Nucleation pathway of core-shell composite nucleus in size and composition space and in component space
The kinetics of nucleation of a core-shell composite nucleus that consists of a core of stable final phase surrounded by a wetting layer of intermediate metastable phase is studied using the kinetic theory of binary nucleation not only in the size and composition space but also in the component space. The steady-state solution of the Fokker-Planck equation is considered. Various formulas for the critical nucleus at the saddle point as well as for the post-critical nucleus are derived. The kinetics of nucleation at the saddle point is more appropriately characterized in the size and composition space, while the kinetics of the post-critical nucleus is more appropriately described in the component space. Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free energy landscape are irrelevant to the kinetics of the post critical nucleus.
Nucleation is a very basic phenomena which plays a vital role in various material processing applications ranging from steel production to food and beverage industries Kelton2010 (). Recently, researchers have focused on the nucleation of complex materials Kelton2010 (). The nucleation of such complex materials can also be complex and often involves intermediate metastable phases Ostwald1897 (); Vekilov2012 (); Erdemir2009 (); Gebauer2008 (); Chung2009 (), which appears during the course of nucleation and growth as predicted from the Ostwald’s step rule Ostwald1897 (). Then, the critical nucleus often has a core-shell structure with stable final phase surrounded by an intermediate metastable phase.
Such a core-shell nucleus appears in various circumstances. It is well known, for example, the nucleation of protein crystal proceeds through the core-shell type nucleus with the final stable crystal surrounded by the metastable dense solution Vekilov2012 (). The model calculation tenWolde1997 () using the Monte Carlo simulation with a simplified intermolecular interaction revealed that the critical nucleus at the saddle point corresponds indeed to the core-shell structure. Subsequent numerical simulation using the Lennard-Jones system Meel2008 () and a model calculation using the capillarity approximation Iwamatsu2011 () confirmed the core-shell structure of critical nucleus which corresponds to the saddle point of the free-energy landscape. A similar composite nucleus is predicted even for a simple metal like aluminum Desgranges2007 () using molecular dynamics simulation. Experimental evidence of such a core-shell structure of critical nucleus is observed not only in protein crystallization Vekilov2012 () but also in colloidal crystallization Zhang2007 (); Savage2009 ().
A similar composite nucleus with core-shell structure is considered in the problem of delquescence Shchekin2008 (); McGraw2009 (), where the condensation of liquid from supersaturated vapor occurs on a soluble core. This core-shell structure also appears in various problems such as semiconductor nano-crystals Fisher2005 (), polymer crystallizations Liu2011 (), and nano-clusters of alloys Colin2012 (). It is also used as a model of linked-flux nucleation or partitioning transformation when the interface-limited growth and diffusion of material is coupled Russell1968 (); Kelton2000 (); Kelton2003 (); Diao2008 (). In this model, a core of the final stable phase is surrounded by a shell of the dense pre-nucleus environment Russell1968 ().
In the previous paper Iwamatsu2012b (), we have studied the nucleation flux of composite nucleus with core-shell structure at the saddle point in the free-energy surface using the theory of decay of metastable phase Kramers1940 (); Langer1969 (); Berezhkovskii2005 (); Peters2009 () and of multicomponent nucleation Trinkaus1983 (); Wilemski1999 (). In this paper, we will supplement our previous study Iwamatsu2012b (), and will study the nucleation pathway for the critical as well as the post-critical nucleus with core-shell structure. We first recapture the Fokker-Planck or the Zeldovich-Frenkel equation Kelton2010 (); Wu1997 () for the composite nucleus Iwamatsu2012b () from the Master equation (Section II) by regarding the composite nucleus as a fully phase-separated two-component binary system. Then the nucleation flux will be characterized more appropriately using the size-composition representation instead of the two-components representation Iwamatsu2012b () (Section III). The nucleation pathway of the post-critical composite nucleus will also be studied using the Zeldovich relation Zeldovich1943 (); Frenkel1955 (). Finally, section IV will contain the conclusion of the study, indicating the outcome as well as suggestions on further research in the field.
Ii Fokker-Planck equation for the composite nucleus
In order to study the nucleation kinetics of the composite nucleus with core-shell structure, the model shown in Fig. 1 has been used Iwamatsu2011 (); Iwamatsu2012b (). The model consists of a core of the stable new phase (number of molecules ) surrounded by an intermediate metastable phase (number of molecules ) nucleated in the metastable parent phase. Kashchiev and coworkers Kashchiev1998 (); Kashchiev2005 () have also considered this model as a model of nucleus when there exists an intermediate metastable phase.
This core-shell nucleus is considered to form by the two-step mechanism: First, the nucleus of the metastable intermediate phase appears within the metastable parent phase. Next, the core of the stable new phase starts to nucleate inside the nucleus of the metastable intermediate phase to form the core-shell structure shown in Fig. 1. However, in contrast to the naïve expectation Kashchiev2005 () the computer simulation tenWolde1997 (), mean-field calculation Iwamatsu2011 () as well as the experimental result in colloidal crystallization Savage2009 () have shown that this two-step mechanism is in fact the single-step nucleation characterized by a single activation energy and, therefore, by a single saddle point. This saddle point corresponds to the core-shell critical nucleus tenWolde1997 (); Iwamatsu2011 (); Zhang2007 () of the specific size and composition. Therefore, we will consider the simplest example of the core-shell nucleus of a single-component system. More complex scenarios will be expected for multi-component system such as the problem of deliquiescence Shchekin2008 (); McGraw2009 () and of semiconductor nano-crystals Fisher2005 ().
Transformation rates and in Fig. 1 are the reaction rates between the stable new phase and the metastable intermediate phase. Attachment rates and are the reaction rate from the metastable parent phase to the intermediate metastable phase. The formation of stable final phase (core) is assume to occur only through the transformation of surrounding metastable phase (shell).
The Master equation for the time-dependence of the concentration of clusters that consists of molecules of the stable phase in the core and molecules of the intermediate metastable phase in the surrounding shell is written generally Kelton2010 (); Wu1997 (); Reiss1950 () in the form
Using the detailed balance condition
and the usual equilibrium cluster distribution given by
where is the work of cluster formation for a cluster with composition , we obtain
which can be written in the form of continuum equation:
where the components of the nucleation flux are given by
and simplified notations , and and are used. where
Therefore, the net flux coming into this composite nucleus is the incoming flux to the wetting layer of the intermediate phase from surrounding parent phase whose magnitude is determined from the rate constant , since all materials are suppled from surrounding parent phase.
which will be written in short
where , and are row vectors, and is a symmetric square matrix defined through Eq. (10). The two flux and are linked Russell1968 () by the non-diagonal rate matrix . By introducing the unit row vector , where the superscript T indicates the transpose vector, the nucleation current vector is written as
using the unit vectors and along the Cartesian coordinate .
Since the Fokker-Planck equation given by Eqs. (5)-(7) for the growth of composite nucleus has the same form as that used to study the binary nucleation, we will extend the theory Reiss1950 (); Stauffer1976 (); Trinkaus1983 (); Temkin1984 (); Greer1990 (); Wu1997 (); Wilemski1999 () developed for the binary nucleation to study the scenario of nucleation and growth of a composite nucleus in the next section.
Iii Critical nucleus and post critical nucleus
iii.1 Critical nucleus in the size and composition space
In this section we extend the theory of nucleation flux of binary nucleation in the size and composition space developed by Fisenko and Wilemski Fisenko2004 () to the composite nucleus shown in Fig. 1 for which the rate matrix is non-diagonal.
At this point we introduce a new coordinate, the size and the composition defined by
where the factor comes from the Jacobian . Then the Fokker-Planck equation (5) in the size-composition space is written as
where the flux components are defined through
and Eq. (11) is transformed into
where the elements of the reaction rate matrix defined through Eq. (19) are given explicitly by
using the elements of the matrix , which will be written in short
where and are the column vectors given explicitly in Eq. (19). Equation (23) is formally the same as Eq. (11). Therefore, various formulas for the steady-state flux in the binary nucleation can be used just by changing the suffices to . Then, it is apparent from Eqs. (20)-(22) that plays the rôle of Brownian diffusion coefficient in the size space, while the Brownian diffusion coefficient in the composition space Fisenko2004 ().
Since as , the fluctuation of the composition will be suppressed and will become constant as the size of the nucleus increases. The growth of nucleus is governed mainly by the reaction rate , and only the size of the nucleus increases. The coupling of the flux in size and that in the composition disappears when , which is attained for the composition
where is defined through Eq. (20). The diffusion coefficient in the composition space given by Eq. (22) is also minimized for the composition given by Eq. (24). Therefore, the composition is the kinetically optimum composition of nucleation, which is solely determined from the kinetic factor .
Next we will consider the nucleation flux at the saddle point in the size and composition space and in the component space characterized by
where these two sets of equations are equivalent.
From Eqs. (11) and (23), it is obvious that we can use the results of the previous work Iwamatsu2012b () simply by replacing the matrix by . First, the angle for the direction of the gradient of in the size and composition space is given by the formula
are the derivative (Hessian) of the free-energy in the size and composition space at the saddle point . These quantities are related to the derivative in the component space
Equations (27)-(29) are slightly different in definition from those used by others Stauffer1976 (); Wilemski1999 () since we have off-diagonal elements . Among the sign , sign must be chosen when otherwise sign must be chosen. For a sufficiently large critical nucleus , we have . Also when the composition is the kinetically optimum composition . Then, we have in Eq. (27), which results in . Therefore, the gradient of () will be nearly parallel to the size axis when the size of the critical nucleus is large. Therefore, only the size rather than the composition is expected to increase.
Using the result of the previous paper Iwamatsu2012b (), we can derive the formula for the angle of the direction of nucleation flux , which is given by
Again, for a sufficiently large critical nucleus , we can approximate
and as . Then the nucleation flux also becomes parallel to the size axis as expected.
Finally, the nucleation rate is given by
where and are the negative and the positive eigenvalue of the matrix product where
It can be shown directly by calculating the eigenvalues and that they are related to the eigenvalues and of the matrix products through
Also, it can be easily shown that . Therefore, the nucleation rate Eq. (37) is also given by the original formula
in the original composition space as expected.
Qualitative assessment of the relative magnitude of the composition fluctuation and the size fluctuation near the saddle point can be possible. We follow the argument of Fisenko and Wilemski Fisenko2004 () and write Eq. (19) as
are the curvatures of the free-energy landscape at the saddle point, and
from Eq. (45), which yields
Therefore, the parameter
will determine the relative magnitude of the relaxation times and . If
calculated from Eqs. (48) and (49) satisfies , then , and the Brownian diffusion along the composition axis is the fastest process and that along the size axis is the slowest process. Then the fluctuation of the composition will be suppressed and the nucleation flux will be narrow along the composition axis. On the other hand, when , the nucleation flux will spread along the composition axis even at the saddle point. In fact, since in the model calculation by Wyslouzil and Wilemski Wyslouzil1995 () which shows ridge crossing, Fisenko and Wilemski Fisenko2004 () argued that we can expect some effect related to the ridge crossing when .
and in usual condition Fisenko2004 () when the saddle point is located within a narrow deep valley, we may expect . Then the nucleation flux at the saddle point along the size axis is much larger than that along the composition axis.
Returning to our original problem of composite nucleus for which the reaction matrix is given by Eq. (10), we find
and , and the kinetically optimum composition is given by from Eq. (24). We need information of the Hessian of the free-energy to calculate the angle of the gradient of from Eq. (27) and the angle of the direction of the nucleation flux from Eq. (35). However, it is apparent that as from Eqs. (57)-(59) and (36). Nucleation rate is proportional to from Eq. (40) Iwamatsu2012b (). The time scale given by Eqs. (51) and (53) can be written more explicitly by
Therefore, the time scale of nucleation in the size space is determined from the curvature of the free energy surface along the size axis and the attachment rate from the parent phase to the intermediate metastable phase. The time scale in the composition space, on the other hand, is determined from the curvature of the free energy surface along the composition axis and as well as the transformation rate from the intermediate metastable phase to the final stable phase. At the kinetically optimum composition , however, the size and the composition is decoupled and is determined solely from and .
Figure 2 shows an example of the free-energy surface calculated from our model Iwamatsu2011 () in the size and the composition space. The size is scaled by the typical size of the critical nucleus . There are two saddle points of almost the same size with different composition indicated by two points in Fig. 2: the one that corresponds to the critical nucleus of the metastable intermediate phase only () and the other that corresponds to the core-shell nucleus (Fig. 1) that consists of a core of the stable phase surrounded by a shell of the intermediate phase (). These two saddle points are not crossed sequentially by a single nucleation pathway. Rather these are crossed in parallel by two nucleation pathways shown by two heavy lines in Fig. 2 independently. These nucleation pathways are the minimum-free-energy path (MFEP) Iwamatsu2009 () that corresponds to the steepest-descent direction of the free-energy surface.
In our modelIwamatsu2011 (), figure 2 seems to suggest since the size is scaled by the typical size of critical nucleus , but we also need the magnitude of the attachment rate and the transformation rate to determine the time scale and of the nucleation quantitatively around these critical points from Eqs. (60) and (61).
iii.2 Post-critical nucleus in the component space
So far, we have considered the critical nucleus at the saddle point. In order to study the nucleation pathway of not only the nucleation but also the growth of post-critical nucleus, it is more convenient to study Eq. (11) in the original component space. This equation is rewritten in the form
where the first term in the right-hand side is termed the diffusion flux and the second term the drift flux. The latter is given explicitly by
which determine the growth of post-critical nucleus though this drift flux is negligible compared to the diffusion flux near the saddle point as . Eq. (63) is known as the Zeldovich relation Zeldovich1943 (); Frenkel1955 ().
Explicitly, Eq. (63) is written by
However, as pointed out by Stauffer Stauffer1976 (), the continuous equation (11) is originally derived from the discrete Master equation (4) of the attachment and the detachment of monomers, therefore we should use the difference equations
For sufficiently large clusters (), we can neglect the surface tension of the spherical nucleus and the free energy of the nucleus is approximately given by Stauffer1976 () , where and are the chemical potential of the stable phase and the metastable intermediate phase relative to the metastable parent phase. Then, Eqs. (68) and (69) are approximately given by
Therefore, the angle of the direction of the nucleation flux is given by
Since the number of molecule changes linearly in time as and , the final composition
given by Eq. (72). For large supersaturations and , we find
Therefore, the final composition becomes the kinetically optimal composition given by Eq. (24).
Returning to our original problem with the reaction matrix given by Eq. (10), we find and . Equation (75) predicts that the final composition is . Then, the post-critical nucleus consists only of the metastable intermediate phase and cannot reach the final stable phase with .
These seemingly unphysical results are due to the assumption that both the stable phase and the metastable intermediate phase are comparably stable (). In fact, we should have . Using the explicit form in Eq. (10), equations (70) and (71) are given by
Therefore, the number of molecules of the stable phase will increase as , while of the intermediate metastable phase may increase or decrease depending not only on the chemical potentials and but also on the reaction rates and In the late stage of the post-critical nucleus when the depletion of monomer starts to occur, the incoming flux will stop and , and the second term of Eq. (77) can be neglected. Then the number of molecules of the metastable intermediate phase will decrease to compensate the increase of the number of molecules of the final stable phase since . Finally the supercritical nucleus will composed only of the molecules of the stable phase , and the final stable phase with will be reached.
In contrast to two minimum-free-energy paths indicated by two heavy curves in Fig. 2, the two real nucleation paths should gradually turn toward axis after they overcome saddle points indicated by two points. They should merge into the axis as the size increases according to Eqs. (76) and (77) since () in the stable bulk phase. This evolution of the supercritical nucleus will not involve saddle points and, therefore, must be a barrier-less gradual change. The time scale of evolution is determined mainly by the reaction rate and from Eqs. (76) and (77). Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free energy landscape are irrelevant to the kinetics of the post critical nucleus.
Of course, if the metastable phase is as stable as the final stable phase (), prediction of Eq. (75) suggests that the nucleus made of metastable phase (upper nucleation route with in Fig. 2) could have a long life time since the kinetically optimum composition is . Then the metastable intermediate phase would be macroscopically observable. The observability of the metastable phase around the core stable phase as the core-shell structure (lower nucleation route in Fig. 2) also depends sensitively on the difference of the chemical potential and from Eq. (77).
In this paper, we have studied the nucleation pathway of the critical and the post critical composite nucleus with core-shell structure not only in the size and composition space but also in the component space. By extending the results of the previous paper Iwamatsu2012b (), we could study the critical nucleus at the saddle point in the size and composition space. Our results suggested that the critical nucleus can be more appropriately characterized in the size and composition space. However, the kinetics of post-critical nucleus can be studied more easily in the original component space.
Recent theoretical Iwamatsu2011 (); tenWolde1997 (); Meel2008 (); Desgranges2007 () as well as experimental Savage2009 (); Liu2011 () results suggest that the appearance of the composite core-shell nucleus does not necessarily mean two successive activations by crossing two saddle points sequentially Kashchiev1998 (); Kashchiev2005 (); Zhang2007 (); Vekilov2012 (). Rather, the word two-step simply means that the nucleation pathway takes a roundabout course on the free energy landscape via the single saddle point which corresponds to the composite nucleus with core-shell structure. Our result in this paper will be useful to understand the nucleation pathway of such a two-step nucleation not only at the saddle point but also at the late stage of growth after crossing the saddle point.
It must be noted, however, our analysis is completely confined to the steady-state process. The transient properties can only be studied numerically by solving coupled Master Equations Kelton2000 (); Diao2008 (); Wyslouzil1996 (). In addition, our analysis is conducted under the assumption that the nucleation flux goes through the saddle point. Saddle point avoidance Trinkaus1983 (); Zitserman1990 (); Berezhkovskii1995 (); Wyslouzil1995 () will be important if the anisotropy of the reaction matrix is large or the ridge between saddle point is low, which can occur at high temperatures or near the spinodal point. In such a case, the ridge-crossing rather than the saddle-crossing may occur. Then the nucleation flux will spread over the whole phase space and the picture used in this study may break down. Also the growth of composite nucleus will be expected to be more complex. Finally, it should be noted that the recent progress to extract the reaction matrix from simulation data Ma2006 (); Peters2009 () makes it possible to apply our formulation directly to a more realistic situation.
Acknowledgements.This work was supported by the Grant-in-Aid for Scientific Research [Contract No.(C)22540422] from Japan Society for the Promotion of Science (JSPS) and the MEXT supported program for the Strategic Research Foundation at Private Universities, 2009-2013.
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