Multiphase Phase-Field Approach for Virtual Melting: A Brief Review

Arunabha Mohan Roy

University of Michigan, Materials Science and Engineering, Ann Arbor, MI, 48109, USA

Corresponding Author E-mail: arunabhr.umich@gmail.com 

DOI : http://dx.doi.org/10.13005/msri/180201

ABSTRACT:

A short review on a thermodynamically consistent multiphase phase-field approach for virtual melting has been presented. The important outcomes of solid-solid phase transformations via intermediate melt have been discussed for HMX crystal. It is found out that two nanoscale material parameters and solid-melt barrier term in the phase-field model significantly affect the mechanism of PTs, induces nontrivial scale effects, and changes PTs behaviors at the nanoscale during virtual melting.

KEYWORDS: Ginzburg-Landau equations; Multiphase Phase-field theory; Virtual melting

Copy the following to cite this article: Roy A. M. Multiphase Phase-Field Approach for Virtual Melting: A Brief Review. Mat. Sci. Res. India;18(2).

Copy the following to cite this URL: Roy A. M. Multiphase Phase-Field Approach for Virtual Melting: A Brief Review. Mat. Sci. Res. India;18(2). Available from: https://bit.ly/3f87WwN

Phase-field (PF) approach ^{1, 2} has been widely used to captures various solid-solid phase transitions (PTs) ^3–11. Lately, it has been discovered that the finite-width interface plays a crucial role in controlling PTs for the different material systems ^{12, 13} such as PTs via intermediate molten state (IM ) which has been observed experimentally hundreds of degrees below the thermodynamic melting temperature in HMX ^{14, 15}. Such a transitional, metastable interface is called a virtual melt ^16–19. Additionally, it has been found that such virtual melt induces nontrivial scale effects, and changes phase transformation behaviors at the nanoscale ²⁰. Previously, a PF model was developed to describe solid-solid PT via IM in hyperspherical order parameter which is limited for n = 3 phase-system ²⁰. More recently, a multiphase phase-field (MPF) theory has been proposed for generalized n phase- system to capture such intriguing PT mechanism during virtual melting ²¹. This MPF model is thermodynamically consistent and satisfies all thermodynamic stability conditions ^21–24. One of the advantages of the MPF model is that, for each of the propagating solid-melt and solid-solid interfaces, the analytical solutions for width, energy, and velocity can be derived ^{21, 22, 25}. Thus, interface material properties can be fully calibrated and characterized for all interfaces. In the aforementioned MPF model, two dimensionless parameters at the nanoscale [e.g., ratios of width and energy of two different interfaces, kE (or ΔΨ) and kδ (or ΔΓ)] can be explicitly defined and controlled during PT. These parameters significantly affect the formation and stability of virtual melt during solid-solid PT in HMX ^21–24. The MPF approach has been employed to investigate the appearance and corresponding thermodynamic, and structure of IM for a three-phase system ^{22, 24}. Additionally, a detailed study on barrierless melt nucleation in HMX has been reported for propagating IM ²³. The kinematics and energetics of appearance of IM have been detailed in ²⁵. It is found out that the nanoscale material parameters and solid-melt barrier term in the MPF model significantly affect the mechanism of PTs, induces nontrivial scale effects, and changes PTs behaviors at the nanoscale during virtual melting ^{22–24, 26}.

In the abovementioned MPF model, order parameter’s (ηi) evolution has been described by Ginzburg-Landau (G-L) equations [21, 22] , where
ψl = ψ˜θ + ψ˘θ + ψp is the local part of the Helmholtz free energy ψθ = ψl + ψ∇ (see [21, 22] for detail). In Fig. 1, different scale effects and non-trivial phase transformation mechanism has been observed when the influence of kE (or ΔΨ) which characterizes the energy of two different interfaces on the appearance and disordering of IM has been explored for two different non-equilibrium temperatures where ξϕ = (η1 + η2)min indicates the disordering. For different critical values of kE (i.e., kc  _E ) and depending on the energy barrier of the solid-melt interface K12 (or Σδβ ), two different solutions exist for kE < kc _E  for relatively low kδ (or ΔΓ) : one issolid-melt-solid interface solution with high disordering of IM at the interface , and another one is solid-solid interface solution with low disordering of IM at the interface. At first critical value k_E = (k_Ec )I , jump occurs between solid-melt-solid initial condition (SMS) solution to less disordered IM solution. Whereas, at second critical value k_E = (k_Ec )II , second jump occurs from solid-solid initial condition (SS) solution to high disordered IM solution. Hence, the solution of propagating inter facial melt can be either continuous-reversible without the hysteresis or jump-like first order discontinuous transformation with hysteresis. In Fig. 2, the influence of temperature on the appearance of propagating inter facial melt has been explored where ξϕ indicates the disordering. From the simulation results, it is clear that increasing temperature θ increases the disordering ξϕ of inter facial melt for all ΔΨ, ΔΓ (or kE, kδ ). For relatively small ΔΓ, the solution of ξϕ is continuous-reversible for both solid-melt-solid and solid-solid initial conditions. However, for relatively large ΔΓ, one is a solid-melt-solid interface solution with a high disordering of IM at the interface during solidification and another one is a solid-solid interface solution with low disordering of IM at the interface during melting at some critical value of temperature. These two different solutions correspond to two different nanostructures that produce a ”hysteretic region”. From the numerical result, it is evident that the appearance of nucleated melt can form much below thermodynamic melting temperature and different ΔΨ and ΔΓ control the width of the temperature hysteresis curve and melt formation temperature. The appearance of such nontrivial multiple solutions of IM could not be captured by the simplified thermodynamic descriptions which did not consider interface width as a scale parameter (i.e., kδ ), thus, can only predict a single solution of IM and the formation of IM (or melt) can only possible for kE > 2 close to thermodynamic equilibrium melt temperature.

Figure 1: ξϕ = (η1 + η2)min has been shown as a function of kE at (a) θ = 402 K and (b) θ∗ = 422 K for K12 = 1010 J/m3. Reprinted from ²²  with the permission of AIP Publishing, 2021.

Click on image to enlarge

Summarizing, the numerical results from the MPF model indicate a new perspective of solid-solid PT via transitive virtual melt in HMX. The penalizing potential in MPF formalism significantly controls the existence of virtual melt by limiting the pure solid-solid interface solution in order parameter space. Hence, these two scale parameters and penalizing term K12 (or Σδβ) influence the formation of virtual melt much below the thermodynamic melting temperature. The presented MPF model demonstrates the general applicability of this formalism to capture first-order jump-like PT as well as second-order continuous PTs. In addition, such MPF approach can be utilized to capture various PTs ²⁷ such as martensitic PTs ^{9–11, 28– 40}, evolution of nanovoids [41–44], surface-induced melting ^45–47, grain boundary premelting [48–53], interface modeling in composite [55], and crack propagation ^{56, 57}.

Figure 2:  ξϕ  = (η1 + η2)min  has been plotted as a function   of θ/θe for different  ∆Γ  for Σδβ = 1 kg/ (nm.s2).  Reprinted from ²³ with the permission of Elsevier, 2021.

Click on image to enlarge

Acknowledgment

The author is grateful to Dr. V. I. Levitas from Iowa State University, IA, USA for his kind guidance and discussion.

Funding Source

National Science Foundation (Grant No. CMMI-0969143).

Competing interests

The authors declare no competing interests.

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