Smoluchowski coagulation equation

In

Coagulation kernel

The

kernel

and describes the rate at which particles of size ${\displaystyle x_{1}}$ coagulate with particles of size ${\displaystyle x_{2}}$.

${\displaystyle K=1,\quad K=x_{1}+x_{2},\quad K=x_{1}x_{2},}$

known as the constant, additive, and multiplicative kernels respectively.^[7] For the case ${\displaystyle K=1}$ it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the dynamic scaling property.^[8] This self-similar behaviour is closely related to scale invariance which can be a characteristic feature of a phase transition.

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,

${\displaystyle K={\sqrt {\frac {\pi k_{B}T}{2}}}\left({\frac {1}{m(x_{1})}}+{\frac {1}{m(x_{2})}}\right)^{1/2}\left(d(x_{1})+d(x_{2})\right)^{2}.}$

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

${\displaystyle K={\frac {2}{3}}{\frac {k_{B}T}{\eta }}\left(x_{1}^{1/y_{1}}+x_{2}^{1/y_{2}}\right)\left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),}$

or Reaction-limited aggregation:

${\displaystyle K={\frac {2}{3}}{\frac {k_{B}T}{\eta }}{\frac {(x_{1}x_{2})^{\gamma }}{W}}\left(x_{1}^{1/y_{1}}+x_{2}^{1/y_{2}}\right)\left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),}$

where ${\displaystyle y_{1},y_{2}}$ are fractal dimensions of the clusters, ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle T}$ is the temperature, ${\displaystyle W}$ is the Fuchs stability ratio, ${\displaystyle \eta }$ is the continuous phase viscosity, and ${\displaystyle \gamma }$ is the exponent of the product kernel, usually considered a fitting parameter.^[9] For cloud, the kernel for coagulation of cloud particles are usually expressed as:

${\displaystyle K=\pi [r(x_{1})+r(x_{2})]^{2}|v(x_{1})-v(x_{2})|E_{coll}(x_{1},x_{2}),}$

where ${\displaystyle r(x)}$ and ${\displaystyle v(x)}$ 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

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

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.^[18][19] The CDA model can be understood by the following reaction scheme

${\displaystyle A_{x}(t)+A_{y}(t){\stackrel {v(x,t)}{\longrightarrow }}A_{(\alpha +1)(x+y)}(t+\tau ),}$

where ${\displaystyle A_{x}(t)}$ denotes the aggregate of size ${\displaystyle x}$ at time ${\displaystyle t}$ and ${\displaystyle \tau }$ is the elapsed time. This reaction scheme can be described by the following generalized Smoluchowski equation

${\displaystyle {\Big [}{{\partial } \over {\partial t}}+{{\partial } \over {\partial x}}v(x,t){\Big ]}n(x,t)=-n(x,t)\int _{0}^{\infty }K(x,y)n(y,t)dy+{{1} \over {2}}\int _{0}^{x}dyK(y,x-y)n(y,t)n(x-y,t).}$

Considering that a particle of size ${\displaystyle x}$ grows due to condensation between collision time ${\displaystyle \tau (x)}$ equal to inverse of ${\displaystyle \int _{0}^{\infty }K(x,y)n(y,t)dy}$ by an amount ${\displaystyle \alpha x}$ i.e.

${\displaystyle v(x,t)={{\alpha x} \over {\tau (x)}}=\alpha x\int _{0}^{\infty }dyK(x,y)n(y,t).}$

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

${\displaystyle n(x,t)\sim t^{-(2+2\alpha )}e^{-{{x} \over {t^{1+2\alpha }}}},}$

which exhibits dynamic scaling. A simple fractal analysis reveals that the condensation-driven aggregation can be best described fractal of dimension

${\displaystyle d_{f}={{1} \over {1+2\alpha }}.}$

The ${\displaystyle d_{f}}$th moment of ${\displaystyle n(x,t)}$ is always a conserved quantity which is responsible for fixing all the exponents of the dynamic scaling. Such conservation law has also been found in Cantor set too.

See also

• Einstein–Smoluchowski relation
• Flocculation
• Smoluchowski factor
• Williams spray equation

References