View source for Smoluchowski coagulation equation

{{Short description|Population balance equation in statistical physics}}
[[File:Smoluchowski Aggregation Kinetics.svg|thumb|This diagram describes the aggregation kinetics of discrete particles according to the Smoluchowski aggregation equation.]]

In [[Physics:Statistical physics|statistical physics]], the '''Smoluchowski coagulation equation''' is a [[Population balance equation|population balance equation]] introduced by Marian Smoluchowski in a seminal 1916 publication,<ref name=Smoluchowski1916>{{cite journal
|last=Smoluchowski
|first=Marian
|title=Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen
|journal=Phys. Z.
|year=1916
|volume=17
|pages=557–571, 585–599
|language=German
|bibcode=1916ZPhy...17..557S}}</ref> describing the [[Time evolution|time evolution]] of the [[Physics:Number density|number density]] of particles as they coagulate (in this context "clumping together") to size ''x'' at time ''t''.

Simultaneous coagulation (or aggregation) is encountered in processes involving [[Chemistry:Polymerization|polymerization]],<ref name="BlatzTobolsky1945">{{cite journal|last1=Blatz|first1=P. J.|last2=Tobolsky|first2=A. V.|title=Note on the Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena|journal=The Journal of Physical Chemistry|volume=49|issue=2|year=1945|pages=77–80|issn=0092-7325|doi=10.1021/j150440a004}}</ref> [[Chemistry:Coalescence|coalescence]] of aerosols,<ref>{{cite book
|last1=Agranovski|first1=Igor|title=Aerosols: Science and Technology|publisher=John Wiley & Sons|year=2011|pages=492|isbn=978-3527632084}}</ref> [[Chemistry:Emulsion|emulsication]],<ref name="DanovIvanov1994">{{cite journal|last1=Danov|first1=Krassimir D.|last2=Ivanov|first2=Ivan B.|last3=Gurkov|first3=Theodor D.|last4=Borwankar|first4=Rajendra P.|title=Kinetic Model for the Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems|journal=Journal of Colloid and Interface Science|volume=167|issue=1|year=1994|pages=8–17|issn=0021-9797|doi=10.1006/jcis.1994.1328|bibcode=1994JCIS..167....8D}}</ref> [[Chemistry:Flocculation|flocculation]].<ref name="ThomasJudd1999">{{cite journal|last1=Thomas|first1=D.N.|last2=Judd|first2=S.J.|last3=Fawcett|first3=N.|title=Flocculation modelling: a review|journal=Water Research|volume=33|issue=7|year=1999|pages=1579–1592|issn=0043-1354|doi=10.1016/S0043-1354(98)00392-3}}</ref>

==Equation==

The distribution of particle size changes 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|integral]]:

: <math>\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.</math>

If ''dy'' is interpreted as a discrete [[List of integration and measure theory topics#Intuitive foundations|measure]], i.e. when particles join in discrete sizes, then the discrete form of the equation is a [[Summation|summation]]:

: <math>\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).</math>

There exists a unique solution for a chosen kernel function.<ref name="Melzak1957">{{cite journal|last1=Melzak|first1=Z. A.|title=A scalar transport equation|journal=Transactions of the American Mathematical Society|volume=85|issue=2|year=1957|pages=547–560|issn=0002-9947|doi=10.1090/S0002-9947-1957-0087880-6|doi-access=free}}</ref>

==Coagulation kernel==

The operator, ''K'', is known as the coagulation kernel and describes the rate at which particles of size <math>x_1</math> coagulate with particles of size <math>x_2</math>.  [[Closed-form expression|Analytic solutions]] to the equation exist when the kernel takes one of three simple forms:

: <math>K = 1,\quad K = x_1 + x_2, \quad K = x_1x_2,</math>

known as the [[Constant (mathematics)|constant]], [[Additive map|additive]], and multiplicative kernels respectively.<ref>{{Cite journal | doi = 10.1016/j.physd.2006.07.024| title = An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach| journal = Physica D: Nonlinear Phenomena| volume = 222| issue = 1–2| pages = 1–20| year = 2006| last1 = Wattis | first1 = J. A. D. |bibcode = 2006PhyD..222....1W | url = http://eprints.nottingham.ac.uk/934/1/tut11.pdf}}</ref> For the case <math>K = 1</math> it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the [[Dynamic scaling|dynamic scaling]] property.<ref name=constkernel>{{Cite journal | doi = 10.1007/BF02186868| title = Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel| journal = Journal of Statistical Physics| volume = 75| issue = 3| pages = 389–407| year = 1994| last1 = Kreer | first1 = Markus | last2=Penrose | first2 = Oliver | bibcode = 1994JSP....75..389K| s2cid = 17392921}}</ref> This self-similar behaviour is closely related to [[Scale invariance|scale invariance]] which can be a characteristic feature of a [[Physics:Phase transition|phase transition]].

However, in most practical applications the kernel takes on a significantly more complex form. For example, the free-molecular kernel which describes [[Physics:Collision|collision]]s in a dilute [[Physics:Gas|gas]]-[[Physics:Phase (matter)|phase]] system,

: <math>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.</math>

Some coagulation kernels account for a specific [[Fractal dimension|fractal dimension]] of the clusters, as in the [[Diffusion-limited aggregation|diffusion-limited aggregation]]:
: <math>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),</math>
or Reaction-limited aggregation:
: <math>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),</math>
where <math>y_1,y_2</math> are [[Fractal dimension|fractal dimension]]s of the clusters, <math>k_B</math> is the Boltzmann constant, <math>T</math> is the temperature, <math>W</math> is the Fuchs stability ratio, <math>\eta</math> is the continuous phase viscosity, and <math>\gamma</math> is the exponent of the product kernel, usually considered a fitting parameter.<ref>{{Cite journal | doi = 10.1002/mats.201300140| title = Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems| journal = Macromolecular Theory and Simulations| volume = 23| issue = 3| pages = 170| year = 2014| last1 = Kryven | first1 = I. | last2 = Lazzari | first2 = S. | last3 = Storti | first3 = G. | url = https://pure.uva.nl/ws/files/2139145/149285_Aggregation_and_Coalescence_12.pdf| url-access = | url-status = | archive-url = | archive-date = }}</ref> For cloud, the kernel for coagulation of cloud particles are usually expressed as:
: <math>K = \pi [r(x_1)+r(x_2)]^2 |v(x_1)-v(x_2)| E_{coll}(x_1,x_2), </math> 
where <math> r(x)</math> and <math> v(x)</math> are the radius and fall speed of the cloud particles usually expressed using power law. 

Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to [[Numerical analysis|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 (probability theory)|method of moments]]<ref>{{cite journal |last1=Marchisio |first1=D. L. |last2=Fox |first2=R. O. |year=2005 |title=Solution of Population Balance Equa- tions Using the Direct Quadrature Method of Moments |journal=J. Aerosol Sci. |volume=36 |issue=1 |pages=43–73 |doi=10.1016/j.jaerosci.2004.07.009 |bibcode=2005JAerS..36...43M }}</ref><ref>{{cite journal |last1=Yu |first1=M. |last2=Lin |first2=J. |last3=Chan |first3=T. |year=2008 |title=A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion |journal=Aerosol Sci. Technol. |volume=42 |issue=9 |pages=705–713 |doi=10.1080/02786820802232972 |bibcode=2008AerST..42..705Y |hdl=10397/9612 |s2cid=120582575 |hdl-access=free }}</ref><ref>{{cite journal |last=McGraw |first=R. |year=1997 |title=Description of Aerosol Dynamics by the Quadrature Method of Moments |journal=Aerosol Sci. Technol. |volume=27 |issue=2 |pages=255–265 |doi=10.1080/02786829708965471 |bibcode=1997AerST..27..255M |doi-access=free }}</ref><ref>{{cite journal |last=Frenklach |first=M. |year=2002 |title=Method of Moments with Interpolative Closure |journal=Chem. Eng. Sci. |volume=57 |issue=12 |pages=2229–2239 |doi=10.1016/S0009-2509(02)00113-6 }}</ref><ref>{{cite journal |last1=Lee |first1=K. W. |last2=Chen |first2=H. |last3=Gieseke |first3=J. A. |year=1984 |title=Log-Normally Preserving Size Distribution for Brownian Coagulation in the Free-Molecule Regime |journal=Aerosol Sci. Technol. |volume=3 |issue=1 |pages=53–62 |doi=10.1080/02786828408958993 |bibcode=1984AerST...3...53L |doi-access=free }}</ref> and sectional methods.<ref>{{cite journal |last1=Landgrebe |first1=J. D. |last2=Pratsinis |first2=S. E. |year=1990 |title=A Discrete-Sectional Model for Particulate Production by Gas-Phase Chemical Reaction and Aerosol Coagulation in the Free-Molecular Regime |journal=J. Colloid Interface Sci. |volume=139 |issue=1 |pages=63–86 |doi=10.1016/0021-9797(90)90445-T |bibcode=1990JCIS..139...63L }}</ref>  In the [[Multivariable calculus|multi-variate]] case, however, when two or more properties (such as size, shape, composition, etc.) are introduced, one has to seek special approximation methods that suffer less from [[Curse of dimensionality|curse of dimensionality]]. Approximation based on Gaussian radial basis functions has been successfully applied to the coagulation equation in more than one dimension.<ref name=Kryven2013>{{cite journal |last1=Kryven |first1=I. |last2=Iedema |first2=P. D. |title=Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding
|journal=Polymer |year=2013 |volume=54 |issue=14 |pages=3472–3484 |doi=10.1016/j.polymer.2013.05.009|arxiv=1305.1034 |s2cid=96697123 }}</ref><ref>{{Cite journal | doi = 10.1002/mats.201300121| title = Topology Evolution in Polymer Modification| journal = Macromolecular Theory and Simulations| volume = 23| pages = 7–14| year = 2014| last1 = Kryven | first1 = I. | last2 = Iedema | first2 = P. D. }}</ref>

When the accuracy of the solution is not of primary importance,  [[Monte Carlo method|stochastic particle (Monte Carlo) methods]] are an attractive alternative.{{citation needed|date=May 2016}}

==Condensation-driven aggregation==

In addition to aggregation, particles may also grow in size by condensation, deposition or by 
accretion. Hassan and Hassan recently proposed a condensation-driven aggregation (CDA) model in which aggregating particles keep growing continuously between merging upon collision.<ref>M. K. Hassan and M. Z. Hassan, “Condensation-driven aggregation in one dimension”, Phys. Rev. E '''77''' 061404 (2008), https://doi.org/10.1103/PhysRevE.77.061404</ref><ref>M. K. Hassan and M. Z. Hassan, “Emergence of fractal behavior in condensation-driven aggregation”, Phys. Rev. E '''79''' 021406 (2009), https://doi.org/10.1103/PhysRevE.79.021406</ref> The CDA model can be understood by the following reaction scheme

:<math>A_x(t) + A_y(t) \stackrel{
v(x,t)}{\longrightarrow} A_{(\alpha + 1)(x + y)}(t + \tau),</math>

where <math>A_x(t)</math> denotes the aggregate of size <math>x</math> at time <math>t</math> and <math>\tau</math> is the elapsed time. This reaction scheme can be described by the following generalized Smoluchowski equation

:<math>\Big[{{\partial }\over{\partial t}} +  {{\partial}\over{\partial x}} v(x,t) \Big]n(x,t)
=-n(x,t)\int_0^\infty K(x,y)n(y,t)dy + {{1}\over{2}}\int_0^x dy K(y,x-y)
n(y,t)n(x-y,t).</math>

Considering that a particle of size <math>x</math> grows due to condensation between collision time <math>\tau(x)</math> equal to inverse of <math>\int_0^\infty K(x,y)n(y,t)dy</math> by an amount <math>\alpha x</math> i.e.

:<math>v(x,t)={{\alpha x}\over{\tau(x)}}=\alpha x\int_0^\infty dyK(x,y)n(y,t).</math>

One can solve the generalized Smoluchowski equation for constant kernel to give 

:<math>n(x,t)\sim t^{-(2+2\alpha)}e^{-{{x}\over{t^{1+2\alpha}}}},</math>
which exhibits [[Dynamic scaling|dynamic scaling]]. 
A simple [[Fractal|fractal]] analysis reveals that the condensation-driven aggregation can be best described fractal of dimension

:<math>d_f={{1}\over{1+2\alpha}}.</math>
 
The <math>d_f</math>th moment of <math>n(x,t)</math> is always a conserved quantity which is responsible for fixing all the exponents of the [[Dynamic scaling|dynamic scaling]]. Such conservation law has also been found in [[Cantor set]] too.

==See also==
*[[Physics:Einstein relation (kinetic theory)|Einstein–Smoluchowski relation]]
*[[Chemistry:Flocculation|Flocculation]]
*[[Physics:Smoluchowski factor|Smoluchowski factor]]
*[[Williams spray equation]]

==References==
{{Reflist|30em}}

{{DEFAULTSORT:Smoluchowski Coagulation Equation}}
[[Category:Differential equations]]
[[Category:Statistical mechanics]]

{{Sourceattribution|Smoluchowski coagulation equation}}