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What is Spinodal Decomposition?

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Evangelos P. Favvas at National Center for Scientific Research Demokritos
Evangelos P. Favvas
A. Ch. Mitropoulos
A. Ch. Mitropoulos
A. Ch. Mitropoulos
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Abstract

Phase separation may occur in a way that the growth is not in extent but in amplitude. Only in the unstable region such a procedure is thermodynamically feasible. In a phase diagram the unstable region is defined by the spinodal. When a system has crossed this locus, phase separation occurs spontaneously without the presence of a nucleation step. This process is known as spinodal decomposition and commonly results to a high interconnectivity of the two phases. The Cahn-Hilliard equation describes the kinetics of the process. In this note both processes (nucleation and spinodal) are depicted schematically.
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25
Journal of Engineering Science and Technology Review 1 (2008) 25- 27
Lecture Note
What is spinodal decomposition?
E. P. Favvas*, 1 and A. Ch. Mitropoulos2
1Institute of Physical Chemistry, NCSR “Demokritos”, 153 10 Ag.Paraskevi, Attikis,Greece.
2Department of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece.
Received 17 January 2008; Accepted 20 February 2008
___________________________________________________________________________________________
Abstract
Abstract: Phase separation may occur in a way that the growth is not in extent but in amplitude. Only in the unstable region such a
procedure is thermodynamically feasible. In a phase diagram the unstable region is defined by the spinodal. When a system has
crossed this locus, phase separation occurs spontaneously without the presence of a nucleation step. This process is known as spinodal
decomposition and commonly results to a high interconnectivity of the two phases. The Cahn-Hilliard equation describes the kinetics
of the process. In this note both processes (nucleation and spinodal) are depicted schematically.
Keywords: spinodal decomposition, nucleation and growth.
___________________________________________________________________________________________
A pair of partially miscible liquids, i.e. liquids that do not
mix in all proportions at all temperatures, shows in a tem-
perature-composition diagram a miscibility gap where phase
separation occurs. Gibbs [1] showed that the condition for
stability (or metastability) in respect to continuous change of
phase is that the second derivative of the free energy of mix-
ing to be positive. If negative, the system is unstable. If zero,
the spinodal is defined. The free energy of mixing, ΔGmix,
has the following general form:
mixmixmix S T HG Δ −Δ =Δ . (1)
For the regular solution model [2] the entropy of mixing is
the same as for the ideal mixing; ΔSmix=-R(XAlnXA+XlnXB),
where XA and XB are the molar fractions of components A
and B in the mixture (XA+XB=1). However, the enthalpy of
mixing may be written as ΔHmix=XAXBβ, where β is an in-
teraction parameter lumping the energy of mixing contribu-
tion [3]. Under these assumptions, eq.(1) becomes:
()
BB A ABA
mix XX X X R TX XG lnln ++= Δ
β
. (2)
Figure 1. A phase diagram with a miscibility gap (lower frame) and a
diagram of the free energy change (upper frame). The phase diagram is
the temperature versus the molar fraction of a component e.g. xB. Note
that the diagram is symmetrical around xB=0.5 which is the case of the
regular solution model. Line (1) is the phase boundary. Above this line
the two liquids are miscible and the system is stable (s). Below this line
there is a metastable region (m). Within that region the system is stable
to small fluctuations but is unstable to large fluctuations. Line (2) is the
spinodal. Below this line the system is unstable (u). Regions (m) and (u)
constitute the miscibility gap. Within that gap the system turns from one
phase to a two-phase system. Temperature (Tc) is the upper consolute
temperature. Above this temperature the two liquids are miscible in all
proportions. At a given temperature (T) the tie line (4) cuts the phase
boundary and the spinodal at points (a, c) and (b, d), respectively. The
changes in the free energy of mixing (ΔG), at this given temperature, in
respect to xB are shown by line (3) in the upper frame. Segments (ab)
and (cd) correspond to a positive second derivative of ΔG, ∂2ΔG/xB2>0,
while segment (bd) to a negative one, ∂2ΔG/xB2<0. At points (b) and (d)
∂2ΔG/xB2=0.
J
estr
JOURNAL OF
Engineering Science and
Technology Review
www.
j
estr.or
g
______________
* E-mail address: favvas@chem.demokritos.gr
ISSN: 1791-2377 © 2008 Kavala Institute of Technology. All rights reserved.
Journal of Engineering Science and Technology Review 1 (2008) 25- 27
Lecture Note
What is spinodal decomposition?
E. P. Favvas*, 1 and A. Ch. Mitropoulos2
1Institute of Physical Chemistry, NCSR “Demokritos”, 153 10 Ag.Paraskevi, Attikis,Greece.
2Department of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece.
Received 17 January 2008; Accepted 20 February 2008
___________________________________________________________________________________________
Abstract
Abstract: Phase separation may occur in a way that the growth is not in extent but in amplitude. Only in the unstable region such a
procedure is thermodynamically feasible. In a phase diagram the unstable region is defined by the spinodal. When a system has
crossed this locus, phase separation occurs spontaneously without the presence of a nucleation step. This process is known as spinodal
decomposition and commonly results to a high interconnectivity of the two phases. The Cahn-Hilliard equation describes the kinetics
of the process. In this note both processes (nucleation and spinodal) are depicted schematically.
Keywords: spinodal decomposition, nucleation and growth.
___________________________________________________________________________________________
A pair of partially miscible liquids, i.e. liquids that do not
mix in all proportions at all temperatures, shows in a tem-
perature-composition diagram a miscibility gap where phase
separation occurs. Gibbs [1] showed that the condition for
stability (or metastability) in respect to continuous change of
phase is that the second derivative of the free energy of mix-
ing to be positive. If negative, the system is unstable. If zero,
the spinodal is defined. The free energy of mixing, ΔGmix,
has the following general form:
mixmixmix S T HG Δ −Δ =Δ . (1)
For the regular solution model [2] the entropy of mixing is
the same as for the ideal mixing; ΔSmix=-R(XAlnXA+XlnXB),
where XA and XB are the molar fractions of components A
and B in the mixture (XA+XB=1). However, the enthalpy of
mixing may be written as ΔHmix=XAXBβ, where β is an in-
teraction parameter lumping the energy of mixing contribu-
tion [3]. Under these assumptions, eq.(1) becomes:
()
BB A ABA
mix XX X X R TX XG lnln ++= Δ
β
. (2)
Figure 1. A phase diagram with a miscibility gap (lower frame) and a
diagram of the free energy change (upper frame). The phase diagram is
the temperature versus the molar fraction of a component e.g. xB. Note
that the diagram is symmetrical around xB=0.5 which is the case of the
regular solution model. Line (1) is the phase boundary. Above this line
the two liquids are miscible and the system is stable (s). Below this line
there is a metastable region (m). Within that region the system is stable
to small fluctuations but is unstable to large fluctuations. Line (2) is the
spinodal. Below this line the system is unstable (u). Regions (m) and (u)
constitute the miscibility gap. Within that gap the system turns from one
phase to a two-phase system. Temperature (Tc) is the upper consolute
temperature. Above this temperature the two liquids are miscible in all
proportions. At a given temperature (T) the tie line (4) cuts the phase
boundary and the spinodal at points (a, c) and (b, d), respectively. The
changes in the free energy of mixing (ΔG), at this given temperature, in
respect to xB are shown by line (3) in the upper frame. Segments (ab)
and (cd) correspond to a positive second derivative of ΔG, ∂2ΔG/xB2>0,
while segment (bd) to a negative one, ∂2ΔG/xB2<0. At points (b) and (d)
∂2ΔG/xB2=0.
J
estr
JOURNAL OF
Engineering Science and
Technology Review
www.
j
estr.or
g
______________
* E-mail address: favvas@chem.demokritos.gr
ISSN: 1791-2377 © 2008 Kavala Institute of Technology. All rights reserved.
Journal of Engineering Science and Technology Review 1 (2008) 25- 27
Lecture Note
What is spinodal decomposition?
E. P. Favvas*, 1 and A. Ch. Mitropoulos2
1Institute of Physical Chemistry, NCSR “Demokritos”, 153 10 Ag.Paraskevi, Attikis,Greece.
2Department of Petroleum Technology, Kavala Institute of Technology, 65 404 St.Lucas, Kavala, Greece.
Received 17 January 2008; Accepted 20 February 2008
___________________________________________________________________________________________
Abstract
Abstract: Phase separation may occur in a way that the growth is not in extent but in amplitude. Only in the unstable region such a
procedure is thermodynamically feasible. In a phase diagram the unstable region is defined by the spinodal. When a system has
crossed this locus, phase separation occurs spontaneously without the presence of a nucleation step. This process is known as spinodal
decomposition and commonly results to a high interconnectivity of the two phases. The Cahn-Hilliard equation describes the kinetics
of the process. In this note both processes (nucleation and spinodal) are depicted schematically.
Keywords: spinodal decomposition, nucleation and growth.
___________________________________________________________________________________________
A pair of partially miscible liquids, i.e. liquids that do not
mix in all proportions at all temperatures, shows in a tem-
perature-composition diagram a miscibility gap where phase
separation occurs. Gibbs [1] showed that the condition for
stability (or metastability) in respect to continuous change of
phase is that the second derivative of the free energy of mix-
ing to be positive. If negative, the system is unstable. If zero,
the spinodal is defined. The free energy of mixing, ΔGmix,
has the following general form:
mixmixmix S T HG Δ −Δ =Δ . (1)
For the regular solution model [2] the entropy of mixing is
the same as for the ideal mixing; ΔSmix=-R(XAlnXA+XlnXB),
where XA and XB are the molar fractions of components A
and B in the mixture (XA+XB=1). However, the enthalpy of
mixing may be written as ΔHmix=XAXBβ, where β is an in-
teraction parameter lumping the energy of mixing contribu-
tion [3]. Under these assumptions, eq.(1) becomes:
()
BB A ABA
mix XX X X R TX XG lnln ++= Δ
β
. (2)
Figure 1. A phase diagram with a miscibility gap (lower frame) and a
diagram of the free energy change (upper frame). The phase diagram is
the temperature versus the molar fraction of a component e.g. xB. Note
that the diagram is symmetrical around xB=0.5 which is the case of the
regular solution model. Line (1) is the phase boundary. Above this line
the two liquids are miscible and the system is stable (s). Below this line
there is a metastable region (m). Within that region the system is stable
to small fluctuations but is unstable to large fluctuations. Line (2) is the
spinodal. Below this line the system is unstable (u). Regions (m) and (u)
constitute the miscibility gap. Within that gap the system turns from one
phase to a two-phase system. Temperature (Tc) is the upper consolute
temperature. Above this temperature the two liquids are miscible in all
proportions. At a given temperature (T) the tie line (4) cuts the phase
boundary and the spinodal at points (a, c) and (b, d), respectively. The
changes in the free energy of mixing (ΔG), at this given temperature, in
respect to xB are shown by line (3) in the upper frame. Segments (ab)
and (cd) correspond to a positive second derivative of ΔG, ∂2ΔG/xB2>0,
while segment (bd) to a negative one, ∂2ΔG/xB2<0. At points (b) and (d)
∂2ΔG/xB2=0.
J
estr
JOURNAL OF
Engineering Science and
Technology Review
www.
j
estr.or
g
______________
* E-mail address: favvas@chem.demokritos.gr
ISSN: 1791-2377 © 2008 Kavala Institute of Technology. All rights reserved.
26
E. P. Favvas and A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 25-27
27
E. P. Favvas and A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 25-27
growth whereas if it is in the unstable region by spinodal
decomposition.
Figure 2 depicts the nucleation and growth mechanism
[4]. Gibbs (Ref.1 p.254) discussed the possibility of the for-
mation of a fluid of different phase within any homogeneous
fluid and showed that the work W required to form (by a
reversible process) an heterogeneous globule in the interior
of a very large mass having initially the uniform phase of the
exterior mass will be:
P rr WΔ − = 32 4
4
πγ π
3, (4)
where γ is the surface ten
d ΔP is the hydrostatic pressure. In the case where a criti-
sion, r is the radius of the nucleus,
an
cal nucleus is formed the first derivative of eq.(4) becomes
zero while the hydrostatic equilibrium is preserved;
ΔP=2γ/rcrit, where rcrit is the radius of the critical nucleus.
When r>rcrit the nucleus grows and when r<rcrit it collapses
because the pressure exerted by the surface is greater than
ΔP. By introducing rcrit in eq.(4) the minimum work Wmin
required to form that critical nucleus is given as:
γπ
2
4r=.
min 3crit (5)
Figure 3 depict
[5]. Cahn and Hilliard [6] utili e the van der Waals equation
W
s the spinodal decomposition mechanism
z
for the energy of binary mixture [7] to develop a model for
spinodal decomposition through Fick’s law of diffusion.
BB
D
B
BC MC
C
f
C2⎪
⎫
⎪
⎧⎞⎛ ∂
∂
where ƒ is the free energy density of homogeneous material
of composition e.g. CB, M is a positive constant [8], and κ is
a positive parameter. The first term on the right hand side of
eq.(6) is the classical diffusion equation where the coeffi-
cient of ∇2CB is the diffusion coefficient D. Since M>0 the
sign of D will be determined by the sign of ∂2ƒ/∂C2B. When
∂2ƒ/∂C2B>0 the solution is stable, D>0, and diffusion (if any)
occurs downhill. When ∂2ƒ/∂C2B<0 the solution has crossed
the spinodal and is unstable, D<0, and diffusion takes place
uphill. It is noted, however, that the second term on the right
hand side of eq.(6) (i.e. the fourth-order term) stabilizes the
system against short distance scale fluctuations when
∂2ƒ/∂C2B<0. Uphill diffusion and consequently spinodal de-
composition requires a large enough distance scale fluctua-
tion (i.e. to move mass over long distances).
In this note an elementary review on the concept of spi-
nodal decomposition was given. The nucleation and growth
mechanism was also presented. Comparing the two proc-
esses it is evident that nucleation is large in degree and small
in extent while spinodal is small in degree and large in ex-
tent. In Fig.2 and 3 the evolution of the density profiles at
various stages of the processes was drawn. It is noted that
spinodal decomposition results to a high interconnectivity of
the two phases. However, the morphology of the resulted
separation is only an indication that spinodal decomposition
has taken place [9]. Nucleation may also produce high inter-
connectivity [10]. Besides, fractal morphology is possible to
be induced by both processes. For instance a fractal flake
that grows is an example of a fractally nucleation process
whereas a twin-dragon Peano curve that thickens is an ex-
ample of a fractally spinodal process [11].
M
t
42
22∇−∇ ⎪
⎭
⎬
⎪
⎩
⎨⎟
⎟
⎠
⎜
⎜
⎝∂
=
∂=
κ
, (6)
______________________________
References
1. J.W.Gibbs, Collected Works, Yale University Press, New H
pp.105-115 and pp.252-258 (1948).
ed, follow
t.Phys. 38, 707 (1985); V.M.Agishev and K.M.Yamaleev,
yss.U.Zav. Fizika 10, 124 (1975); K.R.Mecke and V.Sofonea,
gstaff, and R.J.Charles, J.
Acknowledgments: The authors would like to thank Ar-
chimedes research Project and INTERREG-III ‘Hybrid
Technology for separation’ for funding this work.
aven, J.Sta
Iz.V
2. E.A.Guggenheim, Proc.Roy.Soc.A 148, 304 (1935); J.B.Thompson,
Jr., in Researches in Geochemistry, Ed. by P.H.Abelson. John Wiley
and Sons Inc., New York, Vol.2 p.340 (1967).
3. For the ideal solution the enthalpy of mixing is zero.
4. IUPAC definition of nucleation and growth: A process in a phase
transition in which nuclei of a new phase are first form ed by delingen der Koninklijke Nederlandsche Akademie van Wetenschap-
pen te Amsterdam 1, 1 (1893).
8. M is related to the interdiffusion coefficient. For a binary system
JB=−M·∇(μB-μA), where J and μ are respectively the flux and the
the propagation of the new phase at a faster rate; J.B.Clark,
J.W.Hastie, L.H.E.Kihlborg, R.Metselaar, and M.M.Thackeray, Pure
App.Chem. 66, 577 (1994).
5. IUPAC definition of spinodal decomposition: A clustering reaction in
a homogeneous, supersaturated solution (solid or liquid) which is un-
stable against infinitesimal fluctuations in density or composition. The
solution therefore separates spontaneously into two phases, starting
with small fluctuations and proceeding with a decrease in the Gibbs
energy without a nucleation barrier; ibid.
6. J.W.Chan and J.E.Hilliard, J.Chem.Phys. 28, 258 (1958).; J.W.Chan
and J.E.Hilliard, J.Chem.Phys. 31, 688 (1959); see also J.W.Chan,
J.Chem.Phys. 30, 1121 (1959); R.B.Heady and J.W.Chan,
J.Chem.Phys. 58, 896 (1973); J.W.Cahn, Trans.A.IM.E. 242, 166
(1968); J.W.Cahn, Acta Met. 9, 795 (1961); J.W.Cahn and
R.J.Charles, Phys.Chem. Glasses 6, 181 (1965); V.S.Stubican and
A.H.Schultz, J. Amer. Cerum. Soc., 51 290 (1968); A.Noviek-Cohen,
Phys.Rev.E 56, R3761 (1997); J.Zhu, L-Q.Chen, J.Shen, and
V.Tikare, Phys.Rev.E 60, 3564 (1999); C.P.Grant, Com-
mun.Part.Differ.Eq. 18, 453 (1993).
7. J.D.van der Waals, The thermodynamic theory of capillarity flow
under the hypothesis of a continuous variation in density, Verhan-
11.
chemical potential of e.g. the B component and M is the mechanical
mobility. Changes in the concentration are given by taking the diver-
gence of the flux: ∂CB/∂t=−∇·JB.
9. J.W.Cahn, J.Chem.Phys. 42, 93 (1965).
10. W.Haller, J.Chem.Phys. 42, 686 (1965); see also W.Haller,
D.H.Blackburn, J.H.Simmons, F.E.Wa
Amer. Ceram. Soc. 53, 34 (1970); G.R.Srinivasan, I.Tweer,
P.B.Macedo, A.Sarkar, and W.Haller, J.Non-Cryst. Solids 6, 221
(1971); W.Haller, D.H.Blackburn, and J.H.Simmons, J. Amer. Cerum.
Soc., 57 126 (1974).
F.Katsaros, P.Makri. A.Ch.Mitropoulos, N.Kanellopoulos,
U.Keiderling, and A.Wiedenmann, Physica B 234-236, 402 (1997).
27
E. P. Favvas and A. Ch. Mitropoulos/ Journal of Engineering Science and Technology Review 1 (2008) 25-27
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Article
  • Feb 1985
The behavior of the nonlinear Cahn-Hilliard equation for asymmetric systems,c t =2(c+Bc 2+c 3-2 c) within the unstable subspinodal region is explored. Energy considerations and amplitude equation methods are employed. Evidence is given for a transition from periodically structuredspinodal behavior to nucleation behavior somewhere within the traditional spinodal. A mechanism for describing a time-dependent lengthening of the dominant wavelength is explored.
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Spinodal Decomposition For The Cahn-Hilliard Equation
Article
  • Dec 1998
The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is known as spinodal decomposition. A simple linear analysis about an unstable homogeneous equilibrium of the one-dimensional Cahn-Hilliard equation gives heuristic evidence that most solutions that start with initial data near such an equilibrium exhibit a behavior corresponding to spinodal decomposition. In this paper we formulate this conjecture in a mathematically precise way, using geometric and measure-theoretic techniques, and prove its validity. We believe that this is the first rigorous treatment of this phenomenon.
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For a binary system J B=−M·∇(μB-μA), where J and μ are respectively the flux and the the propagation of the new phase at a faster rate
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M is related to the interdiffusion coefficient. For a binary system J B=−M·∇(μB-μA), where J and μ are respectively the flux and the the propagation of the new phase at a faster rate; J.B.Clark, J.W.Hastie, L.H.E.Kihlborg, R.Metselaar, and M.M.Thackeray, Pure App.Chem. 66, 577 (1994).
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