Approach critical dimensionless parameters

Finally, the particle volume fraction, 4), is a critical dimensionless parameter. For monodisperse suspensions of spheres of radius a, = n(47t/3)a with n the number of particles per volume. As noted in the introduction, with gravitational effects eliminated when the difference in phase densities vanishes, Ap = 0, a suspension with any volume fraction 0 < ( ) < ( )max be formed and studied, a feature unique to suspensions. When interest is in the properties approaching the maximum packing limit, ( )niax/ it is common to describe the properties of the suspension in terms of a deviation from the limiting value i.e., in terms of 1 -< )/( )max-... 

The existence of a critical point in the pressure-volume-temperature (PVT) diagram (actually, a point in the planar PV projection, but a critical line in a three-dimensional representation), a critical point (Curie temperature) in ferromagnetism, a critical point (Neel point) in antiferromagnetism, a critical temperature in superconductivity, and a critical point (lambda point) in liquid 2He4 are physical descriptions of the onset of a sudden macroscopic collective transition. If one approaches the critical point very closely, dimensionless parameters, defined to describe this approach, are common to all these disparate phenomena the approach to criticality, or to a phase transition, are really the same. 

Fig. 5.2 The dimensionless critical reaction rate, calculated analytically (solid lines), obtained numerically (circles), and obtained from the diffusion approach, is plotted vs the dimensionless parameters q or y. Reprinted with permission from [286]. Copyright 2007 by the American Physical Society...

Of course, we should identify the particular path chosen to approach the critical limit, and for this purpose it is convenient to introduce a dimensionless path parameter such as ... 

First-principle calculations of the thermodynamic properties are more or less hopeless enterprise. One of the most famous phenomenological approaches was suggested by van der Waals [6, 8, 9]. Using the dimensionless pressure it = p/pc, the density v = n/nc and the temperature r = T/Tc, the equation of state for the ideal gas reads it = 8zzr/(3 -u) — 3zA Its r.h.s. as a function of the parameter v has no singularities near u = 1 v = it = t = is the critical point) and could be expanded into a series in the small parameter 77 = [n — nc)/nc with temperature-dependent coefficients. Solving this... 

The connection between the present approach and that of the cooperative helical aggregation model of Oosawa and Kasai [13,39] described in Chapter 2 is now easily established. Our dimensionless mass action X is related to the dimension-bearing concentration of monomers Co [M] and equilibrium constant Xh [M ] through X = 0exp(/3G) = CoXh. Hence, om polymerization condition 0p exp(jSG) = 1 corresponds to that of C Xh = 1 given in Chapter 2, where C denotes the critical polymerization concentration. It also follows that our dimensionless equilibrium constant Xa takes on the role of the cooperativity parameter a in the helical aggregation model [39]. 

The heuristic approach described in this paper utilizes linear statistical methods to formulate the basic hyperbolic non-linear model in a particularly useful dimensionless form. Essential terms are identified and others rejected at this stage. Reaction stoichiometry is combined with the inherent mathematical characteristics of the dimensionless rate expression t< reduce the number of unknown parameters to the critical few that must be evaluated by non-linear estimation. Typically, only four or five parameters remain at this point, and initial estimates are available for these. The approach is equally applicable to cases where the rate-limiting mechanism is known and where it is not. 

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