The effect of shear on persistence in coarsening systems

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by  N. P. Rapapa

The effect of shear on persistence in coarsening systems
N. P. Rapapa and A. J. Bray
Department of Physics and Astronomy, The University, Oxford Road, Manchester M13 9PL, UK
(Dated: February 6, 2008)
6 Weanalyticallystudytheeffectofauniformshearflowonthepersistencepropertiesofcoarsening
0 systems. ThestudyiscarriedoutwithintheanisotropicOhta-Jasnow-Kawasaki(OJK)approxima-
0 tionforasystemwithnonconservedscalar orderparameter. Wefindthatthepersistenceexponent
2 θ has a non-trivial value: θ=0.5034... in space dimension d=3, and θ=0.2406... for d=2, the
n latterbeingexactly twice thevaluefoundfor theunshearedsystem in d=1. Wealso findthat the
a autocorrelation exponent λ is affected byshear in d=3 but not in d=2.
J
6 PACSnumbers: 82.20.Mj,64.75.+g,05.70.Ln
1

] I. INTRODUCTION growth exponents for the structure along and perpen-
h dicular to the flow. At present it is not clear whether
c
shear leads to a stationary steady state, or whether do-
e Thephenomenonofpersistenceinnonequilibriumsys-
m tems has attracted considerable interest in recent years maingrowthproceedsindefinitelyatasymptoticallylarge
[1],boththeoretically[2,3,4]andexperimentally[5,6,7, times [12]. Shear may also induce phase transitions: for
-
t 8,9]. Thepersistenceprobability,P (t), ofafluctuating, example,shear-inducedshiftofthephasetransitiontem-
a 0
t spatially homogeneous nonequilibrium field is the prob- perature inthe microphase separationof diblock copoly-
s ability that the field X(t) at a given space point has not mers has been observed [13].
.
t changedsignup to time t. This probability typically de-
a
m caysasapowerlaw, P (t) t−θ atlatetimes,wherethe
0
∼
persistence exponent θ has in general a nontrivial value. In this paper we analytically study the effect of an
-
d Persistence has been studied in a considerable number imposed uniform shear flow on persistence for the sim-
n of systems such as simple diffusion from random initial plest case of a nonconservedscalar order parameter. We
o conditions,phase-orderingkinetics,fluctuatinginterfaces exploit a version of the Ohta-Jasnow-Kawasaki (OJK)
c
and reaction-diffusion processes [1]. approximation in phase-ordering kinetics [10], modified
[
Experiments to determine persistence exponents have to account for the anisotropy induced by the shear [14].

1 been carried out in the context of breath figures [5], liq- Persistenceisdefinedhereastheprobabilitythatapoint
v uid crystals [6], soap froths [7], diffusion of Xe gas in comoving with the flow has remained in the same phase
2
one dimension [8] and fluctuating monatomic steps on a up to time t. We employ an approach called the in-
5
metal/semiconductoradsorptionsystemSi-Alsurface[9]. dependent interval approximation (IIA) which has been
3
1 Many ofthese casesareexamples ofcoarseningphenom- successfullyusedtoobtainratheraccuratevaluesforper-
0 ena, where a characteristic length scale increases with sistence exponents inunshearedsystems [1]. This proce-
6 time as the system relaxes towards an equililibrium that dureassumesthattheintervalsbetweenzerosofthepro-
0 it attains only after infinite time in the thermodynamic cess X(t) are statistically independent when measured
t/ limit. The experimental results are generally in good in the mapped time variable T = lnt. We find that the
a quantitativeagreementwith(exactorapproximate)the- persistence exponent θ is nontrivial and dimensionality
m
oretical predictions. dependent. For d = 3 we find θ 0.5034, compared to
≃
- A classic example of a coarsening phenomenon is the 0.2358 in the unsheared case[3], while θ 0.2406 for
d ≃ ≃
n dynamics of phase ordering, where a system is quenched d = 2 compared to θ 0.1862 without shear [3]. Re-
≃
o from a disordered high-temperature phase into an or- markably, the value of θ in d = 2 is exactly twice the
c dered low-temperature phase. In the simplest case of a value obtained for the unsheared system in d = 1 [3]
v: two-phasesystem,domainsofthetwoequilibriumphases using similar methods. There is a technical subtlety in
i formandgrowwithtime. Thecharacteristiclengthscale d = 2 which requires a careful definition of the persis-
X
atagiventimeisthetypicalscaleofthedomainstructure tence probability. In both d = 2 and d = 3 the shear
r that has formed at that time. The coarsening dynamics increases the persistence exponent.
a
is usually characterised by a form of dynamical scaling,
inwhichthesystemlooksstatisticallysimilaratdifferent
times apart from an overallchange of scale [10]. The paper is organised as follows. In the next section
Recently there has been interest in the effect of shear the OJK theory is introduced and the autocorrelation
in a variety of systems such as macromolecules, binary function, which is a necessary input to the IIA calcula-
fluids and self-assembled fluids [11]. Shear introduces tion,isobtainedford=3andd=2. SectionIIIcontains
anisotropyintothespatialstructure. Forsystemsunder- a brief outline of the IIA, the results of which for the
goingphaseorderinginthepresenceofshear,thedomain shearedproblemarepresentedinsectionIV. Concluding
growth becomes anisotropic and this results in different remarks are given in section IV.
2
II. THE OJK THEORY In k-space, Eq. (4) can be written as
We consider a nonconserved scalar order parame- ∂m(~k,t) ∂m(~k,t)
γk =
x
ter φ(~x,t) evolving via the time-dependent Ginzburg- ∂t − ∂k
y
Landau equation [10]
d d
k 2+ D (t)k k m(~k,t). (7)
∂φ(~x,t) = 2φ(~x,t) V′(φ), (1) − a ab a b
∂t ∇ − Xa=1 aX,b=1
 
whereV(φ)isasymmetricdouble-wellpotential. Theas-
sumption thatthe thickness ξ of the interface separating A. The case d=3
thedomainsismuchsmallerthanthesizeofthedomains
allows one to write an equation of motion for the inter-
We nowconsider the above equationin dimensiond=
face, calledthe Allen-Cahnequation[15]. The velocityv
3 and solve it via the following change of variables [14],
oftheinterfaceisproportionaltothelocalcurvatureand
given by (k ,k ,k ,t) (q ,q γk τ,q ,τ), (8)
x y z x y x z
→ −
v(~x,t)= ~.~n(~x,t), (2)
with the introduction of an equivalent field
−∇
where ~n(~x,t) is the unit vector normal to the interface,
µ(~q,τ)=m(~k,t). (9)
defined in the direction of increasing order parameter.
The normal vector can be written in general as
The left hand side of (7) now becomes ∂µ/∂τ and as
~m(~x,t) a result equation (7) can be integrated directly to give
~n(~x,t)= ∇ (3) (after transforming back to original variables)
~m(~x,t)
|∇ |
where m(~x,t) is the smooth field that has the same sign m(~k,t)= m(kx,ky+γkxt,kz,0)
as the order parameter φ and vanishes at the interfaces 3
1
(wheretheorderparametervanishes). Itiseasiertowork exp k M (t)k , (10)
a ab b
with an equation of motion for m(~x,t) than for φ(~x,t), × "−4ab=1 #
X
an idea that is exploited in the OJK theory [16].
with non-vanishing matrix elements
Byconsideringaframelocallycomovingwiththeinter-
face, with a space-uniform shear in the y-direction and
M (t) = R (t)+2γtR (t)+γ2t2R ,
flow in the x-direction (i.e. the fluid velocity profile is 11 11 12 22
given by ~u = γy~ex), where γ is the constant shear rate M12(t) = R12(t)+γtR22(t),
and ~e is the unit vector in the flow direction, the OJK M (t) = R (t),
x 22 22
equation for the field m(~x,t) becomes [14]
M (t) = R (t), (11)
33 33
∂m(~x,t) ∂m(~x,t)
+ γy = where
∂t ∂x
d ∂2m(~x,t) t ′ ′ ′ ′
2m(~x,t) D (t) , (4) R11(t) = 4 dt [1 D11(t)]+2γtD12(t)
∇ − ab ∂xa∂xb Z0 { −
aX,b=1 +γ2t′2[1 D (t′)] ,
22
−
where t
′ ′ (cid:9) ′ ′
R (t) = 4 dt D (t) γt[1 D (t)] ,
12 12 22
D (t)= n n , (5) {− − − }
ab h a bi Z0
t
and ... denotesaverageoverinitialconditions(or,equiv- R (t) = 4 dt′ 1 D (t′) ,
h i 22 { − 22 }
alently, over space). The correctequation for m involves Z0
an unaveraged D , but the equation is then nonlinear t
ab ′ ′
R (t) = 4 dt 1 D (t) . (12)
and intractable. The essence of the OJK approximation 33 { − 33 }
isthereplacementoftheproductn n byitsaverage. For Z0
a b
anisotropicsystemthisgives,bysymmetry,D =δ /d, Due to the symmetry of the original OJK equation (4),
ab ab
andtheequationformreducestothediffusionequation. the terms R , R , M and M all vanish. The as-
13 23 13 23
For the anisotropic sheared system, however, Dab(t) has sumption thatthe initial condition, m(~k,0), has a Gaus-
to be determined self-consistently [14]. From Eq. (5), it siandistribution,appropriatetoaquenchfromthe high-
follows that temperature phase, is used throughout the paper.
In order to use the IIA to investigate the persistence
d
D (t)=1. (6) properties of the coarsening system, it is first neces-
aa
sary [1, 3] to compute the autocorrelation function of
a=1
X
3
therescaledfieldX(t)=m(~x,t)/ [m(~x,t)]2 1/2,whichis Turning now to the two-time correlator of X(t), we
h i
constructed to have unit variance, using the initial cor- recall that we want to calculate this correlator not at a
relator fixedpointinspace,but ata pointthatis advectedwith
the shear flow. Due to the shear, the field at the space-
m(~x,0)m(x~′,0) =∆δd(~x x~′). (13) time point (x +γyt ,y,z,t ) at time t will be at the
h i − 1 1 1
space-timepoint(x+γyt ,y,z,t )attime t . The auto-
2 2 2
The quantity [m(~x,t)]2 1/2 can easily be evaluated to correlationfunction a(t ,t )= X(t )X(t ) is therefore
give h i given by 1 2 h 1 2 i
1/2
[m(~x,t)]2 1/2 = ∆ 1 . (14)
"(2π)3/2 Det M(t)#
(cid:10) (cid:11)
p
(2π)3 1/2
a(t ,t )= DetM(t )DetM(t ) m(x+γyt ,y,z,t )m(x+γyt ,y,z,t ) . (15)
1 2 ∆2 1 2 h 1 1 2 2 i
(cid:20) (cid:21)
p
The next step is to evaluate the term in the above equation. We note that average over initial conditions in
h···i
k-space implies
m(k ,k +γk t ,k ,0)m(k′,k′ +γk′t ,k′,0) =(2π)d∆δ(k +k′)δ(k +γk t +k′ +γk′t )δ(k +k′). (16)
h x y x 1 z x y x 2 z i x x y x 1 y x 2 z z
2
Using Eqs. (10) and(16) we can evaluate the term M (t) = γt2,
12
5
3 4
1 M (t) = t,
m(1)m(2) = ∆ exp kaBabkb 22 5
h i −2 
k a,b=1 16
X X M (t) = t. (20)
33
∆  1  5
= , (17)
(2π)3/2 Det B(t ,t ) Using equations (20), the determinants of M(t) and
1 2
B(t ,t ) can now be evaluated leading to
where p 1 2
64
B11(t) = M11(t1)+M11(t2)+γ2M22(t2) (t2 t1)2 Det M(t)= 375γ2t5,
× −
h−2γ(t2−t1)M12(t2)]/2, DetB(t1,t2)= 8γ2(t122+5 t1)5 4 31 − (t2t+22tt1)3
B12(t) = [M12(t1)+M12(t2) γ(t2 t1)M22(t2)]/2, (cid:20) (cid:26) 2 1 (cid:27)
− −
B (t) = [M (t )+M (t )]/2, 2t 2 2
22 22 1 22 2 1 2 . (21)
B33(t) = [M33(t1)+M33(t2)]/2, −(cid:26) − (t2+t1)2(cid:27) #
B (t) = B (t)=0. (18)
13 23 The autocorrelation function follows:
The notation (1) and (2) in (17) denotes space-time 1 W5/4(t ,t )
1 2
tpiovienlyts. (Txh+eγpyrto1b,lye,mz,its1)naonwdr(exdu+ceγdytt2o,ye,vza,ltu2a)tirnegsptehce- a(t1,t2) = 2 1− 43W2(t1,t2) 1/2,
determinants of the matrices M(t) and B(t1,t2), as the 1 (cid:2) sech5/2(T/2)(cid:3)
autocorrelationfunction a(t ,t ) can now be written as = , (22)
1 2 2 1 3sech4(T/2) 1/2
− 4
[DetM(t )DetM(t )]1/4
a(t ,t )= 1 2 . (19) where W(t ,t )=4t t /((cid:2)t +t )2, T =T(cid:3) T and the
1 2 1 2 1 2 2 1 1 2
DetB(t ,t ) −
1 2 finalformfollowsafterintroducingthenewtimevariable
T = lnt . After this change of the time variable, the
ThetermsM (t)cannotbpecomputedexplicitlyforgen- i i
ab autocorrelationfunctiondepends onlyonthetimediffer-
eralt;onlyinthescalinglimit(i.e. t )canonemake
progress. In this limit it can be sho→wn∞that (to leading ence T1 −T2. Since the process X(T) is also gaussian,
the process X(T) is a gaussian stationary process. This
order for large t) [14]
willbe the casewheneverthe autocorrelationfunctionof
4 X depends on t1 and t2 only through the ratio t1/t2, i.e.
M11(t) = γ2t3, when it exhibits a scaling form.
15
4
B. The case d=2 terms of the new time variable T = ln(t2/t1), one ob-
tains
Ford=2,wefollowthesameanalysisasford=3but
a(t ,t )= sech(T), (28)
with the change of variables 1 2
p
where T = T T , i.e. the process X(T) becomes sta-
(kx,ky,t)→(qx,qy−γkxτ,τ). (23) tionaryin the1d−efin2ed scaling limit. We will use Eq. (28)
rather than Eq. (27) to extract θ for d = 2, but one
ThesolutionofEq.(7)inthelarge-tlimitisgivenbythe
mustnotethespeciallimittakentoderive(28)wherethe
d=2 analogue of Eq. (10):
persistence probbaility P(t ,t ) is the probability that a
1 2
pointmovingwiththe flowhas stayedinthe same phase
m(~k,t)= m(k ,k +γk t,0)
x y x between times t and t .
1 2
2
1
exp k M (t)k . (24)
a ab b
× −4 
a,b=1 III. THE INDEPENDENT INTERVAL
X
  APPROXIMATION
The matrix elements M (t) can be evaluated for larget
ab
using asymptotic analysis along the lines outlined in ref. Theaboveanalysisinbothd=3andd=2showsthat
[14], with the result X(t) is stationary in the new time variable T (with the
caveatnotedaboveford=2). Wenotethattheexpected
M (t) = 4γt2 lnγt 3γt2 form for the probability, P0(t), of X(t) having no zeros
11 − √lnγt between t and t , namely P (t /t )θ for t t ,
1 2 0 1 2 2 1
p 2t becomes exponential decay P0 ∼e−θ(T2−T1) in th≫e new
M (t) = 4t lnγt ∼
12 time variable. This reduces the problem of calculating
− √lnγt
the persistence exponent to the calculation of the decay
4p
M (t) = lnγt, (25) rates [17].
22
γ
The order parameter field in the OJK theory is given
p
by φ=sgn(X). The autocorrelationfunction
where we have retained just the leading subdominant
terms, of relative order 1/ln(γt).
A(T)= φ(0)φ(T) = sgnX(0)sgnX(T) , (29)
The subleading terms in M (t) and M (t) are nec- h i h i
11 12
essary as there are cancellations to leading terms in the for the field φ at a space point moving with the flow, is
determinant of M(t), which is given by DetM(t) = 4t2. given by
Using Eq. (24) the following averages can be calculated:
2
A(T)= sin−1a(T), (30)
1/2 π
[m(~x,t)]2 1/2 = ∆ 1 = 1 ∆,
"2π DetM(t)# 2tr2π whichfollowsfromthefactthatφisaGaussianfield[18].
(cid:10) (cid:11) We will determine the persistence probabilityP (t) from
∆ 1 0
p
m(1)m(2) = . (26) A(T).
h i (2π) Det B(t ,t )
1 2 We briefly discuss the IIA [1] and use it to obtain ap-
proximate values for the exponent θ following the de-
p
where the matrix elements B are given by the expres-
ab velopment in ref.[3]. In the scaling limit, the interfaces
sionsinEq.(18)butwiththecorrespondingM (t)given
ab occupyaverysmallvolumefractionandasaresultφ(T)
theirbytheird=2equivalentsinEq.(25). Theautocor-
takesvalues 1almosteverywhere. ThecorrelatorA(T)
relationfunctiona(t1,t2)ford=2cannowbe evaluated can be writte±n as
using the set of equations (26) to give
∞
1/2 A(T)= ( 1)nPn(T), (31)
−
a(t ,t )= 4t1t2 . nX=0
1 2  
t2 1+ lnγt2 +t2 1+ lnγt1 where P (T) is the probability that the interval T con-
1 lnγt1 2 lnγt2 n
  tains n zeros of φ(T). For n 1, P (T) is approximated
 (cid:16) q (cid:17) (cid:16) q (cid:17) (27) ≥ n
by, assuming that the intervals between zeros of X are
independent,
Notethata(t ,t )givenbythatEq.(27)doesnothave
1 2
ascalingform,i.e.itisnotsimplyafunctionoft /t ,due T T T
to the logarithms. However it does a scaling re1gim2e. In Pn(t) = T −1 dT1 dT2 dTn
h i ···
the limit t , t , with t /t fixed but arbi- Z0 ZT1 ZTn−1
1 2 1 2
trary, the ra→tio∞of loga→rith∞ms can be replaced by unity Q(T1)P(T2 T1) P(Tn Tn−1)Q(T Tn),
× − ··· − −
and a(t ,t ) depends only on t /t in this regime. In (32)
1 2 1 2
5
where T isthe meanintervalsize,P(T)isthe distribu- (logarithmic) timescale T. It follows that the relation
tionofhintiervalsbetweensuccessivezerosandQ(T)isthe θd=2 =2θd=1 does notrequire the IIA but only that the
sh unsh
probability that an interval of size T to the right or left underlyingfieldm (or,equivalently,X)be Gaussian,i.e.
′
of a zero contains no further zeros, i.e. P(T) = Q(T) it requires use of the OJK theory but not the IIA. It is
−
wheretheprimeindicatesaderivative. TheIIAhasbeen interesting to speculate that it might even hold beyond
madeinEq.(32)bywritingthejointdistributionofzero- the OJK approximation, in which case one might imag-
crossing intervals as the product of the distribution of ine thatthere is a verysimple explanationfor it. As yet,
single intervals. The Laplace transformof Eq.(32) leads however,we have been unable to find one.
to P˜(s)=[2 F(s)]/F(s) where The autocorrelation function A(t ,t ) is also interest-
1 2
−
ing. In the limit t t that defines the autocorrelation
2 1
F(s)=1+ hTis[1 sA˜(s)] (33) exponent λ [10], vi≫a A(t1,t2) ∼ (t1/t2)λ, the quantity
2 − a(t ,t )issmallandEq.(30)canbelinearisedina(t ,t )
1 2 1 2
to give, from Eq. (35) with T =ln(t /t ),
2 1
and A˜(s) is the Laplace transform of A(T).
Itisstraightforwardtoshowthatthemeanintervalsize
(t /t )5/4, d=3,
fisorhTlairg=eT−2i/mAp′l(i0e)s.aTshimepelxeppeoctleatiinonP˜t(hs)atatPs0(=T) ∼θ.eT−θhTe A(t1,t2)∼(cid:26)(t11/t22)1/2, d=2. (36)
−
persistenceexponentθ isthereforegivenbythefirstzero
These results give λ = 5/4 for the sheared system in
on the negative axis of the function
d = 3, compared to λ = 3/4 in the unsheared system
∞ [10], whereas for d = 2 the autocorrelation exponent
s 2s
F(s)=1 1 dT exp( sT) takes the same value, λ=1/2, in both cases. We should
− A′(0) − π −
(cid:20) Z0 repeat the caveat that, for d = 2, the simple power-law
sin−1a(T) . (34) form (36) requires the limit t , t with t /t
× 1 →∞ 2 →∞ 2 1
fixed but large. If t for fixed t , Eq.(27) gives
For further analysis is is useful to first ext(cid:3)ract the a(t ,t ) (t /t )1/2[2ln(→γt∞)/ln(γt )]1/4,1which does not
1 2 1 2 2 1
asymptoticbehavioroftheautocorrelationfunctiona(T) havea si∼mple scaling form(it is notsimply a function of
of the field X(T). From Eqs. (27) and (22) we find, for t /t ).
1 2
T ,
→∞
exp( T/2), d=2,
a(T)∼ exp(−5T/4), d=3. (35) V. CONCLUSION
(cid:26) −
We now turn to the results. We have studied the effect of shear flow on the persis-
tence exponent θ, for a system with nonconservedscalar
order parameter, using an approximate analytical ap-
IV. RESULTS proachbased on the OJK theory and exploiting the “in-
dependent interval approximation”. The persistence is
′ ′
ThetermA(0)caneasilybeevaluatedtogiveA(0)= defined in a frame locally moving with the flow.
17/2/π in d = 3 and √2/π in d = 2. From (34) The exponent θ is nontrivial and is increased by the
− −
F(0) = 1, and from (35) F(s) diverges to for s presenceofshear. Thisimpliesthattheshearaccelerates
p −∞ →
5/4 and 1/2 in d = 3 and 2 respectively. Therefore, the change of sign of the fluctuating field. In dimension
− −
the zero of F(s) lies in the interval (-5/4,0) and (-1/2,0) d=2wefindtheintriguingresultthatθhasavalueequal
for d = 3 and 2 respectively. Solving (34) numerically totwicethatoftheunshearedsystemind=1,withinthe
for this zero, we get the IIA values for the persistence 0JKtheory. The autocorrelationexponentλincreasesin
exponent as θ = 0.5034... for d = 3 and θ = 0.2406... the presence of shear for d = 3 but is unchanged by the
in d = 2. In the absence of shear the IIA gives [3] θ = shear in d=2.
0.2358... in d = 3 and 0.1862... in d = 2, which agree For nonconserved dynamics in the absence of shear,
quite well with simulations [19]. experiments on liquid crystals have been performed to
A very interesting feature of the d = 2 result for θ measure both θ [6] and λ [20]. There is also a recent ex-
is that it is exactly twice the value of the exponent ob- periment on the measurement of a two-time correlation
tained within the same approximation (i.e. using OJK functioninorder-disorderphasetransitioninCu Au[21].
3
theory and the IIA) for the unsheared problem in one Liquid crystal experiments are a possible candidate for
space dimension: θd=2 = 2θd=1. That this must be so testing ourpredictions ina modelwith nonconservedor-
sh unsh
is easily seen directly from the form (28) for a(t1,t2) for derparameter,andnumericalsimulationsmayalsoprove
the sheared problem in d = 2. The equivalent result useful. On the analytical front, the method of the cor-
for the unsheared system in general space dimension is relator expansion [4] might be used to obtain a more
a(t ,t ) = sechd/2(T/2) [3]. For d = 1 this is identi- accurateresultforθ ind=3thancanbe obtainedusing
1 2
cal to Eq. (28) apart from an overall factor 2 in the the IIA.
6
Acknowledgments
The work of NR was supported by a Commonwealth
Fellowship – UK.
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