American Institute of Mathematical Sciences

Advanced
February  2017, 37(2): 963-982. doi: 10.3934/dcds.2017040

On the mathematical modelling of cellular (discontinuous) precipitation

Oliver Penrose 1, and  John W. Cahn 2,

1. 

Department of Mathematics and the Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK
2. 

John W. Cahn died on 14 March 2016. There is an obituary at https://www.washingtonpost.com/local/obituaries/john-w-cahn-who-fled-nazi-germanyand-became-a-foremost-materials-scientist-dies-at-88/2016/03/15/890fe246-eac6-11e5-a6f3-21ccdbc5f74estory.html

The first author is not supported by any grant

Received  June 2015 Published  November 2016

Cellular precipitation is a dynamic phase transition in solid solutions (such as alloys) where a metastable phase decomposes into two stable phases : an approximately planar (but corrugated) boundary advances into the metastable phase, leaving behind it interleaved plates (lamellas) of the two stable phases.

The forces acting on each interface (thermodynamic, elastic and surface tension) are modelled here using a first-order ODE, and the diffusion of solute along the interface by a second-order ODE, with boundary conditions at the triple junctions where three interfaces meet. Careful attention is paid to the approximations and physical assumptions used in formulating the model.

These equations, previously studied by approximate (mostly numerical) methods, have the peculiarity that $v,$ the velocity of advance of the interface, is not uniquely determined by the given physical data such as $c_0$, the solute concentration in the metastable phase. It is hoped that our analytical treament will help to improve the understanding of this.

We show how to solve the equations exactly in the limiting case where $v=0$. For larger $v$, a successive approximation scheme is formulated. One result of the analysis is that there is just one value for $c_0$ at which $v$ can be vanishingly small.

Keywords: Phase transformations in solids, cellular precipitation, discontinuous precipitation, grain boundary motion, diffusion, mathematical modelling, triple junction, surface energy.
Mathematics Subject Classification: Primary:74N20 Dynamicsofphaseboundaries; Secondary:34B15Nonlinearboundaryvalueproblems, 74A50 Structuredsurfacesandinterfaces, coexistentphases, 74G15Numericalapproximationofsolutions, 74N25Transformationsinvolvingdiffusio.
Citation: Oliver Penrose, John W. Cahn. On the mathematical modelling of cellular (discontinuous) precipitation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 963-982. doi: 10.3934/dcds.2017040
References:
[1]

L. Amirouche and M. Plapp, Phase-field modeling of the discontinuous precipitation reaction, Acta materialia, 57 (2009), 237-247.  doi: 10.1016/j.actamat.2008.09.015.  Google Scholar

[2]

R. W. Balluffi and J. W. Cahn, Mechanism for diffusion induced grain boundary migration, Acta Met., 29 (1981), 493-500.  doi: 10.1016/0001-6160(81)90073-0.  Google Scholar

[3]

E. A. Brener and D. Temkin, Theory of diffusional growth in cellular precipitation, Acta Materialia, 47 (1999), 3759-3765.  doi: 10.1016/S1359-6454(99)00238-4.  Google Scholar

[4]

E. A. Brener and D. Temkin, Theory of discontinuous precipitation: Importance of the elastic strain, Acta Materialia, 51 (2003), 797-803.  doi: 10.1016/S1359-6454(02)00471-8.  Google Scholar

[5]

J. W. Cahn, The kinetics of cellular segregation reactions, Acta Met, 7 (1959), 18-28.  doi: 10.1016/0001-6160(59)90164-6.  Google Scholar

[6]

J. W. Cahn, P. C. Fife and O. Penrose, A phase-field model for diffusion-induced grain boundary motion, Acta Mater., 45 (1997), 4397-4413.  doi: 10.1016/S1359-6454(97)00074-8.  Google Scholar

[7]

P. C Fife, J. W. Cahn and C. M. Elliott, A free-boundary model for diffusion-induced grain boundary motion, Interfaces and Free Boundaries, 3 (2001), 291-336.  doi: 10.4171/IFB/42.  Google Scholar

[8]

P. Fratzl, O. Penrose and J. L. Lebowitz, Modeling of phase separation in alloys with coherent elastic misfit, J. Stat Phys, 95 (1999), 1429-1503.  doi: 10.1023/A:1004587425006.  Google Scholar

[9]

M. Hillert, On the driving force for diffusion induced grain-boundary migration, Scripta Met., 17 (1983), 237-240.  doi: 10.1016/0036-9748(83)90105-9.  Google Scholar

[10]

L. M. Klinger, Y. J. M. Brechet and G. R. Purdy, On velocity and spacing selection in discontinuous precipitation. 1. simplified analytical approach, Acta Mater., 45 (1997), 5005-5013.  doi: 10.1016/S1359-6454(97)00171-7.  Google Scholar

[11]

F. C. Larché and J. W. Cahn, The effect of self-stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.   Google Scholar

[12]

IUPAC Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford, 1997. XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi: notenoalianjie. Last update: 2014-02-24; version: 2. 3. 3. DOI of this term: doi: notenoalianjie. Google Scholar

[13]

W. W. Mullins, The effect of thermal grooving on grain boundary motion, Acta Metall., 6 (1958), 414-427.  doi: 10.1016/0001-6160(58)90020-8.  Google Scholar

[14]

O. Penrose and J. W. Cahn, A mathematical model for diffusion-induced grain boundary motion, Proceedings of conference on free boundary problems, Trento 2002, International Series of Numerical Mathematics (ISNM), 147 (2004), 237-254.   Google Scholar

[15]

O. Penrose, On the elastic driving force in diffusion-induced grain boundary motion, Acta Materialia, 52 (2004), 3901-3910.  doi: 10.1016/j.actamat.2004.05.004.  Google Scholar

[16]

http://mathworld.wolfram.com/GreensFunction.html Google Scholar

show all references

References:
[1]

L. Amirouche and M. Plapp, Phase-field modeling of the discontinuous precipitation reaction, Acta materialia, 57 (2009), 237-247.  doi: 10.1016/j.actamat.2008.09.015.  Google Scholar

[2]

R. W. Balluffi and J. W. Cahn, Mechanism for diffusion induced grain boundary migration, Acta Met., 29 (1981), 493-500.  doi: 10.1016/0001-6160(81)90073-0.  Google Scholar

[3]

E. A. Brener and D. Temkin, Theory of diffusional growth in cellular precipitation, Acta Materialia, 47 (1999), 3759-3765.  doi: 10.1016/S1359-6454(99)00238-4.  Google Scholar

[4]

E. A. Brener and D. Temkin, Theory of discontinuous precipitation: Importance of the elastic strain, Acta Materialia, 51 (2003), 797-803.  doi: 10.1016/S1359-6454(02)00471-8.  Google Scholar

[5]

J. W. Cahn, The kinetics of cellular segregation reactions, Acta Met, 7 (1959), 18-28.  doi: 10.1016/0001-6160(59)90164-6.  Google Scholar

[6]

J. W. Cahn, P. C. Fife and O. Penrose, A phase-field model for diffusion-induced grain boundary motion, Acta Mater., 45 (1997), 4397-4413.  doi: 10.1016/S1359-6454(97)00074-8.  Google Scholar

[7]

P. C Fife, J. W. Cahn and C. M. Elliott, A free-boundary model for diffusion-induced grain boundary motion, Interfaces and Free Boundaries, 3 (2001), 291-336.  doi: 10.4171/IFB/42.  Google Scholar

[8]

P. Fratzl, O. Penrose and J. L. Lebowitz, Modeling of phase separation in alloys with coherent elastic misfit, J. Stat Phys, 95 (1999), 1429-1503.  doi: 10.1023/A:1004587425006.  Google Scholar

[9]

M. Hillert, On the driving force for diffusion induced grain-boundary migration, Scripta Met., 17 (1983), 237-240.  doi: 10.1016/0036-9748(83)90105-9.  Google Scholar

[10]

L. M. Klinger, Y. J. M. Brechet and G. R. Purdy, On velocity and spacing selection in discontinuous precipitation. 1. simplified analytical approach, Acta Mater., 45 (1997), 5005-5013.  doi: 10.1016/S1359-6454(97)00171-7.  Google Scholar

[11]

F. C. Larché and J. W. Cahn, The effect of self-stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.   Google Scholar

[12]

IUPAC Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford, 1997. XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi: notenoalianjie. Last update: 2014-02-24; version: 2. 3. 3. DOI of this term: doi: notenoalianjie. Google Scholar

[13]

W. W. Mullins, The effect of thermal grooving on grain boundary motion, Acta Metall., 6 (1958), 414-427.  doi: 10.1016/0001-6160(58)90020-8.  Google Scholar

[14]

O. Penrose and J. W. Cahn, A mathematical model for diffusion-induced grain boundary motion, Proceedings of conference on free boundary problems, Trento 2002, International Series of Numerical Mathematics (ISNM), 147 (2004), 237-254.   Google Scholar

[15]

O. Penrose, On the elastic driving force in diffusion-induced grain boundary motion, Acta Materialia, 52 (2004), 3901-3910.  doi: 10.1016/j.actamat.2004.05.004.  Google Scholar

[16]

http://mathworld.wolfram.com/GreensFunction.html Google Scholar

Figure 1.  Schematic cross-section of a cellular precipitation front, advancing upwards at velocity $v$. The as yet undisturbed metastable phase is labelled 0; the two (relatively) stable phases into which it separates after the front has passed are labelled $\alpha$ and $\beta$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Notation for the coordinate axes and for angles and arc lengths
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The thermodynamic construction for phase equilibrium. The common tangent has slope $\mu_{TJ}$ and touches the curve at the points with abscissas $ c_\alpha^{eq}$ and $c_\beta^{eq}.$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Dihedral angles $\delta$, inclinations of arcs $\theta$ and surface tensions $\gamma$ at a triple junction
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[2]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[3]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[5]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565
[6]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
[7]

Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491
[8]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[9]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[10]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[11]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035
[12]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[13]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[14]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[15]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[16]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
[17]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[18]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[19]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[20]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (75)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences