In the paper we submit the non-local modification of
kinetic Smoluchowski equation for binary aggregation applying to
dispersed media having memory. Our supposition consists in that that
intensity of evolution of clusters is supposed to be a function of the
product of concentrations of the lowest orders clusters at different
moments. The new form of kinetic equation for aggregation is
derived on the base of the transfer kernels approach. This approach
allows considering the influence of relaxation times hierarchy on
kinetics of aggregation process in media with memory.
[1] M. von Smoluchowski, "Drei vorträge über diffusion Brownsche
molecular bewegung und koagulation von kolloidteichen", Physik. Z. 17
1916, pp. 557-571.
[2] J.A.D. Wattis, "An introduction to mathematical models of coagulation-
fragmentation processes: A discrete deterministic mean-field approach ",
Physica D 222, 2006, 1.
[3] H. Sontag, K. Strenge, Koagulation und Stabilitat Disperser Systeme,
Veb Deutsher Verlag der Wissenschaften, Berlin 1970, p. 147.
[4] H. Stechemesser, B. Dobias (eds.), Coagulation and Flocculation,
Taylor & Francis, Boca Raton, 2005, p. 850.
[5] J.C. Neu, J.A. Canizo, L.L. Bonilla, "Three eras of micellization", Phys.
Rev. E 66, 2002, 061406.
[6] V. Volterra, Le├ºons sur la théorie mathématique de lalutte pour la vie ,
Gauther-Villars et Cie , Paris, 1931, p.286.
[7] V.Y. Rudyak, Statistical Theory of Dissipative Processes in Gases and
Liquids, (Nauka, Novosibirsk, 1987, p. 272. (In Russian).
[8] A. M. Brener, "Nonlocal Equations of the Heat and Mass Transfer in
Technological Processes", Theor. Found. Chem. Eng 40, 2006, pp. 564-
573.
[9] A. M. Brener, "Nonlocal Model of Aggregation in Polydispersed
Systems", Theor. Found. Chem. Eng 45, 2011, pp. 349-353.
[10] D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth
Heinemann, Oxford, 2000, p. 525.
[11] L. Onsager, "Reciprocal Relations in Irreversible Processes. I", Phys.
Rev. 37, 1931, pp. 405-415.
[12] H.B.G. Casimir, "On Onsager's principle of microscopic reversibility",
Rev. Mod. Phys. 1945, 17, 343.
[13] D. Jou, J. Casas-Vazquez, and M. Criado-Sancho, Thermodynamics of
Fluids under Flow ,Springer, Berlin, 2001, p. 231.
[14] S.R. de Groot, Thermodynamics of Irreversible Processes North-
Holland Publ. Comp., 1952, p. 280.
[15] A. Brener, B. Balabekov and A. Kaugaeva, "Non-Local Model of
Aggregation in Uniform Polydispersed Systems", Chem. Eng. Trans.
2009, pp. 783-789.
[16] A. Brener, A. Muratov, B. Balabekov, "Non-Local Model of
Aggregation Processes", Jour. of Mater. Sci. and Eng. A, 1, No3, 2011,
pp. 451-456.
[17] J. Kevorkian and J. Cole, Multiple Scale and Singular Perturbation
Methods, Springer, New York, 1996, p. 412.
[18] J. Kozak, V. Basios, and G. Nicolis, "Geometrical Factors in Protein
Nucleation", Biophys. Chem., 2003, 105, 495.
[1] M. von Smoluchowski, "Drei vorträge über diffusion Brownsche
molecular bewegung und koagulation von kolloidteichen", Physik. Z. 17
1916, pp. 557-571.
[2] J.A.D. Wattis, "An introduction to mathematical models of coagulation-
fragmentation processes: A discrete deterministic mean-field approach ",
Physica D 222, 2006, 1.
[3] H. Sontag, K. Strenge, Koagulation und Stabilitat Disperser Systeme,
Veb Deutsher Verlag der Wissenschaften, Berlin 1970, p. 147.
[4] H. Stechemesser, B. Dobias (eds.), Coagulation and Flocculation,
Taylor & Francis, Boca Raton, 2005, p. 850.
[5] J.C. Neu, J.A. Canizo, L.L. Bonilla, "Three eras of micellization", Phys.
Rev. E 66, 2002, 061406.
[6] V. Volterra, Le├ºons sur la théorie mathématique de lalutte pour la vie ,
Gauther-Villars et Cie , Paris, 1931, p.286.
[7] V.Y. Rudyak, Statistical Theory of Dissipative Processes in Gases and
Liquids, (Nauka, Novosibirsk, 1987, p. 272. (In Russian).
[8] A. M. Brener, "Nonlocal Equations of the Heat and Mass Transfer in
Technological Processes", Theor. Found. Chem. Eng 40, 2006, pp. 564-
573.
[9] A. M. Brener, "Nonlocal Model of Aggregation in Polydispersed
Systems", Theor. Found. Chem. Eng 45, 2011, pp. 349-353.
[10] D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth
Heinemann, Oxford, 2000, p. 525.
[11] L. Onsager, "Reciprocal Relations in Irreversible Processes. I", Phys.
Rev. 37, 1931, pp. 405-415.
[12] H.B.G. Casimir, "On Onsager's principle of microscopic reversibility",
Rev. Mod. Phys. 1945, 17, 343.
[13] D. Jou, J. Casas-Vazquez, and M. Criado-Sancho, Thermodynamics of
Fluids under Flow ,Springer, Berlin, 2001, p. 231.
[14] S.R. de Groot, Thermodynamics of Irreversible Processes North-
Holland Publ. Comp., 1952, p. 280.
[15] A. Brener, B. Balabekov and A. Kaugaeva, "Non-Local Model of
Aggregation in Uniform Polydispersed Systems", Chem. Eng. Trans.
2009, pp. 783-789.
[16] A. Brener, A. Muratov, B. Balabekov, "Non-Local Model of
Aggregation Processes", Jour. of Mater. Sci. and Eng. A, 1, No3, 2011,
pp. 451-456.
[17] J. Kevorkian and J. Cole, Multiple Scale and Singular Perturbation
Methods, Springer, New York, 1996, p. 412.
[18] J. Kozak, V. Basios, and G. Nicolis, "Geometrical Factors in Protein
Nucleation", Biophys. Chem., 2003, 105, 495.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53587", author = "A. Brener and B. Balabekov and N. Zhumataev", title = "Kinetics of Aggregation in Media with Memory", abstract = "In the paper we submit the non-local modification of
kinetic Smoluchowski equation for binary aggregation applying to
dispersed media having memory. Our supposition consists in that that
intensity of evolution of clusters is supposed to be a function of the
product of concentrations of the lowest orders clusters at different
moments. The new form of kinetic equation for aggregation is
derived on the base of the transfer kernels approach. This approach
allows considering the influence of relaxation times hierarchy on
kinetics of aggregation process in media with memory.", keywords = "Binary aggregation, Media with memory, Non-local
model, Relaxation times", volume = "6", number = "1", pages = "29-4", }
