# Detection of Capillary-Mediated Energy Fields on a Grain Boundary Groove: Solid–Liquid Interface Perturbations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Variational Grain Boundary Grooves

#### 2.1. Characteristic Size of Grain Boundary Grooves

#### 2.2. Microstructure Free Energy

#### 2.3. Variational Grooves

#### 2.4. Equilibrated and Variational Grain Boundary Grooves

## 3. Thermo-Potential on Variational Grooves

#### 3.1. Curvature of Variational Profiles

#### 3.2. Gradient of the Gibbs–Thomson Thermo-Potential

#### 3.3. Capillary-Mediated Fluxes

#### 3.4. Capillary Flux Divergence

#### 3.5. Cooling Distribution

## 4. Detecting Interfacial Energy Fields

#### 4.1. Proportionality of Potential and Heat Rate

#### 4.2. Sharp and Diffuse Interfaces

## 5. Numerical Model and Results

#### 5.1. Multiphase-Field Model

#### 5.2. Fidelity of the Groove Equilibration

- We set the ratio of the grain boundary’s energy density equal to twice that of the crystal/melt boundary, so chosen to produce after steady-state equilibration the desired dihedral angle of $\mathsf{\Psi}=0$. (Refer to Section 2.2 and reference [21] for further details on this point.).
- We checked carefully that the required uniform 1D thermal gradient and its linear temperature distribution developed fully along the phase-field Y-grid ordinate scale of the evolved grain boundary groove.
- We measured the dihedral angle at each time-step [22]. After the dihedral angle approached its expected equilibrium value of zero, we continued further equilibration for an additional ${10}^{4}$ time-steps. This assured that any further relaxation would not occur on the equilibrated groove profile that could otherwise alter its shape and thermo-potential distribution surrounding the triple-junction region.

#### 5.3. Post-Processing Residuals and Interface Fields

#### 5.4. Interface Potential Residuals

#### 5.5. Nonlinear Residuals of the Thermo-Potential

## 6. Conclusions

- The present study shows that even stationary microstructures, such as equilibrated grain boundary grooves, support persistent capillary-mediated energy fields that allow their precise measurement via multiphase-field numerics. Deterministic bias fields were shown previously as capable of stimulating complex pattern formation on moving interfaces that evolve during the entire course of solidification [1].
- The existence of capillary-mediated interface fields is demonstrated here on the basis of simulated isoline measurements along the solid–liquid interface of an equilibrated grain boundary groove. Phase-field measurements permit calculation of nonlinear residuals by subtracting from the isoline potential measurements the linear distribution imposed by the applied thermal gradient. Residuals are shown to be proportional to the capillary-mediated field strength at the interface.
- Symmetric grain boundary grooves provide well-studied examples of stable microstructures, which remain in constrained thermodynamic equilibrium in the presence of a uniform thermal gradient. The variational groove profile, which approximates the shape of the equilibrated grain boundary groove, is described by a closed-form transcendental solution to the Euler–Lagrange equation. The groove profile predicted from variational calculus yields linear extrema (absolute minima) of its free energy functional. This profile allows an accurate analytic estimate for the capillary field resident on a comparable equilibrated grain boundary groove.
- The equilibrated grain boundary groove profile proves to be a nonlinear energy minimizer because of interactions between its shape and its persistent interface field. The equilibrated grain boundary groove differs in shape only slightly from its variational profile. This allows the latter to be used to estimate the fields that actually develop on the former.
- The distribution of capillary-mediated interface fields on grain boundary grooves may be theoretically estimated from their variational solutions as the surface Laplacian of their Gibbs–Thomson thermo-potential, or, equivalently, from the divergences of their capillary-mediated tangential fluxes. The zero-dihedral angle variational groove, with isotropic solid–liquid energy density, yields a negative polynomial expression of cubic order for its bias field. This interface field exhibits persistent heat removal rates (cooling) that peak sharply in their intensities near the triple junction of the groove’s cusp.
- Phase-field thermo-potential residuals are quantities measured on a simulated grain boundary groove that are proportional to the strengths of the interface’s local cooling rate. Initial simulations and potential measurements presented here support theoretical predictions of cooling rates derived from sharp interface thermodynamics. It is doubtful that comparable experimental measurements of interface potential residuals can be accomplished with laboratory equilibrated grain boundary grooves by using current thermal measurement technology.
- Additional simulations of equilibrated grooves exhibiting larger dihedral angles are of interest also, as the interface fields expected for such grain boundary grooves should differ markedly from that found here for a grain boundary groove with $\mathsf{\Psi}=0$.
- Capillary-mediated interface fields found here on a stationary grain boundary groove might have practical importance and applications to achieve a better understanding and control of solidification microstructures, considering that the energy rates for capillary-mediated interface fields might be manipulated through easily applied physical and chemical means. Such process controls applied in solidification could potentially lead to improved cast microstructures in alloys.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Glicksman, M.E. Capillary-mediated interface perturbations: Deterministic pattern formation. J. Cryst. Growth
**2016**, 450, 119–139. [Google Scholar] [CrossRef] - Glicksman, M.; Voorhees, P.; Setzko, R. The Triple-Point Equilibria of Succinonitrile—Its Assessment as a Temperature Standard. In Temperature, Its Measurement and Control in Science and Industry; Schooley, J.F., Ed.; American Institute of Physics: New York, NY, USA, 1982; Volume 5, pp. 321–326. [Google Scholar]
- Rubinstein, E.; Tirmizi, S.; Glicksman, M. Long–Term Purity Assessment in Succinonitrile. J. Cryst. Growth
**1990**, 106, 167–178. [Google Scholar] [CrossRef] - Huang, S.; Glicksman, M. Fundamentals of Dendritic Solidification: Part I—Steady-State Tip Growth. Acta Metall.
**1981**, 29, 701–716. [Google Scholar] [CrossRef] - Huang, S.; Glicksman, M. Fundamentals of Dendritic Solidification: Part II—Development of Sidebranch Structure. Acta Metall.
**1981**, 29, 717–734. [Google Scholar] [CrossRef] - Schaefer, R.J.; Glicksman, M.E.; Ayers, J.D. High-confidence measurement of solid/liquid surface energy in a pure material. Philos. Mag.
**1975**, 32, 725–743. [Google Scholar] [CrossRef] - Bolling, G.; Tiller, W. Growth from the Melt. I. Influence of Surface Intersections in Pure Metals. J. Appl. Phys.
**1960**, 31, 1345–1350. [Google Scholar] [CrossRef] - Weinstock, R. Calculus of Variations; Dover Publications, Inc.: New York, NY, USA, 1974; pp. 20–31. [Google Scholar]
- Glicksman, M.E. Principles of Solidification; Springer: New York, NY, USA, 2011; p. 287. [Google Scholar]
- Antczak, G.; Ehrlich, G. Jump processes in surface diffusion. Surf. Sci. Rep.
**2007**, 62, 39–61. [Google Scholar] [CrossRef] - Basu, A.S.; Gianchandan, Y.B. Virtual microfluidic traps, filters, channels and pumps using Marangoni flows. J. Micromech. Microeng.
**2008**, 18, 115031. [Google Scholar] [CrossRef] - Carslaw, H.; Jaeger, J. Conduction of Heat in Solids, 2nd ed.; Oxford Science Publications, Clarendon Press: Oxford, UK, 1959; pp. 17–19. [Google Scholar]
- Xu, J.J. Interfacial Wave Theory of Pattern Formation in Solidification, 2nd ed.; Springer Series in Synergetics (Complexity); Springer: Cham, Switzerland, 2017; pp. 16–21. [Google Scholar]
- DeHoff, R. Thermodynamics in Materials Science; Materials Science and Engineering Series; McGraw-Hill Inc.: New York, NY, USA, 1993; pp. 52–55. [Google Scholar]
- Kellogg, O. Foundations of Potential Theory; Dover Publications: New York, NY, USA, 1953; p. 175. [Google Scholar]
- Glicksman, M. Diffusion in Solids, Field Theory, Solid-State Principles, and Applications; Interscience Series; John Wiley & Sons, Inc.: New York, NY, USA, 2000; pp. 76–94. [Google Scholar]
- Glicksman, M.E. Mechanism of Dendritic Branching. Metall. Mater. Trans. A
**2012**, 43, 391–404. [Google Scholar] [CrossRef] - Garcke, H.; Nestler, B.; Stinner, B. A Diffuse Interface Model for Alloys with Multiple Components and Phases. SIAM J. Appl. Math.
**2004**, 64, 775–799. [Google Scholar] - Nestler, B.; Garcke, H.; Stinner, B. Multicomponent alloy solidification: Phase-field modeling and simulations. Phys. Rev. E
**2005**, 71, 041609. [Google Scholar] [CrossRef] [PubMed] - Nash, G.; Glicksman, M.E. A general method for determining solid–liquid interfacial free energies. Philos. Mag.
**1971**, 24, 577–592. [Google Scholar] [CrossRef] - Ankit, K.; Xing, H.; Selzer, M.; Nestler, B.; Glicksman, M.E. Surface rippling during solidification of binary polycrystalline alloy: Insights from 3-D phase-field simulations. J. Cryst. Growth
**2017**, 457, 52–59. [Google Scholar] [CrossRef] - Hötzer, J.; Tschukin, O.; Said, M.; Berghoff, M.; Jainta, M.; Barthelemy, G.; Smorchkov, N.; Schneider, D.; Selzer, M.; Nestler, B. Calibration of a multi-phase field model with quantitative angle measurement. J. Mater. Sci.
**2016**, 51, 1788–1797. [Google Scholar] [CrossRef]

**Figure 1.**Photomicrograph of a stationary grain boundary groove in ultra-pure (7–9s+) [2,3] succinonitrile, a body-centered cubic organic crystal [4,5]. The black area is melt phase, and gray areas are crystallites separated by a vertical grain boundary. This equilibrated groove was photographed in situ in a steady thermal gradient of 4.0 K/m. The material’s melting point, 58.082 ± 0.001 C [2], is realized along the outer flat regions of the groove’s profile. The solid–liquid–grain boundary triple junction is located about 150 microns below the flatter regions, surrounded by a sharp cusp of melt undercooled less than 1 mK. Points added along the solid–liquid interface were analyzed in a prior study to estimate the solid–liquid interface energy along a grain boundary groove equilibrated under various thermal gradients. Micrograph adapted from reference [6].

**Figure 2.**Variational grain boundary groove profile with its dihedral angle, $\mathsf{\Psi}=0$. The profile’s dimensionless Cartesian coordinates, ($\mu ,\eta $) and its normal ($\overrightarrow{n}$) and tangent ($\overrightarrow{\tau}$) vectors are displayed. The $\mu $-axis is coincident with the system’s melting point isotherm, $T={T}_{m}$, insuring that stable melt (light gray) exists where $\eta >0$, and stable solid (white areas) exists where $\eta <0$. Undercooled melt (indicated by increasingly darker grays) and curved solid co-exist within the lower half-plane beneath the $\mu $-axis. Local equilibrium is maintained by matching the thermo-chemical potential at each point along the solid–liquid interface. Both curvature and temperature vary with depth for this groove in the limited sub-space between the $\mu $-axis ($\eta =0$) and the groove’s triple junction located at ${\eta}^{\star}=-\sqrt{2}/2$, which is the deepest cusp allowed for a variational groove in 2D.

**Figure 3.**Plot of the magnitudes of the capillary-mediated dimensionless fluxes, ${\widehat{\mathbf{\Phi}}}_{\tau}\left(\eta \left(\mu \right)\right)$, along the left and right profiles of a grain boundary groove with zero dihedral angle. The directions of the $\mu $-components of these fluxes oppose each other. Left and right thermal flux vectors rotate clockwise and anti-clockwise, respectively, as they descend into the steeper portion of the cusp. These components become parallel as the triple junction is approached, where their $\eta $-components combine and enter the grain boundary with a flux magnitude of 8.

**Figure 4.**Plot of capillary-mediated energy rates, $\mathfrak{B}\left(\eta \right(\mu \left)\right)$ (Equation (18)) for an equilibrated grain boundary groove with zero dihedral angle, specified by Equation (4). Energy rates are negative at all points along the groove’s solid–liquid interface, indicating that heat removal, i.e., cooling, occurs everywhere along the interface. The most intense cooling rate occurs at $\eta =-1/\sqrt{6}$, roughly half-way between the groove’s triple junction at ${\eta}^{\star}=-\sqrt{2}/2$ and the $\mu $-axis, where $\eta =0$.

**Figure 5.**Theoretical cooling distribution for an equilibrated grain boundary groove with zero dihedral angle. Maximum cooling rates occur at ${\mu}_{max}\approx \pm 0.13$. This unique distribution of capillary-mediated heat removal provides a robust target for independent verification, by comparing these theoretical cooling rates against those measured by direct equilibration of an equivalent grain boundary groove simulated using multiphase-field numerics.

**Figure 6.**Phase-field residual potential measurements (⃝-symbols) along the simulated right isoline of an equilibrated grain boundary groove with zero dihedral angle. Data points plotted with the bias field distribution, $\mathfrak{B}\left(\mu \right(\eta \left)\right)$, for its counterpart variational groove profile. The scale-factor 300:1 was chosen between the phase-field’s abscissa, $\mathrm{X}$-grid (minus 1000), and the dimensionless abscissa coordinate, $\mu $. A fixed ratio, $\xi =2.50$, applied to the left-ordinate is used here to compare simulated residuals ($\times {10}^{4}$) with the corresponding right-ordinate $\mathfrak{B}\left(\mu \right(\eta \left)\right)$-distribution. The $\xi $-value proportioning these simulation data was chosen as the ratio of the largest phase-field residual, $-1.740\times {10}^{-4}$, to the largest $\mathfrak{B}$-value, $-4.35$. Residual measurements from phase-field simulation and the corresponding predicted distribution, $\mathfrak{B}\left(\mu \right(\eta \left)\right)$, from sharp-interface theory represent independent estimates of capillary-mediated cooling perturbations along a stationary grain boundary groove.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Glicksman, M.; Ankit, K.
Detection of Capillary-Mediated Energy Fields on a Grain Boundary Groove: Solid–Liquid Interface Perturbations. *Metals* **2017**, *7*, 547.
https://doi.org/10.3390/met7120547

**AMA Style**

Glicksman M, Ankit K.
Detection of Capillary-Mediated Energy Fields on a Grain Boundary Groove: Solid–Liquid Interface Perturbations. *Metals*. 2017; 7(12):547.
https://doi.org/10.3390/met7120547

**Chicago/Turabian Style**

Glicksman, Martin, and Kumar Ankit.
2017. "Detection of Capillary-Mediated Energy Fields on a Grain Boundary Groove: Solid–Liquid Interface Perturbations" *Metals* 7, no. 12: 547.
https://doi.org/10.3390/met7120547