Reaction-diffusion process nonequilibrium thermodynamics

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

A microanalysis study of the eutectoid decomposition of austenite into ferrite and M2C (to bainite) at the bay in Fe-0.24C-4Mo is reported by [2003Hacl]. It was concluded that alloy element partition between ferrite and alloy carbides at the reaction front is largely responsible for the slow kinetics in this and related alloys. A thermodynamic analysis showed that ferrite-carbide interfacial energy and nonequilibrium carbide compositions reduce the thermodynamic driving force for diffusion processes (Mo partition) by up to 20% further slowing the kinetics. 

Modeling of spatiotemporal evolution may serve as a powerful complementary tool for studying experimental nonisothermal reaction-diffusion systems within a porous catalyst particle and a membrane. The linear nonequilibrium thermodynamics approach may be used in modeling coupled nonisothermal reaction-diffusion systems when the system is in the vicinity of global equilibrium. In the modeling, the information on coupling mechanisms among transport processes and chemical reactions is not needed. 

These equations display the spatial order with the thermodynamically coupled heat and mass flows. Here, the coupling between chemical reactions and transport processes of heat and mass is excluded. The analysis of reaction-diffusion systems would be more complete if we consider heat effects and coupling among fluxes of mass and heat. The nonequilibrium thermodynamics approach is useful for iucorporating the coupling phenomena into reaction-diffusion systems (Demirel, 2006). 

The equation above is the Fokker-Planck equation to estimate the evolution of the probability density in space. Various forms of the Fokker-Planck equations result from various expressions of the work done on the systems, and are used in diverse applications, such as reaction diffusion and polymer solutions (Rubi, 2008 Bedeaux et al., 2010 Rubi and Perez-Madrid, 2001). A process may lead to variations in the conformation of the macromolecules that can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics, and applied to transport and relaxation phenomena and polymer solutions (Santamaria-Holek and Rubi, 2003). 

Based on our previous studies of the dissolution and crystallization kinetics of potassium inorganic compounds based on linear nonequilibrium thermodynamics (Ji et al, 2010 Liu et al, 2009 Lu et al, 2011), we proposed to assume that the kinetic process of CO2 absorption by ILs comprised two steps surface reaction and diffusion, as shown in Fig. 17. Figure 17 demonstrates that when CO2 in the vapor phase and the ILs were in contact, the chemical reaction of CO2 with ILs occurred for the chemical absorption process of CO2 by ILs in the first step, which was named as the surface reaction layer, while for the physical mass transport process of CO2 by ILs in the first step, CO2 in the vapor phase would be transported into the IL phase, which was also named as the assumed surface reaction layer. As for the surface reaction layer, the driving force of the surface reaction was the chemical potential gradient of CO2 between CO2 at the vapor—Hquid interface and gas CO2. After that, in the second step, CO2 in the IL phase would... 

As mentioned before, nonequilibrium thermodynamics could be used to study the entropy generated by an irreversible process (Prigogine, 1945, 1947). The concept ofhnear nonequilibrium thermodynamics is that when the system is close to equilibrium, the hnear relationship can be obtained between the flux and the driving force (Demirel and Sandler, 2004 Lu et al, 2011). Based on our previous linear nonequihbrium thermodynamic studies on the dissolution and crystallization kinetics of potassium inorganic compounds (Ji et al, 2010 Liu et al, 2009 Lu et al, 2011), the nonequihbrium thermodynamic model of CO2 absorption and desorption kinetics by ILs could be studied. Figure 17 shows the schematic diagram of CO2 absorption kinetic process by ILs. In our work, the surface reaction mass transport rate and diffusion mass transport rate were described using the Hnear nonequihbrium thermodynamic theory. 

Effects of the non-ideality of adsorbate are incorporated here through the introduction of a dependence of potential V, diffusion coefficient and rate constants of chemical reactions in the operator X. on the distribution function gc- These dependencies can be found from dynamical models of elementary processes, statistical thermodynamics of equilibrium and nonequilibrium processes, and from experimental data (see, e.g., (Croxton 1974)). 

Phase transition occurs at a state of thermodynamic equilibrium, inducing a change in the microstructure of atoms. However, corrosion is a typical nonequilibrium phenomenon accompanied by diffusion and reaction processes. We can also observe that this phenomenon is characterized by much larger scales of length than an atomic order (i.e., masses of a lot of atoms), which is obvious if we can see the morphological change in the pitted surface. 

Recently, kinetic models have been combined with the equilibrium data of the interfacial processes, taking into account that soils and rocks are heterogeneous and consequently have different sites. These models are called nonequilibrium models (Wu and Gschwend 1986 Miller and Pedit 1992 Pedit and Miller 1993 Fuller et al. 1993 Sparks 2003 Table 7.2). These models describe processes when a fast reaction (physical or chemical) is followed by one or more slower reactions. In these cases, Fick s second law is expressed—that the diffusion coefficient is corrected by an equilibrium thermodynamic parameter of the fast reaction (e.g., by a distribution coefficient), that is, the fast reaction is always assumed to be in equilibrium. In this way, the net processes are characterized by apparent diffusion coefficients. However, such reactions can be equally well described using Equation 1.126. 

Another method used to make amorphous solids is thermal amorphization by interdiffusing solid crystalline reactants. In certain cases (if formation of the crystal is kinetically frustrated) an amorphous reaction product is formed from crystalline reactants. This occurs if (a) the two reactants have a high affinity for each other or, in other words, if the reaction has a high reaction free energy (b) one of the reactants only is able to diffuse easily in the other at the reaction temperature, which must be low (50-200°C), much lower than the crystallization temperature of the product. Couples of elements from groups 4 (Zr, Hf) and La, B, or H with group 10 (Ni, Pd) elements and Au, Co, and Rh from the periodic table are suitable. The end product of this thermal process is not the crystalline state with the lowest possible thermodynamic potential but an amorphous nonequilibrium modification. The product cannot crystallize because one of the reactants is immobile at the reaction temperature. 

---