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]

Ken Shirakawa, Hiroshi Watanabe. Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 139-159. doi: 10.3934/dcdss.2014.7.139
[2]

Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824
[3]

Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks & Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669
[4]

Akio Ito, Nobuyuki Kenmochi, Noriaki Yamazaki. Global solvability of a model for grain boundary motion with constraint. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 127-146. doi: 10.3934/dcdss.2012.5.127
[5]

T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013
[6]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431
[7]

Ken Shirakawa, Hiroshi Watanabe. Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint. Conference Publications, 2015, 2015 (special) : 1009-1018. doi: 10.3934/proc.2015.1009
[8]

Shannon Dixon, Nancy Huntly, Priscilla E. Greenwood, Luis F. Gordillo. A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1155-1164. doi: 10.3934/mbe.2018052
[9]

Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007
[10]

Eduardo Ibarguen-Mondragon, Lourdes Esteva, Leslie Chávez-Galán. A mathematical model for cellular immunology of tuberculosis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 973-986. doi: 10.3934/mbe.2011.8.973
[11]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225
[12]

Juan Pablo Aparicio, Carlos Castillo-Chávez. Mathematical modelling of tuberculosis epidemics. Mathematical Biosciences & Engineering, 2009, 6 (2) : 209-237. doi: 10.3934/mbe.2009.6.209
[13]

Benjamin Wincure, Alejandro D. Rey. Growth regimes in phase ordering transformations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 623-648. doi: 10.3934/dcdsb.2007.8.623
[14]

Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427
[15]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[16]

Maciek D. Korzec, Hao Wu. Analysis and simulation for an isotropic phase-field model describing grain growth. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2227-2246. doi: 10.3934/dcdsb.2014.19.2227
[17]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[18]

Geoffrey Beck, Sebastien Imperiale, Patrick Joly. Mathematical modelling of multi conductor cables. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 521-546. doi: 10.3934/dcdss.2015.8.521
[19]

Nirav Dalal, David Greenhalgh, Xuerong Mao. Mathematical modelling of internal HIV dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 305-321. doi: 10.3934/dcdsb.2009.12.305
[20]

John M. Ball, Carlos Mora-Corral. A variational model allowing both smooth and sharp phase boundaries in solids. Communications on Pure & Applied Analysis, 2009, 8 (1) : 55-81. doi: 10.3934/cpaa.2009.8.55

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (28)
  • HTML views (19)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences