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On the kinetics of spinodal decomposition: a numerical solution of a dissipative wave equation
Volume
76, number
3
CHEhlICAL
PHYSICS
ON THE KINETICS OF SPINODAL ~ECO~POS~D~ DISSIPATIVE WAVE EQUATION
Recened
15 December
LETTERS
A ~~~RIC~
SOL~O~
1980
OF A
19 August 1980
A n~mencai so~urmn of a recentI) dented d:sslpatlbe waw equation gavernmg the kmetlcs of spmodal decomposltmn of a Lennard-Jones Ruld 1s presenred In addltron. the r.zsults are compared wrth those of Cahn’s and Abraham’s generaked dtffusion theories for the case of rhe early stages of the coarsemng process
1. Introduction The problem of phase separation mechanism
of spinodal
decomposltIon
caused by the was subject of
many in\eestigations, both expenmentally and theoretically El], during the last two decades Studies on binary mixtures, where short-range Ising systems had been dacussed, indicate that non-hnear effects are responsible for the coarsening of the morphology even at the earliest stages of the decomposition process.
Long-rangeLennard-Jonessystems,however,may show deviations from the behaviour of short-range Ismg models, as has been pointed out by Abraham [l-41, who derived a generahzed diffusion equation for a non-homogeneous fluid system of pauwlse interactmg particles. The only available “expenmental” data of thrs physical situation are the molecular dynamtcs computer slmulattons of MNnk et al. [4] who quenched a Lennard-Jones fluid (argon) at a rate of 10” K/s deep mto the mstable region of the hquid-vapour phase diagram In contrast to the observattons and calculattons on bmary mixtures they observed a hnear range of spmodal decomposition durmg
the first S ps
For later tunes the decom-
position enters the non-linear regime and finally the wavelength A, of the fastest growing wave goes as rQ, where a = l/2.6 1s reasonably consrstent with the Lifshltz-Slyozov law (a = l/3).
Calculating the amphfication factor o as a function of the wavenumber k of the density fluctuations from the hnearrzed version of the generalized diffusion theory and comparing It to the results of the computer study, Abraham [I] found serious quantnatrve discrepancies for the full range of wavenumbers. Stimulated by the results of a kmetrc theory of spmodal decomposition developed very recently [S], we would expect that the disagreement between molecular dynamics computer simulations and the predlctions of pure dlffuslon theories m the hnear range of decomposinon may be based on the fact that delusion approximation ignores kmetlc effects. In order to support this suggestion we shall discuss the influence of kinetx effects on the problem of spmoda1 decomposmon.
2. A dissipativewaveequation It is well known that dlffuslon theory is based on the assumption that or < 1 (and, in addlhon to
kl -=s1, where I IS a mean free path related to the colhslon time 7 by I= Gr where 6 is the mean velocIty, and k = ~T/A IS the wave number). But at frequencres comparable to the collision frequency 7-l of the atoms or molecules the restrictron wr 4 1 is no longer
vahd and the study of spinodal
decomposition
CHEhllCAL
Volume 76. number 3
has to take mto account which
can be dealt
PHYSICS
- in this case - kmetlc effects
wxth by solmg
the
Boltzmann
equation
aflat + u - vf= (af/ai),
(1)
for the non-equilibrium dlstrlbution function f(r, u, t). The first serious attempt to discuss the kinetics of spmodal decomposition in thrs way was due to one of the authors very recently [5] The Idea is to develop on the basis of the relavatlon time approximation (RTA) - a perturbation theory leadmg to a hierarchy of dlfferentlal equations for the local density p(r, t) Usmg RTA, 1 e. the ansatz f(r. u, f) = F(E(u),
T, cl(r, tN+dr.
u, t)
(1) takes the simple
form
(O,+O,)fJ=rZ.
(3)
Here, we have introduced o,=
(If
7a/at)ap/ar = M
VP
,
where III is the mobMy,
6)
defined
Ca Fick‘s law
j=-hIFLL.
(7)
Here, p is a function of T and p(r, I). Eq. (6j is our non-linear dlsslpative wave equation, which, in the hmit 7 + 0, reduces to the generalized diffusion equatlon discussed by Abraham [3] very recently.
3. The amplificationfactor
(dfla[L = -d/i. equation
Eqs (4) and (5) define our perturbation theory [S]. The first-order result [take into account the first two terms m parentheses of the nght-hand side of eq. (5) only] leads to [S]
(2)
and replacing the scattermg term by the Boltzmann
i 5 December 1980
LETTERS
the operator
To evaluate eq. (6), we follow Abraham’s approxlmatlon [l] for the generalized chemical potential g of a non-uniform fluId with density vanations in one dlmenslon
notation
l+;_iJ/C%, 0,=TU lF,
Z ISgiven by --ip
z = -(lS=/l3/L)(a/f/ar + u - Vp) with
and F IS a local equilibrium dlstnbutlon depending the local chemical potential p (r, t) Expandmg F
iij
on
into a Taylor
series up to first-order
F = Fo-I-L - d~aF~a,L.
terms
*
where
the Index 0 refers to the equlhbrlum po, we can replace - if necessary dF/+
=%F/ap
I-1
value p =
lo .
Tha approxunation
makes dFli+~ independent of r and r. The formal operator solution of the Boltzmann
equation m RTA ISgiven by 4 = (1+
0;‘0,)-‘0~’
-TO;‘(l-O~lOx+
side
(rZ)
_. .)Z
where f’ IS the analytIca continuation of the free energy density of the homogeneous fluid into the unstable region, I((&) IS the interactlon potentiai of the pauwlse interactmg particles, gS IS their radial dlstributlon function, and p. is the density of the umform fluId The second term on the nght-hand of eq. (8) describes
fluId m a microscopic (4)
Here we have assumed that the colhslon time 7 IS independent of ,u From the particle conservation law
the inhomogeneity
way. It contains
of the
no
phenomenologlcal parameters Next, eq (8) is lmearized by expanding r@(z. t)) up to second order into a Taylor series about po. According
to the observed
exponential
growth
[a] of
we conclude that
the dens@ fluctuations during the early stage of
J d”uCf-F)=Jd”~~(r,u,t)-(l,~)=o.(3
decomposition
we use the familiar
ansatz [l]
563
Volume
76. number
3
CHEMICAL
Insertion into the hnearized version to the characteristic equation (1 +or)o
PHYSICS
of eq. (8) leads
= -~~‘[(a2~/ap2),+A(k)],
(9)
aJ I
cos (kw - 1) fI( II*,po) dw ,
0 which factor of the ing to
we solved numencally for the amphfication w(k). The values of K PO, and the parameters Lennard-Jones potential were chosen accordthe computer study [4].
4. Results
and discussion
In fig. 1 we have plottEd the quantrties w*/q2 versus q2. where W* = w,, and q = k/k:. Here,
MOLECULAR
d
1980
WC,, IS the maximum of the amphfication factor is the critical wavelength, both refer to Cahn’s theory [6]. Also, for comparison, the amplification factors of Cahn and Abraham, respectively, are shown. To compare our theory with the molecular dynamics data (shaded region) we had to fur the parameter r* = 7w(icmax)_ Takrng l/W(k,,) = 6.02 ps from the computer study [4] and estimatmg the microscopic collision tune T as given in ref. [5] we obtain r* = 0 56. This value of or, which is a measure for the influence of kinetic effects, is obviously not “very small”. As a consequence of ths, the kinetic theory reproduces at least at large wavelengths (small k values) the molecular dynamics results more closely than the diffusion type theories of Cahn and Abraham supporting the suggestion that kinetic effects should not be neglected in the descnption of the early stages of spmodat decompositron. WC(k) and k:
with A(k)=4n
15 December
LETTERS
DYNAMCS
References
CAHN ABRAHAM
[I]
Fig 1 Companson of w’(q)/q’ between theory and expenment The temperature T after the quench IS 0 57 T,
564
F F Abraham.
Phys
therem [2] F.F Abraham,
Chem
Rept
[3]
W E Langlcxs and F F 52 (1977) 129.
[a]
M R. hfruzzk, FF.
Phys
53 (1979)
Letters
Abraham.
Abraham
93. and references
57 (1977) Chem.
179
Phys. Letters
and G hi. Pound,
Phys 69 (1978) 3462 [S] Th F. Nonnenmacher. Z Physlk B38 (1980) [6] J W. Cahn and J E Hdhard. J Chem. Phys 258
J Chem.
327 28 (1958)
76, number
3
CHEhlICAL
PHYSICS
ON THE KINETICS OF SPINODAL ~ECO~POS~D~ DISSIPATIVE WAVE EQUATION
Recened
15 December
LETTERS
A ~~~RIC~
SOL~O~
1980
OF A
19 August 1980
A n~mencai so~urmn of a recentI) dented d:sslpatlbe waw equation gavernmg the kmetlcs of spmodal decomposltmn of a Lennard-Jones Ruld 1s presenred In addltron. the r.zsults are compared wrth those of Cahn’s and Abraham’s generaked dtffusion theories for the case of rhe early stages of the coarsemng process
1. Introduction The problem of phase separation mechanism
of spinodal
decomposltIon
caused by the was subject of
many in\eestigations, both expenmentally and theoretically El], during the last two decades Studies on binary mixtures, where short-range Ising systems had been dacussed, indicate that non-hnear effects are responsible for the coarsening of the morphology even at the earliest stages of the decomposition process.
Long-rangeLennard-Jonessystems,however,may show deviations from the behaviour of short-range Ismg models, as has been pointed out by Abraham [l-41, who derived a generahzed diffusion equation for a non-homogeneous fluid system of pauwlse interactmg particles. The only available “expenmental” data of thrs physical situation are the molecular dynamtcs computer slmulattons of MNnk et al. [4] who quenched a Lennard-Jones fluid (argon) at a rate of 10” K/s deep mto the mstable region of the hquid-vapour phase diagram In contrast to the observattons and calculattons on bmary mixtures they observed a hnear range of spmodal decomposition durmg
the first S ps
For later tunes the decom-
position enters the non-linear regime and finally the wavelength A, of the fastest growing wave goes as rQ, where a = l/2.6 1s reasonably consrstent with the Lifshltz-Slyozov law (a = l/3).
Calculating the amphfication factor o as a function of the wavenumber k of the density fluctuations from the hnearrzed version of the generalized diffusion theory and comparing It to the results of the computer study, Abraham [I] found serious quantnatrve discrepancies for the full range of wavenumbers. Stimulated by the results of a kmetrc theory of spmodal decomposition developed very recently [S], we would expect that the disagreement between molecular dynamics computer simulations and the predlctions of pure dlffuslon theories m the hnear range of decomposinon may be based on the fact that delusion approximation ignores kmetlc effects. In order to support this suggestion we shall discuss the influence of kinetx effects on the problem of spmoda1 decomposmon.
2. A dissipativewaveequation It is well known that dlffuslon theory is based on the assumption that or < 1 (and, in addlhon to
kl -=s1, where I IS a mean free path related to the colhslon time 7 by I= Gr where 6 is the mean velocIty, and k = ~T/A IS the wave number). But at frequencres comparable to the collision frequency 7-l of the atoms or molecules the restrictron wr 4 1 is no longer
vahd and the study of spinodal
decomposition
CHEhllCAL
Volume 76. number 3
has to take mto account which
can be dealt
PHYSICS
- in this case - kmetlc effects
wxth by solmg
the
Boltzmann
equation
aflat + u - vf= (af/ai),
(1)
for the non-equilibrium dlstrlbution function f(r, u, t). The first serious attempt to discuss the kinetics of spmodal decomposition in thrs way was due to one of the authors very recently [5] The Idea is to develop on the basis of the relavatlon time approximation (RTA) - a perturbation theory leadmg to a hierarchy of dlfferentlal equations for the local density p(r, t) Usmg RTA, 1 e. the ansatz f(r. u, f) = F(E(u),
T, cl(r, tN+dr.
u, t)
(1) takes the simple
form
(O,+O,)fJ=rZ.
(3)
Here, we have introduced o,=
(If
7a/at)ap/ar = M
VP
,
where III is the mobMy,
6)
defined
Ca Fick‘s law
j=-hIFLL.
(7)
Here, p is a function of T and p(r, I). Eq. (6j is our non-linear dlsslpative wave equation, which, in the hmit 7 + 0, reduces to the generalized diffusion equatlon discussed by Abraham [3] very recently.
3. The amplificationfactor
(dfla[L = -d/i. equation
Eqs (4) and (5) define our perturbation theory [S]. The first-order result [take into account the first two terms m parentheses of the nght-hand side of eq. (5) only] leads to [S]
(2)
and replacing the scattermg term by the Boltzmann
i 5 December 1980
LETTERS
the operator
To evaluate eq. (6), we follow Abraham’s approxlmatlon [l] for the generalized chemical potential g of a non-uniform fluId with density vanations in one dlmenslon
notation
l+;_iJ/C%, 0,=TU lF,
Z ISgiven by --ip
z = -(lS=/l3/L)(a/f/ar + u - Vp) with
and F IS a local equilibrium dlstnbutlon depending the local chemical potential p (r, t) Expandmg F
iij
on
into a Taylor
series up to first-order
F = Fo-I-L - d~aF~a,L.
terms
*
where
the Index 0 refers to the equlhbrlum po, we can replace - if necessary dF/+
=%F/ap
I-1
value p =
lo .
Tha approxunation
makes dFli+~ independent of r and r. The formal operator solution of the Boltzmann
equation m RTA ISgiven by 4 = (1+
0;‘0,)-‘0~’
-TO;‘(l-O~lOx+
side
(rZ)
_. .)Z
where f’ IS the analytIca continuation of the free energy density of the homogeneous fluid into the unstable region, I((&) IS the interactlon potentiai of the pauwlse interactmg particles, gS IS their radial dlstributlon function, and p. is the density of the umform fluId The second term on the nght-hand of eq. (8) describes
fluId m a microscopic (4)
Here we have assumed that the colhslon time 7 IS independent of ,u From the particle conservation law
the inhomogeneity
way. It contains
of the
no
phenomenologlcal parameters Next, eq (8) is lmearized by expanding r@(z. t)) up to second order into a Taylor series about po. According
to the observed
exponential
growth
[a] of
we conclude that
the dens@ fluctuations during the early stage of
J d”uCf-F)=Jd”~~(r,u,t)-(l,~)=o.(3
decomposition
we use the familiar
ansatz [l]
563
Volume
76. number
3
CHEMICAL
Insertion into the hnearized version to the characteristic equation (1 +or)o
PHYSICS
of eq. (8) leads
= -~~‘[(a2~/ap2),+A(k)],
(9)
aJ I
cos (kw - 1) fI( II*,po) dw ,
0 which factor of the ing to
we solved numencally for the amphfication w(k). The values of K PO, and the parameters Lennard-Jones potential were chosen accordthe computer study [4].
4. Results
and discussion
In fig. 1 we have plottEd the quantrties w*/q2 versus q2. where W* = w,, and q = k/k:. Here,
MOLECULAR
d
1980
WC,, IS the maximum of the amphfication factor is the critical wavelength, both refer to Cahn’s theory [6]. Also, for comparison, the amplification factors of Cahn and Abraham, respectively, are shown. To compare our theory with the molecular dynamics data (shaded region) we had to fur the parameter r* = 7w(icmax)_ Takrng l/W(k,,) = 6.02 ps from the computer study [4] and estimatmg the microscopic collision tune T as given in ref. [5] we obtain r* = 0 56. This value of or, which is a measure for the influence of kinetic effects, is obviously not “very small”. As a consequence of ths, the kinetic theory reproduces at least at large wavelengths (small k values) the molecular dynamics results more closely than the diffusion type theories of Cahn and Abraham supporting the suggestion that kinetic effects should not be neglected in the descnption of the early stages of spmodat decompositron. WC(k) and k:
with A(k)=4n
15 December
LETTERS
DYNAMCS
References
CAHN ABRAHAM
[I]
Fig 1 Companson of w’(q)/q’ between theory and expenment The temperature T after the quench IS 0 57 T,
564
F F Abraham.
Phys
therem [2] F.F Abraham,
Chem
Rept
[3]
W E Langlcxs and F F 52 (1977) 129.
[a]
M R. hfruzzk, FF.
Phys
53 (1979)
Letters
Abraham.
Abraham
93. and references
57 (1977) Chem.
179
Phys. Letters
and G hi. Pound,
Phys 69 (1978) 3462 [S] Th F. Nonnenmacher. Z Physlk B38 (1980) [6] J W. Cahn and J E Hdhard. J Chem. Phys 258
J Chem.
327 28 (1958)