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abstract: 'We revisit the scaling theory of the Smoluchowski equation with special emphasis on the dimensional analysis to derive the scaling ansatz and to give an insightful foundation to it. It has long been argued that the homogeneity exponent $\lambda$ of the aggregation kernel divides the aggregation process into two regimes (i) $\lambda\leq 1$ nongelling and (ii) $\lambda>1$ gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if $\lambda<1$. We show that beyond this limit i.e. at $\lambda\geq 1$, it breaks down and hence it fails to describe a gelation transition. This also happens to be accompanied by violation of scaling.'
author:
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    [M. K. Hassan$^{1,2}$ and J. Kurths$^1$]{}\
    [$~^1$ University of Potsdam, Department of Physics, Am Neuen Palais, D-14415, Potsdam, Germany]{}\
    [$~^2$ University of Dhaka, Department of Physics, Theoretical Physics Division, Dhaka 1000, Bangladesh]{}
title: ' Can Smoluchowski equation account for gelation transition? '
---

PACS number(s): 05.20.Dd,02.50.-r,05.40-y

The kinetics of irreversible and sequential aggregation of particles occurs in a variety of physical processes and it is of wide interest in physics, chemistry, biology and in many other disciplines of science and technology [@kn.has1; @kn.has2; @kn.has3]. Due to its cross-disciplinary importance, statistical physics has offered a number of models: diffusion limited aggregation (DLA) [@kn.has4], percolation [@kn.has5], diffusion limited cluster cluster aggregation (DLCA), ballistic aggregation (BA) [@kn.has6] etc., to name just a few, and these are systems far from equilibrium. There hardly exists any systematic standard theoretical framework for describing the systems out of equilibrium, which is indeed in sharp contrast to its equilibrium counterpart. However, stochastic processes seem to some extent to rescue this shortcoming and appear to capture a wide class of non-equilibrium phenomena [@kn.frag]. The dynamics of these processes often evolves in time following some conservation rules and can be expressed in the form of a master equation that constitutes a relation between the spatial and its temporal variables.

As far as kinetics of cluster-cluster aggregation is concerned, much of the theoretical understanding is provided by the rate equation approach proposed by von Smoluchowski [@kn.has7] almost a century ago which reads as $$\partial_t \psi(x,t)  = -  a(x,t)\psi(x,t)+{{1}\over{2}}\int_0^x dy K(y,x-y)\psi(y,t)\psi(x-y,t).$$ This is a one-dimensional mean-field model, where $\psi(x,t)$ is the concentration of particles of size $x$ at time $t$. We note for the first time that the kernel $K(x,y)$ itself is not the aggregation rate but rather it is the $a(x,t)=\int_0^\infty K(x,y)\psi(y,t)dy$ which is the rate at which particles aggregate and hence it has to be obtained self-consistently. Equation $(1)$ is studied for a large class of aggregation kernels that satisfy the power-law condition $K(bx,by)=b^\lambda K(x,y)$, which has physical importance too. It ensures that the aggregating particles are homogeneously distributed, i.e. that the system has a perfect mixing of particles with continuous distribution of sizes and hence we call $\lambda$ the homogeneity exponent. This $\lambda$ value appears to play a significant role in controlling the dynamics of the process. However, to the best of our knowledge, the limit on the $\lambda$ value up to which Eq. $(1)$ is valid, or if such limits exist at all, has never been studied.

In this article we, therefore, revisit the scaling theory of Smoluchowski equation with a special emphasis on the dimensional analysis in an attempt to obtain a deep insight into the dynamics of the process and find the limits on the $\lambda$ value upto which the Smoluchowski equation is valid. We get several results that are in contradiction with existing predictions, especially those referring to the gelling-nongelling transition. It has long been argued that the homogeneity exponent $\lambda$ of the aggregation kernel divides the aggregation process into two regimes [@kn.has8; @kn.has14]. First, the nongelling model described by $\lambda\leq 1$, where the Smoluchowski equation admits scaling and obeys mass conservation. Second, the gelling model at $\lambda>1$, where it is believed that the gelation transition occurs after a certain time $t>t_{cr}$ and it is accompanied by the violation of scaling and mass conservation. However, it is claimed that at $t<t_{cr}$ the equation admits scaling and obeys mass conservation. In this letter, we question the validity of the equation in the gelling regime and show that the Smoluchowski equation is valid if and only if $\lambda<1$. Beyond this bound, including $\lambda=1$, we show that Eq. (1) violates the basic principle of aggregation for the whole range of the time axis. To further support our argument, we consider the best known gelling model and show that the results obtained from this model are either unphysical or self-contradictory. Note that the sol-gel transition is in fact a kinetic phase transition that is characterized by a singularity of typical quantity, such as enthalpy accompanied by a sharp rise in viscosity and appearance of non-zero shear modulus at the critical point. It is important to emphasize that Smoluchowski’s equation is a one dimensional mean-field model. It does not take into account the detailed nature of cluster-cluster interactions, their spatial correlation and does not include any thermodynamical or mechanical behaviour, instead it only describes some average behaviour of the underlying mechanism through the coagulation kernel. Therefore, we argue that like many other one dimensional mean-field models in statistical physics it cannot account for phase transition of any order and sol-gel transition in particular.

We start with appreciating that Eq. (1) describes a stochastic process; the particle size $x$ and the time $t$ are the only two governing parameters that can completely specify the governed parameter $\psi(x,t)$. However, it is very instructive to note that $x$ and $t$ are intertwined via the dimensional consistency of Eq. $(1)$. Of course, the exact relation will depend on $a(x,t)$ through the precise choice of $K(x,y)$. As a consequence of this, either of the two can be taken to be an independent parameter when the other governing parameter and the governed parameter $\psi(x,t)$ is expressible in terms of this. For example, if $t$ is chosen to be the independent parameter, then using the fact that the dimension of a physical quantity is always expressed as power monomial, we can define the dimensionless quantity $\xi={{x}\over{t^z}}$ [@kn.barenblatt]. Then, it is obvious that the dimension of the governed parameter $\psi(x,t)$ can also be written only in terms of $t$, and hence we can define another dimensionless quantity $\Pi \equiv {{\psi(x,t)}\over{t^{\alpha}}} \sim t^{-\alpha} \psi(\xi t^z,t)\equiv
F(\xi,t)$. Since $\xi$ and $\Pi$ are both dimensionless quantities, upon transition from one system of units of measurement to another, inside a given class, their numerical values must remain unchanged. In other words, we can pass to a system of unit of measurement where $t$ is changed by an arbitrary factor and, upon such a transition, the function $F$ or $\Pi$ must remain unchanged, i.e. ${{\partial F}\over{\partial t}}=0$. This implies that the quantity $\Pi$ is independent of $t$ and can be completely expressed in terms of $\xi$ alone. Thus we can define $\Pi=\phi(\xi)$ to enable us to write the temporal scaling ansatz $\psi(x,t)\approx t^\alpha \phi({{x}\over{t^z}})$. Note that $z$ and $\alpha$ can only take values for which $t^z$ and $t^{\alpha}$ bear the dimension of $x$ and $\psi$ respectively. Had we chosen $x$ to be the independent parameter, a similar argument would lead us to write the spatial scaling ansatz $\psi(x,t)\approx x^{-\theta}\Phi({{t}\over{x^{-\nu}}})$, where $x^{-\nu}$ and $x^{-\theta}$ must bear the dimension of $t$ and $\psi(x,t)$ respectively. The existence of scaling or a self-similar solution actually means that we can choose the self-similar coordinates ${{\psi}\over{t^\alpha}}$ (or ${{\psi}\over{x^{- \theta}}}$) and $x/t^z$ (or $t/x^{-\nu}$) such that their plots for any initial condition collapse into one single curve, which is typically known as [*data collapse*]{} formalism [@kn.barenblatt; @kn.stanley].

To find the scaling exponents and for further use, we define the $n$th moment of $\psi(x,t)$ as $M_n(t)=\int_0^\infty dx x^n \psi(x,t)$ with $n\geq 0$. We then get a rate equation for $M_n(t)$ upon multiplying both sides of Eq. $(1)$ by $x^n$ and integrate over the whole range of $x$; this yields $${{dM_n(t)}\over{dt}}={{1}\over{2}}\int_0^\infty \int_0^\infty dx 
dy[(x+y)^n-x^n-y^n]K(x,y)\psi(x,t)\psi(y,t).$$ This immediately reveals that $M_1(t)$, the mass or size of the system is a conserved quantity, since Eq. $(1)$ describes aggregation of particles in a closed system. This conserved quantity in fact is the intrinsic agent responsible for fixing the numerical value of the mass exponent $\alpha=-2z$ (or $\theta=2$). On the other hand, the $z$ (or $\nu$) value is fixed by the aggregation rate $a(x,t)$ and hence it is called the kinetic exponent. It is quite easy to realize that a general explicit solution for $M_n(t)$ by solving a non-linear equation, like Eq. $(2)$, can be extremely difficult for any choice of $K(x,y)$. Therefore, we will resort to an indirect way of obtaining it, namely the dimensional analysis and find the asymptotic behaviour of $M_n(t)$ instead. Note that the kinetic exponent $z$ can only be calculated if the aggregation kernel is specified with respect to its argument. For example, we select $K(x,y)=(xy)^\beta$ (i.e. $\lambda=2\beta$) and call it the product model. For this choice of aggregation kernel, we substitute the temporal scaling ansatz into the rate equation $(1)$ and get $$-zt^{z-2\beta z-1}={{-\xi^\beta\phi(\xi)\phi_\beta+{{1}\over{2}}\int_0^\xi d\eta 
\eta^\beta (\xi-\eta)^\beta 
\phi(\eta)\phi(\xi-\eta)}\over{2\phi(\xi)+\xi\phi^\prime(\xi)}}.$$ Since the left hand side of the above equation is a function of $t$ only, whereas the right hand side is independent of $t$ and a dimensionless quantity, the exponent of $t$ must be equal to zero to give $z={{1}\over{1-2\beta}}$. This means $t^{{{1}\over{1-2\beta}}}$ bears the dimension of the quantity $x$, in other words $x^{-(2\beta-1)}$ contains the dimension of $t$ i.e. $\nu=2\beta-1$. For this product model the quantity $a(x,t)= x^\beta M_\beta(t)$, being the rate of aggregation, must bear the dimension inverse of time. Combining all this dimensional consistency we get $M_\beta \sim x^{\beta-1}$ or $a(x,t)\sim x^{2\beta-1}$ and a trivial change in variable gives $M_n(t)\sim t^{{{n-1}\over{1-2\beta}}}$.

To further generalize our argument, we consider $K(x,y)=(x+y)^\gamma$ and $K(x,y)=x^\gamma+y^\gamma$ (i.e. $\lambda=\gamma$) and call both the sum model for which the aggregation rates $a(x,t)$ are $\sum_{r=0}^\gamma 
\left( \begin{array}{c}\gamma \\ r\end{array}\right )x^rM_{\gamma-r}$ and $(x^\gamma M_0(t)+ M_\gamma(t))$ respectively. Substituting the scaling ansatz into Eq. $(1)$, we obtain an equation similar to Eq. $(3)$ for both models and both give $z={{1}\over{1-\gamma}}$. Doing further dimensional analysis gives $a(x,t) \sim x^{\gamma-1}$ and $M_n(t) \sim t^{{{n-1}\over{1-\gamma}}}$. Comparing the results of all the three models, we can single out one exponent $\lambda$ and generalize the expression for the moments $M_n(t)$ and the kinetic exponent $z$. That is, for any homogeneous kernel with a homogeneity exponent $\lambda$ we postulate $z={{1}\over{1-\lambda}}$ and $M_n(t)\sim t^{(n-1)z}$. We now define the simple and weighted mean cluster size as $s(t)=M_1(t)/M_0(t)$ and $w(t)=M_2(t)/M_1(t)$ respectively. However, both the definitions give the same result $s(t)=w(t)=t^z$ and this again confirms that $t^z$ bears the dimension of the particle size $x$. Note that the kinetic exponent $z$ suffers a singularity at $\lambda=1$, otherwise $z>0$ if $\lambda<1$ and $z<0$ if $\lambda>1$. Therefore, it is obvious that the behaviour of $s(t)$ should also change with the change of the $\lambda$ value. The expression for $s(t)$ in fact tells us how the mean cluster size should evolve in the system. The basic principle of the aggregation process in a closed system is that the number of particles present in the system must be a decreasing function of time and therefore $s(t)$ should be an ever growing quantity. Clearly, this sets a physical constraint on the choice of the $\lambda$ value in determing the bound beyond which $s(t)$ violates the basic principle. This is what happens when $\lambda>1$ and $s(t)$ becomes a decreasing quantity in time.

Before discussing the so called gelling regime $\lambda>1$, we shall first show that at $\lambda=1$ the scaling is violated, which contradicts with the known results [@kn.has10]. In this case $z$ suffers a singularity and gives $a(x,t)=1$. This means that the relation between the two governing parameters $x$ and $t$ no longer exists and therefore $x$ cannot be expressed in terms of $t$ and vice versa. In this situation, on the one hand, the power monomial character of the system is lost and, on the other, one can no longer define the self-similar coordinates which is indispensible for any process to show scaling; this proves that at $\lambda=1$ the scaling is violated [@kn.barenblatt; @kn.stanley]. This can be further supported by the explicit known solution for $\gamma=1$ (i.e. $\lambda=1$) for which one obtains $s(t)=e^t$ [@kn.has10; @kn.has11]. This clearly shows that both $s(t)$ and $t$ have lost their dimensional character. We argue that to admit scaling (i) $s(t)$ must show a power-law behaviour and (ii) owing to the nature of the process in question (kinetics of aggregation), the exponent has to be a positive and a non-zero finite quantity [@kn.barenblatt]. Therefore, at $\lambda=1$ not only the scaling is violated but also Eq. (1) fails to describe a physically meaningful aggregation process since the parameters that govern the system lose their dimensional (stochastic) character.

We now show that the scaling is also violated for $\lambda>1$, irrespective of whether we are below or above $t_{cr}$. First, we argue that the solution of Eq. (1) must be of a stretched exponential decay type so that all the moments exist and show a power-law behaviour. The justification of this rests on the loss term which dictates that $\psi(x,t)$ must decay exponentially. The argument of the exponential term in the solution must be $\xi \sim a(x,t)t$. That is, it can be either $\xi\sim t/x^{1-\lambda}$ or $\xi\sim x/t^{1/(1-\lambda)}$ since $a(x,t)\sim x^{\lambda-1}$. The important point here is that, this exponential decay term must be present even in the long time behaviour of the solution (scaling solution) so that all the moments exist and show a power-law behaviour. Otherwise, the Eq. (1) itself becomes invalid since the aggregation rate $a(x,t)$ itself is a function of $M_n(t)$, where $n$ depends on the specific choice for the kernel. The scaling solutions of the aggregation process are essentially the solutions in the long-time ($t \longrightarrow \infty$) and large-size ($x \longrightarrow
\infty$) limit so that in this regime Eq. (1) reduces itself to an ordinary differential equation for the scaling function $\phi(\xi)$ [@kn.has15]. That is, the two governing parameters $x$ and $t$ must combine together to form a dimensionless quantity $\xi$ in such a way that in the scaling regime $\xi$ may stay finite. This is possible if and only if $\xi=x/t^z$ or $\xi=t/x^{1/z}$ with $z>0$ or $\lambda<1$. It is important to mention here the close connection between the rate equation approach of fragmentation [@kn.has15] and the aggregation process described by Eq. (1). In both cases, the mean cluster size and the concentration have the same functional relation with time i.e. $s(t)\sim t^z$ and $\psi(x,t)\sim t^{-2z}\phi(x/t^z)$ respectively. The mean cluster $s(t)$ must bear the dimenion of $x$ while the concentration $\psi(x,t)$ must bear the dimension of $M_0(t)/s(t)\sim t^{-2z}$, since it is defined as the number of particles per unit length, irrespective of whether $s(t)$ and $\psi(x,t)$ describe the aggregation or the fragmentation process. The two opposing phenomena keep the signature of their respective processes through their $z$ value i.e. $z<0$ for the fragmentation and $z>0$ for the aggregation process. All the existing solutions in both phenomena agree with this provided we are in the valid regime. The scaling solutions of the fragmentation equation are on the other hand solutions in the long-time ($t \longrightarrow \infty$) and the small-size ($x\longrightarrow 0$) limit. In this case, the two variables can couple together to form a dimensionless finite quantity $\xi$ provided $z<0$ (where $z=-1/(\alpha+1)$ in the case of fragmentation and $\alpha$ is the homogeneity exponent of the breakup kernel). Recently, we have shown using similar arguments that the fragmentation equation breaks down at $\alpha\leq -1$ [@kn.has13], where it is believed to show a shattering transition which is essentially described as the opposite phenomena of the gelation transition[@kn.has15]. Returning to the Smoluchowski equation, it is clear that $z<0$ if $\lambda>1$ and hence the two variables couple together in the product form to make $\xi$; therefore it cannot be a finite quantity in the scaling regime. We thus argue that the scaling is violated for the whole time axis at $\lambda\geq 1$.

Let us check the prototypical case $\lambda=2$ for the product model since this is the best known model and claimed to exhibit gelation transition. We show that, with this model, one cannot obtain even one single quantity which is physically meaningful and correct at the same time. The exact solution for the concentration $\psi(x,t)$ is found to be $\psi(x,t)\sim t^{x-1}x^{x-2}{\rm exp}[-xt]/x!$ at $t<t_{cr}$ [@kn.has8; @kn.has11]. First, note that $xt$ is the dimensionless quantity which clearly means that $x$ bears the dimension inverse of time and $z<0$. Trying to look for the dimensional consistency in the rest of the term of the solution will only yields frustrating results. On the other hand, $t_{cr}=1$ for monodisperse initial condition. It is too short a time to seek scaling solution since it is the solution in the asymptotic regime where the initial condition is irrelevant. Therefore, the scaling cannot hold at $t<1$. Furthermore, one can check and find out that it cannot satisfy Eq. (1), which is essential for a faithful solution. We show further inconsistencies of this model below.

For the monodisperse initial condition, the equation for $M_0(t)$ in Eq.(2) has the solution $M_0(t)= 1-t/2$, when $M_1(t)$ is a conserved quantity. This is believed to be true at $t<1$ if we assume, for arguments sake, that gelation occurs at $t>t_{cr}$. This shows that the number of particles present in the system decreases in a much slower fashion than the corresponding non-gelling model, where it decays in the power-law form. Note that, should it actually be the gelling model then there would already be an indication of it in the faster decrease of $M_0(t)$ at least when it is close to $t_{cr}=1$ since at $t>1$, more and more particles are believed to be lost in the gel phase, which is held responsible for mass violation. In addition, the solution for $M_0(t)$ also implies that after a definite time, the number density becomes a negative quantity, which is unphysical. In this model, we find that the aggregation rate $a(x,t)=xM_1(t)$. This means that if the mass is a conserved quantity then $x$ must bear the dimension inverse of time and hence $xt$ is the dimensionless quantity. We already argued that $xt$ cannot stay as a finite quantity in the scaling regime, hence scaling is violated. On the other hand, if the mass is not assumed to be a conserved quantity, but decays as $1/t$, which is claimed to be the case at $t>t_{cr}$, then we get $a(x,t)\sim x/t$. This is again unphysical since $x$ becomes a dimensionless quantity and Eq. (1) loses its stochastic nature. This means that Eq. (1) breaks down for the whole time axis. Solving Eq. $(2)$ for the second moment with monodisperse initial condition, we obtain $M_2(t)=1/(1-t)$. If the mass is a conserved quantity, then we obtain $w(t)=M_2(t)$ and $s(t)=1/M_0(t)$ and hence $s(t)\neq w(t)$ if $z<0$. This contradicts our previous observation that $s(t)=w(t)$ for all $z$. We find that both $s(t)$ and $w(t)$ suffer a singularity but at two different points along the time axis. However, the singularity of the simple mean has never been noticed before. Note that below $t_{cr}$, the decrease in the number density $M_0(t)$ and the increase in the mean cluster size $w(t)$ are not commensurable. For example, below but close to $t_{cr}$, the number density decreases very slowly, while $w(t)$ increases faster than any power-law can predict.

Most important of all, it is believed that at $t>1$ the mass is no longer a conserved quantity and it decreases as $1/t$. Where is the mass going? It has been argued that the finite size particle (sol) is lost to the infinite cluster (gel) [@kn.has8] and hence gelation transition. That is, the system contains two different kinds of particles: sol and gel. The transition of sol particles to the gel particle was held responsible for the violation mass conservation and for the scaling [@kn.has8]. If it is so, then it can only mean that $\psi(x,t)$, and hence its moment $M_1(t)$ or $w(t)$, is incapable of taking into account the gel particle despite being present in the system. Therefore, $M_2(t)$ cannot bear any information about the gel particle, and hence its divergence should not be taken as the appearence of infinite gel but quite the opposite. This is again self-contradictory. It has been argued that at $t\geq t_{cr}$, the concentration $\psi(x,t)$ has asymptotically a power-law behaviour $\psi(x,t) \sim x^{-5/2}/t$ [@kn.has8]. Firstly, if this is to be the solution of Eq. (1), then it must satisfy it in the first place, at least it must remain dimensionally faithful. One can insert this power-law form of the solution into Eq. (1) and finds that it cannot satisfy it. Secondly, the solution implies that $x^{-5/2}/t$ must bear the dimension of the concentration. However, one cannot find any physical basis to support it. Finally, one cannot obtain any physically meaningful expression for the moments if $\psi(x,t)$ has the power-law form. This holds true in all closely connected problems like the kinetics of fragmentation or the random sequential adsorption process [@kn.has13]. Note that the aggregation rate for both the sum models with $\lambda=2$ contains $M_2(t)$ and so does Eq. (1). So, if the divergence of weighted mean $w(t)=M_2(t)$ is considered to detect the appearence of gelation, then Eq. (1) itself becomes invalid at the singular point. Hence solving Eq. (1) at $t>t_{cr}$ to find $\psi(x,t)$ and the different criterion for gelation i.e. different moments of $\psi(x,t)$ are no longer valid. Note further that $M_2(t)$ diverges only in one particular case i.e. at $\lambda=2$ of the product model. There do not exist another $\beta$ vis-a-vis a $\lambda$ value for which one can show that the second moment diverges at a finite time. Therefore, one cannot generalise that for $\lambda>1$ the second moment diverges at a finite time. It is thus claer that one cannot obtain a self consistent solution in the so called gelling regime rather one only obtains results which are either unphysical or self-contradictory.

Note that Eq. $(1)$ is a one dimensional mean-field model that ignores fluctuation, detailed nature of cluster-cluster interaction and their spatial correlation and the shape of the aggregating cluster. It also does not bear any parameter that could explain thermodynamic or mechanical properties of the system. Instead, it assumes that the system is highly diluted so that merging of two clusters into one is not influenced by the presence of other clusters and they merge into one without failure as soon as they meet. Therefore, it is quite natural that as many other one dimensional mean-field theories in statistical physics it cannot describe phase transition such as sol-gel. In fact, the sol-gel transition is characterized by a sudden rise in viscosity and enthalpy accompanied by the appearance of non-zero shear modulus near the gel point. This the Smoluchowski’s equation can neither explain quantitatively nor qualitatively. This does not mean that we underestimate the importance of this model. It still remains one of the few theoretical approaches through which one can obtain a quantitative comparison of experimental data or data obtained from extensive numerical simulation. It is important to mention that the form of the scaling [*ansatz*]{} and its exponents obtained from Smoluchowski’s equation are found to be the general property of all growth phenomena especially the cluster-cluster aggregation process.

In conclusion, we have given an insightful foundation to the meaning of the scaling ansatz of the highly non-trivial Smoluchowski equation. We hope this will enrich our understanding on the scaling theory. We have demonstrated that the Smoluchowski equation cannot describe the gelation transition, instead it breaks down for all time at $\lambda \geq 1$, which is also accompanied by violation of scaling. To further support our arguments we have considered the best known gelling model and shown that this model cannot give any quantity which is physically meaningful and self-consistent. Finally, we hope this work will be useful in other stochastic processes described by rate equation to gain deep insight of the systems especially of the scaling theory.

MKH is grateful to Dr. G. J. Rodgers for useful correspondence and acknowledges the Alexander von Humboldt Foundation for granting the fellowship.

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