Thermodynamics Interaction Studies Solids, Liquids and Gases 2011 Part 8 ppt

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In analyzing the current state of thermodynamic theory, we indicate critically important refinements necessary to use non-equilibrium thermodynamics and statistical mechanics in the application to material transport in non- isothermal mixtures. 2. Thermodynamic theory of material transport in liquid mixtures: Role of the Gibbs-Duhem equation The aim of this section is to outline the thermodynamic approach to material transport in mixtures of different components. The approach is based on the principle of local equilibrium, which assumes that thermodynamic principles hold in a small volume within a non-equilibrium system. Consequently, a small volume containing a macroscopic number of particles within a non-equilibrium system can be treated as an equilibrium system. A detailed discussion on this topic and references to earlier work are given by Gyarmati (1970). The conditions required for the validity of such a system are that both the temperature and molecular velocity of the particles change little over the scale of molecular length or mean free path (the latter change being small relative to the speed of sound). For a gas, these conditions are met with a temperature gradient below 10 4 K cm -1 ; for a liquid, where the heat conductivity is greater, the speed of sound higher and the mean free path is small, this condition for local equilibrium is more than fulfilled, provided the experimental temperature gradient is below 10 4 K cm -1 . Thermodynamic expressions for material transport in liquids have been established based on equilibrium thermodynamics (Gibbs and Gibbs-Duhem equations), as well as on the principles of non-equilibrium thermodynamics (thermodynamic forces and fluxes). For a review of these models, see (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969). Thermodynamics – Interaction Studies – Solids, Liquids and Gases 344 Non-equilibrium thermodynamics is based on the entropy production expression                1 1 N i ei i JJ TT (1) where  e J is the energy flux,  i J are the component material fluxes, N is the number of the components,  i are the chemical potentials of components, and T is the temperature. The energy flux and the temperature distribution in the liquid are assumed to be known, whereas the material concentrations are determined by the continuity equations      i i n J t (2) Here i n is the numerical volume concentration of component i and t is time. Non- equilibrium thermodynamics defines the material flux as             1 i iii iiQ JnL nL TT (3) where L i and L iQ are individual molecular kinetic coefficients. The second term on the right- hand side of Eq. (3) represents the cross effect between material flux and heat flux. The chemical potentials are expressed through component concentrations and other physical parameters (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999):          2 1 kk lk l l nvP T nT (4) Here P is the internal macroscopic pressure of the system and     kk vPis the partial molecular volume, which is nearly equivalent to the specific molecular volume k v . Substituting Eq. (4) into Eq. (3), and using parameter  iiQi q LL, termed the molecular heat of transport, we obtain the equation for component material flux:                         1 N ii ii i i iki k k q nL JnvPT Tn TT (5) Defining the relation between the heat of transport and thermodynamic parameters is a key problem because the Soret coefficient, which is the parameter that characterizes the distribution of components concentrations in a temperature gradient, is expressed through the heat of transport (De Groot, 1952; De Groot, Mazur, 1962). A number of studies that offer approaches to calculating the heat of transport are cited in (Pan S et al., 2007). Eq. (5) must be augmented by an equation for the macroscopic pressure gradient in the system. The simplest possible approach is to consider the pressure to be constant (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau , Lifshitz, 1959), but pressure cannot be constant in a system with a non-uniform temperature and concentration. This issue is addressed with a well-known expression referred to as the Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 345 Gibbs-Duhem equation (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Landau, Lifshitz, 1959; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):             11 NN ii ik k ik Pn n T nT (6) The Gibbs-Duhem equation defines the macroscopic pressure gradient in a thermodynamic system. In equilibrium thermodynamics the equation defines the potentiality of the thermodynamic functions (Kondepudi, Prigogine, 1999). In equilibrium thermodynamics the change in the thermodynamic function is determined only by the initial and final states of the systems, without consideration of the transition process itself. In non-equilibrium thermodynamics, Eq. (5) plays the role of expressing mechanical equilibrium in the system. According to the Prigogine theorem (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969), pressure gradient cancels the volume forces expressed as the gradients of the chemical potentials and provides mechanical equilibrium in a thermodynamically stable system. However, in a non-isothermal system, the same authors considered a constant pressure and the left- and right-hand side of Eq. (6) were assumed to be zero simultaneously, which is both physically and mathematically invalid. Substituting Eq. (6) into Eq. (5) we obtain the following equation for material flux:                                     11 1 NNN ii i i ik k k iik lii ik kl kkil Lv T JTTq vT T v T T (7) In Eq. (7), the numeric volume concentrations of the components are replaced by their volume fractions   iii nv , which obey the equation     1 1 N i i (8) Using Eq. (8) and the standard rule of differentiation of a composite function                      1 1 1 , ,, 2 kl l kk k ll l l llll (9) we can eliminate  1 and obtain Eq. (7) in a more compact form:                               1 2 NN ii ik ik ikl ii il kl L T JTq Tv T T (10) Here  1 is expressed through the other volume fractions using Eq. (8), and the following combined chemical potential is introduced:     i ik i k k v v (11) We note that the volume fraction selected for elimination is arbitrary (any other volume fraction can be eliminated in the same manner), and that in subsequent mathematical Thermodynamics – Interaction Studies – Solids, Liquids and Gases 346 expressions, we express the volume fraction of the first component through that of the others using Eq. (8). Equations for the material fluxes are usually augmented by the following equation, which relates the material fluxes of components (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969; Ghorayeb, Firoozabadi, 2000; Pan S et al., 2007):     1 0 N ii i vJ (12) Eq. (12) expresses the conservation mass in the considered system and the absence of any hydrodynamic mass transfer. Also, Eq. (12) is used to eliminate one of the components from the series of component fluxes expressed by Eq. (10). That material flux that is replaced in this way is arbitrary, and the resulting concentration distribution will depend on which flux is selected. The result is not significant in a dilute system, but in non-dilute systems this practice renders an ambiguous description of the material transport processes. In addition to being mathematically inconsistent with Eq. (12) because there are N+1 equations [i.e., N Eq. (10) plus Eq. (12)] for N-1 independent component concentrations, Eq. (10) predicts a drift in a pure liquid subjected to a temperature gradient. Thus, at   1 i Eq. (10) predicts        ii i i i q L T J Tv T (13) This result contradicts the basic principle of local equilibrium, and the notion of thermodiffusion as an effect that takes place in mixtures only. Moreover, Eq. (13) indicates that the achievement of a stationary state in a closed system is impossible, since material transport will occur even in a pure liquid. The contradiction that a system cannot reach a stationary state, as expressed in Eq. (13), can be eliminated if we assume    ii q (14) With such an assumption Eq. (10) can be cast in the following form:                    1 2 NN iik ik ik il il kl L JT Tv T (15) Because the kinetic coefficients are usually calculated independently from thermodynamics, the material fluxes expressed by Eq. (15) cannot satisfy Eq. (12) for the general case. But in a closed and stationary system, where   0 i J , Eqs. (12) and (15) become consistent. In this case, any component flux can be expressed by Eq. (15) through summation of the other equations. The condition of mechanical equilibrium for an isothermal homogeneous system, as well as the use of Eqs. (l) – (6) for non-isothermal systems, are closely related to the principle of local equilibrium (De Groot, 1952; De Groot, Mazur, 1962; Kondepudi, Prigogine, 1999; Haase, 1969). As argued in (Duhr, Braun, 2006; Weinert, Braun; 2008), thermodiffusion violates local equilibrium because the change in free energy across a particle is typically comparable to the thermal energy of the particle. However, their calculations predict that Statistical Thermodynamics of Material Transport in Non-Isothermal Mixtures 347 even for large (micron size) particles, the energy difference is no more than a few percent of kT. But the local equilibrium is determined by processes at molecular level, as will be discussed below, and this argumentation cannot be accepted. 3. Dynamic pressure gradient in open and non-stationary systems: Thermodynamic equations of material transport with the Soret coefficient as a thermodynamic parameter Expressing the heats of transport by Eq. (14), we derived a set of consistent equations for material transport in a stationary closed system. However, expression for the heat of transport itself cannot yield consistent equations for material transport in a non-stationary and open system. In an open system, the flux of a component may be nonzero because of transport across the system boundaries. Also, in a closed system that is non-stationary, the component material fluxes  i J can be nonzero even though the total material flux in the system,     1 N ii i JvJ , is zero. In both these cases, the Gibbs-Duhem equation can no longer be used to determine the pressure in the system, and an alternate approach is necessary. In previous works (Schimpf, Semenov, 2004; Semenov, Schimpf, 2005), we combined hydrodynamic calculations of the kinetic coefficients with the Fokker-Planck equations to obtain material transfer equations that contain dynamic parameters such as the cross- diffusion and thermal diffusion coefficients. In that approach, the macroscopic gradient of pressure in a binary system was calculated from equations of continuity of the same type as expressed by Eqs. (2) and (8). This same approach may be used for solving the material transport equations obtained by non-equilibrium thermodynamics. In this approach, the continuity equations [Eq. (2)] are first expressed in the form                    1 2 N iii i i ki k k L vP T tT T (16) Summing Eq. (16) for each component and utilizing Eq. (8) we obtain the following equation for the dynamic pressure gradient in an open non-stationary system:                        11 1 2 NN N ii ii k iii k ik i PJT L T Lv T (17) Substituting Eq. (17) into Eq. (16) we obtain the material transport equations:                                      11 1 2 NN N ij ij iii jjj kkkk k jk k L JT v L T v L T tT (18) Comparing Eq. (18) with Eq. (15) for a stationary mixture shows that former contains an additional drift term       1 iii N kkk k vLJ vL proportional to the total material flux through the open Thermodynamics – Interaction Studies – Solids, Liquids and Gases 348 system. The term     1 N kkk k JT vL in Eq. (17) describes the contribution of that drift to the pressure gradient. This additional component of the total material flux is attributed to barodiffusion, which is driven by the dynamic pressure gradient defined by Eq. (17). This dynamic pressure gradient is associated with viscous dissipation in the system. Parameter  J is independent of position in the system but is determined by material transfer across the system boundaries, which may vary over time. If the system is open but stationary, molecules entering it through one of its boundary surfaces can leave it through another, thus creating a molecular drift that is independent of the existence of a temperature or pressure gradient. This drift is determined by conditions at the boundaries and is independent of any force applied to the system. For example, the system may have a component source at one boundary and a sink of the same component at opposite boundary. As molecules of a given species move between the two boundaries, they experience viscous friction, which creates a dynamic pressure gradient that induces barodiffusion in all molecular species. The pressure gradient that is induced by viscous friction in such a system is not considered in the Gibbs-Duhem equation. Equations (6), (7), and (15) describe a system in hydrostatic equilibrium, without viscous friction caused by material flux due to material exchange through the system boundaries. Unlike the Gibbs-Duhem equation, Eq. (17) accounts for viscous friction forces and the resulting dynamic pressure gradient. For a closed stationary system, in which   0J and     0 t , Eq. (18) is transformed into                 11 1 20 NN N ij ij k k kj j T T (19) There are thermal diffusion experiments in which the system experiences periodic temperature changes. An example is the method used described by (Wiegand, Kohler, 2002), where thermodiffusion in liquids is observed within a dynamic temperature grating produced using a pulsed infrared laser. Because this technique involves changing the wall temperature, which changes the equilibrium adsorption constant, material fluxes vary with time, creating a periodicity in the inflow and outflow of material. A preliminary analysis shows that material fluxes to and from the walls have relaxation times on the order of a few microseconds until equilibrium is attained, and that such non-stationary material fluxes can be observed using dynamic temperature gratings. The Soret coefficient is a common parameter used to characterize material transport in temperature gradients. For binary systems, Eq. (19) can be used to define the Soret coefficient as              2 21 21 22 21 P T P T S (20) [...]... al, 1995), and the curvature of the particle surface can be ignored in calculating the respective integrals This corresponds to the assumption that r '  R and dv  4 R 2 dr in Eq (36) To calculate the Hamaker potential, the expression calculated in (Ross, Morrison, 1 988 ), which is based on the London potential, can be used: 356 Thermodynamics – Interaction Studies – Solids, Liquids and Gases  i*1... dilution, where the standard expression for osmotic pressure is used These results contradict empirical observation Using Eq (27) with the notion of a virtual particle outlined above, and substituting the expression for interaction potential [Eqs (24, 28) ], we can write the combined chemical * potential at constant volume  V as 362 Thermodynamics – Interaction Studies – Solids, Liquids and Gases 3 Z2  m2... Thermodynamics – Interaction Studies – Solids, Liquids and Gases Pan S et al (2007) Theoretical approach to evaluate thermodiffusion in aqueous alkanol solutions The Journal of Chemical Physics, Vol 126, No 1 (January 2007), 014502 (12 pages) Parola, A., Piazza, R (2004) Particle thermophoresis in liquids The European Physical Journal, Vol.15, No 11(November2004), 255-263 Ross, S and Morrison, I D (1 988 ) Colloidal... platform relying on the thermodynamics of surfaces (Linford, 1973) and configurational mechanics (Maugin, 1993) 370 Thermodynamics – Interaction Studies – Solids, Liquids and Gases for the treatment of surface growth phenomena in a biomechanical context A typical situation is the external remodeling in long bones, which is induced by genetic and epigenetic factors, such as mechanical and chemical stimulations... are referred to as Hamaker potential, and are used in studies of interactions between colloidal particles (Hunter, 1992; Ross, Morrison, 1 988 ) In this and the following sections, vi is the specific molecular volume of the atom or molecule in a real or virtual particle, respectively For a colloidal particle with radius R >>  ij , the temperature distribution at the particle surface can be used instead... water with certain alcohols, where a change of sign was observed (Ning, Wiegand, 2006) 9 Conclusion Upon refinement, a model for thermodiffusion in liquids based on non-equilibrium thermodynamics yields a system of consistent equations for providing an unambiguous 364 Thermodynamics – Interaction Studies – Solids, Liquids and Gases description of material transport in closed stationary systems The macroscopic... progress of the jth chemical reaction with J k the flux of species k and 372 Thermodynamics – Interaction Studies – Solids, Liquids and Gases r  i nk    kj j t j 1 (7) The two previous equalities enter into Gibbs relation as      u   se   si  σ : ε    k k  M divJ k    k k  M  kjj (8) j with  the temperature and  k the chemical potential of constituent k The chemical  affinity... (2000) Molecular, pressure, and thermal diffusion in nonideal multicomponent mixtures AIChE Journal, Vol 46, No 5 ( May 2000), 88 3 89 1 Giddings, J C et al (1995) Thermophoresis of Metal Particles in a Liquid The Journal of Colloid and Interface Science Vol 176, No 454-4 58 Gyarmati, I (1970) Non-Equilibrium Thermodynamics Springer Verlag, Berlin, Germany Haase, R (1969) Thermodynamics of Irreversible... n  2 R      2  r '  e dr '   kT 2  (63) Here n is again the ratio of particle to solvent thermal conductivity For low potentials (  e  kT ), where the Debye-Hueckel theory should work, Eq (63) takes the form 360 Thermodynamics – Interaction Studies – Solids, Liquids and Gases  r e  2  r '  2 P 8 ns kR e   dr   kT 2 dr ' T n  2 R  (64) Using an exponential distribution... M., Braun, D (20 08) Observation of Slip Flow in Thermophoresis Physical Review Letters Vol 101 (October 20 08) , 1 683 01(4 pages) Semenov, S N., Schimpf, M E (2011) Internal degrees of freedom, molecular symmetry and thermodiffusion Comptes Rendus Mecanique, doi:10.1016/j.crme .2011. 03.011 Semenov, S N., Schimpf, M E (2011) .Thermodynamics of mass transport in diluted suspensions of charged particles in non-isothermal . expression calculated in (Ross, Morrison, 1 988 ), which is based on the London potential, can be used: Thermodynamics – Interaction Studies – Solids, Liquids and Gases 356           3 * 1 21 1 2 11 ln 622 i i y y v yy. 1962; Kondepudi, Prigogine, 1999; Haase, 1969). Thermodynamics – Interaction Studies – Solids, Liquids and Gases 344 Non-equilibrium thermodynamics is based on the entropy production expression. fraction can be eliminated in the same manner), and that in subsequent mathematical Thermodynamics – Interaction Studies – Solids, Liquids and Gases 346 expressions, we express the volume