Steady-state cluster size distribution in stirred suspensions

1990 ◽  
Vol 86 (12) ◽  
pp. 2133 ◽  
Author(s):  
Ruben D. Cohen
Keyword(s):  
Steady State ◽  
Size Distribution ◽  
Cluster Size ◽  
Cluster Size Distribution

scholarly journals Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 558 ◽  
Author(s):  
Alexander S. Abyzov ◽  
Jürn W. P. Schmelzer ◽  
Vladimir M. Fokin ◽  
Edgar D. Zanotto
Keyword(s):  
Free Energy ◽  
Steady State ◽  
Size Distribution ◽  
Gibbs Free Energy ◽  
Nucleation Rate ◽  
Cluster Size ◽  
Cluster Formation ◽  
Cluster Size Distribution ◽  
Crystal Nucleation ◽  
Self Consistency

Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, Δ G ( n ) = − n Δ μ + γ n 2 / 3 , of crystal clusters of size n and, in particular, on the work of critical cluster formation, Δ G ( n c ) . The first term in the expression for Δ G ( n ) describes changes in the bulk contributions (expressed by the chemical potential difference, Δ μ ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, σ : γ = Ω d 0 2 σ , Ω = 4 π ( 3 / 4 π ) 2 / 3 , where d 0 is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and n = n c defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a “cluster” of size n = 1 , no work is required. However, the commonly used relation for Δ G ( n ) given above leads to a finite value for n = 1 . By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of Δ G ( n ) an expression of the form Δ G ˜ ( n ) = Δ G ( n ) − Δ G ( 1 ) . Such self-consistency correction is usually omitted assuming that the inequality Δ G ( n ) ≫ Δ G ( 1 ) holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, Li 2 O · 2 SiO 2 ), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination.

Transient nucleation in isothermal crystallization of poly(3- hydroxybutyrate)

e-Polymers ◽  
2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Pawel Sajkiewicz ◽  
Maria Laura Di Lorenzo ◽  
Arkadiusz Gradys
Keyword(s):  
Relaxation Time ◽  
Steady State ◽  
Time Dependence ◽  
Size Distribution ◽  
Nucleation Rate ◽  
Isothermal Crystallization ◽  
Cluster Size ◽  
Cluster Size Distribution ◽  
Temperature Range ◽  
Transient Nature

AbstractThe time dependence of nucleation rate in isothermal crystallization of poly(3-hydroxybutyrate) was experimentally shown, both in heterogeneous and homogeneous nucleation. The time dependence of nucleation rate is one of the important limitations for the applicability of the simplified form of Kolmogoroff- Avrami-Evans model with time independent kinetic characteristics. The presented results are interpreted in terms of non-steady-state cluster size distribution underlying transient nature of nucleation. The relaxation time needed for reaching a steady-state cluster size distribution and thus steady-state nucleation rate is relatively long, exceeding the time of exhaustion of heterogeneities. The relaxation time estimated from homogeneous process was tens of seconds in the temperature range between 83 and 120 oC. Application of Arrhenius law allows estimation of relaxation time in broader temperature range, showing an increase of relaxation time with decreasing temperature.

Evolution of cluster size distribution in nucleation and growth processes

1991 ◽  
Vol 136 (3) ◽  
pp. 181-197 ◽  
Author(s):  
J. Bartels ◽  
U. Lembke ◽  
R. Pascova ◽  
J. Schmelzer ◽  
I. Gutzow
Keyword(s):  
Size Distribution ◽  
Cluster Size ◽  
Nucleation And Growth ◽  
Cluster Size Distribution ◽  
Growth Processes

Bimodality of cluster-size distribution and condensation in a finite Lennard-Jones system

1981 ◽  
Vol 24 (6) ◽  
pp. 2893-2902 ◽  
Author(s):  
Julian H. Gibbs ◽  
Biman Bagchi ◽  
Udayan Mohanty
Keyword(s):  
Size Distribution ◽  
Cluster Size ◽  
Cluster Size Distribution ◽  
Lennard Jones
Sign in / Sign up

Export Citation Format

Share Document