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Algebraic approach to the solution of the relaxation problem with a non-conserved order parameter
Physics LettersA 176 (1993) 313—316 North-Holland
PHYSICS LETTERS A
Algebraic approach to the solution of the relaxation problem with a non-conserved order parameter A.F. Izmailov Department of Chemical Engineering, Polytechnic University, Six MetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA
K. Levon Department ofChemistry, Polytechnic University, Six MetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA
and A. Margolina Polymer Research Institute, Polytechnic University, SixMetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA Received 16 July 1991; revised manuscript received 23 March 1993; accepted for publication 26 March 1993 Communicated by A.P. Fordy
The relaxation ofa non-conserved order parameter after a single shot-like disturbance is studied in the framework of the timedependent Ginzburg—Landau model. It is demonstrated that the differential equation describing the growth kinetics can be transformed into a transcendental algebraic equation. A late stage asymptotic solution is obtained.
There exist three well-defined shapes in the relaxation of a metastable state of matter into a thermodynamically stable state: nucleation, growth and coalescence [1]. While in the first stage nucleus formation and decay are governed mainly by thermal fluctuations and result in the appearance of a stable (supercritical) nucleus of the new phase, the second stage describes the monotonic growth of such a stable nucleus with practically no decrease in size or decay. This means that the growth kinetics may be reasonably described in a deterministic fashion, neglecting thermal fluctuations. The first stage has been extensively treated in both the classical theory of nucleation and the linearized time-dependent Ginzburg—Landau (TDGL) approach [2]. However, not so much is known about the late stages of the growth process [2,31. Notably in ref. [3] the deterministic growth of a single nucleus was proposed in the form of a soliton-like solution. This lution was obtained under the assumption that the width of the nucleus boundary is small compared to its size. -
In this Letter we address the problem of deterministic growth for a single stable nucleus with the center at the coordinate origin. This nucleus is formed due to an initial shot-like disturbance. Such an unusual formulation of the problem allowed us to solve it exactly in the vicinity of the certain points, with no other approximations involved. Thus, we have found exact asymptotics of the solution for early and late stages of the growth process. We consider the usual relaxation equation for the non-conserved scatar order parameter field q’ (x, 1) [21, without the “noise” term, but including instead the “source” term, describing the initial shot-like disturbance: ô
49(X,t)
=—M 6F(ço) +aô(x)ö(t),
i?~O. (1)
ö~(x,I) In this equation M is the mobility and F(ç~)is the “coarse-grained” Ginzburg—Landau (GL) functional [2],
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
313
‘H S SItS Ll~ ft ERS 5
Volume I 7b, number S
-‘
0.t.fL.-oI 4(~. I
.-~
.fiqi’i.v. I) + ~uip
~
~~-‘
=
.t1[ (I~V~ + r Fp(x.f I
~.
~-i~ ing Out the action of he exponential operaioi cxp I ~.i’t .v. on the “source’ term t~(x)~(I cc cluires some care and makc’~use of the initial con
it matches tie
elI X. (I)
expt,±(x. / ( JO(xio( I
lion IS possible in our parti lilac ease. Thus. the suIt of I — is as follossv .
We choose the initial condition so thai “source terni q(x.
)Olxo)l:l
dition in form 1. \‘~c~sill lirst present the R’SLI!i H such action and the resulting transcendental alge braic equation and then ss c ~ ill explaii why such
i
(II
f~)(,r)~(
ci,.
~‘
where tile parameters r and u define the (iinzburg~ Landau ( (iL ) double—well potential.. ‘iI/~is the dii— tusion constant, and rt IS the amplitude of the initial disturbance. Thus, the equation we solve has the form
LI.
/
=
dt exp:/. 1/I
it
iiy~ ix.
=~(x)
Thus. tile order parameter itself recei~es an initial shot—like disturbance. To deal with such a singular smg initial condition it is necessars to supplement its dclinition in the inhnitesimal region around the initial lime instant 1=0. Such a supplemented dehnition gives us the opportunit\ to define anc function dc pending on q(x. I) and its derisatises in tile S icnlu\ of this point. We choose 10 do it as follows,
e\~) ~/)( x.
Jl(( .1
110(5 the l-ouriec expansion of ~P.r I and al/ uid e iii’, in~out the integi 01011 05 ri the p ii imHi ~.. we get tinahl\ thai
(
/)
.~
2it )
dqexp(iqx—.tIikq’)
/0-A
It
qexp(
y,(x. 11=
00 /
iq’x- i/iI~q
Lq uat ion I I) ) eonst it utes the algebraic form of eq. 3) with the initial condition I I se were striving
f/see
101-.
(‘leads, initial condition (4 1 is consislent with such a supplementary definition as
\t tills point we ~ ill explain ho~the action of the operator exp [).I’( .v. I ) ssas carried out in expression 7). First. we decompose tIns operator according to
ip(x. 0)
ihe following general scheme.
=
urn p~(x.I)
=C~d(X
.
expl/.I’(x. We now proceed to rewrite the partial differential equation (3) in the form of a transcendental algebraic equation. To this end we presenl (3) as follows. l’f x. I h’p(x.
I) =
x )d( II
.
P1 X. I) = ~I1ti~(i. I) r] ±1)~,~ i)(x. I) =iJ/~9~’— ;~1kV2. k)rmally introducing the inverse operator P (.v. one can write eq. (3 ) in the following farni. ~l4
I
)
/
J =expH1l[
~ ex~~ ~~DI.v. /11/’
OH LI. I
(.1/, / I
if I
Here the factor 1 ( x. g ,‘. is not iiecessaril~equal to unit~as the differential operator D( x. I I does not commute with the order parameter field cit x. i I. I )i Ilerentiating eq. (10) once with respect to the pai’amdc) (observing the designated order of operators I yields a simple operator differential equation iiivol\ ing the factor FIx. I: ) The solution of this equation is found in the farm
Volume 176, number 5
PHYSICS LETTERS A
F(x, t;A)=T[exP(Jd~G(x~ t;
Thus, the supplemented definition (5) of the sin-
~))]
0
G(x, t; t~)=A~(x, I; t~)D(x,t) A(x, t; ~) —D(x,t)
(11)
,
17 May 1993
where A(x, 1; ~) =exp[t~Muç92(x, I)] exp[~D(x, t)] In this expression T[exp( )] is the chronologically ordered exponent with respect to the parameter In our particular case this chronologically ordered exponent is equal to the conventional exponent exp( since the differential operator G(x, I; ~) obeys the following relation, ~.
gular initial condition (4) results in such an effective action of F(x, I; A) on the “source” term as if F(x, I; A) were, in fact, equal to unity. We turn now to solving eq. (9) exactly. It is possible to construct a solution of the transcendental algebraic equation (9) in the vicinity of the points defined by the condition w(x, t)= (Mtu)~2p(x,t)=0. This solution takes the form of the following power series, ç~(x,I) = (Mtu )
—
/2 2
~ C~[L (x, exp(Mtr)
L(x, t) =a(Mtu)i/
x
I)]”,
(2it)3
(15)
Jdqexp(iq.x_Mtkq2).
[G(x, t~if), G(x, 1; ~‘)] 1t=00,
where [ ] is the commutator sign. To explain that let us note that expression (11) for the function G(x, 1; ~) can be rewritten in the form ,
G(x, t; t~)=exp[—~AIuI~(x, 1) ]D(x,
I)
Having at our disposal this representation for the solution of eq. (9) we can use the Burman—Lagrange formalism [4] in order to find exact relations for the coefficients c,, (n = 0, 1, ...): c 2~= 0,
Xexp[t~M’ub~(x,t)]—D(x,t)
2~
(12)
,
where
C 2fl~1=
tl~~(x,t)=exp[ —~D(x,t) ]~o(x,I) exp[t~D(x, t)] ~
[D(x,
t), [...,[D(x,
t),
(p(X,
I) ]...]
](n)’
= fl~o n! Thus, calculating the functions G(x, 1; ~) and F(x, t; A) using (12) is reduced to various combinations of the basic expressions D~(x,t)~(x, t) (n= 1,..., cc) and Vqi(x, I). Taking now into account the supplemented initial condition (5) it is easy to demonstrate that
D~(x,t)q(x, t) 1=~=0, Vq(x, 1)11=0=0.
(13)
In its turn this means that the following holds,
J
(d
________
—
(2n+l)! lim~~~exp[_(2n+l)z2])
(—1 )n(2n+
1
)fl_i
(16) n! Thus,byformulae (15), (16) wehavefoundtheexact solution for the above described problem of single nucleus growth. This solution is obtained in the form of a power series around the region where ~u(x, 1) =0. It can be demonstrated from the d’Alembert test that this power series is convergent under the condition L(x, 1) < 1 /,~/iBefore starting the analysis of the exact solution (15), (16) we note that all —
our results obtained until now are valid for arbitrary dimension. However, in this Letter all further discussions will be restricted to the most relevant three-
A
T[exp(
d~G(x, ~
case. Integration qdimensional in expression (15) gives for over L(x, the I) wave vector
A
L(x,
0
—a (u)
o ~=o =exP(Jd~G(x~t;~))~=1, F(x,
1;
A)5(t) =F(x, 0; A)ô(t) =ô(i).
1) =L(x, t)
(14)
x=
1/2
2/4Mtk) 8mMtk exp(Mtr—x
‘
lxI. 315
PHYSICS LETTERS .A
Volume 176, numberS
Let us now find asymptotic solutions of the transcendental algebraic equation (9) for the two important limiting cases: for early and late stages of the growth process. The limiting case of short times, ,h,1t~
~(x.
I) = ~(.v. I) = (.‘tItu) - “~L0s.I) 2/4?vJtk) exp(Mtr—.v =a~~ -(4it4.Itk)3~ ‘
(17)
For =0 we automatically get the result
which is consistent with initial condition (4). Thus. the spherically symmetrical solution (1 7) describes the growth of a single nucleus which has instantly appeared at the time instant t=0. The limiting case of long times, Mi>~I, is of parocular interest as it corresponds to late stages of the growth process. In this limiting case we arranged a special asymptotic analysis [4]of the transcendental algebraic equation (9) and have obtained the following asymptotically exact expression for the order parameter field t(x. I).
I
the growth process the order parameter ~(v. il is a spatial constant, i.e. ii does not depend on the initial shot-like perturbation and evolves to the value zero. This means that the system under consideration tends to its equilibrium state in which the expectation value çQ(x. I) ,~ is reached. In summary. sve have suggested a new method for the solution of the deterministic TDGL equation with a singular “source” term. We have presented, for simplicity, the usage of this method for a single disturbance corresponding to the physical situation dealing with the growth of a single stable nucleus. A generalization of this method for multiple nuclei of random size, randomls distributed in space and. .
~(x. 0)=a’5(x)
i 7 May
-
possibly, randomly appearing in time, is readily ohtamed and will be discussed elsewhere [5]. We note that a similar (non-singular) initial fluctuation at the time instant r=0 has been introduced into the deterministic TDGL equation in ret’. [6], where tills equation has been solved approximately. The qualitative conclusion presented in ref.becomes [6]states that at late stages the order parameter a spatial constant. This is in agreement with our quantitative result and with that of ref. 131. In conclusion, we note that the above described mathematical approach for solving the TDGL model with a singular “source” term may be useful in searching for the Green funclion in the p~field-theoretical problem.
p(.v. I)=2Mtr—21n(MI)+,v2/21t!Ik+const. const=ln[a2u/(4~k)1]
We wish to thank P.G. de (jennes. V.Ya. Fainberg. M. Fisher. .1. Lebowitz. AZ. Patashinsky and V.
‘
ço(x. t)=ço(.v. t) In /2 [p(x 1)] [1 —
—
(2Mtu)~2
Privman for stimulating discussions. We acknowledge the warm support froni our Polymer Research Institute.
In In p(.v, ~ Inp(.v,i)
+O(lnP(~v~t))]~
(18)
\ lnp(x,t)
This expression once again gives the spherically symmetrical solution for the order parameter field çq(x. 1) in the case of late stages of the growth process. For sufficiently large Mt this solution tends to zero with time as follows. ~(s-. /) =~(t)
(In ( 2Mtr)’\ 2Mtu
~
)
ill l..D. Landau and EM. Lifshitz. (ounse of iheoretical physics. Vol. I 0. Physical kineiics I l’ergamon, Oxford, 1986)
[21 iD. Ciunion,
M. San Miguel and P.S. Sahni. in: Phase
transiiions and criiicai phenomena, Vol. 8. edt. C’. Domb and .i. ~ (Academic Press, New York, 1983).
131
‘..Z. Patashinsk\ and Ri. Shumilo. Soy, Phvs.JETP SI)
141
F.W.J. Oliver, Introduction lo asympiotics and special functions (Academic Press, New York, 1974).
i979) 712
(19)
Thus, one may conclude that at very late stages of
316
References
151
A.F. lzmailov, K. Levon and A. Margolina. in preparation. [6] W. Klein. Phys. Res. Ecu. 65 1 i990) 1462.
PHYSICS LETTERS A
Algebraic approach to the solution of the relaxation problem with a non-conserved order parameter A.F. Izmailov Department of Chemical Engineering, Polytechnic University, Six MetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA
K. Levon Department ofChemistry, Polytechnic University, Six MetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA
and A. Margolina Polymer Research Institute, Polytechnic University, SixMetroTech Center, 333 Jay Street, Brooklyn, NY 11201, USA Received 16 July 1991; revised manuscript received 23 March 1993; accepted for publication 26 March 1993 Communicated by A.P. Fordy
The relaxation ofa non-conserved order parameter after a single shot-like disturbance is studied in the framework of the timedependent Ginzburg—Landau model. It is demonstrated that the differential equation describing the growth kinetics can be transformed into a transcendental algebraic equation. A late stage asymptotic solution is obtained.
There exist three well-defined shapes in the relaxation of a metastable state of matter into a thermodynamically stable state: nucleation, growth and coalescence [1]. While in the first stage nucleus formation and decay are governed mainly by thermal fluctuations and result in the appearance of a stable (supercritical) nucleus of the new phase, the second stage describes the monotonic growth of such a stable nucleus with practically no decrease in size or decay. This means that the growth kinetics may be reasonably described in a deterministic fashion, neglecting thermal fluctuations. The first stage has been extensively treated in both the classical theory of nucleation and the linearized time-dependent Ginzburg—Landau (TDGL) approach [2]. However, not so much is known about the late stages of the growth process [2,31. Notably in ref. [3] the deterministic growth of a single nucleus was proposed in the form of a soliton-like solution. This lution was obtained under the assumption that the width of the nucleus boundary is small compared to its size. -
In this Letter we address the problem of deterministic growth for a single stable nucleus with the center at the coordinate origin. This nucleus is formed due to an initial shot-like disturbance. Such an unusual formulation of the problem allowed us to solve it exactly in the vicinity of the certain points, with no other approximations involved. Thus, we have found exact asymptotics of the solution for early and late stages of the growth process. We consider the usual relaxation equation for the non-conserved scatar order parameter field q’ (x, 1) [21, without the “noise” term, but including instead the “source” term, describing the initial shot-like disturbance: ô
49(X,t)
=—M 6F(ço) +aô(x)ö(t),
i?~O. (1)
ö~(x,I) In this equation M is the mobility and F(ç~)is the “coarse-grained” Ginzburg—Landau (GL) functional [2],
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
313
‘H S SItS Ll~ ft ERS 5
Volume I 7b, number S
-‘
0.t.fL.-oI 4(~. I
.-~
.fiqi’i.v. I) + ~uip
~
~~-‘
=
.t1[ (I~V~ + r Fp(x.f I
~.
~-i~ ing Out the action of he exponential operaioi cxp I ~.i’t .v. on the “source’ term t~(x)~(I cc cluires some care and makc’~use of the initial con
it matches tie
elI X. (I)
expt,±(x. / ( JO(xio( I
lion IS possible in our parti lilac ease. Thus. the suIt of I — is as follossv .
We choose the initial condition so thai “source terni q(x.
)Olxo)l:l
dition in form 1. \‘~c~sill lirst present the R’SLI!i H such action and the resulting transcendental alge braic equation and then ss c ~ ill explaii why such
i
(II
f~)(,r)~(
ci,.
~‘
where tile parameters r and u define the (iinzburg~ Landau ( (iL ) double—well potential.. ‘iI/~is the dii— tusion constant, and rt IS the amplitude of the initial disturbance. Thus, the equation we solve has the form
LI.
/
=
dt exp:/. 1/I
it
iiy~ ix.
=~(x)
Thus. tile order parameter itself recei~es an initial shot—like disturbance. To deal with such a singular smg initial condition it is necessars to supplement its dclinition in the inhnitesimal region around the initial lime instant 1=0. Such a supplemented dehnition gives us the opportunit\ to define anc function dc pending on q(x. I) and its derisatises in tile S icnlu\ of this point. We choose 10 do it as follows,
e\~) ~/)( x.
Jl(( .1
110(5 the l-ouriec expansion of ~P.r I and al/ uid e iii’, in~out the integi 01011 05 ri the p ii imHi ~.. we get tinahl\ thai
(
/)
.~
2it )
dqexp(iqx—.tIikq’)
/0-A
It
qexp(
y,(x. 11=
00 /
iq’x- i/iI~q
Lq uat ion I I) ) eonst it utes the algebraic form of eq. 3) with the initial condition I I se were striving
f/see
101-.
(‘leads, initial condition (4 1 is consislent with such a supplementary definition as
\t tills point we ~ ill explain ho~the action of the operator exp [).I’( .v. I ) ssas carried out in expression 7). First. we decompose tIns operator according to
ip(x. 0)
ihe following general scheme.
=
urn p~(x.I)
=C~d(X
.
expl/.I’(x. We now proceed to rewrite the partial differential equation (3) in the form of a transcendental algebraic equation. To this end we presenl (3) as follows. l’f x. I h’p(x.
I) =
x )d( II
.
P1 X. I) = ~I1ti~(i. I) r] ±1)~,~ i)(x. I) =iJ/~9~’— ;~1kV2. k)rmally introducing the inverse operator P (.v. one can write eq. (3 ) in the following farni. ~l4
I
)
/
J =expH1l[
~ ex~~ ~~DI.v. /11/’
OH LI. I
(.1/, / I
if I
Here the factor 1 ( x. g ,‘. is not iiecessaril~equal to unit~as the differential operator D( x. I I does not commute with the order parameter field cit x. i I. I )i Ilerentiating eq. (10) once with respect to the pai’amdc) (observing the designated order of operators I yields a simple operator differential equation iiivol\ ing the factor FIx. I: ) The solution of this equation is found in the farm
Volume 176, number 5
PHYSICS LETTERS A
F(x, t;A)=T[exP(Jd~G(x~ t;
Thus, the supplemented definition (5) of the sin-
~))]
0
G(x, t; t~)=A~(x, I; t~)D(x,t) A(x, t; ~) —D(x,t)
(11)
,
17 May 1993
where A(x, 1; ~) =exp[t~Muç92(x, I)] exp[~D(x, t)] In this expression T[exp( )] is the chronologically ordered exponent with respect to the parameter In our particular case this chronologically ordered exponent is equal to the conventional exponent exp( since the differential operator G(x, I; ~) obeys the following relation, ~.
gular initial condition (4) results in such an effective action of F(x, I; A) on the “source” term as if F(x, I; A) were, in fact, equal to unity. We turn now to solving eq. (9) exactly. It is possible to construct a solution of the transcendental algebraic equation (9) in the vicinity of the points defined by the condition w(x, t)= (Mtu)~2p(x,t)=0. This solution takes the form of the following power series, ç~(x,I) = (Mtu )
—
/2 2
~ C~[L (x, exp(Mtr)
L(x, t) =a(Mtu)i/
x
I)]”,
(2it)3
(15)
Jdqexp(iq.x_Mtkq2).
[G(x, t~if), G(x, 1; ~‘)] 1t=00,
where [ ] is the commutator sign. To explain that let us note that expression (11) for the function G(x, 1; ~) can be rewritten in the form ,
G(x, t; t~)=exp[—~AIuI~(x, 1) ]D(x,
I)
Having at our disposal this representation for the solution of eq. (9) we can use the Burman—Lagrange formalism [4] in order to find exact relations for the coefficients c,, (n = 0, 1, ...): c 2~= 0,
Xexp[t~M’ub~(x,t)]—D(x,t)
2~
(12)
,
where
C 2fl~1=
tl~~(x,t)=exp[ —~D(x,t) ]~o(x,I) exp[t~D(x, t)] ~
[D(x,
t), [...,[D(x,
t),
(p(X,
I) ]...]
](n)’
= fl~o n! Thus, calculating the functions G(x, 1; ~) and F(x, t; A) using (12) is reduced to various combinations of the basic expressions D~(x,t)~(x, t) (n= 1,..., cc) and Vqi(x, I). Taking now into account the supplemented initial condition (5) it is easy to demonstrate that
D~(x,t)q(x, t) 1=~=0, Vq(x, 1)11=0=0.
(13)
In its turn this means that the following holds,
J
(d
________
—
(2n+l)! lim~~~exp[_(2n+l)z2])
(—1 )n(2n+
1
)fl_i
(16) n! Thus,byformulae (15), (16) wehavefoundtheexact solution for the above described problem of single nucleus growth. This solution is obtained in the form of a power series around the region where ~u(x, 1) =0. It can be demonstrated from the d’Alembert test that this power series is convergent under the condition L(x, 1) < 1 /,~/iBefore starting the analysis of the exact solution (15), (16) we note that all —
our results obtained until now are valid for arbitrary dimension. However, in this Letter all further discussions will be restricted to the most relevant three-
A
T[exp(
d~G(x, ~
case. Integration qdimensional in expression (15) gives for over L(x, the I) wave vector
A
L(x,
0
—a (u)
o ~=o =exP(Jd~G(x~t;~))~=1, F(x,
1;
A)5(t) =F(x, 0; A)ô(t) =ô(i).
1) =L(x, t)
(14)
x=
1/2
2/4Mtk) 8mMtk exp(Mtr—x
‘
lxI. 315
PHYSICS LETTERS .A
Volume 176, numberS
Let us now find asymptotic solutions of the transcendental algebraic equation (9) for the two important limiting cases: for early and late stages of the growth process. The limiting case of short times, ,h,1t~
~(x.
I) = ~(.v. I) = (.‘tItu) - “~L0s.I) 2/4?vJtk) exp(Mtr—.v =a~~ -(4it4.Itk)3~ ‘
(17)
For =0 we automatically get the result
which is consistent with initial condition (4). Thus. the spherically symmetrical solution (1 7) describes the growth of a single nucleus which has instantly appeared at the time instant t=0. The limiting case of long times, Mi>~I, is of parocular interest as it corresponds to late stages of the growth process. In this limiting case we arranged a special asymptotic analysis [4]of the transcendental algebraic equation (9) and have obtained the following asymptotically exact expression for the order parameter field t(x. I).
I
the growth process the order parameter ~(v. il is a spatial constant, i.e. ii does not depend on the initial shot-like perturbation and evolves to the value zero. This means that the system under consideration tends to its equilibrium state in which the expectation value çQ(x. I) ,~ is reached. In summary. sve have suggested a new method for the solution of the deterministic TDGL equation with a singular “source” term. We have presented, for simplicity, the usage of this method for a single disturbance corresponding to the physical situation dealing with the growth of a single stable nucleus. A generalization of this method for multiple nuclei of random size, randomls distributed in space and. .
~(x. 0)=a’5(x)
i 7 May
-
possibly, randomly appearing in time, is readily ohtamed and will be discussed elsewhere [5]. We note that a similar (non-singular) initial fluctuation at the time instant r=0 has been introduced into the deterministic TDGL equation in ret’. [6], where tills equation has been solved approximately. The qualitative conclusion presented in ref.becomes [6]states that at late stages the order parameter a spatial constant. This is in agreement with our quantitative result and with that of ref. 131. In conclusion, we note that the above described mathematical approach for solving the TDGL model with a singular “source” term may be useful in searching for the Green funclion in the p~field-theoretical problem.
p(.v. I)=2Mtr—21n(MI)+,v2/21t!Ik+const. const=ln[a2u/(4~k)1]
We wish to thank P.G. de (jennes. V.Ya. Fainberg. M. Fisher. .1. Lebowitz. AZ. Patashinsky and V.
‘
ço(x. t)=ço(.v. t) In /2 [p(x 1)] [1 —
—
(2Mtu)~2
Privman for stimulating discussions. We acknowledge the warm support froni our Polymer Research Institute.
In In p(.v, ~ Inp(.v,i)
+O(lnP(~v~t))]~
(18)
\ lnp(x,t)
This expression once again gives the spherically symmetrical solution for the order parameter field çq(x. 1) in the case of late stages of the growth process. For sufficiently large Mt this solution tends to zero with time as follows. ~(s-. /) =~(t)
(In ( 2Mtr)’\ 2Mtu
~
)
ill l..D. Landau and EM. Lifshitz. (ounse of iheoretical physics. Vol. I 0. Physical kineiics I l’ergamon, Oxford, 1986)
[21 iD. Ciunion,
M. San Miguel and P.S. Sahni. in: Phase
transiiions and criiicai phenomena, Vol. 8. edt. C’. Domb and .i. ~ (Academic Press, New York, 1983).
131
‘..Z. Patashinsk\ and Ri. Shumilo. Soy, Phvs.JETP SI)
141
F.W.J. Oliver, Introduction lo asympiotics and special functions (Academic Press, New York, 1974).
i979) 712
(19)
Thus, one may conclude that at very late stages of
316
References
151
A.F. lzmailov, K. Levon and A. Margolina. in preparation. [6] W. Klein. Phys. Res. Ecu. 65 1 i990) 1462.