Homogeneous nucleation calculation

Figure 1. Calculated homogeneous nucleation rate for various values of Tyg as a function of temperature.

This paper presents the physical mechanism and the structure of a comprehensive dynamic Emulsion Polymerization Model (EPM). EPM combines the theory of coagulative nucleation of homogeneously nucleated precursors with detailed species material and energy balances to calculate the time evolution of the concentration, size, and colloidal characteristics of latex particles, the monomer conversions, the copolymer composition, and molecular weight in an emulsion system. The capabilities of EPM are demonstrated by comparisons of its predictions with experimental data from the literature covering styrene and styrene/methyl methacrylate polymerizations. EPM can successfully simulate continuous and batch reactors over a wide range of initiator and added surfactant concentrations. 

The particle generation rate was calculated by a step mechanism, namely formation of primary precursor particles by homogeneous nucleation (JLQ.) followed by coagulation to latex particles (8-9). This homogeneous nucleation mechanism is often referred to as the HUFT mechanism for its originators Hansen, Ugelstad, Fitch, and Tsai. 

The rates of propagation and termination in the aqueous phase were also calculated. The radical entry rate, radical generation rate, and aqueous propagation rate were then used to develop an algebraic equation for the rate of formation of primary precursors. This equation is an extension to copolymers of the homogeneous nucleation equation derived by Hansen and Ugelstad (7.) for a homopolymer. 

Having described the equilibrium structure and thermodynamics of the vapor condensate we then re-examine homogeneous nucleation theory. This combination of thermodynamics and rate kinetics, in which the free energy of formation is treated as an activation energy in a monomer addition reaction, contains the assumption that equilibrium thermodynamic functions can be applied to a continuum of non-equilibrium states. For the purpose of elucidating the effects of the removal of the usual approximations, we retain this assumption and calculate a radially dependent free energy of formation. Ve find, that by removing the conventional assumptions, the presumed thermodynamic barrier to nucleation is absent. 

FIGURE 9.30 Theoretically predicted and experimentally measured concentrations of H2S04 required for homogeneous nucleation of sulfuric acid at a rate of 1 particle cm 3 s 1 (adapted from Hoppel et al., 1994 based on theoretical calculations of Jaecker-Voirol and Mirabel (1989) and experimental data of Wyslouzil et al. (1991). 

Fi gure 19.2 Calculated nucleus shapes for homogeneous nucleation of an f.c.c. phase in... 

The homogeneous nucleation of martensite in typical solids is too slow by many orders of magnitude to account for observed results. Calculations of typical values of AQc using the classical nucleation model of Section 19.1.4 (see Exercise 19.3) yield values greatly exceeding 76 kT. Furthermore, nearly all martensitic transformations commence at very sparsely distributed sites. Small-particle experiments [14] have yielded typical nucleation densities on the order of one nucleation event per 50 pm diameter Fe-Ni alloy powder particle.3 Thus, nucleation of martensite is believed to occur at a small number of especially potent heterogeneous nucleation sites. 

Homogeneous Nucleation (a) Using Eq. 8.6-2, calculate the rate of homogeneous nucleation of styrene as a function of temperature at atmospheric pressure and a temperature range from 145°C to 325°C. In calculating the pressure in the bubble, assume that it equals the vapor pressure (extrapolate it from lower temperature values). Use the Eotvos equation a = 2.1 (p/M)1 (Tc — T — 6), where the surface tension is in erg/cm3, temperature is in °C, and density in g/cm3, to evaluate the surface tension as a function of temperature. The critical temperature... 

Figure 4.6 shows the dependence on r for both contributions with small values of r its square is predominant and AG increases with increasing r the nucleus will stop growing and (with homogeneous nucleation) it disappears. From a certain value of r, the critical nucleus size, rk, AG decreases upon growth the nucleus is then stable and continues growing. The value of rk can be easily calculated at rk ... 

FIGURE 7.6 Critical nudeus radius of silicon calculated by homogeneous nucleation theory for 0.2 atm and 0.7 atm. Atomic radius of silicon is 0.146 run. From Sawano [18]. 

In calculated intermolecular frequencies specifically carried out to point to the differences of cluster and liquid structural mismatch, Plummer (1973) showed that the free energy change for a 20-molecule clathrate is 57 kcal/mol, while the ice Ih of 20-molecule size has a 72 kcal/mol difference. The conclusion that this may be a reason for difficulties associated with Ih ice formation via homogeneous nucleation is worthy of further consideration. 

Upon substitution of this condition into the expression for w, one can calculate the value of Wmax by differentiating the expression with respect to r as before. In fact rc can be found to be equal to AGy, which is the same as that for homogeneous nucleation. However, Wmax for heterogeneous nucleation Wmax hetero) works out to be,... 

The dependence of / on S is known from homogeneous nucleation theory discussed earlier in this chapter. Thus, there are four differential equations in five unknowns. A, M, S,N, and /, with a relationship between / and S. The set of equations for the dynamics of the stable aerosol is remarkable because the calculation of the important moments, A and N, does not require the detemiination of the size distribution of the stable particles. 

Figure 2. Temperature dependence of the nucleation rate in superheated water at atmospheric pressure. Data of dynamic experiments 1 — [ ], 2 - [ ], 3 - fj, 4 - [ ], 5 - [ ]. The solid line shows calculation by homogeneous nucleation theory.

Figure 3. B oundary of limiting superheats of water and solutions of acetone with water 1 — water, 2 - water + 5 % acetone, 3 — water + 15 % acetone. Solid line — line of liquid-vapor phase equilibrium, C - critical point, dashed line — calculation by homogeneous nucleation...

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