Nucleation and growth by diffusion under Ostwald-Freundlich boundary condition
The critical radius of a nucleus grown by diffusion in a solution is studied thermodynamically as well as kinetically. The thermodynamic growth equation called Zeldovich equation of classical nucleation theory (CNT) and the kinetic diffusional growth equation combined with the Ostwald-Freundlich boundary condition lead to the same critical radius. However, it should be pointed out that the diffusional equation may lead to a kinetic critical radius that is different from the thermodynamic critical radius, thus indicating the possibility of kinetically controlling the critical radius of a nucleus.
Nucleation and growth are basic phenomena that play vital roles in the processing of various materials across industries Kelton2010 . In particular, the growth of various fine particles such as semiconductor quantum dots Gorshkov2010 , bio-minerals Meldrum2008 and other molecular crystals in solution Sear2012 has attracted considerable interest recently. In those materials, the material transport by diffusion plays fundamental role. This growth problem by diffusion has been studied, for example, in the precipitation from solution Zener1949 , the liquid droplet nucleation from supersaturated vapor Frank1950 , the vapor bubble nucleation from supersaturated Epstein1950 solution, and the colloidal particle formation LaMer1950 ; Reiss1951 from solution. However, a more complex scenario has been suggested recently, for example, to explain the mono-dispersed colloidal particles Privman1999 , where primary particles produced by the nucleation and subsequent growth will aggregate to form mono-dispersed secondary particles Robb2008 . In most of previous studies, however, either nucleation or growth is studied separately Robb2008 ; Kuchma2009a ; Grinin2011 and a fully consistent formulation of nucleation and growth is missing.
In fact, nucleation and growth can be studied on the same footing using the general formulation using master equation called Becker-Döring equation Kalikmanov2013 or using Fokker-Planck equation Kalikmanov2013 . Although, they can be used to study complex nucleation in general, the result is mostly numerical Wilemski1995 ; Wyslouzil1995 ; Kathmann2004 in the abstract phase space and cannot give a clear picture of nucleation by fluctuation and growth by diffusion in real space.
In this communication, we will look at the nucleation and growth by diffusion Zeldovich1943 ; Zener1949 ; Reiss1951 of the post-critical nucleus. We closely follow the discussions of Slezov Slezov2009 and Peter Peter2011 , and observe the relationship between the Ostwald-Freundlich boundary condition and the critical radius of a nucleus, which is usually assumed to be determined solely by the thermodynamical growth equation Iwamatsu2012 .
We concentrate on the one-component system and consider nucleation from solution within the context of the classical nucleation theory (CNT) Kelton2010 . We start from the fundamental kinetic equation for the evolution of the cluster distribution function , which represents the number of clusters consisting of monomers. This is written as
where is the rate of attachment of a monomer onto a cluster consisting of monomers. Similarly, is the rate of detachment of a monomer from a cluster that consists of monomers. By introducing the virtual state, that is equilibrium with the cluster consisting of monomers Slezov1998 , this master equation is usually transformed into the well-known Fokker-Planck equation that describes probability flow in a space of size () :
where denotes continuum version of the rate of attachment , denotes the usual inverse temperature, and denotes the free energy of a cluster consisting of monomers. Within the CNT, it is given by
is the chemical potential of a solution with the concentration that is higher than the saturation concentration (). Therefore, the solution is supersaturated and . is the surface tension of the cluster (nucleus), and is a shape factor that connects the radius of the cluster, , to the number of monomers, , in the cluster through for spherical clusters.
Because the second term on the right-hand side of Eq. (2) is similar in form to the usual diffusion equation in real space, this term describes diffusion in space and is called the diffusion term. While the first term describes cluster growth through
and is called the drift term, wherein we have used the chemical potential of the cluster with monomers defined as
The chemical potential of a solution that is in equilibrium with a cluster of size at the surface is given by
is the radius of critical nucleus with its free energy that corresponds to the energy barrier of nucleation given by
determined thermodynamically from in Eq. (3), and is the molar volume defined as and is related to in Eq. (3) as . In Eq. (8) we have expressed the concentration at the surface, , as a function of the radius, , instead of the number of monomers, . The boundary condition (8) assumes that the chemical equilibrium is reached so quickly as the radius of cluster growths. This equation is also known to represent the Gibbs-Thomson effect Saito1996 which is directly observable even on solid surface McLean1997 .
From Eq. (7) we have
that can be used for determining cluster growth Shneidman2001 given the rate of attachment, .
Slezov Slezov2009 showed that the rate of attachment, , for the post-rcritical nucleus could be determined in the case the post-critical nucleus grows by diffusion and material attachment Reiss1951 ; Grinin2004 , by considering the Zeldovich equation (12) as the diffusional growth equation as follows.
is the diffusion flux at the surface of growing droplet, where is the solute diffusivity. The concentration field obeys the diffusion equation Grinin2004
for spherical symmetry, the steady state () solution of which is given by
Then, the diffusion flux is simply given by
and Eq. (13) is given by
which can be written as
by expanding in Eq. (8) around . Therefore, the diffusional growth equation can be written in the form
which is similar to the Zeldovich equation (12), in which the kinetic critical radius coincides with the thermodynamic critical radius defined by Eq. (9). The rate of attachment in Eq. (12) is now given by
for the post-critical nucleus Slezov2009 . We should note in passing that the specific form Eq. (19) is not directly related to the Ostwald-Freundlich equation given by Eq. (8), but is related indirectly through the specific form of the steady state diffusion flux given by Eq. (17).
Therefore, as soon as the embryo crosses the thermodynamic saddle point at as the critical nucleus, the solution concentration at the surface of the nucleus with radius starts to decrease (Fig.. 1), and the concentration around the supercritical nucleus starts to decrease (Fig. 1), which in turn, induces a diffusion flux according to Eq. (14). Then, the nucleus continue to grow according to Eq. (19) as if it just crossed the kinetic critical point at . Therefore, the critical radius, , in Eq. (12) should be interpreted as the thermodynamic critical radius fixed by the Ostwald-Freundlich boundary condition, whereas the in the kinetic equation Eq. (19) should be interpreted as the kinetic critical radius determined from the diffusion flux. In Fig. 1, we can observe that the size of the depletion zone or diffusion shell around the nucleus increases as the size of the nucleus increases. In fact, the self-similar theory Adzhemyan2006 predicts that the size of the depletion zone scales with the size of a droplet. Furthermore, the size depends on the supersaturation Kuchma2009 , and Fig. 1 corresponds to weak supersaturation, where the depletion zone is wider than the nucleus radius.
Thus far, we have assumed that the diffusion is very fast so that not only does the concentration at the surface of the nucleus quickly reach equilibrium as defined by the Ostwald-Freundlich boundary condition (8) but the concentration field and flux also reach the steady state defined by Eqs. (16). In fact, there must be a time delay in reaching the steady state, and the time-dependent concentration field is approximately given by Kuni2003 ; Grinin2004
so that the resulting growth velocity at the beginning of growth is faster than that predicted from the steady state flux obtained using Eq. (19). However, the kinetic critical radius remains the same as the thermodynamic critical radius ().
As has been noted below Eq. (21), the specific form Eq. (19) is not ascribed to the Ostwald-Freundlich boundary condition but solely to the diffusion flux given by Eq. (17). Any subtle change in the nucleation flux given by Eq. (14) may lead to a kinetic critical radius defined using the diffusional growth equation (20), which would be different from the thermodynamic critical radius defined using Eq. (12). For example, a temporal depletion or superabundance of materials in the mother phase leads to the flux
When the mother phase is depleted (), the kinetic critical radius, , becomes larger than the thermodynamic critical radius, (), from Eq. (26). Then, the thermodynamic post-critical nucleus turns to the kinetic pre-critical nucleus, and it shrinks kinetically Peter2011 . In contrast, the kinetic critical radius is smaller than the thermodynamic critical radius () when the mother phase is superabundant (). Then, the thermodynamically pre-critical nucleus turns to a post-critical nucleus, and it grows kinetically. Thereafter, the nucleation rate is enhanced. Such enhanced nucleation in a concentrated solution is typically found Erdemir2009 ; Sear2012 , for example, in the crystallization of proteins Vekilov2012 and small molecules Bonnett2003 .
It is well known that a rapid decrease of supersaturation due to the depletion of monomer causes a drastic effect to the growth of nucleus LaMer1950 ; Gorshkov2010 ; Nanev2011 . Our simple model has predicted that the thermodynamic critical nucleus turns to that of the kinetic pre-critical nucleus from Eq. (26) and the nucleation will be hindered. Also, the thermodynamic critical radius in Eq. (9) and the thermodynamic energy barrier in Eq. (10) increases as the chemical potential in Eq. (4) decreases due to the depletion of monomer. In fact, the increase of the thermodynamic critical radius defined by Eq. (9) is given by the same formula Eq. (26). Therefore, the nucleation becomes less probable not only kinetically but also thermodynamically and will be hindered in later time Gorshkov2010 ; Nanev2011 . Our kinetic picture is consistent to this scenario. Moreover, our discussion has clearly indicated that it can happen even temporally and locally due to the temporal fluctuation of diffusion flux by the depletion of local concentration.
Our discussion, however, has concentrated on the stage where the critical nucleus just turns to the growing nucleus by diffusion. Therefore, subsequent Ostwald ripening Slezov2009 ; Iwamatsu1999 of single nucleus or the aggregation of multiple nuclei Gorshkov2010 during the growth stage is beyond the scope of the present study. Also, since we have considered only the material diffusion and neglected the heat flow, the instability and fractal growth Mullins1963 of nucleus cannot be discussed within our present formalism.
In conclusion, we showed semi-analytically that any subtle change in the diffusion flux leads to a change in the kinetic critical radius, , from the thermodynamic critical radius, , even when Ostwald-Freundlich boundary condition is maintained. The same conclusion has already been reached by Peter Peter2011 using numerical simulation. Incidentally, a recent development in self-similar solutions Adzhemyan2006 ; Grinin2011 to diffusional growth assumes a large limit for the growth equation Eq. (19) and is unsuitable for discussing nucleus growth near the critical radius, .
Acknowledgements.This work was supported under a project for strategic advancement of research infrastructure for private universities, 2009-2013, operated by MEXT, Japan.
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