Theory

Expressions for the nonequilibrium fluctuations in liquids can be obtained on the basis of fluctuating hydrodynamics [ 11,121. Fluctuating hydrodynamics assumes that the fluctuations in the local temperature T, the local fluid velocity u, and the local concentration c satisfy the linearized hydrodynamic equations supplemented with random force terms [7,12]. Crucial in its application to fluctuations in nonequilibrium states is the assumption that the correlation functions of the random force terms retain their local equilibrium value. The excellent agreement obtained between theory and experiment for the nonequilibrium temperature and fluctuations in one-component liquids has confirmed the validity of this approach [5,9]. Hence, it is expected that fluctuating hydrodynamics provides also a suitable approach for dealing with non-equilibrium concentration fluctuations, although it should be noted that complete agreement between theory and experiment in the case of liquid mixtures has not yet been obtained [9, 13,14].

Law and Nieuwoudt [7] first derived an expression for the concentration fluctuations in a liquid mixture subjected to a temperature gradient from fluctuating hydrodynamics. The theory was further developed by Segre and Sengers [15] to include the effects of gravity and by Vailati and Giglio [16] to include time-dependent states. The theory can also be extended to colloidal suspensions [ 17] and to polymer solutions provided that in the latter case the solutions have concentrations where entanglement effects on the hydrodynamics equations can be neglected [18].

A temperature gradient AT induces a concentration gradient A w through the Soret effect such that [ 19,20]

where ST is the Soret coefficient and where w is the concentration expressed as mass fraction of the polymer. Fluctuating hydrodynamics predicts that the concentration fluctuations will couple with the transverse-velocity fluctuations through the induced concentration gradient. This coupling leads to an enhancement of the concentration fluctuations when the transverse-velocity fluctuations have a component in the direction of the concentration gradient. Hence, the non-equilibrium enhancement reaches a maximum for concentration fluctuations with a wave vector k perpendicular to the concentration gradient. Since the concentration fluctuations do not propagate but are diffusive, they will be insensitive to the upward or downward direction of the perpendicular concentration gradient; the enhancement of the concentration fluctuations will be proportional to the square of the concentration gradient and, hence, proportional to ( A T)2 and S 2xin Eq. (1). Moreover, as in the case ofnonequilibrium temperature and viscous fluctuations observed in liquids and liquid mixtures [8,9],the enhancement ofthe concentration fluctuations will be inversely proportional to k4, where k is the wave number of the concentration fluctuations. The divergence of the nonequilibrium enhancement ofthe fluctuations at small k means that the fluctuations become long ranged. It is now believed that one will always encounter long-rangepower-law correlations in stationary nonequilibrium states [6]. Inpractice there will be a low wave-number cutoffdue to gravity or to finite-size effects [ 10,15,21].

The nonequilibrium concentration fluctuations can be measured experimentally by dynamic light scattering. Fluctuating hydrodynamics predicts that the time-dependent correlations function C(k,t) ofthe scattered light is given by

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