##
1Geometrical *vs.* statistical models for describing phase transition kinetics in thin films
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Geometrical

*vs.*statistical models for describing phase transition kinetics in thin films
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We present calculations and simulations to investigate different theories describing phase transitions in thin films, with special emphasis on the growth of the new phase after nucleation.

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In particular, models with geometric and statistical growth rules are compared.

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It is demonstrated that the commonly employed

*geometrical*approach, which assumes nucleation and subsequent radial growth of the newly formed phase, has distinct limitations for thin film systems.
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More realistic statistical Monte Carlo simulations that are governed by

*statistical*growth rules, predict that a non-spherical (prolate) shape may develop after nucleation at or near a surface or interface.
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In addition, the predicted kinetics of the phase transformation is notably different for the geometric

*vs.*the statistical model, for similar parameters.
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The simulation results are compared to recent experiments on the crystallization of thin amorphous solid water films.

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## Introduction

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The transformation of materials from the amorphous to the crystalline phase and its theoretical description have received considerable attention due to its relevance for areas as diverse as polymer science,

^{1,2}interstellar physical chemistry,^{3–5}semiconductor technology^{6}and metallurgy.^{7}
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Particular attention has been paid to the kinetics of this transition in thin films, for reasons of experimental sample geometry and to mimic finite-size effects in naturally occurring systems.

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For example, the crystallization of amorphous solid water in confined geometries has received significant attention the past decade.

^{5,8–23}
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There has recently been renewed interest in theories that predict the crystallized fraction as a function of time (as this constitutes the commonly measured quantity), and its sensitivity to the various parameters associated with the phase transition: the nucleation probability, the rate of subsequent growth and the desorption rate of material from the thin films, if pertinent.

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These analytical theories (see

*e.g.*^{refs. 16, 24–26}) are based on Avrami’s seminal contribution^{27}and assume that nucleation sites are randomly distributed throughout the volume (or over the surface) of the initial phase and appear with a fixed probability per unit volume (per unit surface area) per unit time.
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The subsequent growth is radial, with a constant velocity.

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In these models, the growth criterion is therefore

*non-local*and geometric: if a point within the layer is sufficiently close to a nucleation center, it will have changed phase.
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At any point in the layer, the probability for conversion therefore does not depend on the local environment of that point, but depends on how far it is removed from a crystallization nucleus and how much time has elapsed after nucleation, hence the denotation geometric, or non-local.

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The geometric model has recently been extended to include the different desorption rates from the different phases in simulations by Ahlström

*et al.*^{28}
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Although this geometrical model has been successfully applied in a number of cases,

^{12,16,19,20,23–26,28}it does not account for the finite size of the (molecular) elements in the system.
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As a continuum model, the geometrical approach implicitly relies on a sample size well exceeding the size of its molecular constituents; this approximation therefore must break down when the sample size is on the order of typical molecular diameter.

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19

For very small samples, a

*local*, statistical description for the growth would seem more appropriate.
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In such a statistical model, the probability for, say, a water molecule to become part of the crystalline phase, is determined by the phase state of its immediate surroundings in a statistical manner; if the surrounding is predominantly crystalline, the chance of phase transformation will be large.

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To investigate possible shortcomings of the non-local geometrical model in finite-size systems, we compare the results of geometrical models with the results of the most basic statistical (local) model that predicts spherical growth in bulk media: an atomistic lattice model simulation

^{29,30}based on a Monte Carlo approach.
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As opposed to the non-local growth rule of the geometric model, the Monte Carlo (MC) simulations are characterized by statistical growth behavior in the following manner: a film is subdivided into cells, and the individual cells can crystallize depending on the state of their direct neighbors.

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The growth criteria for the MC simulation are the simplest that reproduce three-dimensional spherical growth in the bulk, without any prior knowledge of the energetics of the process.

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Although both approaches therefore in principle give rise to the same shape of crystalline domains, a comparison of the MC simulations of the crystallization and growth process (using local, statistical criteria) with the Avrami-type (non-local) geometrical approach for finite sized systems reveals distinct differences between the two methods.

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First of all, we demonstrate that the statistical model predicts that the growth rate depends on the size of the converted region, and therefore on time, as opposed to the geometric model that assumes time-independent growth rate.

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This effect is not limited to thin films.

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Additional discrepancies appear when nucleation occurs at or near the surface or an interface of a thin film.

^{16,24–26}
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Whereas, again, for the geometrical model the growth rate is constant, for the statistical model the presence of the surface decelerates crystal growth near the surface.

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As a result, for the MC simulation, the transformed region does not exhibit the shape of a truncated sphere, as predicted by geometrical models, but rather by a truncated prolate spheroid.

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## Approach

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As we are interested in very thin films, the film dimensions will be expressed in monolayers (ML) throughout this paper, corresponding to approximately one molecular diameter.

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Time will simply be expressed in seconds.

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The phase transformation in a thin film, such as the crystallization of amorphous solid water, requires the nucleation and growth of small crystalline domains in the amorphous phase.

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In the two models described here, nucleation occurs in the same manner, but the growth rules differ.

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Nucleation occurs at random times at random positions within the layer, or on its surface.

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The nucleation probability for bulk nucleation is defined per unit volume per unit time

*J*_{bulk}(expressed in ML^{−3}s^{−1}).
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For nucleation occurring at the surface, it is defined per unit surface area per unit time

*J*_{surf}(expressed in ML^{−2}s^{−1}).
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The energy barrier that must be overcome to form a crystalline nucleus is determined by, amongst other, a competition between the surface tension that arises between the two phases upon nucleation (counteracting nucleation) and the difference in chemical potential between the two phases (favoring nucleation, as the crystalline phase is thermodynamically most stable).

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As a result, the instantaneously converted volume (critical nucleus size) that appears upon nucleation will contain a significant number of molecules (previously reported to be ∼100 for water crystallization

^{19}).
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The size of the, assumed to be spherical, critical nucleus is defined by its radius

*r** (expressed in ML).
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The subsequent growth of crystalline domains is modeled in two different manners, as introduced above: through a geometric criterion, and by statistical growth rules, that are explained in the following.

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### A. Statistical growth

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For the MC simulations, a cubic lattice of (100 × 100 × 50) cells is employed, and the different cells are distinguished by their spatial coordinates (

*x*,*y*,*z*).
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The choice of 50 for the thickness was motivated by recent experiments on the crystallization of amorphous solid water in which a layer thickness of approximately this number of molecules was employed.

^{23}
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Note, however, that the results described here are not specific to this thickness.

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In any case, this comparison to experiment implies that each cell corresponds to one water molecule.

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Any of the cells may be untransformed (amorphous) or transformed (crystalline), defined with cell values

*Θ*= 0 and*Θ*= 1, respectively.
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The phase state of each cell is evaluated in every time step Δ

*t*employed in the simulations.
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The nucleation process is treated in the following manner: For each time step, the discrete number of nucleation events is determined by the nucleation probability

*J*_{bulk}, in conjunction with the system volume*V*and time step Δ*t*employed in the simulations.
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This number is obtained from a simulated Poisson distribution, whose mean is given by the product

*J*_{bulk}*V*Δ*t*.
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For surface nucleation, the Poisson mean is given by

*J*_{surf}*A*Δ*t*, with*A*the surface area on which nucleation occurs.
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The critical nucleus size, whose size is defined by its radius

*r**, subsequently grows using local, statistical growth rules: a cell at position (*x*,*y*,*z*) has a certain probability of crystallizing either as a result of a nucleation event, or a growth event if one of its 26 neighboring cells is crystalline.
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This means that the probability for conversion of the cell is determined by its immediate surroundings (local), as opposed to the geometrical (non-local) growth rule, which simply considers the distance from a nucleation center (see below).

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Note that, as depicted in Fig. 1, a cell has three types of neighbors:

*N*_{1}= 6 direct neighbors that share a face, at a distance of the unit cell size*l*,*N*_{√2}= 12 neighbors that share an edge at a distance of √2*l*, and*N*_{√3}= 8 neighbors that share a corner point, at a distance of √3*l*.
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The probability

*P*_{trans}for the crystallinity to propagate to an adjacent cell is inversely proportional to the geometric distance, and linearly proportional to elapsed time.
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It is important to note that these growth criteria are the simplest that ensure spherical growth in three dimensions (growth through just nearest-neighbor faces results in a diamond-shaped volume).

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It is evident that the criteria for spherical growth such as defined here for a cubic lattice, with hopping probability inversely proportional to the geometric distance to a neighbor, are readily extrapolated to other types of lattices.

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Ideally, one would like the growth criteria to be determined by the energetics associated with the growth process.

^{32}
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Unfortunately, however, for most systems (as is also the case for the transition from amorphous solid water to crystalline ice), very little is known about these energetics.

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Therefore we adopt this statistical approach.

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At every time step, for each cell that has not yet been transformed, the state of its 26 neighbors is evaluated.

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The probability that during this time step this cell will transform equals:with summations over the cell values

*Θ*of the different types of neighboring cells:*Θ*_{N1},*Θ*_{N√2}and*Θ*_{N√3}(*Θ*= 1 (or 0) for an (un)transformed cell).
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*P*

_{prop}represents the probability to propagate

*via*a face, which is proportional to the time step size and the growth rate.

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A random number between zero and one is then computer generated, and if this number is smaller than

*P*_{trans}, the value of the cell*Θ*changes from zero to one as a result of crystallization.
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Otherwise, it remains unchanged.

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Care was taken to ensure that the results were independent of the size of the time step used in the simulations,

*i.e.*to keep*P*_{prop}sufficiently small.
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From a set of simulations with varying time step size, it is clear that, if

*P*_{prop}is set to be smaller than 0.015 per time step, the results converge.
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In the simulations reported here, we set

*P*_{prop}= 0.01 per time step.
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For convenience, time is expressed in seconds in the remainder of this report.

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To account for contributions of spheres nucleated outside, but grown into, the considered (100 × 100 × 50) volume, the simulation volume is embedded in a larger volume (typically (300 × 300 × 50)),

*i.e.*the lateral dimensions (but not the thickness) of the simulation volume are enlarged by an amount determined by the product of the maximum growth rate and the total simulation time.
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The (100 × 100 × 50) volume is then analyzed to determine the time-dependent converted fraction.

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We did not use periodic boundary conditions, as the statistical fluctuations in nucleation and growth rates are amplified in that way.

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Desorption was included in some of the simulations in a non-statistical manner, simply by reducing the layer thickness from the top as a function of time.

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The presented results are averages of 30 simulation runs, to reduce the effects of statistical fluctuations in the growth process.

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A set of 30 simulations typically takes 10 h on a desktop personal computer.

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The converted volume as a function of time can be obtained by evaluating the summation of cell values over all cells:

*V*_{conv}= ∑_{x,y,z}*Θ*_{x,y,z}.
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The time-dependent crystalline fraction is given by dividing

*V*_{conv}by the total considered simulation volume.
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The time-dependent crystallinity at the surface or at the sample–support interface can be obtained analogously (experimentally, the thin water layers are deposited on a support).

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### B. Geometric growth

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For the model that assumes geometrical (non-local) growth, nucleation occurs as described above, but the description of the growth of the nucleus is governed by geometric considerations: growth simply occurs radially from a nucleation center at geometric rate

*G*_{geo}(in units ML s^{−1}).
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Thus, if nucleation of the new phase occurs at time

*τ*, and the nucleus has an initial radius*r**, then the radius*r*of the sphere at time*t*>*τ*equals*r*=*r** +*G*_{geo}(*t*−*τ*).
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It is therefore evident that any cell at position (

*x*,*y*,*z*) is crystalline if the inequality:is satisfied, for any of the nucleation sites (*x*_{i},*y*_{i},*z*_{i}).
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If one is interested in the overall kinetics (

*i.e.*the converted crystalline fraction as a function of time) one can derive analytical expressions describing the kinetics, as reported in detail in^{refs. 16 and 26}.
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The essence of the derivation of these expressions is that the converted (crystalline) volume is calculated as a function of time for all nucleated regions.

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Because the nucleation probability per unit volume and unit time,

*J*_{bulk}, is independent of elapsed time and position, so-called ‘phantom nuclei’^{27}may appear in already transformed regions and transformed regions may overlap.
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This approach results in the calculation of a time-dependent

*extended*volume*N*(*t*) (*i.e.*doubly counting overlapping regions), from which the real crystalline volume*X*(*t*) follows through the relation*X*(*t*) = 1 − exp[−*N*(*t*)]^{27}.*N*(*t*) is calculated by considering all possible nucleation events,*i.e.*by integrating the function*N*(*t*,*τ*) over all times*τ*up to time*t*.
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The result can be directly related to the converted fraction of bulk material.

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Following

^{26}(neglecting the desorption of material from film), this results in the following expression for*N*(*t*), for a layer of thickness*d*:Similar expressions for the surface and sample–support interface crystallinities can be readily derived following the procedure in^{ref. 26}, as well as the corresponding expressions for surface nucleation.
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These expressions including desorption can be found in the same reference.

^{26}
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In analogy to the statistical growth criterion, one can also perform simulations using a non-local, geometric criterion, as demonstrated previously in .

^{ref. 28}
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The results of such simulations for the non-local geometric approach are in perfect agreement with the analytical results, the only difference being the discrete nature of the lattice in the simulation that gives rise to a quantized increase in the converted fraction.

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## Results and discussion

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Two types of investigations were performed: one with fixed nucleation site, and one with random nucleation throughout the film.

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The first allows us to investigate the differences between statistical and geometric growth rules regarding the

*shape*of the transformed volume.
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Two limiting cases are considered: nucleation in the center of the film at cell (

*x*,*y*,*z*) = (50,50,25), and on the surface at (50,50,1), at time*t*= 0 s.
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In the second case (random nucleation throughout the film) we investigate the effect of the different types of simulated growth on the time-dependent fraction of converted material, to mimic typical experimental approaches.

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For each of these cases (fixed and random nucleation), statistical and geometric growth criteria were employed.

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The results for fixed nucleation site will be discussed first.

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### Fixed nucleation site

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We first consider nucleation in the precise center of the film, at cell (

*x*,*y*,*z*) = (50,50,25), with a critical nucleus consisting of only one cell.
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The parameters used in the simulation were Δ

*t*= 0.1 s and*P*_{prop}= 0.01 per time step Δ*t*.
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Although both models indeed produce spherical growth, a number of striking differences appear between the two models.

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The first is that the rate of radial growth, which is defined as constant in the non-local geometric model, increases in the course of time in the MC simulations for statistical growth by a factor of 6.

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For the non-local geometrical model, the growth rate

*G*_{geo}is simply defined as the rate of change in the sphere radius*r*,*i.e. G*_{geo}= ∂*r*/∂*t*.
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In the MC simulation, for statistical growth, the sphere is not necessarily perfectly round and compact, but we can still define an

*effective*radius*r*_{eff}by calculating the radius corresponding to the converted volume*V*_{conv}= ∑_{x,y,z}*Θ*_{x,y,z}, as if it were a perfect sphere: .
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Thus, we can compare the geometric non-local growth rate with that observed in the statistical MC simulations by comparing

*G*_{geo}with the time-dependent radial growth rate in the MC simulations defined as:*G*_{MC}= ∂*r*_{eff}/∂*t*.
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These quantities are plotted

*vs.*time in Fig. 2.
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The time range (0–14 s) is sufficiently small so that the sphere does not reach the sample edges.

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For the parameter set Δ

*t*= 0.1 s and*P*_{prop}= 0.01, the long-time growth equals ∼1.58 ML s^{−1}for the MC simulations.
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106

For the non-local geometric model, the growth rate is set to be

*G*_{geo}= 1.58 ML s^{−1}in eqns. (2) and (3) and is, of course, time-independent.
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The increase in the growth rate observed in the statistical MC model can be understood as follows: consider an untransformed cell at the edge of the transformed volume.

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In the very beginning, there is only one pathway for the considered cell to become transformed, through the face of the adjacent, initially transformed cell at (50,50,25).

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In other words, the curvature of the crystallized ‘sphere’ is so large, that only one pathway is available.

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When the radius becomes very large (and the curvature small) a cell bordering the transformed volume will experience the oncoming transformed front as a flat front, and there will be

*nine*pathways for the cell to become transformed: one as before, and an additional eight from cells sharing cube edges and corner points (four and four, respectively).
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The growth probability per time step therefore increases from

*P*_{prop}at small times to (*P*_{prop}+ 4*P*_{prop}/√2 + 4*P*_{prop}/√3) at large times.
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112

For growth along a linear, one-dimensional chain, it is clear that the growth rate is given by

*G*_{MC}= 2*P*_{prop}/Δ*t*, the factor of two accounting for the fact that growth will occur in two directions.
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113

This should also be the

*initial*growth rate for our system, as borne out in the simulations:*G*_{MC}= 0.2 ML s^{−1}for*t*→ 0 (see Fig. 2).
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114

For long times, one might expect a growth rate of

*G*_{MC}(*t*→ ∞) = 2(*P*_{prop}+ 4*P*_{prop}/√2 + 4*P*_{prop}/√3)/Δ*t*= 1.23 ML s^{−1}, following the reasoning above.
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115

The observed long-time growth rate is larger than this simple estimate, amounting to ∼1.58 ML s

^{−1}.
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116

This is due to the fact that there are additional, indirect pathways through the three-dimensional system (not considered in the simple argument presented above) that contribute to the growth rate.

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117

We find the phenomenological relationship

*G*_{MC}(*t*→ ∞) ≈15.8*P*_{prop}/Δ*t*, which will be used below.
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118

The black line in Fig. 2 is the prediction of the time-dependent growth rate from the aforementioned simple considerations: this calculated rate is related to the curvature of the sphere, 1/

*r*_{eff}through*G*= 1.58 − 1/*r*_{eff}.
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119

It provides a reasonable description of the simulation results.

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120

Hence it is evident that the main contribution to the time-dependent growth rate originates from the time-dependent curvature of the transformed sphere.

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121

It should be noted that the limiting large-radius value for

*G*_{MC},*G*_{MC}(*t*→ ∞), is reached relatively soon, in terms of the transformed volume, as demonstrated in the inset of Fig. 2.
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122

For example, for a critical nucleus containing 100 cells (

*i.e.*water molecules),^{19}the initial growth rate is 0.7*G*_{∞}, rather than 0.13*G*_{∞}for a nucleus of size one.
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123

Besides the different growth rates, a second marked difference between the statistical and the geometrical model is that the transformed region reaches the surface appreciably earlier for the statistical MC results,

*despite the slower initial growth*.
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124

This is illustrated in Fig. 3, which shows the radii of the two differently growing spheres.

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125

The plot depicts

*r*_{bulk}− the bulk radius derived from the volume of the sphere – and*r*_{surf}– the radius of the circle of the exposed area of the sphere at the surface.
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126

The expressions for

*r*_{bulk}and*r*_{surf}read: , as introduced above, and, analogously, ,*i.e.*proportional to the square root of the area associated with cells that have crystallized at the surface.
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127

It is evident from Fig. 3 that for identical long-time growth rates (

*i.e. G*_{geo}=*G*_{MC}(*t*→ ∞) = 1.58 ML s^{−1}), the transformed sphere reaches the surface at*t*= 13 s for the MC simulation, and at*t*= 16 s for the non-local geometrical model, despite the fact that the radius of the geometrical sphere is ∼20% greater than the effective radius of the (statistical) MC result at this time, owing to the delayed initial growth for the MC sphere.
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128

Whereas the geometrically grown sphere is fully compact, the MC sphere is not; it has a fill factor of roughly ∼80%, with a strongly non-uniform crystalline density (the center is mostly crystalline).

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129

As a result, the sphere may reach the surface before

*r*_{bulk}= 25 ML, which is the geometric criterion for reaching the surface.
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130

The non-compact nature of the MC sphere is illustrated in the right panel of Fig. 3, which depicts a slice through the

*xy*-plane at*z*= 25 for a sphere that nucleated at point (50,50,25),*i.e.*through the center of the sphere, from one randomly chosen simulation run.
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131

It is evident that in real life, the precise shape of a cross section such as presented in Fig. 3 is the result of many competing factors, such as entropy (favoring a ‘rough’ sphere surface), surface tension (favoring a perfect circle) by possible variations in local growth rate due to

*e.g.*density fluctuations and varying growth rates along different crystallographic axes.
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132

Neither of the models presented here takes these effects into account, as very little is known about these effects.

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133

In addition, potential effects of the underlying support that is present in the experiments are neglected, as these effects can be suppressed for appropriate substrates.

^{23}
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134

A third difference between crystal growth determined by statistical and geometric rules is the shape of the crystalline domain in the proximity of the surface(s).

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135

Whereas for the non-local (geometrical) case, this shape is always a (truncated) sphere, for the statistical growth criterion the domain deviates from spherical growth, resulting in a deformed crystalline domain shape.

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136

This is most clear for nucleation immediately at the surface (center placed at point (50,50,1)).

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137

This is illustrated in Fig. 4, which depicts the simulation volume projected onto one of its sides,

*i.e.*integrated over one of the coordinates parallel to the surface:*I*(*y,z*) = ∑_{x}*Θ*_{x,y,z}, for both the non-local geometrical and the statistical MC model.
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ConceptID: Obs15

138

In addition to the non-spherical behavior of the MC model, it is also clear from Fig. 4 that the

*apparent*nucleation site is situated ∼5 cells below the surface (at*z*= 5), although the real site is at*z*= 1.
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139

This is clearly observed from the contour lines (also plotted in the graph).

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140

This somewhat surprising result can be understood in the following manner: Consider 2 cells, at the same distance from the surface nucleation site (N), but one straight down into the bulk (A) and one in the plane of the surface (B).

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141

In the non-local, geometrical growth model, as the distances N–A and N–B are identical, A and B become crystalline at the same point in time, and lie on the same contour line.

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142

With the statistical (local) growth rules, crystallization of cells A and B can occur through one of the many pathways by which the crystalline front can propagate from cell N. However, due to the broken symmetry at the surface, it is clear that there are roughly twice as many pathways from N to A, as there are from N to B, as the crystalline phase cannot propagate through vacuum.

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143

As a result, growth of the nucleus along the coordinate parallel to the surface plane is slower than growth along the coordinate perpendicular to the surface.

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144

A consequence of this behavior is that the exposed surface area of the converted volume is smaller than what one would expect from the non-local geometric model.

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145

This effect can be quantified by considering the ratio of

*r*_{bulk}(now defined as , since only half a sphere is formed) and*r*_{surf}, defined as above.
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146

The anisotropy in

*r*_{bulk}/*r*_{surf}is a good measure for the deviation from spherical behavior, as, in case of surface nucleation, it always equals 1 for the geometrical model.
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147

Fig. 5 represents the time-dependent anisotropy in apparent radii in the bulk and at the surface, and demonstrates that this effect is largest for short times.

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148

As mentioned already briefly above, it should be stressed that both the models described here are phenomenological and neither explicitly takes into account the thermodynamic quantities that are relevant for the crystallization and growth process.

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149

In particular, for the crystallization of water, which is driven by the difference in chemical potential between the two phases, the nucleation and growth will be determined by surface tension between the two phases and density effects.

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ConceptID: Bac32

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151

For crystalline grains reaching the surface, additional contributions arise: first, the three-phase line tension must be taken into account, which will raise the free energy as a larger part of the surface is converted, and therefore will slow down the growth along the surface.

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152

Secondly, there is a surface tension difference between the amorphous–vacuum interface and the crystalline–vacuum interface.

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153

Unfortunately, the magnitude, and sometimes even the sign, of many of these quantities are unknown.

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154

This is one of the motivations for taking the most simple approaches to both the statistical and geometric growth.

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### Random nucleation

155

To investigate to what extent the aforementioned effects influence the interpretation of experimental results, we investigate the quantity that is generally determined experimentally: the time-dependent fraction of converted material.

Type: Goal |
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156

Although both surface and bulk fractions are experimentally accessible, for simplicity we will first restrict ourselves to the bulk fraction, for nucleation of the new phase occurring in the bulk of the material.

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157

We use values for the nucleation and growth parameters that have been reported previously for crystallization kinetics of thin water films:

*J*_{bulk}= 2.0 × 10^{−7}ML^{−3}s^{−1}and*G*= 5.0 × 10^{−2}ML s^{−1}.^{28}
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158

Whereas

*J*is uniquely defined in both the non-local geometric and the statistical MC simulations, the growth rate*G*is only in the former, as in the MC simulations,*G*depends both on time and the initial critical nucleus size.
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159

In the simulations, we choose the numerical parameters

*P*_{prop}and Δ*t*such that GMC (*t*→ ∞) = 5.0 × 10^{−2}ML s^{−1}, since*G*_{MC}(*t*→ ∞) = 15.8*P*_{prop}/Δ*t*(see above).
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160

To be as realistic as possible, we chose a critical nucleus size of not only

*r** = 0.5 ML (corresponding to a nucleus of one cell in diameter) but also*r** = 3 ML, the latter corresponding to a critical nucleus containing 93 cells (the specific value of 93 is dictated by the discrete character of the lattice), in agreement with previous estimates of the critical nucleus.^{28}
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161

The Monte Carlo approach is further detailed in the section ‘approach’ above.

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162

The results of these simulations for both the non-local geometric and statistical MC models are depicted together in Fig. 6; there is a clear discrepancy between the results of the two models, despite the identical nucleation parameters.

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163

It is interesting to note that in both cases the shapes of the two curves are nearly identical, but the MC result is offset in time due to the time-dependent growth rate.

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164

As expected, the temporal offset is largest for

*r** = 0.5 ML, as the initial growth rate for the small nucleus is relatively low (see Fig. 2).
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165

For the case of

*r** = 3.0 ML, the geometrical model describes the kinetics observed in the MC simulations best with parameter values*J*= 6.0 × 10^{−8}ML^{−3}*s*^{−1},*G*= 7.0 × 10^{−2}ML s^{−1}(keeping*r** fixed at 3.0 ML),*i.e.*significantly different from the parameter values used in the simulations.
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166

Finally, we make a direct comparison to experiments recently reported in .

^{ref. 23}
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167

In these experiments, surface crystallization of a thin amorphous water film was reported.

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168

Here, we compare results of the geometric and statistical models with previously inferred

^{26}crystallization parameters:*J*_{surf}= 4.5 × 10^{−7}ML^{−3}s^{−1},*G*= 0.17 ML s^{−1},*r*_{des}= 0.04 ML s^{−1}and*r** = 3.0 ML.
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169

The results of the calculations, performed for a layer of 45 ML thickness (instead of 50 ML as in the previous part of this paper), are plotted in Fig. 7, from which it is immediately evident that the results of the two models are not very different.

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170

The main difference is the smaller temporal separation between the calculated surface and bulk fractions for the statistical model compared to the geometrical one.

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171

This is a consequence of the non-spherical shape that appears for surface nucleation in the statistical model, with growth lagging in the surface plane; As a result, the surface crystallinity is retarded compared to the geometrical case.

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172

The observation that there is no significant time lag for all three curves (as opposed to the results presented in Fig. 6) can be understood by noting that, for the

*surface*nucleation parameters used in the calculations, the crystalline growth advances through the layer as one front, as depicted in the inset in Fig. 7.
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ConceptID: Res15

173

As nucleation occurs in one plane, the converted regions appear close to each other, allowing them to merge relatively soon.

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174

The long-time growth rate

*G*_{MC}(*t*→ ∞) is therefore reached at an early time, also due to the relatively large*r** = 3.0 ML.
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175

It is evident that for these nucleation parameters, the difference between statistical and geometrical growth is rather small; this, however, is not necessarily true for other systems, and depends intricately on the values of the nucleation parameters.

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## Conclusions

176

We have investigated the effect of different crystallization models on the observed phase transition kinetics in thin films.

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177

A simple statistical Monte Carlo routine is presented that allows us to follow the phase transition in space and time.

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178

A comparison between the result of these Monte Carlo simulations and the routinely employed geometric models reveal a time-dependent growth rate for statistical growth, whereas the growth rate is assumed constant for geometric models.

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179

Moreover, the assumed spherical shape for geometric models is not observed for statistical growth for nucleation at or near a surface.

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180

A comparison to recent experimental results demonstrates that the consequences of these effects on the analysis of experimental data in thin films depend on the precise nucleation parameters, and may be small.

Type: Conclusion |
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ConceptID: Con5