# American Institute of Mathematical Sciences

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

## On the mathematical modelling of cellular (discontinuous) precipitation

 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.

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]

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##### 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]
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 $\beta$
Notation for the coordinate axes and for angles and arc lengths
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 $c_\beta^{eq}.$
Dihedral angles $\delta$, inclinations of arcs $\theta$ and surface tensions $\gamma$ at a triple junction
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