https://orcid.org/0000-0002-6487-0457
Figures
Abstract
In this study, we focus on random terms with a significant effect on the calculation time and the sidebranch structure of dendrites. We quantitatively evaluate the effect of the random terms on dendrites by changing the term introduced into the evaluation equation, the distribution of the random term, and variance. We introduced the terms: and
. In the phase-field equation, we use
to represent interfacial noise. In the heat conduction equation, we use
to represent thermal noise. We compare the results of calculation using only
with those using only
and analyze the effect of each random term introduced on the shape of dendrites. In terms of the probability distribution for generating random numbers, the uniform and Gaussian distributions are used for comparison. The magnitude of the noise Fu, which controls the variance of noise, is used with 3 or 4 patterns of values to the extent that the calculation results do not diverge. Each value of Fu differs by a factor of 10. Sidebranch length, contour length, and area are used as indices for evaluating dendrites. The succinonitril is used for comparing the simulation and experimental results for dendrites to design the optimal random number model. On the basis of the introduced terms, we clarify that each term requires a different order of the variance. The probability density function of random numbers does not affect the shape of dendrites. On the other hand, the calculation time which is used the random numbers following the uniform distribution is more than twice as fast as when using random numbers that follow a Gaussian distribution. The variance of a random term has the greatest effect on the shape of dendrites.
Citation: Fujiwara S, Okada K, Endo K, Hirayama K, Muramatsu M (2025) Effect of random terms on the shape of dendrites analyzed by the phase-field model. PLoS One 20(8): e0324281. https://doi.org/10.1371/journal.pone.0324281
Editor: Antonio Riveiro Rodríguez, University of Vigo, SPAIN
Received: November 26, 2024; Accepted: April 23, 2025; Published: August 4, 2025
Copyright: © 2025 Fujiwara et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data of Figs 6–11 can be found at the publicly available repository: https://github.com/mmc-research-group/fujiwara random number dendrite/tree/main.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
1.1 Phase-field model for dendrite growth
Materials are often formed in intricately branched crystals in solid form. They are called dendrites, commonly seen in an undercooled melt [1,2]. Predicting the growth of a dendrite is extremely important for improving material properties because the dendrite is a common structure formed in the solidification of many commercial metals, and the shape of dendrites has significant impacts on metal properties [3–5]. Some studies have shown the metallic material solidification by synchrotron X-ray imaging [4,6–8]. Also, the dendritic architecture changes material surface characteristics. The growth induction of dendritic architecture has also attracted attention. Ivantsov’s solution provides a foundational analytical framework for modeling steady-state dendrite growth by solving the heat diffusion equation for an isothermal paraboloid in a uniformly supercooled melt [9]. This solution can be applied to both two-dimensional (cylindrical) and three-dimensional (spherical) geometries, enabling analytical descriptions of dendritic growth in various spatial dimensions. This approach effectively predicts the relationship between undercooling and dendrite tip velocity but lacks a mechanism for selecting the unique tip radius. This limitation was later addressed by the microscopic solvability theory, which introduced the concept that anisotropic surface tension stabilizes dendritic growth and determines the tip radius [10]. While these classical models have significantly contributed to our understanding of dendritic solidification, they rely on simplified assumptions, such as steady-state growth, and do not naturally accommodate the effects of solute diffusion, convection, or complex microstructural interactions. In contrast, the phase-field method provides a more comprehensive framework for simulating dendritic growth, as it can inherently capture the evolution of complex interface morphologies without explicitly tracking the solid-liquid boundary [11]. Unlike Ivantsov’s analytical solution, which assumes a fixed tip shape, or the microscopic solvability theory, which assumes infinitesimally small perturbations, the phase-field method enables dynamic tracking of morphological instabilities and solute interactions in both binary and multi-component systems [12]. Hsu et al. found the relationship between external driving force and character patterns of dendritic architecture by using the phase-field model [13]. Therefore, there has been great discussion about simulating the growth of dendrites, and some numerical techniques for dendrite growth have been established [14–17]. In particular, the phase-field method is the most widely used for simulating dendrite growth [13,18–22].
As shown in Fig 1(a), dendrites can be evaluated on the basis of their two structures. The first is the primary branch, which has a tree-trunk-like structure. The second is the sidebranches, which are derived from the primary branch. The sidebranches are used for analyzing dendrites [23,24]. In the phase-field method, random numbers are used to represent sidebranches. In this study, we use the sidebranches to evaluate the morphological characteristics of dendrites, and we analyze the relationship between the random terms and calculation times.
1.2 Random terms in the phase-field model
The phase-field method is characterized by the adoption of an order parameter referred to as the phase-field variable . The phase-field method enables us to calculate the time variation by using driving and diffusion terms. We utilize the Allen-Cahn equation, which is used for nonconserved value, to minimize the total energy of the system. Furthermore, a heat conduction equation incorporating the latent heat generation of dendrites was formulated based on the conservative law of energy. The basic equations of this model are expressed below.
Here, and
are the random terms referring to the fluctuations of driving force and thermal noise, respectively. Additionally,
is the phase-field mobility coefficient, F is the Gibbs free energy function, T is the temperature,
is the thermal diffusion coefficient, L is the latent heat, and c is the specific heat. Eq (1) is the time evolution equation of the phase-field variable, and
is generally utilized in the phase-field method [25,26]. Karma and Rappel [27] used
instead of
as a random term in the heat conduction equation shown in Eq (2). The random term determines complex shapes in dendrite growth. Fig 1 shows the results of the calculation. Fig 1(a) shows the dendrite obtained by using the random term and Fig 1(b) shows that obtained without using the random term. It indicates that the random term determines the shape of dendrites in terms of branched crystal structures, such as the primary branch and sidebranches. Random terms have an essential role in the development of sidebranches. Kobayashi [25] used a random term that follows a uniform distribution for
to represent the instability of the shape of the interfaces against the noise. This noise is phenomenologically set and used to reproduce the interface dynamics. In Kobayashi’s model, thermodynamic noise based on the Langevin equation is not considered, and therefore
is not used. On the other hand, Karma and Rappel [27] used a random term that follows the Gaussian distribution for
. There have been only a few studies of the types of random term and the distribution of random number, in which the variance of random terms is determined empirically. Little attention has been given to the accuracy of the amplitude of the variance.
In this research study, we focus on the type, distribution, and variance of random terms in the phase-field method. The material simulated in this study is succinonitrile (SCN), whose dendrites have been extensively evaluated [28–32]. The relationship between sidebranch length and the distance between the tip of the primary branch and a arbitrary point z0 on the z axis in the Fig 2 is indicated by the power law. In addition, not only sidebranch length but also contour length and area to the distance between the tip of the primary branch and the arbitrary point z0 are indicated by the power law, as shown in previous studies [33–36]. Moreover, we calculate the coefficient of power law and compare them in the basis of random terms, distribution, and variance. We compare these coefficients across different random term conditions and experimental results in terms of morphological characteristics. Next, we compare computation times to identify a more suitable random number model, as each random number distribution employs a different generation process. We then discuss the suitable model for the random term in the phase-field model. To simplify this problem, we employ the sharp interface model [37], the most fundamental form of the phase-field method.
2 Methods
2.1 Phase-field method
2.1.1 Fundamental phase-field model.
The phase-field method is widely used to simulate the processes associated with meso-scale material compositions, such as interface migration [38], phase transformation [39] and grain growth [40]. Various studies have established the phase-field method to simulate dendritic growth in pure materials [25,41,42]. The phase-field method introduces the order parameter . The phase-field
varies gradually in the solid–liquid interface region as shown in Fig 3. In this simulation,
is solid when
and liquid when
. Owing to
, complicated boundary-following calculations are not necessary. The phase-field method has the advantage of enabling the easy simulation of dendrite growth. The order parameter
in the dendrite growth model represents the phase state. The time evolution of dendrite growth is expressed as
where is the gradient energy density,
is the chemical free energy density, and
is the double-well potential. These are defined as follows.
Here, is the gradient coefficient,
is the monotonically increasing function, fS and fL are the bulk free energies in the solid and liquid, respectively, W is the height of the energy barrier, and
is a double-well function. In this case,
and
are expressed as
The gradient coefficient is expressed as
where is the average of
,
is the strength of anisotropy, k is the anisotropy mode,
is the angle between the x-axis and the normal vector of the interface, and
is the angle between the x-axis and the vector of primary arm growth. Considering Eqs (3)–(6), we obtain the equation.
Here, the free bulk energy is expressed as
. The functional derivative
can be given by Eq (9).
Substituting Eqs (5)–(7) into fdoub of Eq (10), we can obtain the following equation.
Here, is the difference in energy between the solid and liquid phases, and
is the random term. Furthermore,
is expressed as
Eq (12) shows that consists of the latent heat L, the temperature T and the melting point Tm. Substituting Eqs (10) and (12) into Eq (3), we can obtain the following phase-field equation.
In the phase-field method of simulating solidification, the parameters are as follows.
Here, is the interfacial kinetic coefficient,
is the interfacial energy,
is the interface thickness, b is the contant defined as
, and
is the interfacial area determination coefficient. Through the application of the sharp interface model [37], the mobility
is defined and scaled to be consistent with the interface velocity. In the sharp interface model, the interface thickness
is assumed to converge to 0. Then, the phase-field mobility coefficient
is defined to be consistent with the Stephan condition, which defines the interface velocity, and the Gibbs-Thomson effect, which accounts for the influence of interface curvature [43,44].
The heat conduction equation should be solved when we use a pure material in the phase-field method, as shown in Eq (2). The second term on the right-hand side of Eq (2) is the generation term introduced to generate latent heat during the change from the liquid phase to the solid phase. Substituting Eq (7) into Eq (2), we obtain the following heat conduction equation.
In order to determine the initial temperature , we introduce dimentionless undercooling
. The relation is the following equation:
2.1.2 Random terms in the phase-field model.
Karma and Rappel have shown the variances of and
[27].
Here, is Kronecker’s delta,
is the inverse of capillary length, J is the numerical constant, Fu is the magnitude of noise,
is the time step, and
is the lattice of spacing. Various Fu values are used to evaluate the effect on its dendrite shape.
In this study, Mersenne Twister [45] is employed as the random number generator. Mersennne Twister generates a sequence of pseudo-random numbers on the basis of an incremental formula by giving true random numbers as a random number seed. In terms of distribution, Merssenne Twister generates random numbers following a uniform distribution. Random numbers following a Gaussian distribution are generated by Box–Muller transformation [46]. Transformation converts random numbers following a uniform distribution into random numbers following a Gaussian distribution. Fig 4 shows an example of the uniform and Gaussian distributions utilized in this study for generating random numbers.
2.2 Measurement of the characteristics of dendrite
The dendrites are evaluated using three indices, i.e., the sidebranch length Xact, the contour length U, and the area F. These indices are normalized by the tip radius to enable the quantitative comparison of dendrite shapes. First, the dendrite tip shape is approximated as
where z is the distance between the tip of the primary branch and the arbitrary point on the z axis and x is the distance from the central axis to the surface of the dendrite (Fig 5). Eq (20) represents a refined version of Ivantsov’s solution, incorporating the effects of anisotropy and surface tension [47]. The coefficients c1 and c2 are derived by applying this equation to the tip shape depending on the difference in random number. The relationship between the tip radius R and the coefficients is expressed as follows.
The fitting range must be considered in deriving the tip radius. We define the tip as the range from the top of primary branch to the first sidebranch that appears. The tip contour is approximated by Eq (20). As mentioned above, three evaluation indices are compared: the sidebranch length , the contours length U, and the area F in Fig 2. The sidebranch length
is defined as the length from the center axis of the primary branch to the tip of the sidebranch. The indices
, U, F, z, and R can be approximated by the power law [33–36]. The following equations provide the approximate formulas for each indicator.
The coefficients ai and bi(i = X,U,F) are evaluated by comparing the experimental data with the results of all of these analyses with different random numbers. Table 1 shows the experimental values. Table 2 shows the computational parameters and properties of SCN. In this study, several values of the anisotropy strength are analyzed. We simulate dendrites by the phase-field method and evaluate
and bF. In this study, we simulate dendrites 20 times in different random seeds for smoothing. We use Intel Xeon Gold 6136 for calculation.
3 Results and discussion
3.1 Comparison of results obtained using 
and 
The effects of random terms are investigated. Specifically, the analysis results obtained using and those obtained using
are compared. The experimental and analytical results are plotted in Figs 6–11. Dendrites are not generated correctly at several Fu value. The parameters for Fu that form dendrites correctly are shown in Figs 6–11. Fig 6 shows the results of the coefficient aX for the sidebranch length Xact. Fig 7 shows the results of the coefficient aX for the sidebranch length Xact. In Figs 6 and 7, the ranges of aX and bX are almost the same between the results obtained using
and those obtained using
. On the other hand, the range of Fu is different. When
is used, Fu takes a value from 10−10 to 10−8. When
is used, Fu takes a value from 10−3 to 10−0. The main reason for the difference is the difference in range between
and
in Eqs (18) and (19). Figs 8–11 show the same trend as Figs 6 and 7. We can observe the difference in Fu range between the results obtained using
and
Another reason for the difference in Fu range in which normal dendrites are observed is that the phase-field variable ranges from 0 to 1, so the analysis introducing
requires the variance of the random numbers in Eq (18) to be sufficiently small so as not to exceed the
range. On the other hand, the analysis introducing
depends on the melting point. The range taken by the temperature field T is wider than
, so the variance of the random term
becomes larger than that of the random term
.
3.2 Comparison of distributions
As indicated by Figs 6–11, almost all of the circles and x marks show the same value. The value with random numbers following the uniform distribution almost agrees with that with random numbers following the Gaussian distribution in the results of the sidebranch length, contour length, and area. This finding is attributed to the central limit theorem. When random numbers act, the sum of random numbers approaches the Gaussian distribution, even if they follow a uniform distribution. Consequently, the difference between distributions cannot be confirmed.
According to Karma’s study [52], owing to the Langevin formalism, random numbers are governed by the Gaussian distribution in the theoretical formula. However, this result indicates that a uniform distribution can also be used in the numerical analysis by the phase-field method.
3.3 Comparison between numerical and experimental results
In Figs 6–9, the simulated sidebranch and contour lengths are closer to the experimental values with the increase in Fu. When is used for the phase-field method, its better value for Fu is
. When
is used for the phase-field method, its better value for Fu is 1.0. As the magnitude of the noise Fu increases, the variance of the random term increases, which allows a sidebranch to develop. Therefore, the simulated sidebranch length Xact and contour length U are expected to approach the experimental values with increasing Fu. In addition, since aF and bF, which are the coefficients of the area F, do not change when Fu and
are changed as shown in Figs 10 and 11, it is confirmed that the random term does not affect the area F. The reason is associated with the sum of random numbers. The average of the random number term converges to zero. Fig 13 shows the analysis results for different Fu values, and it can be seen that the sidebranch grows as Fu increases. When Fu is small, the sidebranch can hardly be seen, but the primary branch becomes thicker. When Fu increases, the sidebranch grows, but the primary branch becomes thinner. This is the reason why no changes can be observed in the coefficients of the area.
(a) The magnitude of the noise Fu is . (b) The magnitude of the noise Fu is
.
On the other hand, when the anisotropy strength is low, the dendrite shape is highly influenced by random terms and cannot accommodate random numbers with large variance. As shown in Fig 12, the growth along the primary branch axis is more pronounced with higher anisotropy strength. Therefore, the growth in the direction of sidebranches is suppressed, resulting in a stable shape even when random numbers with large variance are applied. However, with low anisotropy strength, the growth becomes more sensitive to random numbers, as it tends to occur in directions perpendicular to the primary branch axis. Consequently, when random numbers with large variance are applied, the influence of random terms becomes excessive, preventing the formation of dendrite-like structures. Moreover, significant differences in the coefficients , and bF for each anisotropy strength are not clearly observed. This is because the sidebranch length decreases,which results in the tip radius becoming smaller as the anisotropy strength increases, as shown in Fig 12. These findings are consistent with previous studies on anisotropy strength and tip radius [43,53].
3.4 Calculation time
The time required for each calculation when varying the random terms and distributions is measured and listed in Table 3. The calculation time is 7000 s when using random numbers following a uniform distribution, whereas it is more than twice that when using random numbers following the Gaussian distribution. This result indicates that the computational cost of generating random numbers following the Gaussian distribution makes up a significant portion of the total computational cost. The Box–Muller transformation is used to generate random numbers following the Gaussian distribution in this study. However, this transformation requires computationally expensive functions such as logarithmic functions, trigonometric functions, and root signs, which require two random numbers following a uniform distribution.
4 Conclusion
In this study, the effect of random terms in the phase-field method on the dendrite shape was investigated by changing the introduced term, distribution, and variance of a random number. By introducing the random terms to the equations of the difference between the order parameter and temperature, we found that the difference in the magnitude of the noise increased by a factor of 108. This discrepancy arises due to the lack of normalization for each variable, suggesting the necessity of appropriately adjusting the magnitude of the noise according to the introduced terms. As for the distribution, there was no clear difference in the shape of the dendrites between the uniform distribution and the Gaussian distribution. This result suggested that similar outcomes can be obtained from Phase-field analyses based on both uniform and Gaussian distributions, providing valuable insights. The magnitude of the noise was set empirically in a previous study by Kobayashi [25]. This measurement clarified the trend of the dendrite shape as the parameters of a random number were changed and achieved a random number design that was close to the experimental value. This result provided a meaningful contribution, as it allows the magnitude of the noise in Phase-field simulations, which has traditionally been set empirically, to be determined through quantitative analysis. Finally, from the viewpoint of computational cost, we conclude that the use of random numbers following a uniform distribution for is better. In particular, the computational cost varied by approximately a factor of two depending on the choice of probability distribution. We believe our research facilitates the development of an efficient approach to accelerate Phase-field simulations.
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