Smoluchowski coagulation equation

This diagram describes the aggregation kinetics of discrete particles according to the Smoluchowski aggregation equation.

In statistical physics, the Smoluchowski coagulation equation is a population balance equation introduced by Marian Smoluchowski in a seminal 1916 publication,[1] describing the time evolution of the number density of particles as they coagulate (in this context "clumping together") to size x at time t.

Simultaneous coagulation (or aggregation) encountered in processes involving polymerization,[2] coalescence of aerosols,[3] emulsication,[4] flocculation.[5]

Equation

The distribution of particle size change in time according to the interrelation of all particles of the system. Therefore, the Smoluchowski coagulation equation is an integrodifferential equation of the particle-size distribution. In the case when the sizes of the coagulated particles are continuous variables, the equation involves an integral:

\frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\int^x_0K(x-y,y)n(x-y,t)n(y,t)\,dy - \int^\infty_0K(x,y)n(x,t)n(y,t)\,dy.

If dy is interpreted as a discrete measure, i.e. when particles join in discrete sizes, then the discrete form of the equation is a summation:

\frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^{i-1}_{j=1} K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \sum^\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t).

There exist a unique solution for a chosen kernel function.[6]

Coagulation kernel

The operator, K, is known as the coagulation kernel and describes the rate at which particles of size x_1 coagulate with particles of size x_2. Analytic solutions to the equation exist when the kernel takes one of three simple forms:

K = 1,\quad K = x_1 + x_2, \quad K = x_1x_2,

known as the constant, additive, and multiplicative kernels respectively.[7]

However, in most practical applications the kernel takes on a significantly more complex form. For example, the free-molecular kernel which describes collisions in a dilute gas-phase system,

K = \sqrt{\frac{\pi k_B T}{2}}\left(\frac{1}{m(x_1)}+\frac{1}{m(x_2)}\right)^{1/2}\left(d(x_1)+d(x_2)\right)^2.

Some coagulation kernels account for a specific fractal dimension of the clusters, as in the Diffusion-limited aggregation:

K = \frac{2}{3} \frac{ k_B T} {\eta} \left(x_1^{1/y_1} +x_2^{1/y_2}\right)\left(x_1^{-1/y_1} +x_2^{-1/y_2}\right),

or Reaction-limited aggregation:

K = \frac{2}{3} \frac{ k_B T} {\eta} \frac{(x_1x_2)^\gamma}{W}\left(x_1^{1/y_1} +x_2^{1/y_2}\right)\left(x_1^{-1/y_1} +x_2^{-1/y_2}\right),

where y_1,y_2 are fractal dimensions of the clusters, k_B is the Boltzmann constant, T is the temperature, W is the Fuchs stability ratio, \eta is the continuous phase viscosity, and \gamma is the exponent of the product kernel, usually considered a fitting parameter.[8]

Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to numerical methods. Most of deterministic methods can be used when there is only one particle property (x) of interest, the two principal ones being the method of moments and sectional methods. In the multi-variate case however, when two or more properties (such as size, shape, composition, etc.) are introduced, special approximation methods that suffer less from curse of dimensionality has to be applied. For instance, approximation based on Gaussian radial basis functions has been successfully applied to the coagulation equation in a few dimensions.[9] [10]

When the accuracy of the solution is not of primary importance, stochastic particle (Monte-Carlo) methods are an attractive alternative.

See also

References

  1. ↑ Smoluchowski, Marian (1916). "Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen". Physik. Z. (in German) 17: 557–571, 585–599. Bibcode:1916ZPhy...17..557S.
  2. ↑ Blatz, P. J.; Tobolsky, A. V. (1945). "Note on the Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena". The Journal of Physical Chemistry 49 (2): 77–80. doi:10.1021/j150440a004. ISSN 0092-7325.
  3. ↑ Agranovski, Igor (2011). Aerosols: Science and Technology. John Wiley & Sons. p. 492. ISBN 3527632085.
  4. ↑ Danov, Krassimir D.; Ivanov, Ivan B.; Gurkov, Theodor D.; Borwankar, Rajendra P. (1994). "Kinetic Model for the Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems". Journal of Colloid and Interface Science 167 (1): 8–17. doi:10.1006/jcis.1994.1328. ISSN 0021-9797.
  5. ↑ Thomas, D.N.; Judd, S.J.; Fawcett, N. (1999). "Flocculation modelling: a review". Water Research 33 (7): 1579–1592. doi:10.1016/S0043-1354(98)00392-3. ISSN 0043-1354.
  6. ↑ Melzak, Z. A. (1957). "A scalar transport equation". Transactions of the American Mathematical Society 85 (2): 547–547. doi:10.1090/S0002-9947-1957-0087880-6. ISSN 0002-9947.
  7. ↑ Wattis, J. A. D. (2006). "An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach". Physica D: Nonlinear Phenomena 222: 1. doi:10.1016/j.physd.2006.07.024.
  8. ↑ Kryven, I.; Lazzari, S.; Storti, G. (2014). "Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems". Macromolecular Theory and Simulations 23 (3): 170. doi:10.1002/mats.201300140.
  9. ↑ Kryven, I.; Iedema, P.D. (2013). "Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding". Polymer 54 (14): 3472–3484. doi:10.1016/j.polymer.2013.05.009.
  10. ↑ Kryven, I.; Iedema, P. D. (2014). "Topology Evolution in Polymer Modification". Macromolecular Theory and Simulations 23: 7. doi:10.1002/mats.201300121.