Mullins equation

When cast in terms of strand concentrations, the Bueche-Mullins equation becomes... 

Many other equations have been applied to rubber reinforcement. Perhaps the most important include the rule of mixtures laws, the Mullins equation, and the upper- and lower-bound relations (Broutman and Krock, 1967). Since these relationships often apply more satisfactorily to plastic rather than elastomeric composites, they will be broadly developed in Section 12.1. 

A third method (Bell and Tiller, 1967) has utilized Mullins equation (eqn (8.12)). 

Here, p resolves the normal velocity in the y direction, and from eq. (1) K = 9zz[p ct(z)]. In the case of grain growth (F = 0) in isotropic system (a(z) = const, M(z) = const), eq.(l) reduces to the familiar Mullins equation of boundary motion. Eq. (2), which generalizes this case to the problem of anisotropic growth with the possible application to recrystallization is the backbone of our present discussion. 

Solubility is also affected by particle size, small crystals (<1 pm say) exhibiting a greater solubility than large ones. This relationship is quantified in the Gibbs-Thomson, Ostwald-Freundlich equation (see Mullin, 2001)... 

As for other mass transfer operations in ehemieal engineering, several authors have proposed equations for the ealeulation of heat and mass balanees used for the estimation of erystal yield, heat load, and evaporation duty in bateh erystal-lizations, e.g. (Mullin, 2001)... 

This equation was then solved numerieally with known values of erystal growth rate (G) and solubility dc jdO) for seeded solutions. It was further assumed that if G f G(L, 6), dc /d6 = eonstant and dAc/d6 = 0 then the simplified equation for ealeulating transient temperature results (Mullin and Nyvlt, 1971)... 

According to Sohnel and Mullin(29), the change in agglomerate size as a function of time may be represented by equations of the form ... 

Whilst the fundamental driving force for crystallisation, the true thermodynamic supersaturation, is the difference in chemical potential, in practice supersaturation is generally expressed in terms of solution concentrations as given in equations 15.1-15.3. Mullin and Sohnel(19) has presented a method of determining the relationship between concentration-based and activity-based supersaturation by using concentration-dependent activity-coefficients. 

A delay occurs between attainment of supersaturation and detection of the first newly created crystals in a solution, and this so-called induction period, t, is a complex quantity that involves both nucleation and growth components. If it is assumed that /, is essentially concerned with nucleation, that is t, ot 1 /J, then Mullin 3 has shown, from equation 15.9, that ... 

The exponents i and s in equations 15.13 and 15.14, referred to as the order of integration and overall crystal growth process, should not be confused with their more conventional use in chemical kinetics where they always refer to the power to which a concentration should be raised to give a factor proportional to the rate of an elementary reaction. As Mullin(3) points out, in crystallisation work, the exponent has no fundamental significance and cannot give any indication of the elemental species involved in the growth process. If i = 1 and s = 1, c, may be eliminated from equation 15.13 to give ... 

As Mullin 3-1 points out, this equation can be used for comparing batch and continuous processing since Vw and nFVL represent the wash liquor requirements for both cases. 

In this relation, 2C2 provides a correction for departure of the polymeric network from ideality, which results from chain entanglements and from the restricted extensibility of the elastomer strands. For filled vulcanizates, this equation can still be applied if it can be assumed that the major function of the dispersed phase is to increase the effective strain of the rubber matrix. In other words, because of the rigidity of the filler, the strain locally applied to the matrix may be larger than the measured overall strain. Various strain amplification functions have been proposed. Mullins and Tobin33), among others, suggested the use of the volume concentration factor of the Guth equation to estimate the effective strain U in the rubber matrix ... 

Equation 16.36 predicts that x(t) oc f1/5 and that the neck growth rate will therefore fall off rapidly with time. The time to produce a neck size that is a given particle-size fraction is a strong function of initial particle size—it increases as R4. Equation 16.36 agrees with the results of a numerical treatment by Nichols and Mullins [11].9... 

Mullins (48-50) published a series of papers detailing the mass-transfer processes which leads to the flattening of corrugated surfaces and thermal grooving. Assuming a nearly flat crystal surface, with surface properties independent of orientation, Mullins showed that the equation... 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

Sbhnel and Mullin, Garside and recently Barlow and Haymet have discussed the molecular Interpretation of induction times from the standpoint of classical nucleation theory. Crystal nuclei with a critical size must be formed before the new solid phase is visible. According to the model there exists a free energy barrier, AG to the formation of the crystal nuclei. AG is proportional to (InS), where S is the supersaturation ratio. The Gibb s free energy, AG of the supersaturated solution is equal to -RTlnS (R=gas constant T=temperature). The induction time is a function of AG and thus AG according to the following equation... 

Figure 16.4. Several ways of recording the same data of crystal size distribution (CSD) Mullin, 1972). (a) The data, (b) Cumulative wt % retained or passed, against sieve aperture, (c) Log-log plot according to the RRS equation P = exp[(-d/d,)"] off this plot, = 850, d = 1000, n = 1.8. (d) Differential polygon, (e) Differential histogram.

Based on the assumptions above and the equation for drift, the maximum rale of entry of PBO into aquatic systems via drift is 0,018 lb PBO per acre (or 0.0084 kg ha 1) water- The results of this calculation arc used as the input rale per acre (surface area) of water and then adjusted for depth, degradation rates, and other factors (Mullins at.. 199.1. ... 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

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