System and experimental setup Studied a wetting film of binary mixture MC/PFMC on Si(100) ,

1. Perfluoro- methylcyclohexane (PFMC) Methylcyclohexane (MC) Outer cell (0.03C) MC + PFMC wetting film on Si(100) at T = Trsv+DT. Inner cell (0.001C) z Saturated MC + PFMC vapor Bulk reservoir: Critical MC + PFMC mixture (x ~ xc = 0.36) atT = Trsv. At Tfilm = 46.2 °C ~ Tc Temperature [C] MC rich PFMC rich From: Heady & Cahn, 1973 [9], Tc = 46.13  0.01 °C xc = 0.361  0.002 x (PFMC mole fraction) DT = 0.020 °C Total film thickness L [Å] Normalized Reflectivity R/RF At Tfilm = 46.2 °C ~ Tc DT = 0.10 °C MFT D+,- (RG) 2D+,- (RG) DT = 0.50 °C +,- = (kBTc)-1 [DmL3 – Aeff/6p] DT = 0.020 °C Tfilm [°C] qz [Å-1] DT = 0.10 °C y = (L/x)1/n = t (L/x0)1/n DT = 0.50 °C D+,- = ½ +,- (y = 0) (+,-) 2D+,- qz= (4p/l)sin(a) MFT scaling functions for Casimir pressure, where the ordinate has been rescaled so that ½+,±(0) = D+,±(RG) at y = 0. (Based on [3]) +, Dm DT = Tfilm – Trsv 47.7 °C Incident X-rays l = 1.54 Å (Cu Ka) Specular Reflection DT = Tfilm – Trsv [K] (+,+) a a 2D+,+ L MC + PFMC Si (100) y = (L/x)1/n = t (L/x0)1/n 46.2 °C 45.6 °C 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 • System and experimental setup • Studied a wetting film of binary mixtureMC/PFMC onSi(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 • that MC-rich liquid wets the liquid/Si interface and PFMC is • favored at the liquid/vapor interface. • Intrinsic chemical potential Dm of the film relative to bulk • liquid/vapor coexistence was controlled by temperature offset • DT between the substrate and liquid reservoir [10]. Thickness measurements by x-ray reflectivity • Comparison with theory • Film thickness L is determined by • Dm = P(L) + pc(L, t) • i.e., a balance between: • (i) Chemical potential (per volume) of film relative to bulk liquid/vapor coexistence: •  Dm > 0tends to reduce film thickness. •  Can be calculated from DT and known latent heat of MC and PFMC. • (ii) Non-critical (van der Waals) disjoining pressure: P = Aeff/[6pL3] •  Effective Hamaker constant Aeff > 0 for the MC/PFMC wetting films (T > Twet). •  P tends to increase film thickness. •  Aeff for mixed films can be estimated from densities in mixture and • constants Aij estimated previously for pairs of pure materials [10]. • (iii) Critical Casimir pressure: pc = [kBTc/L3]+,-(y) •  +,- > 0  pc tends to increase film thickness. •  Scaling variable: y = (L/x)1/n = t(L/x0)1/n, • where n = 0.632 and x0+/x0- = 1.96 for 3D Ising systems [11], and • x0+ = 2.79 Å (T > Tc) for MC/PFMC [12]. • Scaling function can be extracted experimentally from the measured L, using: • Casimir forces in adsorbed fluid films near bulk critical points • (i) Fluctuations: Local order parameterf(r,z) • [e.g., mole fraction x-xc 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 Tc, critical adsorption at each wall. • For sufficiently small t = (T –Tc)/Tc, • correlation lengthx = x0t-n ~ film thicknessL •  Each wall starts to “feel” the presence of the other wall. •  “Casimir effect”: film thinning (attractive) for (+,+) • and film thickening (repulsive) for (+,-) when t ~ 0. Symbols are based on the measured L, Dm = (2.2  10-22 J/Å3)DT/T, and Aeff = 1.2  10-19 J estimated for a homogeneous MC/PFMC film at bulk critical concentration xc = 0.36. The red line (—) is for DT = 0.020 °C. The dashed red line (---) for T < Tc is based on Aeff 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. • 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, x  , andL  ) [2]. • Scaling functions have been calculated using mean field • theory (MFT) (Krech, 1997 [3]). • “Casimir amplitudes” at bulk Tc (t= 0), • for 3D Ising systems: • Summary: • Both the extracted Casimir amplitude D+,- and scaling function +,-(y) appear to converge with decreasing DT (or increasing L). This is consistent with the theoretical expectation of a universal behavior in the critical regime [2]. • The Casimir amplitude D+,- extracted at Tc and small DT agrees well with D+,- ~ 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]. • Thickness enhancement near Tc for small • DT, with a maximum slightly below Tc. •  Qualitatively consistent with theoretically • expected repulsive Casimir forces for (+,-). • 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 4He films on Cu, near the superfluid transition. • (Garcia & Chan, 1999 [7]) • Thickening of binary 3He/4He films on Cu, near the triple point. • (Garcia & Chan, 2002 [8]) 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, 086101 (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, 205405 (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-01-24936.