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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

2
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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

3
In particular, models with geometric and statistical growth rules are compared.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

4
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

5
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

6
In addition, the predicted kinetics of the phase transformation is notably different for the geometric vs. the statistical model, for similar parameters.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

7
The simulation results are compared to recent experiments on the crystallization of thin amorphous solid water films.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

Introduction

8
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 technology6 and metallurgy.7

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

9
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

10
For example, the crystallization of amorphous solid water in confined geometries has received significant attention the past decade.5,8–23

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

11
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

12
These analytical theories (see e.g.refs. 16, 24–26) are based on Avrami’s seminal contribution27 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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

13
The subsequent growth is radial, with a constant velocity.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

14
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

15
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

16
The geometric model has recently been extended to include the different desorption rates from the different phases in simulations by Ahlström et al.28

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

17
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.

Type: Method | Advantage: No | Novelty: Old | ConceptID: Met1

18
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.

Type: Method | Advantage: No | Novelty: Old | ConceptID: Met1

19
For very small samples, a local, statistical description for the growth would seem more appropriate.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

20
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

21
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 simulation29,30 based on a Monte Carlo approach.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

22
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

23
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

24
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

25
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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

26
This effect is not limited to thin films.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

27
Additional discrepancies appear when nucleation occurs at or near the surface or an interface of a thin film.16,24–26

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

28
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

29
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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

30
This is not a purely abstract issue, as there have recently been two independent reports for nucleation of the thermodynamically stable phase at the surface: for the crystallization of amorphous solid water23 and polymers31.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

Approach

31
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

32
Time will simply be expressed in seconds.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

33
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

34
In the two models described here, nucleation occurs in the same manner, but the growth rules differ.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

35
Nucleation occurs at random times at random positions within the layer, or on its surface.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

36
The nucleation probability for bulk nucleation is defined per unit volume per unit time Jbulk (expressed in ML−3 s−1).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

37
For nucleation occurring at the surface, it is defined per unit surface area per unit time Jsurf (expressed in ML−2 s−1).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

38
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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

39
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 crystallization19).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

40
The size of the, assumed to be spherical, critical nucleus is defined by its radius r* (expressed in ML).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

41
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

A. Statistical growth

42
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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

43
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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

44
Note, however, that the results described here are not specific to this thickness.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

45
In any case, this comparison to experiment implies that each cell corresponds to one water molecule.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

46
Any of the cells may be untransformed (amorphous) or transformed (crystalline), defined with cell values Θ = 0 and Θ = 1, respectively.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

47
The phase state of each cell is evaluated in every time step Δt employed in the simulations.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

48
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 Jbulk, in conjunction with the system volume V and time step Δt employed in the simulations.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

49
This number is obtained from a simulated Poisson distribution, whose mean is given by the product JbulkVΔt.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

50
For surface nucleation, the Poisson mean is given by JsurfAΔt, with A the surface area on which nucleation occurs.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

51
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

52
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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

53
Note that, as depicted in Fig. 1, a cell has three types of neighbors: N1 = 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 √2l, and N√3 = 8 neighbors that share a corner point, at a distance of √3l.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

54
The probability Ptrans for the crystallinity to propagate to an adjacent cell is inversely proportional to the geometric distance, and linearly proportional to elapsed time.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

55
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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

56
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

57
Ideally, one would like the growth criteria to be determined by the energetics associated with the growth process.32

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

58
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

59
Therefore we adopt this statistical approach.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

60
At every time step, for each cell that has not yet been transformed, the state of its 26 neighbors is evaluated.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

61
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: ΘN1, ΘN√2 and ΘN√3 (Θ = 1 (or 0) for an (un)transformed cell).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

62
Pprop represents the probability to propagate via a face, which is proportional to the time step size and the growth rate.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

63
A random number between zero and one is then computer generated, and if this number is smaller than Ptrans, the value of the cell Θ changes from zero to one as a result of crystallization.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

64
Otherwise, it remains unchanged.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

65
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 Pprop sufficiently small.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

66
From a set of simulations with varying time step size, it is clear that, if Pprop is set to be smaller than 0.015 per time step, the results converge.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

67
In the simulations reported here, we set Pprop = 0.01 per time step.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

68
For convenience, time is expressed in seconds in the remainder of this report.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

69
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

70
The (100 × 100 × 50) volume is then analyzed to determine the time-dependent converted fraction.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

71
We did not use periodic boundary conditions, as the statistical fluctuations in nucleation and growth rates are amplified in that way.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

72
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

73
The presented results are averages of 30 simulation runs, to reduce the effects of statistical fluctuations in the growth process.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

74
A set of 30 simulations typically takes 10 h on a desktop personal computer.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

75
The converted volume as a function of time can be obtained by evaluating the summation of cell values over all cells: Vconv = ∑x,y,zΘx,y,z.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

76
The time-dependent crystalline fraction is given by dividing Vconv by the total considered simulation volume.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

77
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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

B. Geometric growth

78
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 Ggeo (in units ML s−1).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

79
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* + Ggeo (t − τ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

80
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 (xi,yi,zi).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

81
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

82
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

83
Because the nucleation probability per unit volume and unit time, Jbulk, is independent of elapsed time and position, so-called ‘phantom nuclei’27 may appear in already transformed regions and transformed regions may overlap.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

84
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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85
The result can be directly related to the converted fraction of bulk material.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac20

86
Following26 (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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

87
These expressions including desorption can be found in the same reference.26

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac21

88
In analogy to the statistical growth criterion, one can also perform simulations using a non-local, geometric criterion, as demonstrated previously in .ref. 28

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac22

89
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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

Results and discussion

90
Two types of investigations were performed: one with fixed nucleation site, and one with random nucleation throughout the film.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

91
The first allows us to investigate the differences between statistical and geometric growth rules regarding the shape of the transformed volume.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met3

92
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

93
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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

94
For each of these cases (fixed and random nucleation), statistical and geometric growth criteria were employed.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met4

95
The results for fixed nucleation site will be discussed first.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac23

Fixed nucleation site

96
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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

97
The parameters used in the simulation were Δt = 0.1 s and Pprop = 0.01 per time step Δt.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

98
Although both models indeed produce spherical growth, a number of striking differences appear between the two models.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

99
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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

100
For the non-local geometrical model, the growth rate Ggeo is simply defined as the rate of change in the sphere radius r, i.e. Ggeo = ∂r/∂t.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

101
In the MC simulation, for statistical growth, the sphere is not necessarily perfectly round and compact, but we can still define an effective radius reff by calculating the radius corresponding to the converted volume Vconv = ∑x,y,zΘx,y,z, as if it were a perfect sphere: .

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

102
Thus, we can compare the geometric non-local growth rate with that observed in the statistical MC simulations by comparing Ggeo with the time-dependent radial growth rate in the MC simulations defined as: GMC = ∂reff/∂t.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

103
These quantities are plotted vs. time in Fig. 2.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

104
The time range (0–14 s) is sufficiently small so that the sphere does not reach the sample edges.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac24

105
For the parameter set Δt = 0.1 s and Pprop = 0.01, the long-time growth equals ∼1.58 ML s−1 for the MC simulations.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

106
For the non-local geometric model, the growth rate is set to be Ggeo = 1.58 ML s−1 in eqns. (2) and (3) and is, of course, time-independent.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

107
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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

108
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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

109
In other words, the curvature of the crystallized ‘sphere’ is so large, that only one pathway is available.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

110
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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

111
The growth probability per time step therefore increases from Pprop at small times to (Pprop + 4Pprop/√2 + 4Pprop/√3) at large times.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

112
For growth along a linear, one-dimensional chain, it is clear that the growth rate is given by GMC = 2Pprop/Δt, the factor of two accounting for the fact that growth will occur in two directions.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

113
This should also be the initial growth rate for our system, as borne out in the simulations: GMC = 0.2 ML s−1 for t → 0 (see Fig. 2).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs11

114
For long times, one might expect a growth rate of GMC(t → ∞) = 2(Pprop + 4Pprop/√2 + 4Pprop/√3)/Δt = 1.23 ML s−1, following the reasoning above.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

115
The observed long-time growth rate is larger than this simple estimate, amounting to ∼1.58 ML s−1.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

117
We find the phenomenological relationship GMC (t → ∞) ≈15.8 Pprop/Δt, which will be used below.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac26

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/reff through G = 1.58 − 1/reff.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

119
It provides a reasonable description of the simulation results.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

121
It should be noted that the limiting large-radius value for GMC, GMC (t → ∞), is reached relatively soon, in terms of the transformed volume, as demonstrated in the inset of Fig. 2.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

124
This is illustrated in Fig. 3, which shows the radii of the two differently growing spheres.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

125
The plot depicts rbulk − the bulk radius derived from the volume of the sphere – and rsurf – the radius of the circle of the exposed area of the sphere at the surface.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

126
The expressions for rbulk and rsurf 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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

127
It is evident from Fig. 3 that for identical long-time growth rates (i.e. Ggeo = GMC (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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

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).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

129
As a result, the sphere may reach the surface before rbulk = 25 ML, which is the geometric criterion for reaching the surface.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs14

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac28

132
Neither of the models presented here takes these effects into account, as very little is known about these effects.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac28

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac29

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

136
This is most clear for nucleation immediately at the surface (center placed at point (50,50,1)).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

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) = ∑xΘx,y,z, for both the non-local geometrical and the statistical MC model.

Type: Observation | Advantage: None | Novelty: None | 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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

139
This is clearly observed from the contour lines (also plotted in the graph).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs16

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

145
This effect can be quantified by considering the ratio of rbulk (now defined as , since only half a sphere is formed) and rsurf, defined as above.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

146
The anisotropy in rbulk/rsurf is a good measure for the deviation from spherical behavior, as, in case of surface nucleation, it always equals 1 for the geometrical model.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

150
For instance, the effect of the density difference between the amorphous and the crystalline phase (0.94 g cm−321 and 0.93 g cm−3,13 respectively) will raise the nucleation barrier, and presumably impede growth of larger crystallites due to strain effects.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

152
Secondly, there is a surface tension difference between the amorphous–vacuum interface and the crystalline–vacuum interface.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac33

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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 | Advantage: None | Novelty: None | ConceptID: Goa2

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.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met5

157
We use values for the nucleation and growth parameters that have been reported previously for crystallization kinetics of thin water films: Jbulk = 2.0 × 10−7 ML−3 s−1 and G = 5.0 × 10−2 ML s−1.28

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

159
In the simulations, we choose the numerical parameters Pprop and Δt such that GMC (t → ∞) = 5.0 × 10−2 ML s−1, since GMC (t → ∞) = 15.8 Pprop/Δt (see above).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

161
The Monte Carlo approach is further detailed in the section ‘approach’ above.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac34

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

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).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

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−3s−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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

166
Finally, we make a direct comparison to experiments recently reported in .ref. 23

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

167
In these experiments, surface crystallization of a thin amorphous water film was reported.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac35

168
Here, we compare results of the geometric and statistical models with previously inferred26 crystallization parameters: Jsurf = 4.5 × 10−7 ML−3 s−1, G = 0.17 ML s−1, rdes = 0.04 ML s−1 and r* = 3.0 ML.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac35

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs18

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

173
As nucleation occurs in one plane, the converted regions appear close to each other, allowing them to merge relatively soon.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac36

174
The long-time growth rate GMC (t → ∞) is therefore reached at an early time, also due to the relatively large r* = 3.0 ML.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

Conclusions

176
We have investigated the effect of different crystallization models on the observed phase transition kinetics in thin films.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj6

177
A simple statistical Monte Carlo routine is presented that allows us to follow the phase transition in space and time.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met6

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

179
Moreover, the assumed spherical shape for geometric models is not observed for statistical growth for nucleation at or near a surface.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

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 | Advantage: None | Novelty: None | ConceptID: Con5