1. Critical crossover phenomena in fluids Jan V. Sengers Institute for Physical Science & Technology University of Maryland, College Park, Md 20742 A celebration of 50 years Statistical Physics Rutgers University, December 2008

2. 1955-1962: Graduate student at University of Amsterdam Time of expectations in NONEQUILIBRIUM STATISTICAL PHYSICS: 1872: Boltzmann 1922: Enskog 1946: Bogoliubov 1958: Uhlenbeck

3. CRITICAL SLOWING DOWN OF FLUCTUATIONS Classical Van Hove theory

4. J.V. Sengers and A. Michels (1962) M.I. Bagatskii, A.V. Voronel, V.G. Gusak (1962): C V of argon diverges at critical point W.M. Fairbank, M.J. Buckingham, C.F. Kellers (1960): C V of liquid helium diverges at lambda line M.R. Moldover, W.A. Little (1965): C V of He 3 and He 4 diverges at critical point Coexistence curves are not parabolic

5. Universality of critical phenomena? (M.S. Green, L.P. Kadanoff) Critical exponents (M.E. Fisher) Lattice models (C. Domb et al.) Mode-coupling theory (Fixman,Kadanoff, Swift, Kawasaki) Renormalization-group theory (K.G. Wilson, M.E. Fisher) Scaling in fluids (B. Widom)

7. Critical power laws and universality ξ is correlation length χ = (  M/  H) T is susceptibility τ = (T  T c )/T ξ = ξ 0 τ  ν χ = χ 0  γ Ising like Classical ν  0.63 ν = 0.50 γ  1.24 γ = 1.00

8. Isobutyric acid + water J.G. Shanks Ph.D. Thesis (1986)

9. All fluids, simple and complex, are asymptotically Ising CONSENSUS: Peter Debye at first conference on critical phenomena in 1965 in Washington DC “I would like that the theoretical people tell me when I am so and so far away from the critical point, then my curve should look so and so.” (p. 129).

10. Crossover critical behavior Effective critical exponent*: Ising limit: γ eff = 1.24 Mean-field limit: γ eff = 1.00 *Kouvel and Fisher (1964)

11. Anisimov, Povodyrev, Kulikov, Sengers, Phys. Rev. Lett. 75, 3146 (1995) 3-methylpentane+ nitroethane tetra-n-butyl ammonium picrate+ 1,4-butanediol/ 1-dodecanol xenon isobutyric acid + water

12. Crossover theory TWO crossover parameters: Λ is dimensionless cutoff wave number related to a length ξ D = v 0 1/3 /πΛ Ginzburg number N G  First nonasymptotic correction  Z.Y Chen, P.C. Albright, J.V. Sengers, Phys. Rev. A 41, 3161 (1990) M.A. Anisimov, S.B. Kiselev, J.V. Sengers, S. Tang, Physica A 188, 487 (1992)

13. Gutkowski, Anisimov, Sengers, J. Chem. Phys. 114, 3133 (2001)

14. Y.C. Kim, M.A. Anisimov, J.V. Sengers, E. Luijten, J. Stat. Phys. 109, 591 (2003) [Computer simulations: E. Luijten and K. Binder, Phys. Rev. E (1999)]

15. Phase diagram for polymer solutions Theta-point = limiting point for line of critical points = tricritical point  (almost) mean-field Critical point of demixing  fluctuations induced behavior (Ising) Tc(N)Tc(N) phase separation boundaries   -point T N -1

16. Anisimov, Kostko, Sengers, Yudin, J. Chem. Phys. 123, 164901 (2005)

19. Acknowledgments Mikhail A. Anisimov Andrei F. Kostko Thanks, Joel, for shepherding our statistical-physics community for 50 years And last but not least: