Non-Equilibrium Fluctuations in Liquids

  (In the group of Profs. Jan V. Sengers and Bob Gammon, Physics Dept.,

IPST Building, Univ. of Maryland: Jan.'88-March '93.)

 
RECENT: Enhanced Fluctuations in a Polymer Solution in a Temperature Gradient. (PAPER)
 
                    The project concerns the behavior of fluctuations in density that occur in fluids that are in a non-equilibrium state through the application of a (stabilizing) temperature gradient.

            In thermal equilibrium in a pure fluid, local fluctuations in density (temperature) occur and give rise to a light scattering spectrum consisting of a diffusive 'mode' whose decay rate is given by G~DTq2 and whose amplitude is given by AE~KBT, where DT  is the thermal diffusivity and q the scattering wavevector. When a temperature gradient is applied, the fluid is in a non-equilibrium state and the spectrum changes considerably. First, the thermal 'mode' amplitude increases as AN~AE{1+aÑT2/q4}, and a new viscous mode appears, reflecting the interplay of velocity and temperature fluctuations that become coupled when a temperature gradient is present. It should also be noted that in our experiments the temperature gradient profoundly changed the fluctuation amplitudes, in some cases we found  AN was more than 100 times AE.

                The first experimental verification of the increasing thermal mode amplitude was done in ref. [1], with further refinements and more extensive data presented in [2] and [2a]. (For a schematic of the temperature gradient cell used in all of the experiments, click here). In a binary liquid mixture, there is in addition to a thermal mode, a concentration mode governed by the mass diffusivity D, (usually D<<DT ). Upon application of a temperature gradient, a concentration gradient is induced due to the Soret effect, and a corresponding fluctuation amplitude increase was seen in ref. [3]. Reference [4] details accurate measurements of the refractive index dependencies upon concentration and temperature necessary to interpret the theory for ref. [3]. References [5] and [6] solve the hydrodynamic equations for the fluctuations in both pure fluids and binary mixtures including the influence of gravity. One of the main results of these theory works was the prediction that gravity causes a saturation of the 1/q4 divergence at much lower values of q than we were able to study. This prediction was later confirmed by Vailati and Giglio as described below. 

 

Gravity can Stabilize or De-Stablize Fluids in Temperature Gradients. 

                   Our measurements of nonequilibrium fluctuations were never influenced by gravity, principally because gravity only affects q-vectors of much smaller magntudes than we were able to measure. Other groups however have explored this regime with in custom made annular solid state detector and shadow graph techniques. 
        In the first instance Vailati and Giglio heated the fluid from above and saw how gravity causes a saturation of the long-wavelength fluctuations (Full Paper) so that macroscopically the fluid is always quiescent. 
          In the second case Wu, Ahlers, and Cannel heated the fluid from below and saw how gravity causes a divergence of the long-wavelength fluctuations (Full Paper) that lead to convective motions at a critical Rayleigh number. 
           While at first glance these experiments and ours may look quite different, from a theoretical point of view they in fact are all described from one set of fluctuating hydrodynamics equations originally put forth by Landau and Lifshitz (Fluid Mechanics, L and L, Pergammon press, 1959). For a glance at this look at the final result for the nonequilibrium scattering intensity I(q) shown below (Zaitsev et al., and our ref. [5] below). The Rayleigh number ratio is given by R(q)/Rc ~gÑT/q4, (note the sign change upon heating from above or below), which reflects the importance of gravity. 
 

  

  • In our experiments q1000 cm-1, and R(q)/Rc ~ 0, we found I(q)~(Ac*) ÑT2/q4, strong q-dependence, but no gravity effects.
  • Vailati and Giglio had q100 cm-1  so R(q)/Rc varied from near to 0 to <<-1 and found I(q) crossed over from   I(q) ~(Ac*) ÑT2/q4 to I(q)~(Ac*)ÑT, i.e a saturation of fluctuations at the smallest q values, as shown below.     (They actually used binary liquid mixtures, see their references)
  • Wu, Ahlers, and Cannel had R(q)/Rc ~ +1 and found I(q) diverges at the critical wavenumber q=qc. 
      

  
                 Sengers group, ref. [2]  

     R(q)/Rc ~ 0, so no gravity effects 

  
             Vailati and Giglio, 1997 

Gravity induced saturation as  

      R(q)/Rc < - 1

  
Wu, Ahlers, and Cannel, 1997. 

Gravity induced divergence as  

   R(qc)/Rc approaches + 1.

 

  • V. M. Zaitsev and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583 (1970) [Sov. Phys. JETP 32, 866 (1971)].
  • Vailati A, Giglio M.,  Giant fluctuations in a free diffusion process, NATURE 390: (6657) 262-265 Nov. 20 1997
 

 
 
                   Related references by the Sengers and Gammon Group.
 

1.    "Light-Scattering observations of Long-Range Correlations in a Nonequilibrium Liquid"
                        B.M. Law, R.W. Gammon and J.V. Sengers, Phys. Rev. Lett. 60, 1554 (1988). Full Paper
 

2.    "Rayleigh Scattering in a Liquid Far from Thermal Equilibrium"
                        P.N. Segrè, R.W. Gammon, J.V. Sengers and B.M. Law, Phys. Rev. A 45, 714 (1992). Full Paper

                2 (a).    "Light-Scattering Measurements of Entropy and Viscous Fluctuations in a
                                Liquid Far from Thermal Equilibrium"
                                B.M. Law, P.N. Segrè R.W. Gammon, J.V. Sengers, Phys. Rev. A 41, 816 (1990). Full Paper

3.    "Light-Scattering Measurements of Nonequilibrium Fluctuations in a Liquid Mixture."
                        P.N. Segrè, R.W. Gammon and J.V. Sengers, Phys. Rev. E 47, 1026 (1993). Full Paper

            3 (a).    "Small-Angle Rayleigh Scattering from Nonequilibrium Fluctuations in liquids and
                            liquids and liquid mixtures"
                                W.B. Li, P.N. Segrè, R.W. Gammon and J.V. Sengers, Physica A 204, 399 (1994).   abstract
            3 (b).    "Nonequilibrium Fluctuations in liquids and liquid-mixtures subjected to a stationary
                            temperature gradient"
                               W.B. Li, P.N. Segrè, J.V. Sengers and R.W. Gammon, J. Phys. Cond. Mat. 6, 119 (1994).  Full Paper
            3 (c).    "Measurement of Transport Properties of Liquids with Equilibrium and Non-Equilibrium
                            Rayleigh Scattering"
                               W.B. Li, J.V. Sengers, R.W. Gammon and P.N. Segrè, Int. J. Thermo. 16, 23 (1995). abstract
 
4.    "Determination of the Temperature and Concentration Dependence of the Refractive Index of a Liquid Mixture"
                        W.B. Li, P.N. Segrè, R.W. Gammon, J.V. Sengers and M. Lamvik,
                                J. Chem. Phys. 101, 5058 (1994). abstract

5.    "Fluctuations in Inhomogeneous and Non-Equilibrium Fluids under the Influence
        of Gravity"   (Theory)
            P.N. Segrè, R. Schmitz and J.V. Sengers, Physica A 195, 31 (1993). abstract

6.    "Nonequilibrium Fluctuations in Liquid Mixtures under the Influence of Gravity" (Theory)
               P.N. Segrè and J.V. Sengers, Physica A 198, 46 (1993). abstract