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6Ученые записки Таврического национального университета имени В.И. Вернадского Серия «Физико-математические науки». Том 25 (64). 2012 г. № 1. С. 193-205
УДК 539.213
EVOLUTION OF ANTIPHASE ORDERED DOMAIN STRUCTURES IN CONFINED BINARY ALLOYS Krasnyuk I.B, Melnik T.N, Yurchenko V.M.
Donetsk Institute for Physics and Engineering named after A.A.Galkin of NASU, Donetsk, Ukraine
E-mail: krasnyukigr@rambler.ru
An initial boundary value problem with Dirichlet or Neumann boundary conditions is considered. This problem describes evolution of a nonconserved order parameter in binary alloys near the disordered phase. It is shown that the problem has asymptotically oscillating stationary distributions of the order parameter. This oscillation describes so-called antiphase boundaries (APB) in binary alloys. The corresponding phase diagram is constructed. Particularly, it is shown that the temperature is a bifurcation parameter. The stationary antiphase boundaries with a different number of oscillations between the walls confining the sample arise at cooling in binary alloys on long time scales. Further, it is shown that if there is a small parameter of the system, contrast structures are formed, i.e. step-constant asymptotically stationary distributions of APB. Keywords: binary alloys, antiphase boundaries.
INTRODUCTION
The paper contains analysis of three main aspects of the problem. The first one is the formation of antiphase domain structures from the disordered state (see [1], Photo. 1, described anneal-ageing of an alloy in disordered state). This process was studied by Matsubara (1990) in Fe3Al, by Allen and Krzanowsky [1985], Park and Allen [1986] with microscopic observation (see references in [1]). For example, Matsubara [1989] observed the process of DO3 ordering of Fe3Al, after quenching from B2 state. Here it is assumed that small domains are formed randomly within the disordered (B2) matrix. It was supposed in the experiments that the domains change their shapes and sizes at the early stage of isothermal ageing. In contrary, we consider the case of asymptotic behavior as time t , because the strict asymptotics has not been considered in the previous cited paper.
To analyze the process, the corresponding evolution equation has been constructed [1]. The Allen-Cahn partial differential equation of the second order was supplemented with periodic boundary conditions and random initial conditions. At every moment t>0, there are oscillating solutions of sinusoidal type (see [1], Fig. 1). This figure describes the evolution of nonconserved order parameter, which models so-called antiphase boundaries (APBs). Thus, we obtain oscillating order parameter distributions of sinusoidal type in the disordered phase at early stage of evolution. But as t>0 is large, we obtain (by computer simulation) the step-constant oscillating structures of relaxation type ([1], Fig.2). In mathematics, such structures are called contrast ones (see, for example, [2]).
In the previous papers, the Allen-Cahn equation with periodic boundary conditions has been studied [1]. Here we consider the Dirichlet and Neumann boundary conditions. In the previous papers, random initial conditions were studied with the help of computer simulation. We consider deterministic initial conditions and use the well-known
mathematical result to study of evolution of non-conserved order parameter in confined binary alloys. We show that the typical asymptotic is sinusoidal near the disordered phase. There are also limit distributions of spike-type, if dimensionless diffusion coefficient is small; and limit distributions exist in the form of stationary contrast structures, if there is the transform t ^ t/s, where e>0 is the small parameter that characterizes spatial-temporal "inhomogeneities" in the binary alloy and describes the thickness of antiphase boundary layers (see, for example, [3, 4]). Further, we introduce the function of the inhomogeneities at the disordered phase a(x,s), which changes the amplitude u(x,t)=0 of the disordered phase by the transform u(x, t) ^ u(x, t)- a(x,s) . Coordinates of the
front of phase transition can be established by the coordinates of the intersection of u(x,0,s) and a(x,s) curves.
The bifurcation diagram for initial boundary value problem (IBVP), which determines the properties of antiphase boundaries (APB) in binary alloys, is considered. It is proven that APBs are asymptotically unstable at all temperatures, but "life time" of such metastable oscillating states may be long. It is well-known that in the case of ordered alloys, typical defects are antiphase domains, which are the results of interaction between atoms of sort A and B in the binary lattice. In this paper, we study IBVP for a non-conserved order parameter with the classical Neumann (or Dirichlet) boundary conditions.
We define the order parameter u := pA - pB (pA + pB = 1) as the difference between the densities of atoms of type A and B, correspondingly. We show that as rint ^ +oo, where rint is an exchange time between the jumps of atoms A and B in the lattice, there
exist asymptotically stationary oscillating distributions of the order parameter.
There are many problems, where formation of APB in binary alloys is included into consideration, but in previous papers, in typical cases, periodic boundary conditions are considered, and, usually, computer simulations are provided. The results are phase diagrams for special values of parameters of the problem (see, for example, [1]). In this paper, the strong phase (or bifurcation) diagram is constructed, which describes all admissible parameters of the boundary problem. Two situations may arise: the oscillating asymptotic stationary distributions of "spinodal type" may appear, and, for a special type of parameters, when a small parameter s of the problem exists, there are asymptotically oscillating step-constant distributions.
1. FORMULATION OF THE PROBLEM
Let us consider the thermodynamical potential
(1)
where f(u) is the free energy density per unit volume, as given by
f (u ):=-| A (T )u2 B (T )u4,
1
(2)
where A(T) changes from negative to positive values at the critical temperature T=T for
g
the order-disorder transition; B(T) is positive; K(T) is the positive interfacial energy per unit length; k.B is the Boltzman constant. The temperature Tg is given by the root of
equation A(T) = 0. Evolution of the order parameter may be described by the Landau-Chalatnikov equation [5]:
8uu = -m(tS, (3)
8t V ' Su
where M(T) is the positive reaction rate, SF/Su is the variational derivative. The equilibrium distributions should be given by
SF d ( dF8F
V dux J
"TT = 0- (4)
du
8u dx From (4), we obtain
G[u]:= B(T)((ue(T))2 -u2)u + M(T)K(T)V2u = 0, (5)
where ue =
r ^>2
W)
. Then equation (3) can be rewritten as
du
dt = A[ ], (6)
where G[u] is described by (5). Here, we use the transforms t^ M(T) tand x^ x/l.
Further, we consider the Neumann boundary conditions for solutions of (5)
ux(0,t)=0, u.x(l,t)=0, (7)
or the Dirichlet boundary conditions
u(0,t)=0, u(l,t)=0. (8)
From (7), it follows that thermodynamic flux is zero on the walls, which confine binary alloys. According to (8), the binary alloy is in the disordered phase at the walls. Additionally, we consider an initial condition
u (x,0) = u0 ( x), (0 < x <x). (9)
If f^œ, in Sobolev space H0 (0,^) of distribution [6], the solutions u(-,t) tend to stationary states 9(x), 0 < x< n, which are solutions of equation:
<p"(x) + aç(x)-bp(x) = 0, 0 < x<n, (10)
2
where a=B(T)ue(T) and b = B(T). From [7], it follows that there is a finite number of
22
stationary states in the interval 0 <x < n, i.e. (2^+1), n=0,1,..., if n <a<(n+1) (correspondingly, the transform n ^ nx/1takes place in the interval 0<x<l. If 0<a<1, zero or trivial solution is globally asymptotically stable. If a > 1, this solution is unstable as well as all other solutions, with the exception of two solutions, which we define as
(, ( . These asymptotically stable limit distributions have the property ( < 0 < ( for all 0< x< n. A region of attraction of , ( j is open dense set in H0 (0, tt) .
It is well-known that there is a neighborhood of zero, in H (0,t) , that is positiveinvariant for all small enough |a - 1|. This region is separated in two open sets by stable manifold of the zero solution for small a - 1 > 0. We call this two sets by the attractive
regions of ,(1+j (see Fig.1, [6]). If B is a sufficiently large ball with the center in the
zero in H (0, t) , and u(t, B is the set of all points that are reached by solutions u at the moment of time t, then
K = n t,0 u (UB) (11)
is the maximal invariant set. This set is of finite dimension, compact and connected. The set is a union of unstable manifolds of equilibrium states. We have phase portraits that may be found in [6], Fig. 2. 22
If n <a<(n+1) , the set K has the number of dimensions n, that is a closer of an unstable manifold of zero.
2. THE CHAFEE-INFANTE INITIAL BOUNDARY VALUE PROBLEM
In [7], the following boundary problem is considered:
ut(x,t) = ux(x,t) + Af(u(x,t)), (0<x<t, 0<t, (12)
u(0,t)=u(n,t)=0, (0<f<+»), (13)
u(x,0)=9(x), (0<x<n). (14)
What is the asymptotic solutions of this problem as t^+w? The basic tool to solve this problem is the maximum principle for parabolic partial differential equations. Here, f: R ^ R is a given continuous function; (p: [0,t] ^ R and ^e C1 such that
<P(0 ) = ((T).
2.1 The Chafee-Infante boundary problem for binary solid solutions
In (12), we may define a function f(u) to be proportional to the function £,[u]/^bT.
The parameter X may be defined as E^JkfiT, where Eis so-called mixing energy, and
X describes potential energy (normalized on heat energy) of interaction of atom in the central lattice point with all atoms of the first coordinating sphere, which is equal to Z(Paa®aA+PbB^BB+2PaB^Ab). The potential energy of the lattice is
N7
E :=—(( AA + PBB $ BB + 2 PAB $ AB ), (15)
where Z is the coordination number; N=Nj+Nj, N^ N^, are numbers of atoms of sort A,
B, correspondingly; PAAu PBB, Pab describe ordering in equilibrium systems. Further, O AA, O BB, and O AB are potential energies of pair interactions of two atoms, which are placed on the distance of radius of the first coordination sphere. Here, the first index describes the type of atom placed in the considered lattice point, and the second index denotes the type of atom on the first coordination sphere. If the lattice is stable, we obtain OAA < 0, OBB < 0, OAB < 0, and ®AB=®BA, so that in disordered solid solution, we
have Pj\=ca> PbTcB where c^, cb are the corresponding concentrations of atoms. Then
we can obtain that
PAB = cacb (1 - 2cacb (exp (- EmJkBT)~ l)), (16)
paa = ca (1 + 2c2 (exp ( - EmJkBT)~ l)). (17)
Further, there is a formula [8]: ^&A.B=-ZPAEEmX where Asab is the internal
energy released when a solid solution is formed. It is well-known that [8]
NZ
E :=— (CaOAA + 4oBB + 2PABOAB). (18)
From (16), it follows that
Z
SAB = J (Ca O AA + C2BO BB - 2 PAB (ca , cb , Emx )) , (19)
where PAB is determined by (16).
Without loss of generality, it can be assumed that Emjx/kBT «1, where k£,T is of
order of heat energy. Then from (16),(17), we obtain that
PAB ~ CACB
f ( E w
1 + 2cAcB '
"mix
-1
V kBT yy
P - 2 PAA - CA
( ( E ^^ mix — 1
1 — 2cb 2
(20)
(21)
yy
JbT
Thus E . ¡kjTT describes the correlation between neighbors atoms, i.e., the short mix B °
order. For given Emix, with increasing of the temperature, the corresponding correlations
are reduced. It will be proved that the value x=E . !knT, which characterizes intercalation r K mix B
energy, is the bifurcation parameter. This means that solution of the boundary problem,
which models an evolution of the order parameter in confined binary alloys, tends to
stationary oscillating limit solutions, as time A number of oscillation per interval
0<X<l depends on the parameter %.
If E . ^0, but x<<1, we may use the approximation of ideal solid solution, supposing that As/k\T is sufficiently small. In this case, A//kB T <x ±cA (1 - cA), and we must use
!(-), i
NZ,
the sign (+), if E . >0, and sign(-), if E . <0. Further, we obtain that
° w mix ' ° v " mix
E _■
where
(cAOAA + 4OBB - 2PABEmix),
2EmX
PAB _ CACB +
kBT
Let us suppose u=Cj—Cb where c^+c^fI. Then from (22), we have
E _
NZ
\u + Cb|2 OAA + Iu + Ca|2 OBB - 4Emix
|2 Z7 j l2 E
U + cB\ B U + CA -mix-
kBT
B J
(22)
(23)
(24)
Below we consider perturbations of a nonconserved order parameter at a neighborhood of the disordered phase u=0, where C\fCb It follows from the last equality
that we can use cA — cB —1/2 , where 1/2 plays the role of probability. Now we define U ^ U +1/2 , and rewrite (24) in the form
E [u ] _ NZ
kTT
A\uf + B\uf - 4|U
f E j2 j
mix
kBT
where A _ OAA/kB T, and B _ OBB/kB T. Then
e[u ]
kBT
_ NZ (A + B) u - 6 nz%2u 3.
(25)
(26)
The linear approximation at the point u = 1/2 results in the determination of energy: ET _ 1 NZ (A + B )-f 3 NZZ2 + NZ (A + B)-4 NZ ( u -1 j + O ( u -1 ] . (27)
Transforming U ^ U -1/2, from (27), we obtain for the derivative
_ 2 NZ (A + B )-Au + O (u ) kBT 2
(28)
at the point u = 0, where | NZ%2 + NZ (A + B)- 4 NZ^x2 and
1 K k:_ — NZ ( A + B) .Let us establish U ^ U +—. Then from (28), we obtain that 2 A
m
k\T
_-Au +o (u )
(29)
at disordered phase.
Further, for a disordered solid solution, entropy S^g can be represented in the form [8]:
SAB = Sid + Smix, (30)
where S. is the entropy of an ideal solid solution, and Sis the entropy of mixing. It is well known that for ideal solid solutions, the second term in (30) can be neglected. The
first term is
5(cA, Cb) - c
--^f In cA ln cB, (31)
kBT Na Nb
where N^+Nb^N. Assuming that Ca=Cb=c, we obtain the curve of phase coexistence:
lln (¿j^1 - jC) = (32)
3. THE ALLEN-CAHN EQUATION
Further, we consider the Allen-Cahn equation, which represents a gradient flow for the free energy functional
£[u] = j(Jux\2 -F'(u)jdx, (33)
where F(u) = f(u) and 0 < x< l is the spatial domain, which is occupied by the binary alloys under consideration. Here,
f (u) = Au + ln (jUU-, (34)
where the first term follows from the linear approximation of ordering energy for an ideal solid solution. The second term, at u = 1/2, is
ln f—j = -— u +O ( u. (35)
11 - uj N I 2 J V '
Hence, -F(u) = A--1. We consider the case X> 1/N when there is phase
decomposition of disordered phase on two ordered phases. Further, k ~ Ur0, where U is the interface energy between ordered and disordered phases; r0 is the correlation radius
between atoms of solid solution.
A corresponding gradient flow is [3-5]:
ut = -r(u)f-, (36)
5u
2
where SG/Su is the L functional derivative of £ . The result is
Ut _r(U)
K(U )Uxx - 1 K(U )Ux|2 - f (U)
(37)
2
The linear version (at u = 0, that is, at disordered phase) is
ut _ Du. + au - bu3, (38)
where we use the representation (26) for f(u), so that a = NZ(A+B)u and b=6NZx2u3(xF^-1/N). In the general case, A + B<x (1 - T/Tg), where T is the critical temperature of decomposition of a homogeneous disordered binary solid solution (see
[5]). Here, t ^ r, D _k(0) where k(0)_ (^J .
Then, applying formal results of [7], we obtain following statements: If t^+w, solutions u(-,t) tends to a stationary state 9 in the Sobolev space H0 (0,^) that is a solution of the problem [6]:
( + (x)- b( (x)_ 0, (39)
23
where a=NZ(A+B)/D and b =6NZx2u3/D. It follows from (39) that
Ut _(xx + A((e-()(_ 0, (40)
where !=b and (pe _ a/ b.
The Dirichlet boundary conditions are
((0)_ 0, ((l)_ 0. (41)
Chafee and Infante [7] proved that, if n2 < a <<(n +1) , n_ 0,1,2,..., there are (2n+1) stationary points (0 _ 0, and (± (k _ 1,...,n), where (d/dx)(k > 0 at x = 0, as
2 2 ± n <a<<(n+1) , (djdx)q)- < 0 at x = 0, and the function (± has (k-1) zeros on interval
0 < x < n . Further, if X > 1, (± are asymptotically stable in linear approximation, but (
and (± (2 < k < n) are together asymptotically unstable.
4. SPIKE-TYPE ASYMPTOTICALLY STATIONARY STABLE STRUCTURES IN CONFINED BINARY MIXTURES
The results of [9] show that periodic APBs have two competing annealing processes: the annihilation of APBs trough formation of peak stationary distributions (Fig.1). This
distribution appears as parameter s= K(T)/]2 becomes small, whereas K(T) plays the role of "mobility". Indeed, let us study the boundary-value problem
22
u=M(T)K(T) uxa+M(T)B(T)(ue(T)2-u2) u. (42)
< y/2ue(T)
B
o * i
Fig. 1. Peak stationary distributions of the nonconsereved order parameter in a binary alloy.
Let us assume t ^ M (T)/12 , and x ^ x/1. Then, from (42), we obtain that
ut = eUx + B(T) (ue (T)2 - u2) u - h(u), (43)
where s= M(T)/We suppose s111, and consider the Dirichlet boundary condition
u(0,s)=A, u(1,s)=B, (44)
which means that surface atoms are in "partially ordered" phase. This fact may be interpreted as surface segregation of atoms with surface densities pA (0, t)-pA (B, t) = 0
and pA (1s, t)-pA (B,1) = B, where A,B are constants. In literature, only the case of A=B=0
is considered, where the surface atoms are in the disordered state.
4.1. The degenerated case
At s = 0, the degenerated equation (42), or h(y)=0, has three stationary solutions u13(x) = +ue(T) and ^(x)=0. Since h'(u2)>0, h'(u1)=h'(u3)<0, only the solution u=u2(x) is stable.
Using the integral condition of stability [10] for solutions of boundary problem (42),(44), which has the form
J B (T ){ue (T )2 - 52 )sds > 0
(45)
(see also, [5] and [10], p.53). Simple calculations of integral result in the inequality ue (T^. Hence, from the result of O'Mally [10], the statement follows: if
AI, B <
ue (T^ ; then there are solutions where the limit relation
lim u(x,s) = 0, xei£,1 (46)
is valid.
In addition to solutions (46) of boundary-layer type, as the considered
boundary problem has oscillating solutions of the peak-type layer when a solution tends,
to zero for all points on interval [0,1] as with exception of some number of points
of the interval, which are placed at equal distance from one to another (see Fig. 1). In some neighborhood of every point, for sufficiently small s>0, this solution has a narrow peak. This peak does not tend to zero as s^0.
This result follows from [10], at the point ^(x)^ the integral
U
Y[u]_{B(T)(ue(T)2 -s2)sds (47)
A
has the maximal value, i.e. the potential energy of the system is maximal. We have also ^[u _V2 |ue (T)|]_Y[0]_ 0, if IA <V2 |ue (T)|, and the value u _V2| ue (T)| is not the point of maximum of function Y[u]. This result means that, for any integer n>0, the boundary problem has four solutions u=u(x,s), if |A| < Ue (T) and
B < Ue (T)|, satisfying the limiting relation
lim u (x,e)_ 0, x e[S,1 -S] (48)
with S e (0,1/2) , except the points xi _ i/n, i _ 1, n-1, where the limit relation is satisfied:
lim U(x,e)_±42\ue(T)|. (49)
For n = 2, we have one peak, and for n = 3 we have two peaks. In the general case, amplitudes may not be equal (see, for example, Fig. 1).
4. THE CONTRAST STRUCTURES
Now we consider the bistable Cahn-Hilliard equation [3, 11]
Ut _ M~(T)(K (T)Uxx- f (u)), 0 < x< 1, (50)
where the diffusion coefficient K- ( T)/12 is small. The parameter M- ( T)rdiff _ O is large, so that K- ( T) M- ( T) _ O(1) , where O(1) is a bounded value; zdiff is the typical
time of diffusion of atoms. As shown in [3], we obtain from this condition that there are
-1/2
solutions in the form of internal layers of thickness O(k ). The layers move with the phase velocity
V _-M~(T) K (T) R, (51)
where R is the mean curvature of a layer.
We rescale the time variable t, so that t^ tje2 , where t^ M (T); and we
transform the spatial variable x, so that x^ x/e2 , where x ^ x/1. Then equation (50) can be rewritten as
e2ut = e2K- (T) ux + B(T)(ue(T)2 - u2)(u- a(x,e)), 0 < x< 1. (52)
Further, we introduce new time variable t^ K (T) t and rewrite (52) as
e2ut = e2u„ + (B(T)/K(T))( (T)2 - u2 )(u- a(x, e)), 0 < x< 1. (53)
Here, we introduce the function a ( x, e), which describes the spatial-temporal inhomogeneities in the disordered phase. As e = 0, there are two stationary solutions u (x, t) = ±ue (T ). The points ±ue (T ) correspond to two minimums of the free energy.
Further, let us introduce the point a (x, e ), which corresponds to the maximum of free
energy. This point determines a boundary between the areas of influence of the left stable zone u=-ue(T) and the right one u=ue(T).
It is well-known (see, for example, [2,12]) that equation (52) with the homogeneous
Neumann boundary condition and with the initial state u(x,0) = u0 (x), has
asymptotically stable solutions presented in Fig.2. The phase boundary is established as
the intersection point between the graphics of functions a ( x, e) and ^(x). It is evidently,
how this boundary changes, if a (x, e )=0. Thus we obtained the distributions of APB that may be called contrast structures [2, 12].
Fig. 2. Contrast asymptotically stationary structures in binary alloys.
CONCLUSIONS
We investigated the dynamics of formation of asymptotic stationary domains at isothermal ageing in binary alloys. This dynamics may be observed by a microscop as a dark field image (see [1]). The first considered case is the evolution of APBs, or antiphase ordered domains, in an alloy with so-called contrast structures, which are step-constant oscillating stationary structures described by the nonconserved order parameter with
Neumann's boundary conditions. These structures, as shown in [1], are observed in substitution binary alloys. The obtained theoretical results are compared with the scenario of formation of APBs in Fe^Al at ordering B2 to DO3. The second case is the classical
APD of sinusoidal type. The third case is the phase separation, which results in formation of so-called peak-type structures that can be reduced, if the dimensionless diffusion of atoms is small. The APBs of sinusoidal type have been observed for small oscillations of the order parameter at the disordered phase. But the contrast structures and peak-type structures are obtained for finite amplitudes of order parameters.
References
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Краснюк И. Б. Эволюция антифазных упорядоченных доменных структур в ограниченных бинарных сплавах / Краснюк И. Б., Мельник Т. Н., Юрченко В. М. //Ученые записки Таврического национального университета имени В.И. Вернадского. Серия: Физико-математические науки. - 2012. - Т. 25(64), № 1. - С. 193-205.
Рассмотрена начально-краевая задача с граничными условиями Дирихле и Неймана. Эта задача описывает эволюцию несохраняющегося параметра порядка в бинарных сплавах вблизи неупорядоченной фазы. Показано, что задача имеет асимптотически осциллирующие стационарные распределения параметра порядка. Эта осцилляция описывает так называемые антифазные границы (АФГ) в бинарных сплавах. Построена соответствующая фазовая диаграмма. В частности, показано, что температура является бифуркационным параметром, поэтому при охлаждении в ограниченных бинарных сплавах на больших временах появляются стационарные антифазные границы с различным числом осцилляций между стенками, ограничивающими образец. Далее показано, что если в системе
имеется малый параметр, то наличествуют и контрастные структуры - ступенчатые асимптотически
стационарные распределения АФГ.
Ключевые слова: бинарныесплавы, антифазные границы.
Краснюк I. Б. Еволющя антифазних впорядкованих доменних структур в обмежених бшарних сплавах / Краснюк I. Б., Мельник Т. М., Юрченко В. М. // Вчеш записки Тавршського национального ушверситету iменi В.1. Вернадського. Серш: Фiзико-математичш науки. - 2012. -Т. 25(64), № 1. - С. 193-205.
Розглянуто початково-крайову задачу з граничними умовами Дiрiхле та Неймана, яка описуе еволюцта параметру порядку, що не збертаеться, в бiнарних сплавах поблизу невпорядковано! фази. Показано, що задача мае асимптотично осцилюючi стацiонарнi розподшення параметру порядку. Ця осциляцш описуе так зваш антифазнi межi (АФМ) в бшарних сплавах. Побудовано вдаовщну бифуркацшну д1аграму. Зокрема, показано, що температура е бифуркацшним параметром, тому тд час охолодження в обмежених бшарних сплавах за великого часу з'являються стацюнарш антифазнi межi з рiзним числом осциляцiй мiж стшками.. Далi показано, що якщо в система е малий параметр, то е присутшми i контрастнi структури - асимптотично кусочно-постiйнi стацiонарнi розподшення АФМ. Ключовi слова: бшарш сплави, антифазнi межi.
Поступила в редакцию 14.05.2012 г.
