University of Groningen
Thermodynamically consistent fluid particle modelling of phase separating mixtures
Thieulot, Cedric
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2004
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Thieulot, C. (2004). Thermodynamically consistent fluid particle modelling of phase separating mixtures
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CHAPTER
6
Conclusions and outlook
This thesis is a long journey from the microscopical world to the continuum macroscopic
world for fluid systems that can phase separate. We have further travelled to the discrete
world that allows one to simulate these fluid systems in a thermodynamically consistent
way. We will present here the main results of this work as well as several perspectives
for future research.
In chapter 3, we have provided a microscopic basis for a thermodynamically consistent
set of diffuse interface hydrodynamic equations for a phase separating fluid mixture.
This microscopic derivation has the advantage over phenomenological derivations that
the different thermodynamic variables and, in particular, the internal energy, have a welldefined meaning in terms of microscopic variables, allowing us to dispel some ambiguities
that may appear in phenomenological treatments. Our main motivation was to extend
van der Waals’ ideas from the description of equilibrium states of fluid mixtures to out
of equilibrium situations.
In chapter 4, we have presented a fluid particle model of the sph type for the simulation of the continuum hydrodynamic equations obtained in the second chapter. The
model has been constructed in such a way as to fit within the thermodynamically consistent generic framework. It is the combination of both, the generic and sph methodologies that allowed us to write thermodynamically consistent discrete equations.
The thermodynamic consistency of the discrete fluid particle model ensures that total mass, momentum and energy are exactly conserved while the entropy is a strictly
non-decreasing function of time. The model allows one to study liquid/vapour, miscible/immiscible phase transitions in mixtures in non-isothermal situations with full inclusion of surface tension and cross effects.
In chapter 5, we have presented the validation of the model for the particularly simple,
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Conclusions and outlook
although not trivial, case of fluids where viscous processes are neglected and where only
mass and energy diffusive processes take place. We also restricted ourselves to the case of
a binary mixture that can exhibit only liquid-liquid phase separation. This was achieved
by setting all fluid particles on a regular square lattice (therefore preventing them to
form regions of high and low density responsible for gas/liquid phase separation) and by
carefully selecting meaningful input parameters.
The simulation results have proved to comply to the First and Second Law of thermodynamics: the total energy of the system is conserved and the total entropy of the
system is an increasing function of time. Extended series of test simulations were run
on systems of various sizes, numbers of particles, temperatures and concentrations. In
the absence of surface tension, we were able to observe sub-critical systems that phase
separate, each of the fluid particle becoming either A-rich or B-rich. We observed a strict
compliance of simulation outcome with the theoretically predicted phase diagram.
When surface tension terms were present, we observed that the coexistence curve was
recovered far from the critical point. When the system was set near the critical point
of the system, size effects played a non negligible role due to the interfaces becoming
larger, and we could observe how surface tension affects the dynamics of the system, in
particular the coarse-graining process.
Finally, non-isothermal simulations shed some light on other interesting phenomena:
imposed temperature inhomogeneities (temperature gradients and heat sinks) created a
range of fascinating patterns that certainly deserve further investigations. For instance
one would like to better understand the orientation of the observed patterns and stripes
with respect to the temperature gradient present in the system.
We have validated the model for the case where no viscous processes are present and
particles are at rest. The natural continuation of this work would be to re-introduce
the particle movement: in that case the fluid particles form regions where their spatial
density is high or low compared to the system average. This gives rise to liquid/gas
phase separation phenomena, that would explore fully the phase diagram of the system.
However the sparse distribution of the fluid particles in space in the gas phase would
lead to computational errors of second order derivative estimates and one would want to
address this problem first by going back to the study of single fluids.
In Ref. [1] gas-liquid phase separation has been studied where sph is used to simulate
equations of motions. The authors acknowledge that the presence of interface leads to
(code) instabilities. In their work the surface tension is not an input parameter but
is instead numerically induced by varying the range of support of the kernel function
which at the same time solves the problem of instabilities. Our stable code as well as the
control we have over surface tension through the long range part of the potential, gives us
a superior tool to conduct studies such as liquid drops in their vapour, evaporation and
condensation problems. Then, in-depth studies of the influence of hydrodynamics on the
coarse-graining processes could be carried out which would improve our understanding
of how hydrodynamics affects thermodynamics.
In another approach the fluid particles could simply remain on a regular grid, and
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Conclusions and outlook
a)
b)
c)
d)
Figure 6.1: Spinodal decomposition of a subcritical system after a quench, a) t = 600,
b) t = 4200, c) t = 15200, d) t = 50000.(x = 0.5, NT = 4096, L = 360)
near critical point studies could be conducted. Such studies would turn out to be computationally heavy since the dynamics of coarsening close to the critical point are slow and
reaching equilibrium would therefore require long runs. We also know that the interface
width depends on the (subcritical) temperature [2] and critical exponent measurements
could be carried out. Then, due to the implementation of periodic boundary conditions,
it would be interesting to see how the interplay between system size and interface width
takes place. Indeed, if the system is very large, the amount of fluid particles involved
with interfaces will be small compared to those which are not and one will be able to
speak about bulk values and carry out measurements of bulk pressure far from interfaces
(useful in the case of Laplace Law studies [3]). On the other hand if the system is small,
particles involved in interfaces will form the majority and it is expected that the phase
diagram will be dramatically affected. Finally, generic allows for the inclusion of thermal fluctuations which are expected to take a prominent role close to critical point. This
still represents an open area of research.
When a system is quenched from a disordered homogeneous high temperature phase
into a low-temperature regime, where several phases coexist, a complex domain structure
develops, as observed in chapter 5. The late stages of this phase ordering process is
believed to show universal scaling behaviour: it is found that the kinetics are dominated
by a single length scale, the domain size R(t), which increases with a power law in
time R(t) ∝ t1/z . This fact is illustrated in Fig.(6.1). The domain patterns belong to
three different times. but if the graphs are rescaled by the domain size, quite similar
pictures are obtained. It would therefore be worth pursuing this direction and studying
the domain growth time exponent 1/z which has been subject to multiple debates in the
seventies [4] [5], eighties [6] [7] [8] [9], nineties [10] [11] [12] [13] [14] [15], and even quite
recently [16]. Most of these references also aim at explaining the influence of different
competing processes involved in coarsening and domain growth, while our model allows
for studying the influence of thermal conductivity terms, cross-effects terms and surface
tension terms in the equations on this segregating processes.
In the way we have introduced the surface tension we have assumed that the cαβ
coefficients accounting for surface tension are independent of the thermodynamical state
of the system. It is certainly not the case in the reality and one might want to push further
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Conclusions and outlook
investigations where the surface tension value is coupled to the state of the system.
In this thesis we have presented simulations for the binary case but the model allows for the treatment of multicomponent systems. Chemical reactions can easily be
implemented in our model by means of the addition of reaction rate terms and stoichiometric coefficients, but whenever chemical reactions are present in the system, such as
A + B C, we need to at least consider three components. Obviously, in this case, the
model proves to be much more complex than the binary case.
Bibliography
[1] Liquid drops and surface tension with smoothed particle applied mechanics, S.Nugent
and H.A.Posch, Phys. Rev. E 62, 4968 (2000).
[2] Free energy of a nonuniform system. I-interfacial free energy, J.W. Cahn and J.E.
Hilliard, J. Chem. Phys. 28, 258 (1958).
[3] Investigations of a two-phase fluid model, B.T.Nadiga and S.Zaleski, Eur. J. Mech.
B/Fluids 15, 885 (1996).
[4] Computer simulation of the time evolution of a quenched model alloy in the nucleation region, J.Marro, J.L.Lebowitz and M.H.Kalos, Phys. Rev. Lett. 43, 282
(1979).
[5] Theory for the dynamics of ’clusters’. II-Critical diffusion in binary systems and the
kinetics of phase separation., K.Binder, Phys. Rev. B 15, 4425 (1977).
[6] Dynamics of phase separation in two-dimensional fluids: spinodal decomposition,
S.W.Koch, R.C.Desai, and F.F.Abraham, Phys. Rev. A 27, 2152 (1983).
[7] A dynamics scaling assumption for phase separation, H. Furukawa, Advances in
Physics 34, 703 (2001).
[8] Computationally efficient modelling of ordering of quenched phases, Y.Oono and
S.Puri, Phys. Rev. Lett. 58, 836 (1987).
[9] Numerical study of the late stages of spinodal decomposition, T.M.Rogers, K.R.Elder
and R.C.Desai, Phys. Rev. B 37, 9638 (1988).
[10] deformation, growth, and order in sheared spinodal decomposition, D.H.Rothman,
Phys. Rev. Lett. 65, 3305 (1990).
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Conclusions and outlook
[11] Hydrodynamic spinodal decomposition: growth kinetics and scaling functions,
F.J.Alexander, S.Chen, and D.W.Grunau, Phys. Rev. B 48, 634 (1993).
[12] Theory of phase-ordering kinetics, A.J.Bray, Advances in Physics 43, 357 (1994).
[13] Non-equilibrium ordering dynamics and pattern formation, M.Siegert, in Scale invariance, interfaces and non-equilibrium dynamics, Proceedings of the NATO meeting in Cambridge Jun. 1994, edited by A.J. McKane, Plenum, NY.
[14] Domain growth in computer simulations of segregating two-dimensional binary fluids, S.Bastea and J.L.Lebowitz, Phys. Rev. E 52, 3821 (1995). Spinodal decomposition in binary gases, S.Bastea and J.L.Lebowitz, Phys. Rev. Lett. 78, 3499 (1997).
[15] Phase ordering in fluids, J.M.Yeomans, Ann. Rev. Comp. Phys. VII (1999).
[16] Three-dimensional hydrodynamic lattice-gas simulations of domain growth and selfassembly in binary immiscible and ternary amphiphilic fluids, P.J. Love, P.V.
Coveney and B.M. Boghosian, Phys. Rev. E 64, 021503-1 (2001).
acknowledgements
Merci à mes Parents, à Emmanuelle et Arnaud, à Mamie Coucou, à André et Jany, pour
leur amour, leur aide, et leur support tout au long de ces dernières années mouvementées.
Merci à Aude, à Aurélie et Thomas, à Caroline et Willy, à Cathy, à Céline, à Christelle,
à Delphine B., à Delphine M., à Emmanuelle, à Françoise et Sjoerd, à Gilles, à Hervé, à
Ingrid, à Laurent, à Magali, à Mylène, à Oriane et Fred, à Seb, à Stéphanie, et à Vince.
Parce que je ne serais pas celui que je suis aujourd’hui sans vous, votre amitié m’est
chère.
Because friendship knows no boundaries, I would like to thank Bessel, Charms,
Christa, Dorien, Francesca, Jelske, Lavinia, Maartje, Naomi, Paola, and Uazir for the
wonderful moments spent together these last years.
Mijn Ragdoll muzikale familie : Alex, Dennis, Frank, Irene, Sophie, Gerjan, Jan-Peter,
mijn leraren Jacques Roorda en Winfred Burma.
Ik bedank ook Marcel, Anne, Erwin, Marya, Laurens, Mario, Vincent V, Vincent F,
Jasper voor hun hulp en collegialiteit.
And last but not least, I want to express my gratitude to Prof. Mico Hirschberg who
welcomed me at the Laboratory of Fluid Mechanics in Eindhoven during Spring 2000 for
my Master Thesis internship, then to Prof. Leon Janssen for giving me the chance to
become a PhD student, for trusting me all along the past 4 years, and for leaving me the
freedom to do things my way, and finally to Pep Español for his support, his patience,
his guidance, and his wisdom.
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