Hysteresis, domain theory

Analysis of behavior in single pores is certainly an excellent place to start an understanding of adsorption hysteresis. On the other hand, real porous materials eu e in most cases not simply described in terms of single pore behavior. At the very least a distribution of pores of different sizes should be contemplated. The first analysis of hysteresis loops using a theory of adsorption in single pores together with a pore size distribution was the independent domain theory of Everett and coworkers (Everett, 1967). The most sophisticated application of this kind of approach was made by Ball and Evans (1989) who used density functional theory for adsorption in a distribution of cylindrical pores and compared the hysteresis loops obtained with those for xenon adsorbed in Vycor glass. 

Mualem, Y. 1984. A modified dependent domain theory of hysteresis. Soil Science 137(5) 283-291. 

More generally, the dynamic behavior of domain walls in random media under the influence of a periodic external field gives rise to hysteresis cycles of different shape depending on various external parameters. According to a recent theory of Nattermann et al. [54] on disordered ferroic (ferromagnetic or fe) materials, the polarization, P, is expected to display a number of different features as a function of T, frequency, / = iv/2tt, and probing ac field amplitude, E0. They are described by a series of dynamical phase transitions, whose order parameter Q = uj/2h) Pdt reflects the shape of the P vs. E loop. When increasing the ac... 

The values of X (within the domain of interest) at which the number of solutions of Eq. (1) changes are called bifurcation points. At these points F 9F/3x = 0. Using bifurcation theory it can be shown that the nature of a bifurcation diagram can change only if the parameter values cross one of three hypersurfaces [3]. The first called the Hysteresis variety (H) is the set of all points in the parameter space satisfying... 

Smith, R.C, and Horn, C. L. 1999. Domain wall theory for ferroelectric hysteresis. Journal of Intellegent Material Systems and Structures, 10(3)pp. 195-213. 

Thermodynamic and theoretical models for hysteresis and kinetics of phase reactions in the Pr and Tb oxide systems have been proposed. In the first study by Knittel et al. (1977) a model based on regular solution theory is developed and applied to hysteresis in the Pr and Tb oxide systems. Maren et al. (1984) modeled the Pr70 2-PrgOi6 hysteretic reaction using a thermodynamic formalism. Models assuming both noninteracting and interacting domains were considered. 

In this chapter, some recent studies on moisture transport processes in paper materials are reviewed. The equilibrium aspect of moisture interaction with paper shows significant hysteresis which can be estimated by an application of Everett s theory of independent domain complexions. Thus, when a paper sheet is subjected to arbitrary cycles of humidity all the while allowed to reach equilibrium at each state, the sheet s moisture content evolution may be predicted by an analysis of the sorption isotherms and the interior of the sorption hysteresis loop. It is shown that the theoretical predictions of equilibrium moisture content are in good agreement with experimentally determined values. 

In this chapter, we review some of our recent studies on moisture equilibrium and transport in paper materials. Equilibrium sorption is first studied with particular reference to sorption hysteresis. We found that the evolution of the equilibrium moisture content of a paper sheet subjected to arbitrary humidity changes can be predicted by applying the theory of independent domain... 

The sorption equilibrium of moisture in paper and board has been widely studied. However, only recently has it been recognized that a close examination of the interior of the hysteresis loop is necessary in order to understand the complete sorption behavior of paper materials in order to model the unsteady moisture sorption process under cyclic humidity variations [ 31]. Such an investigation of sorption hysteresis was reported by Chatterjee et al. [32]. In the following, we provide a description of the hysteresis and its representation using the theory of independent domains for the case of paper. 

The detailed calculation from experimental data is presented in [32], Table 4 shows various trajectories estimated by applying the theory of independent domain complexions and using the moisture density function generated in Table 3. Close agreement between experimental and theoretical predictions are found indicating that Everett s theory is a valid description of the hysteresis in paper materials. 

An interesting point of comparison between the theoretical predictions and the experimental data may be made. For the case of trajectories starting from an initial wet state (Tl, T2, and T3), the theory underestimates the actual moisture content. However, when the trajectories start from an initial dry state, the agreement between the theoretical prediction and experimental data is very good. Similar behavior has been found when more such trajectories were investigated [33]. The reason seems to be the fact that the moisture density function is estimated from approximate data for the desorption and adsorption isotherms between 90% and 100% RH. More accurate data in this region should provide better theoretical estimates. It appears that the theory of independent domain complexions advanced by Everett and coworkers is indeed applicable for paper materials showing moisture sorption hysteresis. 

Both cases were observed experimentally using various substances [118]. For a certain set of material parameters theory [118] predicts the bistable behavior of the smectic C phase in a strong field. Such bistability has been observed in the form of the field-induced motion of the domain walls and in a hysteresis in the process of the director reorientation [119]. 

Theories based on regular solutions or on heterogeneous domains predict hysteresis behavior. Knittel et al. (1977) are developing these theories and comparing the results with carefully determined intrinsic loops. Hysteresis in solid state reactions is clearly an important property related to the structure and texture and containing a great deal of information and memory. 

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