# Time-evolution of grain size distributions in random nucleation and growth crystallization processes

## Abstract

We study the time dependence of the grain size distribution during crystallization of a dimensional solid. A partial differential equation including a source term for nuclei and a growth law for grains is solved analytically for any dimension . We discuss solutions obtained for processes described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and growth (RNG). Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates the distribution evolves in time into the logarithmic-normal (lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310, 3135 (2008)]. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.

## I Introduction

The micromorphology of solids impacts in an essential way their mechanical, electronic, optical and magnetic properties. It is thus an important task to characterize properly the granularity and homogeneity of materials. This allows in particular the determination or tailoring of their functionality for the development of new microdevices and nanodevices. Crystallization and treatment processes define the microstructure of a material. These time-dependent processes occurring under adequate thermodynamic conditions are generally spatially inhomogeneous, generating grains with different sizes, shape and orientation.

One of the main physical observable describing the resulting microstructure is the grain size distribution (GSD), , which is a quantitative determination of how many grains of a certain size are found in the sample at a given time.(1) The vector is some entity modeling the size and shape of the grains (e.g. diameter of spherical, or semi-axes of ellipsoidal grains, number of atoms in the grains, mass of the grains, etc.) and is the time. The main purpose of the present work is to determine and analyze the time-evolution of the grain size distribution obtained from the resolution of a partial differential equation (PDE), describing the crystallization of an amorphous solid in dimensions. The structure of this first-order PDE is motivated by the study of crystallization processes and involves two functions and that describe the source of nuclei and the growth of grains, respectively.

The crystallization of very different materials leads to distribution laws such as the normal, the Weibull, the Gamma (see, e.g., Ref. (2)), or most notably the logarithmic-normal (lognormal) distribution.(3); (4); (5); (6); (7); (8) The shape of the distribution depends on the mechanisms involved in the formation of polycrystalline materials. When fitting experimental data the choice of the distribution is often not univocal due to the uncertainty of the sampling size considered and the precision of the measurements performed.(9) Nevertheless, it is remarkable that the great variety of crystallization processes can be described by only a few distribution laws. This points towards the fact that the detailed knowledge of the interactions and mechanisms involved in the crystallization of solids may determine the shape of grains, but is not required for the determination of the GSD (see Refs. (1) and (10) and references therein.)

Many theoretical approaches have been proposed to describe the formation of grains during crystallization and they can be divided in two main groups: those that describe the process from an analytical point of view,(11); (10); (12); (13); (14); (15) mainly in one dimension, and those that are based on numerical first principle calculations.(2); (13); (16); (17); (18) The latter group heavily relies on numerical approaches applied to statistical physics. We take the first, complementary approach and derive a closed analytical solution to a phenomenological model that contains the main ingredients of crystallization in dimensions. The PDE and the obtained solution are rather general, but we limit our discussion in this paper to the time-evolution of the grain size distribution for crystallization determined by random nucleation and growth (RNG) processes. This type of crystallization occurs, for example, in amorphous Si thin films.(5); (6); (8)

The physical picture underlying the present study is motivated by the Kolmogorov-Avrami-Mehl-Johnson (KAMJ) theory.(19); (20); (21) Starting from a dimensional solid in the amorphous phase, we assume that nuclei form randomly and homogeneously over the volume of the sample and over time with a constant microscopic rate . Each nucleus subsequently grows at constant microscopic rate into a grain until it impinges on other growing grains, at which point the growth in the direction perpendicular to the interface stops. We do not include coalescence of grains in our model. The central result of the KAMJ theory is the analytical determination of the volume fraction of untransformed material during crystallization (20) (see Sec. II.3). That the KAMJ result is correct within the assumptions made is well documented,(22); (23) although it was shown to be the large time limit of a more general expression for the transient process.(10) Several interesting extensions and modifications of the theory have also been discussed to account for a variety of other crystallization phenomena.(10); (23) Here we work within the framework of KAMJ’s original model to derive and describe the grain size distribution.

Several groups have studied the grain size distribution during first-order phase transitions taking into account the KAMJ result in their considerations.(12); (13); (14); (15); (16) Most notably, Sekimoto considered a partial differential equation in which appears an effective (sometimes also termed actual, transient or average) nucleation rate for his study of one-dimensional magnetism.(12) Similar work followed that was made in the same spirit.(13); (14); (15); (16) This effective time-dependence is a direct consequence of the time decay of the fraction of available space for nucleation of the new stable phase, derived by KAMJ.(19); (20); (12); (13); (10); (14); (15); (2); (16); (17); (18) On the other hand, the growth rate has been considered constant, in all these models.

While staying in the spirit of the KAMJ model by considering constant microscopic nucleation and growth rates and respectively, our approach differs in an essential way with past work on the grain size distribution. Next to the effective time-dependent nucleation rate first derived by Kolmogorov and Avrami in Refs. (19); (20), we also introduce an effective time-dependent growth rate . The physical justification of this time dependence not considered previously lies in the fact that the reduced fraction calculated by KAMJ not only reduces the actual nucleation rate, but also affects the average growth rate. This is shown schematically in Fig. 1. The left column displays snapshots of the grain distribution found at early, intermediate and later stages of crystallization. Important for the present discussion are the shaded grains, which cannot grow further due to impingement. At a given time some grains (in white) grow at a constant microscopic rate , while others have zero growth rate (shaded grains). In average, the effective growth rate will be less than . Comparing Figs. 1 (b) and 1(c) we note that the number of shaded grains increases with time, implying that the average growth rate decays in time. The right column shows schematically the shape of the grain size distribution at these stages of crystallization. The variable is the grain radius. The goal of the paper is to derive analytically and describe, as a function of time and value of parameters, this grain size distribution. All the following figures provide exact results for the sketched GSD of Fig. 1.

The above considerations can be summarized by stating that we place nucleation and growth on the same footing. As will be seen in the following sections, the introduction of time dependence is necessary to describe properly the crystallization of an amorphous solid. Our model and derivation leads to substantial differences with respect to those studied previously. For example, in contrast to previous work our derivation leads to an explicit truncated lognormal-type distribution.(24) In one dimension it is the lognormal distribution. This result is derived, not postulated. The time-dependence of the average growth rate is essential to obtain this result. The truncation of the GSD is another feature of our model that follows from the effective time dependent growth rate: we obtain a maximal grain size at all times. Previous analytical works do not contain such a physical cutoff; they decay to zero only in the limit of the infinite grain size. Finally, we obtain an analytical solution of the equations for any dimension of the crystal, while previous analytical considerations were limited to one dimension. This is because we are able to calculate the inverse Laplace transform for any dimension .

Using the model just described, this paper provides a detailed derivation and generalization to dimensions of the remarkable result presented in Ref. (24) for three dimensions. We establish classes (subsets) of solutions of our partial differential equation one of which leads to a lognormal distribution in the asymptotic limit of large times. The proposed determination of lognormal-like distributions is interesting in itself because contrary to the usual derivation it does not rely on a probabilistic argument.(25); (26); (17); (9) We discuss in depth the conditions under which such distribution is found in our model, and how it relates to existing derivations of the lognormal distribution in Sec. IV.5. We emphasize at this point already that the lognormal asymptotic form is not obtained for the general solution of the equation we establish, but only for a certain class of functions and . This class is made of functional forms known to be relevant for the description of random nucleation and growth crystallization processes and involves a time-dependence of the actual nucleation and growth rates.

To end this introduction, we point out that the results provided in the following sections may be relevant to a variety of topics and phenomena. Indeed, the concept of a distribution law is very general and occurs in many fields; for example, the fragmentation of solids,(25); (26) gas evaporation,(17), the distribution of mass in galaxies,(27) the growth of biological tissues,(28) the distribution of firms as a function of its number of employees,(30) the production of scientific publications as a function of the number of researchers in a group,(31) etc. All these processes are described in terms of a distribution , where must be appropriately defined and is generally a scalar . In the examples just enumerated, the experimental data can actually all be fitted with more or less accuracy by a lognormal distribution or a composition thereof.(30); (32); (28); (29) Because of the general structure of the partial differential equation discussed here, and the fact that the lognormal distribution emerges from fairly general principles, it cannot be excluded that the results of the present paper might be applicable to other phenomena in nature, such as those mentioned above.

This paper is structured as follows. In Sec. II we describe the partial differential equation for the time-dependent grain size distribution and its physical meaning for RNG crystallization models. In this context we introduce the functional forms for nucleation and growth rates and , obtained in the context of the KAMJ model.(19); (20); (21) In Sec. III we present the general solution of the PDE leading to the central result [Eq. (21) or (48a)]. We also calculate the time-dependence of the maximal grain size [Eq. (28) or (48c)] and provide a simple relation between basic measurable parameters of the model [Eq. (31) or (49)]. The general solution is divided into classes according to the time dependence of the nucleation and growth rates and the dimensionality of the crystallization process. We show that for one specific class, the distribution evolves into a lognormal form in the asymptotic limit .(24) We elaborate on the relevance of this result in the following section. In Sec. IV, we present a detailed analysis of the time- and model-parameter dependences of the GSD and discuss our results. We also provide in Sec. IV.5 a discussion of our model in the context of other PDEs and derivations of the lognormal distribution. Finally, we conclude with Sec. V. An example of application of the theory to the analysis of experimental data can be found in Refs. (24) and (33), where the case of solid phase crystallization of amorphous silicon thin films has been considered.

The theoretical calculations are performed with dimensionless quantities. Since one of the purposes of this paper is to provide simple closed analytical forms of the time-dependent distribution that can be used to analyze data and extract fundamental parameters of a physical system we added two appendixes. Appendix A gives a convenient table of all parameters, variables and relations between constants defined in the paper, and Appendix Sec. B summarizes the main results written in quantities with physical dimensions.

## Ii Model

The theory introduced in this section is potentially applicable to a variety of phenomena in nature. Therefore, we first describe in part (Sec. II.1) the differential equation for the distribution in general terms. We then write in Sec. II.2 the equation specifically for RNG crystallization.

### ii.1 Differential equation for the distribution

We are interested in phenomena describable by a time-dependent distribution of a certain entity defined in terms of a dimensional vector . In the case of crystallization processes, the entity is the crystalline grain and the vector describes the shape and size of a grain. For example, if the entities are grains of ellipsoidal shape, they may be described by the semi-axes () and . Another possible choice of is to describe the grain in terms of the number of constituting atoms or the mass , in which case is a scalar (as discussed in the next section, the latter choice leads to a non-linear differential equation, which cannot be solved analytically, while the first choice leads to a set of linear differential equations that can be solved analytically). These two examples show that, in general, the dimension of the vector may be different from the spatial dimension. In the considerations of the present paper the two dimensions will turn out to be identical. Note that this definition of presupposes that the distribution is spatially homogeneous and does, therefore, not depend on the position vector in the sample.

We assume that the dynamics of the distribution is affected by two processes. One is a source-and-sink term that describes the creation and/or annihilation of the entities. The other describes the dynamic growth and/or shrinking of the entities and is defined by the vectorial quantity . The analytical calculations of this paper can be extended to PDEs that include further terms to account, for example, for the possible coalescence of entities, as long as the equation remains first order and linear. Beyond that, numerical techniques are likely the path to follow, except in one dimension, as shown in Refs. (12); (13); (14).

The contribution of the source-and-sink term to the time-evolution of the grain size distribution is given by

(1) |

In the following we consider the separation of variables . The function may often be modeled by a Gaussian centered on some characteristic quantity (for example, the radius of a spherical nucleus ), which may be replaced by a Dirac delta distribution in the limit of zero variance. The time-dependent part of the nucleation rate, , will be discussed in more details in Sec. II.3 below.

The growth of a fixed number of entities is described in terms of a general continuity equation in the space of the vector

(2) |

The growth rate of the grains, , will also be discussed in more details in Sec. II.3.

Assuming that these are the two only contributions to the rate of change in the distribution , we are led to the following partial differential equation

(3) |

We emphasize the point made above that is not in general related to a spatial coordinate, but characterizes the entities that are created and grow in time.

The PDE introduced here on physical grounds is well known and displays a large variety of more or less complicated solutions, depending on the ingredients introduced to account for a particular phenomenon. The study of this equation leads to an interesting and unexpected outcome when applied to crystallization processes, as shown in later sections (see also Ref. (24).) A formal general solution can be found for this first-order linear partial differential equation and specific solutions can be obtained by quadrature for well-behaved source-and-sink terms and growth rates .(34) We discuss in the next sections closed analytical solutions for growth rates and source terms applicable to random nucleation and growth processes during crystallization of amorphous solids. Note that altered forms of the above equation, some pertinent to other physical situations, have been discussed in the literature (see, e.g., Refs. (11); (35); (36); (12), (13), and (14) and references therein.) The choice of and , the formalism, and the resulting analytical solutions distinguish the present work from others.

### ii.2 Differential equation for RNG processes

To apply the differential equation of the previous section to RNG processes, we need to specify the rates and . Transmission electron microscope(5); (6); (4); (37); (29) or atomic force microscope (38) measurements performed during crystallization of many different materials reveal grains with a great variety of shapes and sizes. These grains are generally characterized by a scalar quantity, such as the number of atoms, the volume or the mass of the grain. In the present paper, we take an alternative approach, and describe the grains as ellipsoids with semi-axes with . For thin films , whereas for bulk solids . The components of the vector defined in the previous section are now given by the semi-axes of the ellipsoid with . In this representation the dimensionality of the vector coincides with the spatial dimension of the nucleation and growth process.

In general, the experimental representation of the grain size distribution is done in terms of the average radius of a dimensional sphere, the volume of which equals the measured volume of the grain. For this reason, we later consider the special case of spherical grains to allow comparison of the theory with existing experimental data.(24); (33) This reduction to spheres in dimensions has the additional advantage of expressing the grain size distribution in terms of two variables only (the radius of the grain and time ) and allows a better understanding of the behavior of the distribution as a function of nucleation and growth rates without the additional difficulty related to the anisotropic growth and orientation of the grains.

In our RNG model, the source describes the formation of nuclei (we do not consider the dissolution and coarsening of grains) that is determined by microscopic interactions and the thermodynamical conditions, under which crystallization of a specific material occurs. We assume that nuclei are formed with a critical volume at a rate . The source term thus takes the form

(4) |

where

(5) |

gives the volume of a dimensional ellipsoid, and is the gamma function. is defined as in Eq. (5) and is the critical volume of a nucleus, that is, the volume of the smallest grain that can be found in the sample. This simplified expression for the nucleation term (4), which states that all nuclei are formed with the same volume , is consistent with the fact that we do not discuss the details of the nucleation process. It is a good first approximation, except for describing early stages of crystallization.(10) This will become apparent in Sec. IV and the application of the theory presented in Ref. (33).

Once a nucleus is formed, crystallization leads to its growth into a grain. One simplification relevant to some RNG processes is the case where the variation in the growth rate only weakly depends on . In this case, Eq. (3) can be approximated by

(6) |

In fact, this equation applies to the case of the solid phase crystallization processes considered in Refs. (24) and (33). In this case, the growth rate is independent of . This non-trivial simplification is one reason for choosing the present formalism in the vector space of rather than writing the grain size distribution in terms of a scalar such as the volume or the number of atoms of a grain. Indeed, in processes such as solid-phase crystallization,(5); (6); (8); (24); (33); (4) atoms are present in the immediate vicinity of nuclei and grains at all time. Thus, their contribution to the growth of grains does not require diffusion processes and the rate does not depend on the size of the grains.(19); (20); (21) It is essential to realize that writing the equation for random nucleation and growth, in terms of a scalar, such as the number of atoms in the grain, would result in a non-linear dependence of the growth rate on that scalar, which is generally untractable analytically. The present formalism allows to circumvent this problem at the cost of a multidimensional PDE, which turns out to be solvable analytically.

As mentioned above, applications(24); (33) assume grains of spherical shape and the grain size distribution depends only on the radius of the dimensional sphere. In the space of semi-axes of the ellipsoid, the vector for a sphere is , and the magnitude of the vector appearing in Eqs. (4)-(6) is . Introducing polar coordinates, where is the radial coordinate, and using the expression for the source term (4), we obtain

(7) |

where is the surface of the hyperspherical nucleus of radius in dimension.

We derive in the next section the exact solution of the above equation for rates relevant to RNG processes, with the boundary conditions

(8a) | |||

and the initial condition | |||

(8b) |

where is the incubation time. We emphasize that contrary to many approaches, the present formalism does not assume prior knowledge of an initial distribution; there are no grains in the samples until the incubation time is reached.

### ii.3 Random nucleation and growth rates

We define in this section the effective nucleation and growth rates and that appear in Eq. (3). The thermodynamics of nucleation and early stages of crystallization is involved and a subject of its own.(10); (39) We assume that the conditions necessary for the creation of nuclei and their subsequent growth into grains are fulfilled. The creation of nuclei is described in terms of the existence of a critical average volume and an effective rate introduced in Eq. (4). We are mainly interested in the time dependence of the grain size distribution, resulting from RNG processes and consider situations where no coarsening occurs.(5); (6); (8) Full crystallization has been completed once each atom of the sample is assigned to a grain. This is a realistic scenario, for example, in solid-phase crystallization of an amorphous sample,(24); (33) and is the one considered in the KAMJ theory for RNG processes.(24); (19); (20); (21)

The model assumes homogeneous nucleation; nuclei appear randomly in the sample (no pre-existing nuclei or nucleation centers) with a constant microscopic rate , irrespective of the presence of already transformed material. To satisfy this condition, the concept of phantom nuclei was introduced in Ref. (20). Avrami obtained an expression relating the effective to the extended volume fraction of transformed material that corrected for the presence of these phantom nuclei and lead him to derive an expression for the fraction of material available at time for further nucleation and growth(19); (20); (21)

(9) |

is the incubation time, is the critical crystallization time, and the integer determines the time dependence of the crystallization process. is the Heaviside function.

Later work on grain formation in various systems to which the KAMJ applies (e.g. Refs. (12); (13); (10); (14); (15); (16); (17); (18) and (39)) has recognized that though the microscopic nucleation rate is constant, the actual (effective) nucleation rate relevant for the grain size distribution decays in time as because of the reduced fraction of available space for nucleation. This effective rate has been confirmed, refined, and extended to include a variety of nucleation phenomena. For example, the expression for has been generalized to account more precisely for the early stages of nucleation.(10) The concept of ”phantom nuclei” introduced by Avrami(20) and the importance of which has been thoroughly discussed(22) has been linked to the concept of spatial correlations of nucleation.(12); (22) Further extensions of the model have been proposed to include simultaneous nucleation, non-random processes, coalescence, or to take into account the symmetry of the crystal structure in the transformed phase.(12); (14); (23); (22)

In the present paper we account for the insight gained in earlier work, but extend the analysis in an essential way by also considering the effective change in the growth rate. This aspect of the growth of grains during crystallization has not been addressed in any previous work. We start with the random and homogeneous nucleation of grains in the sample. As the crystallization process develops, each nucleus grows into a grain at a rate . The growth of each grain eventually comes to a halt, either because it is inhibited by the presence of another grain boundary (e.g., by impingement) or because crystallization has been completed (no further free atoms are available). Thus, in the same spirit as for the analysis of the effective nucleation rate, the growth rate actually decreases with time (see Fig. 1). The proper account of this effective time-dependent growth rate turns out to modify, in important ways, the grain size distribution and leads to the lognormal-like form of the distributions observed in experiment.(24); (33)

Based on Eq. (9), we thus develop further the theory of RNG by introducing different rates and for nucleation and growth, respectively. We postulate that both rates take similar functional form, except, for different critical times and and different power laws in the exponential. As a result, we introduce the following expressions for nucleation and growth rates:

(10a) | |||||

(10b) |

While the effective nucleation rate [Eq. (10a)] has been used in previous work(12); (13); (14); (15); (16); (17); (18) to determine the grain size distribution, the introduction of the effective growth rate [Eq. (10b)] is new. The analytical form of and the choice (exponential time decay) will be justified a posteriori by analyzing the analytical results and comparing the theory with experimental data.(33) It is important to realize that the KAMJ model relies on the fundamental assumption of constant and homogeneous microscopic nucleation and growth rates, denoted and . By introducing the above effective rates [Eqs. (10)], we thus presuppose that the physical picture underlying the present theory is the same as that of the KAMJ model, but we account for the effective time decay of the nucleation and growth rates.

The nucleation and growth rates contain five parameters: , , , and . However, if one shifts the time variable by , it turns out that only two parameters actually affect the normalized distribution, namely, and the ratio . In addition, the integers and determining the power law of the time decay have to be specified. For (or ), the decay is exponential, and for it is Gaussian. For it is super Gaussian. For , the growth rate is constant, and the PDE for reduces to that of Refs. (12) and (14) in the absence of coalescence. The nucleation rate is essentially given by the fraction of material available for crystallization and thus , the dimensionality of the grain.(20) Consequently, our is the critical time in the conventional KAMJ model. On the other hand, the growth rate is determined by another mechanism, namely, the inhibition of the grain growth by their neighbors. We therefore expect the exponent to differ from . Of importance for the following are the values and . The case and has been studied analytically,(12); (14) taking into account coalescence, which is absent in the present model. Note finally that must not necessary be an integer, but for simplicity we consider only this case here.

## Iii Solution for the extended Kolmogorov-Avrami-Mehl-Johnson model

We calculate the grain size distribution for the random nucleation and growth crystallization model described in the previous section by solving Eq. (3) with Eqs. (10). This can be achieved formally by various methods as, for example, the Laplace transform or the methods of characteristics.(34) An explicit expression for is then obtained by quadrature once and are defined. For the nucleation and growth rates defined in Eqs. (10), it is more natural and instructive to use the Laplace transform. Remarkably, we can obtain the inverse Laplace transform in any dimension within the present model. The details of the derivation of are presented in Appendix C. We also determine the time-dependent maximal size of the grains that can be found in the sample, and the moments of the distribution.

### iii.1 Partial differential equation in dimensionless quantities

It is useful to introduce the following dimensionless variables:

(11a) | |||||

and constants | |||||

(11b) |

It is important to bear in mind that a simplified notation is used in Eqs. (11) and the following to avoid overloading expressions with indexes. The dimensionless variable actually depends on through . This choice of dimensionless radius is made because, in general, and the latter is easily measured experimentally while the former is not. Since we consider solutions obtained for different growth laws (distinguished by ) separately, no confusion can arise from this simplified notation. In Sec. IV, we analyze the dependence of the general grain size distribution on the parameters of the model, that is, for various values of the constants defined in Eq. (11b).

To solve the partial differential equation (7) it is useful to write it in dimensionless variables (11) and to introduce the auxiliary dimensionless function , where is the surface of the largest hyperspherical grain of radius found in a dimensional sample at full crystallization. Then, the differential equation. (3) or Eq. (7) takes the form

(12) |

with , and the dimensionless nucleation and growth rates

(13a) | |||||

(13b) |

It is important to remember that the grain size distribution is given by , not by . The latter is only an auxiliary function to bring the partial differential equation in a form that can be solved analytically.

### iii.2 Solution of the partial differential equation (12)

We derive in Appendix C the solution of Eq. (6), written in dimensionless quantities as Eq. (12), for the KAMJ model [Eq. (9)], and for effective nucleation and growth rates [Eqs. (13)]. We obtain the following grain size distribution for any dimension

(14) | |||||

with the constant prefactor defined by

(15) |

with defined above Eq. (12).

The general solution is directly proportional to the ratio and inversely proportional to . The coefficient cancels out when considering normalized distributions. Such is the case in the numerical calculations of the next sections and often also in the description of experimental data.(33) The functions () are solutions of (see appendix C)

(16) |

with defined by

(17) |

The function only depends on the growth rate, not on the nucleation rate. For the KAMJ growth rate (13b), the latter equation has a single solution; the sum over in Eq. (14) and the index in can therefore be dropped. Equation (16) can be rewritten in the form as

(18) |

for , where is the upper incomplete gamma function. This equation yields for

(19a) | |||

and for | |||

(19b) | |||

where erf is the error function. For higher values of , Eq. (16) is implicit and solving it involves special functions. |

To bring to the fore the physics contained in our general solution for the KAMJ model, we transform the time-dependent product of Heaviside functions appearing in Eq. (14) into a size-dependent difference of these functions (see Appendix C)

(20a) | |||||

(20b) |

where is determined below. This difference expresses the fact that at all times, only grains with radius between the minimum () and the maximal size () can be found in the sample. Thus, Eq. (14) now reads as

(21) | |||||

This is the central result of the theory, the general solution of Eq. (12) for spherical grains in dimensions using random nucleation and growth related rates that contain the fraction of material available for further crystallization of the Kolmogorov-Avrami-Mehl-Johson model [Eqs. (13)]. The function appearing in Eq. (21) is a solution of Eq. (16) and is given by Eqs. (19). The maximal grain size is obtained in Eq. (28) below.

This result has several general properties. First, the solutions can be divided into classes characterized by the dimensionality of the crystallization process , and the time-decay of the nucleation and growth rates, which are specified by the values of and . Thus, each triplet of non-negative integers defines another class of the general solution. In the RNG process discussed in the present paper, is identified with the dimension and, therefore, the classes are defined by the doublet . Second, the time dependence of Eq. (21) appears through [Eq. (18)] and [Eq. (28)], which are non-trivial functions of . Third, the nucleation rate appears explicitly only in the prefactor , whereas the growth rate is present in and as well. This is also the case for (constant growth rate) and agrees with Ref. (12). On the other hand, both critical times and appear in the exponential through the ratio . Fourth, the derivation of the grain size distribution provides cutoffs at the radii of the nucleus and the largest grain found in the sample at time . This will be discussed further in the next section. Finally, we emphasize that the explicit analytical solutions were derived here for specific source and growth terms and , namely, Eqs. (10), which are consistent with the KAMJ model. This is important to remember for the discussion of the next sections, especially the result obtained in the limit .

Explicit expressions of the GSD can be written for specific classes of solutions , in particular, for the cases of interest for the description of experimental data, namely, and . For but arbitrary , Eq. (21) reads as

(22a) | |||||

with | |||||

(22b) |

and is derived below [Eq. (29a)].

From the above time-dependent expression we obtain the GSD at full crystallization. For , Eqs. (22) obtained for becomes

(24) | |||||

The decay of the growth rate is exponential in this case. Of particular interest is the case is when . The distribution then takes the form(24)

(25) | |||||

This result is closely related to the lognormal distribution. As shown in Ref. (24) this is seen more clearly by noting that, in general, , which implies that for , the equation simplifies to

(26) |

The close relation of this result to the lognormal distribution [see Eq. (45) in Appendix II for the expression written in physical units] and its application to full crystallization of amorphous silicon were presented in Ref. (24). We emphasize that obtaining a lognormal-like distribution as a solution of a partial differential equation is a quite remarkable result, and we elaborate on its significance in Sec. IV.5.

Contrary to the case , no significant simplification of Eqs. (23) () is obtained in the limit . The above expressions allow studying the behavior of the theoretical distribution in the following sections, and were used to analyze experimental data during,(33) and at full crystallization(24) of amorphous silicon.

### iii.3 Maximal grain size

The solution presented in the previous paragraph contains the maximal grain size that can be observed in a sample undergoing RNG crystallization. This quantity is obtained from . Integrating one immediately obtains in dimensionless variables

(27) |

For given by Eq. (13b), the above equation can be written in terms of the upper incomplete gamma functions, ,

(28) |

Explicit expressions of Eq. (28) with and are

(29a) | |||||

(29b) |

with .

It is instructive to digress from the main path of the paper and determine the maximal grain size in the case , which corresponds to a constant growth rate . We obtain

(30) |

We note that . This is consistent with previous work that showed unbounded maximal grain size but is not physically justified unless coalescence is taken into account. We do not analyze this case further here, as we would have to redefine our dimensionless grain size .

One can also determine the maximal grain size once crystallization is completed (). In the dimensionless formalism used in this section this corresponds to . The expression for can be transformed into an interesting relation between fundamental parameters of the model for any [see Eq. (49) in Appendix B]

(31) |

This equation can be used as a self-consistency check or to determine the value of one parameter once the others have been measured.

The time-dependence of the maximal grain size is depicted in Fig. 2 for and reasonable values of .(33) Within our model given by Eq. (13b) is proportional to . Hence, if follows from Eqs. (28), (48c), and (49) that does not depend on ; the results in Fig. 2 depend only on and . It is also worth pointing out that, except for early stages of crystallization, the inequality is generally satisfied and Fig. 2 essentially displays the ratio . The figure demonstrates that the radius of the largest grain observed in the sample saturates rapidly in time to reach the size observed at full crystallization (calculations of the GSD for the parameters of Fig. 2 show that full crystallization is obtained for if and for .) A decreasing value of (e.g., faster decay of the growth rate at constant nucleation rate) enhances the rate at which reaches . A similar conclusion is reached with increasing . This behavior results from the fact that both a decrease of and increase of accentuate the time decay of the growth rate, which –as discussed in previous sections– reflects the impingement caused by neighboring grains.

The analytical determination of the maximal grain size in terms of fundamental parameters of the model is of particular interest for those interested in a probabilistic approach using computer simulation to determine the grain size distribution because, as discussed in Ref. (9), the proper account of rare events (the size of the largest and smallest grains) is essential for describing the data with the adequate distribution and obtaining the correct value for its average.

## Iv Characterization and time-evolution of

This section is devoted to the characterization of the grain size distribution obtained in Eq. (21). First we analyze the influence of the model parameters and , the ratio of critical times and the growth rate, respectively. This leads to scaling properties for in the limit . We then consider the time-evolution of the distribution. Finally, we show how for certain classes of solutions the distribution takes the lognormal form in the asymptotic limit of large time.

To proceed with the numerical analysis, it is appropriate to consider the normalized GSD

(32a) | |||

where the total number of grains at time is given by | |||

(32b) |

This normalization procedure eliminates the constant factor defined in Eq. (15). As mentioned earlier, this is the only term containing the nucleation rate and the results discussed in the remainder of this paper are therefore independent of the explicit value of . Such normalization is also useful for comparing the theory with experimental data.(24); (33)

The figures in the present paper all depict the normalized GSD . Thus, the area under the curve is one. It is also worth noting for the next sections that none of the distributions discussed here contain divergences.

### iv.1 Moments and maxima of the distribution

The time-dependence of the distribution can be characterized in terms of its moments. In particular, the first three moments, which give the mean , the variance , and the skewness of the distribution. , are central moments and the latter is normalized. The two higher moments give an indication about the spread and asymmetry of the distribution about the mean. All moments are calculated for the normalized GSD [Eq. (32)] and thereby independent of the nucleation rate coefficient . We define

(33a) | |||||

(33b) | |||||

(33c) |

The definitions are given for the dimensionless GSD written in terms of . Moments for are given in Appendix B. The conventional notation for the third moment, (always written with its index), should not be confused with the variable (never written with an index.) We calculate the time dependence of the mean and variance for the case and in Sec. IV.4.

In some cases, it may be of interest to compare the radius () for which the distribution is maximal to the mean of the GSD. Since it turns out that under certain circumstances the GSD [Eq. (21)] has more than one maximum (see Sec. IV.4), we add the index to . We thus define by

(34) |

The analytical form of can be determined from the zero of the derivative

(35) |

For the case and the maximum in the open interval is given by

(36) |

### iv.2 Grain size distribution at

For , the (unormalized) distribution is given by Eq. (24) for . We remind that in this case the time-decay of the growth rate is exponential. The normalized distribution is shown in Fig. 3(a) for and the parameter values specified in the caption. These are physically reasonable choices of parameters as discussed in Refs. (24) and (33).

All curves have identical upper cutoff (the upper cutoff is out of the range of the figure for ; Fig. 3 right). For both the mean value and the variance of the distribution decrease with larger ; the peak of the distribution is sharper and shifts to lower values of for increasing . Since we identify with the dimensionality of the system, we expect the GSD of three-dimensional crystallization to be sharper than that of thin films when the thickness of the film is smaller than the average grain size at full crystallization. Furthermore, the majority of grains have smaller size in three dimensions than in two dimensions.

The distribution is also affected by the choice of . Choosing (Fig. 2, right) instead of (Fig. 2, left) implies a faster time-decay of the growth rate. From Fig. 2 we obtain the physically intuitive result that a stronger time-decay of the growth rate leads to sharper peaks, with a maximum located at smaller grain radii. This is emphasized by the difference in abscissa and ordinate scales between and in Fig. 3. Replacing the exponential decay () of the growth rate by a Gaussian decay () has a dramatic effect on the grain size distribution. This results in the unambiguous choice to describe the experimental data of solid-phase crystallization of amorphous silicon.(24); (33)

### iv.3 Dependence of on and

The grain size distribution is a function of and . However, not all parameters need always be known to determine the GSD. Considering Eqs. (18), (28), and (51), we note that it is possible to reduce the number of parameters that appear in Eq. (21). For example, from Eqs. (18) and (28), we can express in terms of , , , and . Thus, the normalized GSD can be written in terms of the latter parameters, and and are not explicitly needed in the expression. Conversely, it is possible to express the GSD in terms of the two latter quantities, thereby, removing other parameters. For example, for , Eq. (51) implies , which can be used to write in Eqs. (22) and (24). As is often the case, , which implies that the ratio is essentially . Then, in the limit , the GSD [Eq. (24)] only depends on and and the normalized GSD only on . At finite times remains present and is directly proportional to . However, can be determined experimentally, and is not required. To summarize, the choice of which parameters are needed and which can be obtained from the expressions above depends on the particular situation under consideration. A natural choice of parameters to discuss the properties of the normalized distribution is and . We discuss in this section the general dependence of on these two parameters.

To study the generic influence of on the behavior of , it turns out to be sufficient to consider Eq. (21) in the limit when full crystallization is achieved. This limit has been calculated analytically for classes of solutions relevant for experimental studies in Eqs. (22-24). Figures 4 and 5 highlight the influence of and , respectively, on the shape of the distribution for (left column) and [right column; the rows are for (top), and (bottom)].

The first general observation is that in all cases, the distribution displays one maximum, and cutoffs at and . While the latter is obvious from Eq. (21), the former is only true at large times within or model, as will be seen in the next section.

As decreases, the number of small grains increases at the expense of the formation of larger grains. This is confirmed in Fig. 4, which shows that irrespective of the value of and a decrease of results in an increase in amplitude, a sharpening of the peak, and a shift of the maximum to smaller values of . Furthermore, the properties observed on Fig. 3 are also found in Fig. 4. Thus, the qualitative features inferred from Fig. 3 do not strongly depend on the particular value of when taken within a physically reasonable range. It is interesting to observe that the position of the maximum of the distribution for and is essentially insensitive to the value of unless the latter is very close to one. Even then, comparing the order of magnitude of abscissa and ordinate scales of Figs. 4(e) and 4(f) and 4(a) and 4(b), the peak of the distribution barely shifts for and increasing . On the other hand, the amplitude of the maximum strongly varies with . Finally, the distribution for and is rectangular [lower red dotted line in Fig. 4(d)]. This result is derived in the next section.