System and experimental setup Studied a wetting film of binary mixture MC/PFMC on Si(100), in equilibrium with the binary vapor and bulk liquid mixture at critical concentration. Anti-symmetric (+,  ) B.C.: Previous study at 30°C [10] showed MC-rich liquid wets the liquid/Si interface PFMC is that MC-rich liquid wets the liquid/Si interface and PFMC is favored at the liquid/vapor interface favored at the liquid/vapor interface. Intrinsic chemical potential  of the film relative to bulk liquid/vapor coexistence was controlled by temperature offset  T between the substrate and liquid reservoir [10]. Observation of Critical Casimir Effect in a Binary Wetting Film: An X-ray Reflectivity Study Masafumi Fukuto, Yohko F. Yano, and Peter S. Pershan Department of Physics and DEAS, Harvard University, Cambridge, MA What is a Casimir force? A long-range force between two macroscopic bodies induced by some form of fluctuations between them. Two necessary conditions: (i) Fluctuating field (ii) Boundary conditions (B.C.) at the walls Casimir forces in adsorbed fluid films near bulk critical points (i) Fluctuations: Local order parameter  (r,z) [e.g., mole fraction x  x c in binary mixture] (ii) B.C. : Surface fields, i.e., affinity of one component over the other at wall/fluid and fluid/vapor interfaces. As T  T c, critical adsorption at each wall. For sufficiently small t = (T – T c )/T c, correlation length  =  0 t  ~ film thickness L  Each wall starts to “feel” the presence of the other wall.  “Casimir effect”: film thinning (attractive) for (+,+) and film thickening (repulsive) for (+,  ) when t ~ °C 46.2 °C 45.6 °C From: Heady & Cahn, 1973 [9], T c =  0.01 °C x c =  x (PFMC mole fraction) Temperature [  C] PFMC rich MC rich Methylcyclohexane (MC) Perfluoro- methylcyclohexane (PFMC) Inner cell (   C) Outer cell (  0.03  C) Saturated MC + PFMC vapor Bulk reservoir: Critical MC + PFMC mixture ( x ~ x c = 0.36 ) at T = T rsv. MC + PFMC wetting film on Si(100) at T = T rsv +  T. z Incident X-rays = 1.54 Å (Cu K  ) q z = (4  / )sin(  ) Si (100) MC + PFMC L  Specular Reflection  T = 0.50 °C  T = 0.10 °C  T = °C T film [°C] Total film thickness L [Å] y = (L/  ) 1/ = t (L/  0 ) 1/ +,  = (k B T c )  1 [  L 3 – A eff /6  ] MFT 2  +,  (RG) y = (L/  ) 1/ = t (L/  0 ) 1/ +,  (+,  ) (+,+) 2  +,  2  +,+ MFT scaling functions for Casimir pressure, where the ordinate has been rescaled so that ½ +,± (0) =  +,± (RG) at y = 0. (Based on [3]) q z [Å  1 ] Normalized Reflectivity R/R F  T = 0.50 °C  T = 0.10 °C  T = °C At T film = 46.2 °C ~ T c Comparison with theory Film thickness L is determined by  =  (L) + p c (L, t) i.e., a balance between: (i) Chemical potential (per volume) of film relative to bulk liquid/vapor coexistence:   > 0 tends to reduce film thickness.  Can be calculated from  T and known latent heat of MC and PFMC. (ii) Non-critical (van der Waals) disjoining pressure:  = A eff /[6  L 3 ]  Effective Hamaker constant A eff > 0 for the MC/PFMC wetting films ( T > T wet ).   tends to increase film thickness.  A eff for mixed films can be estimated from densities in mixture and constants A ij estimated previously for pairs of pure materials [10]. (iii) Critical Casimir pressure: p c = [k B T c /L 3 ] +,  (y)  +,  > 0  p c tends to increase film thickness.  Scaling variable: y = (L/  ) 1/ = t(L/  0 ) 1/, where = and  0 + /  0  = 1.96 for 3D Ising systems [11], and  0 + = 2.79 Å ( T > T c ) for MC/PFMC [12]. Scaling function can be extracted experimentally from the measured L, using: Theoretical background Finite-size scaling and universal scaling functions (Fisher & de Gennes, 1978 [1]) Casimir energy/area: Casimir pressure: For each B.C., scaling functions  and are universal in the critical regime ( t  0,   , and L   ) [2]. Scaling functions have been calculated using mean field theory (MFT) (Krech, 1997 [3]). “Casimir amplitudes” at bulk T c ( t = 0 ), for 3D Ising systems: Recent observations of Casimir effect in critical fluid films Thickening of films of binary alcohol/alkane mixtures on Si near the consolute point. (Mukhopadhyay & Law, 1999 [6]) Thinning of 4 He films on Cu, near the superfluid transition. (Garcia & Chan, 1999 [7]) Thickening of binary 3 He/ 4 He films on Cu, near the triple point. (Garcia & Chan, 2002 [8]) Method  +,   +,+ RG: Migdal-Kadanoff procedure [4] RG:  = 4 – d expansion [3] 2.39  Monte Carlo simulations [3]  “Local free-energy functional” [5] 3.1  0.42    T = T film – T rsv Thickness measurements by x-ray reflectivity References: [1] M. E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. Paris, Ser. B 287, 209 (1978). [2] M. Krech and S. Dietrich, Phys. Rev. Lett. 66, 345 (1991); Phys. Rev. A 46, 1922 (1992); Phys Rev. A 46, 1886 (1992). [3] M. Krech, Phys. Rev. E 56, 1642 (1997). [4] J. O. Indekeu, M. P. Nightingale, and W. V. Wang, Phys. Rev. B 34, 330 (1986). [5] Z. Borjan and P. J. Upton, Phys. Rev. Lett. 81, 4911 (1998). [6] A. Mukhopadhyay and B. M. Law, Phys. Rev. Lett. 83, 772 (1999); Phys. Rev. E 62, 5201 (2000). [7] R. Garcia and M. H. W. Chan, Phys. Rev. Lett. 83, 1187 (1999). [8] R. Garcia and M. H. W. Chan, Phys. Rev. Lett. 88, (2002). [9] R. B. Heady and J. W. Cahn, J. Chem. Phys. 58, 896 (1973). [10] R. K. Heilmann, M. Fukuto, and P. S. Pershan, Phys. Rev. B 63, (2001). [11] A. J. Liu and M. E. Fisher, Physica A 156, 35 (1989). [12] J. W. Schmidt, Phys. Rev. A 41, 885 (1990). Work supported by Grant No. NSF-DMR  T = T film – T rsv [K]  +,  = ½ +,  (y = 0)  +,  (RG) At T film = 46.2 °C ~ T c Thickness enhancement near T c for small  T, with a maximum slightly below T c.  Qualitatively consistent with theoretically expected repulsive Casimir forces for (+,  ). Symbols are based on the measured L,  = (2.2  10  22 J/Å 3 )  T/T, and A eff = 1.2  10  19 J estimated for a homogeneous MC/PFMC film at bulk critical concentration x c = The red line (—) is for  T = °C. The dashed red line (---) for T < T c is based on A eff estimated for the case in which the film is divided in half into MC-rich and PFMC-rich layers at concentrations given by bulk miscibility gap. Summary: Both the extracted Casimir amplitude  +,  and scaling function +,  (y) appear to converge with decreasing  T (or increasing L ). This is consistent with the theoretical expectation of a universal behavior in the critical regime [2]. The Casimir amplitude  +,  extracted at T c and small  T agrees well with  +,  ~ 2.4 based on the renormalization group (RG) and Monte Carlo calculations by Krech [3]. The range over which the Casimir effect (or the thickness enhancement) is observed is narrower than the prediction based on mean field theory [3].