# Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification

^{1}

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## Abstract

**:**

## 1. Introduction

^{−1/2}, where t is time [11,12]. The growth rate also depends on the curvature of the interface, as predicted by the classic kinetic model based on monomer attachment and detachment [13,14]. For spherical particles:

_{at}the atomic volume, k

_{B}Boltzmann’s constant, T the temperature, Δg the volumetric free energy difference between the liquid and the solid, γ the solid-liquid interfacial free energy, and r the radius of the crystallite. Note that for the critical size, r* = 2γ/Δg, the deterministic growth rate is zero, for larger particles it is positive, whereas for smaller particles it is negative. This expression is strictly valid only for diffusionless processes, where the velocity of the flat interface (r → ∞) is independent of time. The behavior of post-nucleation particles in diffusion-controlled systems is less obvious.

^{−1/2}for overdamped (diffusion controlled) relaxation dynamics (denoted as DPFC [21,22,23]), at small driving forces. The latter model yields diffusionless growth at extreme undercoolings, where the density of the growing crystal is about the same as that of the liquid [22,23]. These molecular-scale models offer a possibility to study early-stage solidification in three dimensions (3D), and to compare nucleation and post-nucleation growth for the two types of dynamics. While structural aspects of crystal nucleation in the DPFC model has been addressed in some detail in previous work [24,25,26,27,28,29,30,31], similar studies are unavailable for the hydrodynamic model.

## 2. Materials and Methods

#### 2.1. Phase-Field Crystal Method

#### 2.1.1. Thermodynamics

_{0}):

_{0})/ρ

_{0}is the scaled density difference, whereas ε is the reduced temperature (normalized distance from the critical point). The latter can be connected to such physical properties as the bulk moduli of the fluid and crystalline phases at a reference density and temperature. The phase diagram of the 3D system is shown in Figure 1 [27,30,34]. Besides the homogeneous liquid, stability domains exist for the body-centered cubic (bcc), hexagonal close-packed (hcp), and face-centered cubic structures. With appropriate quenching procedure in the presence of noise representing the thermal fluctuations, amorphous solids of realistic radial distribution function g(r) can also be obtained in the vicinity of the linear stability limit of the liquid state [26,27,28,29,30].

#### 2.1.2. Equations of Motion (EOMs)

**p**(

**r**, t) is the momentum, ρ(

**r**, t) the mass density,

**v**=

**p**/ρ the velocity, while

**∇⋅R**= −ρ∇{δΔF[ρ]/δρ} ≈ −ρ

_{0}∇{δΔF[ρ]/δρ} is the reversible stress tensor, ρ

_{0}the reference density, and

**D**= μ

_{S}{(∇ ⊗

**v**)+ (∇ ⊗

**v**)

^{T}} + [μ

_{B}− (2/3)μ

_{S}] (∇

**⋅**v) the dissipative stress tensor.

**S**represents stochastic momentum noise, with correlation:

_{S}and μ

_{B}denote the shear and bulk viscosities. To avoid violent interatomic flow in the crystalline phase that would develop due to steep density gradients, coarse-grained momentum and density were used when computing the velocity: $v=\widehat{p}/\widehat{r}$. (For further details see References [19,20].) It has been shown that this model recovers steady-state growth of velocity v ∝ 1/μ

_{S}, the longitudinal and transversal quasi-phonons of proper dispersion, and capillary waves of realistic spectrum at the crystal–liquid interface [19]. While there are other hydrodynamic extensions of the PFC model [37,38,39], they have not been used for studying nucleation, and the amplitude expansion based model [38] cannot be easily formulated for detecting competing amorphous and crystalline polymorphs in the liquid.

_{S}+ μ

_{B}}/ρ, and the free energy functional defined by Equation (1) is used in δΔF[ρ]/δρ, while

**S**is the stochastic momentum tensor used in Equation (3). Note, assuming β = 0, and omitting the noise term, we recover the modified PFC (MPFC) model proposed earlier [41,42]. It can be shown that similarly to the full HPFC, at small undercoolings the sHPFC model recovers steady-state growth of velocity v ∝ 1/μ

_{S}, and that longitudinal and transversal quasi-phonons of proper dispersion occur in the crystal. In the present work we use this simplified hydrodynamic approach to study crystal nucleation and early stage growth in simple liquids in 3D.

**r**,t)ζ(

**r**′ t′)〉 = ξ

^{2}∇

^{2}g(|

**r**−

**r**′|, σ)δ(t − t′), where ξ is the noise strength, and g(|

**r**−

**r**′|, σ) is a high-frequency cutoff function [27,28], to remove wavelengths shorter than the interatomic spacing (σ). In this model different growth modes occur at large and small driving forces for crystallization: (i) at small undercoolings or supersaturations growth is diffusion controlled and v ∝ t

^{−1/2}, whereas (ii) at large driving forces, where diffusion length l

_{D}= D/v becomes comparable to the interface thickness, a transition is observed to a diffusionless fast growth mode displaying steady-state growth (v = const.) [22,23], as expected for colloids [43].

#### 2.2. Numerical Methods

^{3}that, under the conditions used, corresponds to about 3.6 × 10

^{5}atoms. The computations were performed on high-end Graphics Processing Units.

#### 2.3. Materials Parameters

_{0}= −0.2720 and different constant reduced temperatures chosen between the values corresponding to the liquidus line ε

_{L}= 0.1548 and the linear stability limit ε

_{c}= 0.2220. The time and spatial steps used in the numerical computations were Δt = 0.01 and Δx = 1. Note that the single PFC model can be fitted to iron [44,45], yielding realistic structural properties for the amorphous state [26,27,28], and combined with the sHPFC dynamics it can be viewed as a model for bcc metals. A noise strength of ξ = 0.005 was used in all three models. When transforming the results to dimensional form, the diffusion coefficient (DPFC) or the viscosity/damping coefficient (sHPFC and DPFC) sets the physical time scale of the process.

#### 2.4. Structural Analysis

_{b}

^{k}is the number of neighbors of the particle k, n

_{b}

^{i}is the number of the first neighbors of the ith neighbor or particle k, whereas Y

_{lm}is the spherical harmonics of degree l and order m, while

**r**

_{ij}is the vector between the first neighbor atom i and its jth first neighbor.

## 3. Results and Discussion

#### 3.1. Two-Step Nucleation

_{0}= −0.2720 at reduced undercoolings of ε = 0.200, 0.202, and 0.204. Snapshots of these simulations taken at different instances are shown in Figure 3, showing nucleation and post-nucleation growth.

#### 3.2. Structure Evolution during Nucleation

#### 3.3. Post-Nucleation Growth

^{1/3}) was evaluated as a function of time from the number n of density peaks for the DPFC, MPFC, and sHPFC clusters shown in Figure 5, Figure 6 and Figure 7, respectively. The results are shown in Figure 9a, whereas the prediction for pure Fe at 1500 K obtained by integrating Equation (1) is shown in Figure 9b. It is found that predictions of the three models for post-nucleation growth were qualitatively similar: after a plateau corresponding roughly to the critical size, a smooth transition into an apparently steady-state growth was seen. In the case of the DPFC model this probably indicates that the system entered the fast diffusionless growth regime. The transition to steady-state growth was somewhat longer for the sHPFC model than for the other two. The differences in the growth mechanisms, according to the three models, were perceptible only for longer time scales.

#### 3.4. Flat-Front Growth in the sHPFC Model

^{1/2}in the whole-time window covered, indicating a dominantly diffusion-controlled process (see Figure 10a). Deviation from this behavior is expected at much longer times, when the diffusion field ahead of the front interacts with the boundary condition. In contrast, the sHPPC model obtained with α = 0 and β = 1 showed a linear position–time relationship for short times (t < 1000) (Figure 10a), yet for longer times the growth rate decreased due to flow-controlled depletion of the liquid (Figure 10b) and converged towards a stationary state. Note the stepwise change of the number of atoms in all three models, indicating a layerwise growth of the slab.

_{L}plot was somewhat curved (see Figure 11a) as reported for the MPFC model [51], which may follow either from a deviation from Turnbull’s linear relationship for the thermodynamic driving force [52], or may originate from a relaxation effect Ankudinov and Galenko identified in the case of the MPFC model [53]. The growth velocity, in turn, is about inversely proportional with the viscosity term β (see Figure 11b).

## 4. Conclusions

- In all three models the stable bcc phase appears via two-step nucleation.
- During the early stage of nucleation, a disordered solid of liquid-like structure (0.28 > ${\overline{q}}_{6}$) appears, followed by the formation of MRCO (0.28 < ${\overline{q}}_{6}$ < 0.4), which precede the formation of the stable bcc structure.
- The first appearing solid structure is ordered so that it has a concentric ring-like view from certain directions. The number of icosahedral neighborhoods is higher in this domain than in the LJ fluid. This structure may coexist with a disordered solid structure of liquid-like order. Remarkably, these two structures are not distinguished by the ${\overline{q}}_{6}$-based color scheme of Kawasaki and Tanaka.
- The time dependence of grain size emerging from the post-nucleation growth was evaluated from DPFC, MPFC, and sHPFC simulations. At short times the predictions are rather similar to each other and agree reasonably well with the dependence obtained by integrating the size-dependent growth rate given by Equation (1). Differences in the growth mechanism become perceptible only on a longer time scale.
- The steady-state growth velocity the sHPFC predicts increasingly deviates downwards from a linear relationship with increasing undercooling, and it is roughly inversely proportional to the viscosity.
- Finally, we note that nucleation and the post nucleation behavior appear to be less sensitive to the mechanism of density relaxation than the kinetics of crystal growth. This is so, despite the fact that in the present studies the nucleation that took place via intermediate states preferred kinetically relative to direct bcc nucleation.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Phase diagram of the single-mode phase-field crystal model in 3D. (Reproduced with permission from Reference [30]). Besides the 3D crystalline phases, see the presence of 2D periodic phases such as the triangular rod phase and the lamellar phase.

**Figure 2.**${\overline{q}}_{4}$–${\overline{q}}_{6}$ map evaluated from molecular dynamics simulations performed using the Lennard-Jones potential for the bulk bcc, hcp, fcc, and liquid structures. (Reproduced with permission from Reference [46]).

**Figure 3.**Nucleation and growth in the sHPFC model after instantaneous quenching to reduced temperatures ε = 0.200 (top row), 0.202 (central row), and 0.204 (bottom row). Coloring indicates the magnitude of the average bond order parameter ${\overline{q}}_{6}$. White (amorphous): ${\overline{q}}_{6}$ < 0.28; red (MRCO): 0.28 < ${\overline{q}}_{6}$ < 0.4; green (bcc): ${\overline{q}}_{6}$ > 0.4. The deeper the quench, the larger the nucleation rate for the amorphous phase, and the smaller the bcc fraction forming during solidification. The bcc phase forms via heterogeneous nucleation on the surface of amorphous globules. Instances for the snapshots—upper row: t = (

**a**) 800, (

**b**) 1700, and (

**c**) 5900; central row: t = (

**e**) 600, (

**f**) 900, (

**g**) 7700; bottom row: t = (

**i**) 300, (

**j**) 400, and (

**k**) 7700. Panels (

**d**,

**h**,

**l**) show the time dependence of phase fractions. The incubation time is not shown.

**Figure 4.**Snapshots of the ${\overline{q}}_{4}$–${\overline{q}}_{6}$ map during solidification in the sHPFC system for the simulation shown in the upper row of Figure 3. The time elapses from left to right. Maps corresponding to dimensionless times t = (

**a**) 800, (

**b**) 1400, (

**c**) 2000, and (

**d**) 5900 are presented. The transparent, green, red, and yellow circles indicate the ideal bcc, hcp, fcc, and icosahedral neighborhoods, respectively. Note the gradual transition from the liquidlike initial amorphous state towards the bcc structure. The “bridge” between the liquidlike and bcc domains indicates transition states corresponding to the MRCO.

**Figure 5.**Snapshots of a solid cluster forming in a DPFC simulation performed at ε = 0.2 and ψ

_{0}= −0.2720. The snapshots were taken at dimensionless times t = 520, 600, 700, 1000, and 1400. Coloring indicates the magnitude of ${\overline{q}}_{6}$. White (amorphous): ${\overline{q}}_{6}$ < 0.28; red (MRCO): 0.28 < ${\overline{q}}_{6}$ < 0.4; green (bcc): ${\overline{q}}_{6}$ > 0.4. In the upper and central rows atom-size circles denote the density peaks. In the upper row atoms with all three types of neighborhoods are shown, whereas in the central row only particles of MRCO and bcc-type neighborhoods. In agreement with previous results for the DPFC model [28,29,30], the early stage of solidification is dominated by the formation of an amorphous precursor. In the bottom row, we used smaller spheres to indicate the position of the density peaks, which then allows us to see the presence of an ordered but non-crystalline initial structure. The ring-like ordering can only be seen from certain directions. A fraction of a 512

^{3}simulation is shown.

**Figure 6.**Snapshots of a solid cluster forming in an MPFC simulation performed under the same conditions as the simulation shown in Figure 5. In the left three panels, small spheres are used to indicate the position of the density peaks, which allows us to see the presence of an ordered but non-crystalline initial structure. On the right atom-size spheres are displayed. The snapshots were taken at dimensionless times t = 460, 500, 600, 900, and 1100. The same coloring is used as in Figure 3. A fraction of a 512

^{3}simulation is shown.

**Figure 7.**Snapshots of a solid cluster forming in an sHPFC simulation performed under the same conditions as the simulation shown in Figure 5. In the left three panels, small spheres are used to indicate the position of the density peaks, which allows us to see the presence of an ordered but non-crystalline initial structure. On the right atom-size spheres are displayed. The snapshots were taken at dimensionless times t = 450, 500, 600, 800, and 950. The same coloring is used as in Figure 3. A fraction of a 512

^{3}simulation is shown.

**Figure 8.**Different views of an early-stage cluster from the sHPFC simulation shown in Figure 7. The snapshots were taken at a dimensionless time of t = 550. Views in the upper row show some degree of order, combined with disorder, whereas structural order in the bottom row is less obvious. A fraction of a 512

^{3}simulation is shown.

**Figure 9.**Post-nucleation growth: (

**a**) effective linear size vs. time for the clusters shown in Figure 5, Figure 6 and Figure 7; (

**b**) effective linear size vs. time for an iron cluster nucleated at 1500 K as follows from integrating Equation (1). Note the similarity of the curves presented in panels (

**a**,

**b**).

**Figure 10.**Flat-interface slab growth in 3D within the DPFC, MPFC (α = 1 and β = 0), and the sHPFC model (α = 0 and β = 1). The number of atoms in the crystalline slab is shown as a function of dimensionless time for (

**a**) short times and (

**b**) long times. A linear function fitted to the sHPFC result for t < 500, and an Aτ

^{1/2}function fitted to the DPFC result for τ < 5000 are also shown (dotted lines).

**Figure 11.**Dimensionless growth velocity, v as a function of (

**a**) the reduced undercooling, and (

**b**) the inverse of the viscosity parameter β as predicted by the sHPFC model (α = 0).

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Podmaniczky, F.; Gránásy, L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. *Crystals* **2021**, *11*, 437.
https://doi.org/10.3390/cryst11040437

**AMA Style**

Podmaniczky F, Gránásy L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. *Crystals*. 2021; 11(4):437.
https://doi.org/10.3390/cryst11040437

**Chicago/Turabian Style**

Podmaniczky, Frigyes, and László Gránásy. 2021. "Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification" *Crystals* 11, no. 4: 437.
https://doi.org/10.3390/cryst11040437