Exact solutions for a nonstandard viscous Cahn—Hilliard system
The one-dimensional version of a nonstandard viscous Cahn–Hilliard system (proposed by Colli et al.) for the order parameter and chemical potential with a generally asymmetric polynomial double-well potential is considered. For this system, an exact travelling wave solution, which describes the ad...
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nasplib_isofts_kiev_ua-123456789-1267962025-02-10T00:50:16Z Exact solutions for a nonstandard viscous Cahn—Hilliard system Точне розв'язання системи нестандартних в'язких рівнянь Кана—Хілліарда Точное решение системы нестандартных вязких уравнений Кана—Хиллиарда Mchedlov-Petrosyan, P.O. Davydov, L.N. Фізика The one-dimensional version of a nonstandard viscous Cahn–Hilliard system (proposed by Colli et al.) for the order parameter and chemical potential with a generally asymmetric polynomial double-well potential is considered. For this system, an exact travelling wave solution, which describes the advancing front of a phase transformation in an infinite domain, is found. Розглянуто одновимірний варіант нестандартної системи рівнянь Кана—Хілліарда (запропонованої Colli et al.) для параметра порядку і хімічного потенціалу з асиметричним поліномним двоямним потенціалом. Для цієї системи знайдено точне розв’язання вигляду рухомої хвилі, що описує рух фронту фазового перетворення в нескінченній області. Рассмотрен одномерный вариант нестандартной системы уравнений Кана—Хиллиарда (предложенной Colli et al.) для параметра порядка и химического потенциала с асимметричным полиномиальным двуямным потенциалом. Для этой системы найдено точное решение вида бегущей волны, которое описывает движение фронта фазового превращения в бесконечной области. 2017 Article Exact solutions for a nonstandard viscous Cahn—Hilliard system / P.O. Mchedlov-Petrosyan, L.N. Davydov // Доповіді Національної академії наук України. — 2017. — № 7. — С. 37-42. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2017.07.037 https://nasplib.isofts.kiev.ua/handle/123456789/126796 536.42:517.9 en Доповіді НАН України application/pdf Видавничий дім "Академперіодика" НАН України |
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Фізика Фізика Mchedlov-Petrosyan, P.O. Davydov, L.N. Exact solutions for a nonstandard viscous Cahn—Hilliard system Доповіді НАН України |
| description |
The one-dimensional version of a nonstandard viscous Cahn–Hilliard system (proposed by Colli et al.) for the order
parameter and chemical potential with a generally asymmetric polynomial double-well potential is considered. For
this system, an exact travelling wave solution, which describes the advancing front of a phase transformation in an
infinite domain, is found. |
| format |
Article |
| author |
Mchedlov-Petrosyan, P.O. Davydov, L.N. |
| author_facet |
Mchedlov-Petrosyan, P.O. Davydov, L.N. |
| author_sort |
Mchedlov-Petrosyan, P.O. |
| title |
Exact solutions for a nonstandard viscous Cahn—Hilliard system |
| title_short |
Exact solutions for a nonstandard viscous Cahn—Hilliard system |
| title_full |
Exact solutions for a nonstandard viscous Cahn—Hilliard system |
| title_fullStr |
Exact solutions for a nonstandard viscous Cahn—Hilliard system |
| title_full_unstemmed |
Exact solutions for a nonstandard viscous Cahn—Hilliard system |
| title_sort |
exact solutions for a nonstandard viscous cahn—hilliard system |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2017 |
| topic_facet |
Фізика |
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https://nasplib.isofts.kiev.ua/handle/123456789/126796 |
| citation_txt |
Exact solutions for a nonstandard viscous Cahn—Hilliard system / P.O. Mchedlov-Petrosyan, L.N. Davydov // Доповіді Національної академії наук України. — 2017. — № 7. — С. 37-42. — Бібліогр.: 15 назв. — англ. |
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Доповіді НАН України |
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37ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 7
ОПОВІДІ
НАЦІОНАЛЬНОЇ
АКАДЕМІЇ НАУК
УКРАЇНИ
ФІЗИКА
doi: https://doi.org/10.15407/dopovidi2017.07.037
UDC 536.42:517.9
P.O. Mchedlov-Petrosyan, L.N. Davydov
NSC “Kharkov Institute of Physics and Technology” of the NAS of Ukraine
E-mail: ldavydov@kipt.kharkov.ua
Exact solutions for a nonstandard viscous Cahn—Hilliard system
Presented by Academician of the NAS of Ukraine N.F. Shulga
The one-dimensional version of a nonstandard viscous Cahn–Hilliard system (proposed by Colli et al.) for the order
parameter and chemical potential with a generally asymmetric polynomial double-well potential is considered. For
this system, an exact travelling wave solution, which describes the advancing front of a phase transformation in an
infinite domain, is found.
Keywords: Cahn—Hilliard equation, phase transition, travelling wave solution.
Introduction. The Cahn—Hilliard equation [1, 2] is now the well-established phenomenological
model in the theory of phase transitions. The basic underlying idea of this model is that, for an in-
homogeneous system, e.g., a system undergoing a phase transition, the thermodynamic potential
(e.g., free energy) should depend not only on the order parameter (here, the term “order param-
eter” denotes a field, whose values characterize the phase), but also on its gradient as well. The
idea of such dependence was introduced by van der Waals [3] in his theory of capillarity. Due to
this dependence, instead of the usual second order, the resulting diffusion equation for the order
parameter becomes a fourth-order PDE.
The stationary form of this equation was introduced in [1]. In [2], a linearized version of the
time-dependent equation was treated, and the corresponding instability (“spinodal decomposition”)
of the homogeneous state was identified. However, it was only much later on that the intense study
of the fully nonlinear form of this equation was started [4]. The impressive amount of work has now
been done on the nonlinear Cahn–Hilliard equation (see [5]), as well as on its numerous modifica-
tions. One of the most interesting modifications was introduced by Gurtin [6]. Following the gen-
eral trend in the development of nonlinear continuous mechanics, he based his derivation “on the
separation of the basic balance laws (such as those for mass and force), which are general and hold
for large classes of materials, from the constitutive equations (such as those for elastic solids and
viscous fluids), which delineate specific classes of material behavior”, see [6, p. 179]. Additionally,
he introduced a new balance law for microforces and used the second law of thermodynamics in the
form of a dissipation inequality, see [6] for details. Then he obtained the standard Cahn—Hilliard
equation and the viscous Cahn—Hilliard equation [7] as special cases of his model.
© P.O. Mchedlov-Petrosyan, L.N. Davydov, 2017
38 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 7
P.O. Mchedlov-Petrosyan, L.N. Davydov
The next step was done by Podio-Guidugli [8]: keeping the balance of microforces as in [6], he
replaced the dissipation inequality by the imbalance of entropy and the mass balance by the ener-
gy balance. In his derivation, the chemical potential plays the same role as the “coldness” (inverse
absolute temperature) in the derivation of the heat equation. Based on the latter approach, Colli
et al. [9] introduced and studied the following system of governing equations for two unknowns,
the chemical potential μ and the order parameter ρ :
( 2 ) 0
t t
∂μ ∂ρε + ρ + μ − Δμ =
∂ ∂
, (1)
( )f
t
∂ρδ − Δρ+ ρ = μ′
∂
. (2)
Here, ε is a regularization constant introduced to preserve the parabolic structure of Eq. (1)
(see [9]), δ is the characteristic time for the evolution of the order parameter, and ( )f ρ is some
double-well potential. They proved the existence and uniqueness of a global-in-time smooth solu-
tion to the associated initial-boundary-value problem and studied the long-time behavior of the
solution.
In the present communication, we consider the one-dimensional version of this system with a
generally asymmetric polynomial double-well potential ( )f ρ , i.e.
3 2( )f r qρ = ρ − ρ − ρ+ η′ . (3)
For this system, we give the exact travelling wave solution, which describes the advancing
phase transformation in an infinite domain.
Travelling wave solution. Looking for the travelling-wave solution ( ), ( )z zμ ρ , where z =
x vt= − , Eqs. (1) and (2) yield
2
2
( 2 ) 0
d d d
v v
dz dz dz
μ ρ με + ρ + μ + = , (4)
2
3 2
2
d d
v r q
dz dz
ρ ρ− δ − +ρ − ρ − ρ+ η = μ . (5)
Now, we introduce our ansatz, by presuming that μ is the n -th power polynomial of ρ ,
( )nPμ = ρ . Comparing the powers of the nonlinearities, it is easily seen that it should be 3n� ;
postponing the discussion of other possibilities to the further communications, we consider the
simplest case 1n = :
μ = γρ+ω . (6)
Let us first consider Eq. (4). The substitution of (6) into (4) for μ yields
[ ]
2
2
3 ( )
d d
v
dz dz
ρ ργρ+ εγ +ω = −γ . (7)
Integrating Eq. (7) once, we get
2
1
3 2
( )
2 3
d
v C
dz
⎡ ⎤ ρ− ρ + εγ +ω ρ+ =⎢ ⎥γ⎣ ⎦
, (8)
39ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 7
Exact solutions for a nonstandard viscous Cahn—Hilliard system
where 1C is the integration constant. We are looking for the monotone “kink-” or “antikink-”like
solution approaching the limiting values
1 2,z z→−∞ →+∞ρ = ρ ρ = ρ . (9)
At this level of phenomenology, there is no reason to attach any particular value neither to
1ρ , nor to 2ρ . It is important only that the different values are assigned to different homogeneous
phases. For definiteness, we presume 1 2ρ < ρ , i.e. the kink solution. We will get such a solution for
ρ , if Eq. (8) takes the form
1 2 1 2( )( );
d
dz
ρ = −κ ρ−ρ ρ−ρ ρ ρ ρ� � . (10)
For Eq. (8) to take form of (10), the following constraints should be satisfied:
3
2
vκ = , (11)
2
( )
3
Xεγ +ω = −
γ
. (12)
Here, we have denoted 1 2( )X = ρ +ρ for brevity. Because 1C is an arbitrary constant, there
are only two constraints, (11) and (12). The integration of Eq. (10) yields
1 2 2 1
2 1tanh ( )( )
2 2 2
z c
ρ +ρ ρ −ρ κ⎡ ⎤ρ = + ρ −ρ +⎢ ⎥⎣ ⎦
, (13)
where c is the integration constant. It is natural to take the position of the maximal value of the
derivative
d
dz
ρ
(where
2
2
0
d
dz
ρ = ) as 0z = ; then 0c = . The constant κ has the same sign as the
velocity v [see (11)], so solution (13) is a kink for positive v , and an antikink for negative v . In
other words, the state with a lower value of the order parameter is always advancing. The evident
formal reason for this is that Eq. (4) is linear and homogeneous in μ .
Now, let us consider Eq. (5). Using (10), the second derivative
2
2
d
dz
ρ
is easily calculated:
2
2 3 2 2
2
2 3 (2 )
d
X Y X XY
dz
ρ ⎡ ⎤= κ ρ − ρ + + ρ−⎣ ⎦ , (14)
where 1 2Y = ρ ρ . Substituting expressions (6), (10), and (14) for μ ,
d
dz
ρ
, and
2
2
d
dz
ρ
, respectively,
into Eq. (5) and equating the coefficients at all powers of ρ to zero yield four constraints
21 2 0− κ = , (15)
23 0v X rδκ + κ − = , (16)
2 2[2 ]v X Y X q− δκ −κ + − = γ , (17)
2v Y XYδκ +κ + η = ω . (18)
For definiteness, we will presume 0κ > below, i.e., we consider the kink solution.
40 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 7
P.O. Mchedlov-Petrosyan, L.N. Davydov
The substitution of expressions (11) and (15) for v and κ into (16) yields
2
3 3
X r
δ⎛ ⎞= −⎜ ⎟⎝ ⎠
. (19)
Substituting expressions (17) and (18) for γ and ω into constraint (12), we get
3 21
9 6( ) 2(9 2 ) 12( )
4( 3 3 )
Y X X q X q
X
⎡ ⎤= + δ + ε + + δε + ε − η⎣ ⎦δ − − ε
. (20)
The detailed study of the parametric dependence of 1 2,ρ ρ , i.e. of X and Y , will be given
elsewhere. Here, we present the short version only. In system (1—2), the terms proportional
to ε and δ were introduced in [9] for the purpose of regularization of this system, with some
a posteriori justification. Setting 0ε = δ = simplifies expressions (19) and (20) for X and Y
substantially:
2
3
X r= , (21)
21 3 3
3 2 2
Y r q
r
η⎡ ⎤= − + −⎢ ⎥⎣ ⎦
. (22)
Correspondingly, the limiting values of the order parameter at ∞∓ are
2
1, 2
1 4 3 3
3 9 2 2
r r q
r
ηρ = + −∓ . (23)
Using Eqs. (21) and (22), the coefficients γ and ω in ansatz (6) are easily calculated as
21 1 3
;
9 2 2
r q r
r
ηγ = + − ω = −γ . (24)
So, finally, the solution for the chemical potential μ is
21 1 3
( )
9 2 2
r q r
r
η⎛ ⎞μ = + − ρ−⎜ ⎟⎝ ⎠
, (25)
where the order parameter ρ is given by Eq. (13).
Discussion. It is reasonable to presume that the phase with the lower value of the chemical
potential μ is advancing, so it should be 0γ > . This imposes the following condition on the coef-
ficients of the polynomial potential:
21 1 3
9 2 2
r q
r
η+ > . (26)
Here, q is positive. So, if the last inequality is fulfilled, (23) always yields real 1 2,ρ ρ . If 0η= ,
then 1 2,ρ ρ are real independently of the sign of r . So, formally, the solution exists. However, the
case where 0η= results in the negative values of the chemical potential μ . Indeed, the lowest
value of the chemical potential, which corresponds to 1ρ = ρ , is
2
1
2 4 3 3
3 9 2 2
r r q
r
⎛ ⎞ημ = γ − − + −⎜ ⎟⎝ ⎠
. (27)
41ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 7
Exact solutions for a nonstandard viscous Cahn—Hilliard system
For 1μ to be nonnegative, both r and η should
be negative, and it should be additionally
23 3 1 1
2 2 9 2
q r q
r
η< < + . (28)
Figure shows the surfaces A q rη = and B
32 1
27 3
r q rη = + ,
corresponding to the lower limit in the first inequal-
ity of (28), and to the upper limit in the second in-
equality, respectively. For values of the coefficients corresponding to the points ,q r , and η in
the space between these surfaces, the physically reasonable solutions exist.
So, the physically reasonable exact solution exists for an essentially asymmetric potential
( )f ρ only. This is somewhat reminiscent of the situation with the convective Cahn—Hilliard
equation, where the exact travelling-wave solutions exist only for the asymmetric potential [10].
In the sequence of works, Colli et al. [11—15] introduced and studied the following general-
ization of system (1—2):
(1 2 ( )) ( ) div( ( , ) ) 0g g k
t t
∂μ ∂ρ+ ρ + μ ρ − μ ρ ∇μ =′
∂ ∂
, (29)
( ) ( )f g
t
∂ρδ − Δρ+ ρ = μ ρ′ ′
∂
. (30)
If ( )g ρ and ( , )κ μ ρ are polynomials in ρ of the powers m and l , respectively, and if ( , )κ μ ρ
is a polynomial in μ of the power s , for several sets of ,m l , and s , the above method is applicable
finding the exact travelling wave solutions of system (29)—(30) as well. These solutions will be
systematically studied in the following publications.
REFERENCES
1. Cahn, J. W. & Hilliard, J. E. (1958). Free energy of nonuniform systems. I. Interfacial free energy. J. Chem.
Phys., 28, pp. 258-267.
2. Cahn, J.W. (1961). On spinodal decomposition. Acta Metallurgica, No. 9, pp. 795-801.
3. Van der Waals, J. D. (1979). The thermodynamic theory of capillarity under the hypothesis of a continuous
variation of density. (1893). Translated by J. S. Rowlinson. J. Stat. Phys., 20, pp. 197–244.
4. Novick-Cohen, A. & Segel, L. A. (1984). Nonlinear aspects of the Cahn—Hilliard equation. Physica D, 10,
pp. 277-298.
5. Novick-Cohen, A. (2008). The Cahn—Hilliard equation. In C. M. Dafermos and E. Feireisl (Eds.) Handbook
of Differential Equations. Evolutionary Equations, (Vol. 4, pp. 201-228), Amsterdam: Elsevier.
6. Gurtin, M. E. (1996). Generalized Ginzburg—Landau and Cahn—Hilliard equations based on a microforce
balance. Physica D, 92, pp.178-192.
Surfaces A and B correspond to the lower limit in the first
inequality of (28) and to the upper limit in the second
inequality, respectively
42 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 7
P.O. Mchedlov-Petrosyan, L.N. Davydov
7. Novick-Cohen, A. (1988) On the viscous Cahn—Hilliard equation. In J. M. Ball (Ed.) Material Instabilities
in Continuum Mechanics and Related Mathematical Problems (pp. 329-342), Oxford: Oxford Univ. Press.
8. Podio-Guidugli, P. (2006). Models of phase segregation and diffusion of atomic species on a lattice. Ricerche
di Matematica, No. 55, pp. 105-118. doi: https://doi.org/10.1007/s11587-006-0008-8
9. Colli, P., Gilardi, G., Podio-Guidugli, P. & Sprekels, J. (2011). Well-posedness and long-time behavior for a
nonstandard viscous Cahn–Hilliard system. SIAM J. Appl. Math., 71, No. 6, pp. 1849-1870.
10. Mchedlov-Petrosyan, P. O. (2016). The convective viscous Cahn—Hilliard equation: Exact solutions. Euro-
pean Journal of Applied Mathematics, 27, pp. 42-65. doi: https://doi.org/10.1017/S0956792515000285
11. Colli, P., Gilardi, G., Podio-Guidugli, P. & Sprekels, J. (2012). Global existence for a strongly coupled Cahn—
Hilliard system with viscosity. Boll. Unione Math. Ital., 9, No. 5, pp. 495-513.
12. Colli, P., Gilardi, G., Podio-Guidugli, P. & Sprekels, J. (2012). Continuous dependence for a non-standard
Cahn—Hilliard system with nonlinear atom mobility. Rend. Sem. Mat. Univ. Politec. Torino, 70, pp. 27-52.
13. Colli, P., Gilardi, G., Podio-Guidugli, P. & Sprekels, J. (2013). Global existence and uniqueness for a singular/
degenerate Cahn—Hilliard system with viscosity. J. Differential Equations, 254, pp. 4217-4244.
14. Colli, P., Gilardi, G., Krejčí P., & Sprekels, J. (2014). A continuous dependence result for a nonstandard system
of phase field equations. Math. Meth. Appl. Sci., 37, pp. 1318-1324.
15. Colli, P., Gilardi, G., Krejčí P., & Sprekels, J. (2014). A vanishing diffusion limit in a nonstandard system of
phase field equations. Evol. Eqn. Control Theory, No. 3, pp. 257-275.
Received 03.04.2014
П.О. Мчедлов-Петросян, Л.М. Давидов
ННЦ “Харківський фізико-технічний інститут” НАН України
E-mail: ldavydov@kipt.kharkov.ua
ТОЧНЕ РОЗВ’ЯЗАННЯ СИСТЕМИ
НЕСТАНДАРТНИХ В’ЯЗКИХ РІВНЯНЬ КАНА—ХІЛЛІАРДА
Розглянуто одновимірний варіант нестандартної системи рівнянь Кана—Хілліарда (запропонованої Colli
et al.) для параметра порядку і хімічного потенціалу з асиметричним поліномним двоямним потенціалом.
Для цієї системи знайдено точне розв’язання вигляду рухомої хвилі, що описує рух фронту фазового пере-
творення в нескінченній області.
Ключові слова: рівняння Кана—Хілліарда, фазові перетворення, розв’язання виду рухомої хвилі.
П.О. Мчедлов-Петросян, Л.Н. Давыдов
ННЦ “Харьковский физико-технический институт” НАН Украины
E-mail: ldavydov@kipt.kharkov.ua
ТОЧНОЕ РЕШЕНИЕ СИСТЕМЫ НЕСТАНДАРТНЫХ
ВЯЗКИХ УРАВНЕНИЙ КАНА—ХИЛЛИАРДА
Рассмотрен одномерный вариант нестандартной системы уравнений Кана—Хиллиарда (предложенной
Colli et al.) для параметра порядка и химического потенциала с асимметричным полиномиальным двуям-
ным потенциалом. Для этой системы найдено точное решение вида бегущей волны, которое описывает
движение фронта фазового превращения в бесконечной области.
Ключевые слова: уравнение Кана—Хиллиарда, фазовые превращения, решение вида бегущей волны.
|