) is the single photon counter which measures a randomly measured by a single photon varying scattered intensity Isc as a function of time (see counter. The lat - Fig. 2(b)). From this signal one determines the probabi - eral diffusion of lity distribution function P(z) that characterizes the (a) the particle is prevented by a erratic motion of the particle. In thermal equilibrium vertically inci - P(z) is given by the Boltzmann distribution associated dent green with the total potential ⌽(z) to which the particle is sub - optical tweezer. (b) Isc as a func - ϰ ⌽ jected: P(z) exp{- (z)/( kBT)}. is theoretical relation (b) tion of time, allows one to determine the potential ⌽(z) from the reflecting the random motion apparently meaningless random signal Isc (t) associated of the particle in with the Brownian motion of the colloid. the vertical direction. (Taken With this technique at hand one can study the onset of from Ref.[11].) critical Casimir forces on the colloid as the tempera - ture T of the binary mixture is increased towards its

critical value Tc, at fixed critical concentration x=xc. First we consider the case of an hydrophilic colloid of radius R=1.2µm and a hydrophilic wall, corresponding to (––) boundary conditions. (With reference to the water- lutidine mixture we indicate by“+” and“–” the preferential regions, the sphere is repelled. As in the previous experi - adsorption of lutidine and water, respectively.) Far from ment, far below the critical point, ⌽(z) consists only of the

the critical point the measured potential ⌽(z) (apart from electrostatic repulsion, as in Fig.3(b) for Tc-T =0.90K. Upon the contribution of buoyancy and of the optical tweezer, heating the mixture, the repulsive part of the potential which are subtracted from the data presented in Fig.3) is curves shis towards larger values of z due to the fact that, given only by the electrostatic repulsion between the col - as expected, repulsive Casimir forces are acting on the par -

loid and the wall, as shown in Fig.3(a) for Tc-T=0.30K. ticle and yield a positive contribution ⌽c(z) to the total Upon heating the mixture, however, an increasingly deep potential ⌽(z). As in the case of Fig.3(a), one finds a remar - potential well develops, which is due to the attractive Casi - kable agreement with the corresponding theoretical

mir force providing a negative contribution ⌽c(z). As prediction (Eq.(3), solid lines in Fig.3(b)). expected theoretically, the spatial range ξ of this contri - By changing the adsorption preference of the silica bution increases upon increasing T. e set of measured wall from hydrophilic (–) to hydrophobic (+) via a sui -

⌽c(z) can be compared with the corresponding theoreti - table chemical surface treatment, attraction is recovered cal prediction (Eq. (3), solid lines in Fig.3(a)), resulting in (data not shown). In contrast to the smooth onset of

a remarkable agreement. e maximum of the attractive the critical Casimir force, in mixtures with x≠xc and force measured in this case is about 600 fN. (––) or (++) boundary conditions one observes the If one exchanges the hydrophilic colloid with an hydro - abrupt formation of a potential well upon approaching phobic one (of radius R≅1.8µm), the corresponding the transition line. is occurs only on the side of the boundary conditions change from (––) to (+–) and on phase diagram where the mixture is poor in the com - theoretical grounds one expects repulsion. Indeed, in this ponent preferentially adsorbed by both surfaces and is case, areas rich in water form at the wall, whereas those rich due to the sudden formation of a liquid bridge which in lutidine stay close to the colloid. Since it takes free energy spans the space between the particle and the wall and to make contact between these water- and lutidine-rich is rich in the component preferred by the surfaces [11].

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Concluding remarks Max-Planck Fellow. Christopher Hertlein and Laurent We have observed and measured the fluctuation-induced Helden are members of Bechinger’s group as PhD stu - critical Casimir forces between a colloid and a wall [11]. dent and tenured research scientist, respectively. S. Based on the general argument presented above, one Dietrich is Director at the Max-Planck-Institut für expects them to act also between two or more colloidal Metallforschung in Stuttgart and holds a Chair for particles immersed in a near-critical mixture, with eoretical Physics at the University of Stuttgart; Andrea interesting many-body phenomena due to the non- Gambassi joined his group aer the PhD in eoretical additivity of these fluctuation-induced forces. In contrast Physics at the Scuola Normale Superiore in Pisa (Italy). with other types of interactions which are typically acting among colloids ( e.g. , electrostatic) the critical Casimir References force exhibits a striking temperature dependence which [1] H.B.G. Casimir, Proc. Kon. Nederl. Akad. Wet. B 51 , 793 (1948). can be exploited in order to control the phase behaviour [2] S.K. Lamoreaux, Phys. Rev. Lett. 78 , 5 (1997). See also A. Lam - of such particles via minute temperature changes. is brecht and S. Reynaud, Phys. Rev. Lett. 84 , 5672 (2000) and fact could be used, for example, in order to control the S.K. Lamoreaux, Phys. Rev. Lett. 84 , 5673 (2000). aggregation of colloids, which is a central problem in [3] P. Ball, Nature 447 , 772 (2007). many areas of materials science where colloidal particles [4] S.K. Lamoreaux, Physics Today 60 , February 2007, 40; M. Kardar dispersed in a solvent form the basis of diverse subs - and R. Golestanian, Rev. Mod. Phys. 71 , 1233 (1999). tances such as milk or paints. e force we have [5] M. Krech, The Casimir Effect in Critical Systems (World Scientific, measured can be easily turned from attractive to repul - Singapore, 1994); J.G. Brankov, D.M. Dantchev, and N.S. Ton - chev, The Theory of Critical Phenomena in Finite-Size sive by suitable surface treatments. is property might Systems (World Scientific, Singapore, 2000). be used in order to compensate the attractive quantum [6] M.E. Fisher and P.-G. de Gennes, C.R. Acad. Sci. Paris B 287 , mechanical Casimir force which brings MEMS to a 207 (1978). standstill. If these machines would work not in a vacuum, [7] K. Symanzik, Nucl. Phys. B 190 , 1 (1981). but in a liquid mixture close to the critical point, the stic - [8] O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, Europhys. tion could be prevented by coating the machine parts so Lett. 80 , 60009 (2007). that the critical Casimir force has a repelling effect. [9] R. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83 , 1187 (1999); Our combined experimental and theoretical approach A. Ganshin, S. Scheidemantel, R. Garcia, and M.H.W. Chan, Phys. Rev. Lett. 97 , 075301 (2006). demonstrates that the minute forces which result from random thermal fluctuations at the sub-micrometer [10] M. Fukuto, y. yano, and P. Pershan, Phys. Rev. Lett. 94 , 135702 (2005). scale can be harnessed to serve dedicated purposes. I [11] C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechin - ger, Nature 451 ,172 (2008). about the authors [12] D.C. Prieve, Adv. Colloid. Interf. Sci. 82 , 93 (1999). Clemens Bechinger holds a Chair for Experimental Sta - [13] D. Rudhardt, C. Bechinger, and P. Leiderer, J. Phys. Cond. tistical Physics at the University of Stuttgart and he is also Matt. 11 , 10073 (1999).

(a) (b)

᭡ FiG. 3: Critical Casimir potentials for a colloidal particle in a water-lutidine mixture and close to an hydrophilic wall (from Ref. [11]). Upon approaching the critical point, an attractive Casimir force is observed with an hydrophilic particle ( (–– ) boundary con - ditions, (a)), whereas with an hydrophobic particle ( (+– ) boundary conditions, (b)) the force is repulsive. The solid lines in (a) and (b) correspond to the theoretical predictions for the Casimir potentials.

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