Coarsening Dynamics of a Nonconserved Field Advected by a Uniform Shear Flow

4 Pages·0.11 MB·English

by  Alan J. Bray

Coarsening Dynamics of a Nonconserved Field Advected by a Uniform Shear Flow
Alan J. Bray and Andrea Cavagna
Department of Physics and Astronomy, The University, Manchester, M13 9PL, UK
(February 6, 2008)
0
Weconsidertheorderingkineticsofanonconservedscalarfieldadvectedbyauniformshearflow.
0
Using the Ohta-Jasnow-Kawasaki approximation, modified to allow for shear-induced anisotropy,
0
we calculate the asymptotic time dependence of the characteristic length scales, L and L , that
2 k ⊥
describe the growth of order parallel and perpendicular to the mean domain orientation. In space
n dimension d= 3 we find L ∼γt3/2, L ∼ t1/2, where γ is the shear rate, while for d=2 we find
a L ∼γ1/2t(lnt)1/4, L ∼γk−1/2(lnt)−1/⊥4 . Our predictions for d=2 can be tested by experiments
J k ⊥
on twisted nematic liquid crystals.
0
2
The coarsening dynamics of systems quenched from To address these issues it would be helpful to have an

] a disordered phase into a two-phase region are by now exactly soluble model. The large-n limit of the n-vector
h
reasonably well understood [1]. Domains of the ordered model has been solved for a conserved order-parameter
c
e phases form rapidly, and then coarsen with time t, i.e. field, but without hydrodynamics (i.e. the order param-
m thereisacharacteristiclengthscale(‘domainsize’),L(t), eter is simply advected by the shear flow) [7]. Length
- which grows with time, typically as a power law. Fur- scalesLkandL⊥,growingasγ(t5/lnt)1/4and(t/lnt)1/4
at thermore,thereis goodevidence foraformofdynamical respectively,describe correlationsalong and perpendicu-
t scaling in which the domain morphology is statistically lar to the flow. Scaling is not strictly satisfied (there is
s
. scale-invariantwhen all lengths are scaled by L(t) [1]. instead a form of ‘multiscaling’), but this is presumably
t
a In recent years attention has been directed at sys- an artifact of the large-n limit, as in the zero-shear case
m temssubjectedtoexternaldriving,e.g.animposedshear [8]. There is no saturation at late times in this model.
- [2–4]. The shear induces ananisotropy,leading to differ- However,for the large-n model there are no domains, so
d
entcoarseningratesindirectionsparallelandperpendic- the conceptofstretchingandbreakinglosesits meaning.
n
o ular to the flow. This has been observed in experiments What is needed is an analytically tractable model for a
c [3]. scalar order parameter. For a nonconserved field, the
[ Animportantopenquestionforsuchdrivensystemsis Ohta-Jasnow-Kawasaki (OJK) approximation [9] fulfills

1 whetherthecoarseningcontinuesindefinitely(foraninfi- thisrequirement: Intheabsenceofshear,itcapturesthe
v nite system), asin the caseofno driving,or whether the essential features of the coarsening process [1].
9 driving force arrests the coarsening, leading to a steady In this Letter we present the results of applying the
9
state. For the case of a sheared phase-separating binary OJK approach to a nonconserved scalar field advected
2
fluid,ithasbeenargued,onthebasisofthestabilityofa by a uniform shear flow. The equation of motion for the
1
0 singledropofonefluidimmersedinanother,thatthedo- order parameter is
0 mainscalewilleventuallysaturateatamaximumlength ∂ φ+ (uφ)= 2φ V′(φ), (1)
0 scale,Lmax,determinedbytheshearrate: Lmax σ/γη, t ∇· ∇ −
/ ∼
t where σ and η are the surface tension and viscosity re- where V(φ) is a symmetric double well potential, and
a
spectively [2]. In the multi-domain context, it has been u = γyxˆ is the velocity of the imposed shear flow, with
m
argued[5]thatasteadystatewouldbe achievedthrough the flow in the x-direction. Two aspects of real binary
-
the shear-induced stretching and breaking of domains. fluids are neglected in this model: the order parameter
d
n However,the experimental evidence for a steady state is is not conserved, and it is simply advected by the shear
o notcompletelyclear-cut. Inparticular,asemphasizedby rather than being coupled to the fluid velocity through
c Cates et al. [6], saturation of the domain length occurs theNavier-Stokesequation. Thisapproachis,however,a
:
v naturally in a finite-size system when the domain length veryinstructivefirststepwhichyieldsimportantinsights
i becomes of the same order as the system size. concerning the main questions raisedearlier, namely the
X
A second important question concerns the nature of nature the asymptotics and of dynamical scaling, in the
r
a the growth laws, and the nature of scaling – if it exists physically correctcontext of a scalar field. Furthermore,
– in this anisotropic system. Naively, one might expect in d = 2 this model describes the coarsening dynam-
two characteristic length scales, L and L , measuring ics of a twisted nematic liquid crystal, in which discli-
k ⊥
correlations parallel and perpendicular to the flow. In nation lines, separating domains of opposite twist, relax
dimension d = 2, for example, it would be natural to viscously, driven by their line tension [10]. Under shear,
conjecture that the coarsening follows conventionalscal- this system will furnish an experimental test of our pre-
ing when lengths along and perpendicular to the flow dictions. It should be noted that our analysis leads to
are scaled by these characteristic lengths. We will show, very different behavior in d = 2 and d = 3: This is one
however, that the actual scaling is subtly different. of the main results of the present work.
1
The result of the OJK analysis is that, for space di- ∂ m+γy∂ m= 2m D (t)∂ ∂ m (6)
t x ab a b
meneinnsgiornatde=pa3r,aLllekl∼toγtt3h/e2 flanowd Lis⊥e∼nhta1n/2c,edi.eb.ythae cfaocatrosr- Dab =∇∂(am−m∂)b2m . (7)
γt relative to the unsheared system, while the growth of (cid:28) ∇ (cid:29)
L is unchanged (the same features describe the large-n Intheabsenceofshearthecoarseningisisotropic,and
⊥
result [7]). Hence there is no saturation of the coarsen- the matrixD has the simple formD =δ /d, indepen-
ab ab
ing. For d = 3, furthermore, conventional scaling holds. dentoft. Theresultingdiffusionequationformisreadily
For d = 2, however, very different results are obtained: solved, leading to a √t coarsening. For γ =0, the coars-
6
L γ1/2t(lnt)1/4, and L γ−1/2(lnt)−1/4. These re- ening is anisotropic: D is both non-diagonaland time-
k ⊥ ab
∼ ∼
sults imply L L t for d = 2, i.e. the product of the dependent. From the definition (5) of D , though, the
k ⊥ ab
∼
two length scales, or ‘scale area’, is independent of the sum-rule, TrD = 1, is trivially valid. From the symme-
shear rate and has the same form as for the unsheared try of (6) under the combined transformations x x,
→ −
system, where L = L t1/2. We will show that this y yatfixedz,andundertheseparatetransformation
k ⊥ ∼ →−
result can be understood by a topologicalargument. An z z at fixed x,y, we see that D has a block diago-
ab
→−
importantsubtletyind=2isthatL andL havetobe nalform,with D =D the onlynon-zerooff-diagonal
k ⊥ xy yx
defined as characteristic scales parallel and perpendicu- elements.
lar to the mean domain orientation, instead of the flow In Fourier space, (6) reads
direction. This distinction is not important for d = 3,
∂m(k,t) ∂m(k,t)
but crucial for d = 2. Only with the new definition is γk = Ω (t)k k m(k,t) , (8)
x ab a b
dynamicalscalingrecovered. Ford=2,furthermore,L⊥ ∂t − ∂ky − ab
X
decreaseswithtimeasymptotically. Ourapproachbreaks Ω (t)=δ D (t) . (9)
ab ab ab
down, however, when L becomes comparable with the −
⊥
width,ξ,oftheinterfacesbetweendomains,whichoccurs This is readilysolvedbythe changeofvariablesq=Ak,
after a time of order exp(const/γ2ξ4), when we expect τ =t, and µ(q,τ)=m(k,t), where A has elements
the domains to break, as observed in simulations with
conserved dynamics in d=2 [5]. Aab =δab+γtδa2δb1. (10)
The OJK approach starts from the Allen-Cahn equa-
In the new variables, the left-hand side of (8) becomes
tion [11] relating the normal component of the inter-
∂ µ(q,τ). Integrating the equation, and transforming
face velocity, v , to the local curvature of the interface, τ
n back to the original variables, gives
K = n, where n is the normal to the interface,
∇·
1
vn =−∇·n+u·n, (2) m(k,t)=m(k˜(t),0)exp −4 kaMab(t)kb!, (11)
ab
where the final term is the drift due to the shear. The X
derivation of this equation from (1) follows the same where k˜(t) = (k ,k + γk t,k ) and the matrix M is
x y x z
route as the zero-shear case [1]. The next step is to given by
introduce a smooth auxiliary field m(x,t) whose zeroes
coincide with those of φ. In a frame comoving with the M(t)=AT(t)R(t)A(t) (12)
interface one has dm/dt = 0 = ∂ m+v m. Combin- t
t n|∇ | R(t)=4 dt′[AT(t′)]−1Ω(t′)[A(t′)]−1. (13)
ing this with (2), and using
0
Z
n= m/ m, (3) Equations(11–13)determinethefunctionm(k,t)com-
∇ |∇ |
pletelyifthematrixD (t)isknown. However,Disitself
ab
yields the following equation for m,
determined from the distribution for m, via (7), so we
have to solve these equations self-consistently. We take
d
∂ m+u m= 2m n n ∂ ∂ m . (4) the initial condition, m(k,0) to be a gaussian random
t a b a b
·∇ ∇ − variable with correlator m(k,0)m(k′,0) =∆δ(k+k′).
aX,b=1 Then the real-space corrhelation functioniof m, G(r,t) =
So far this is exact. Equation (4) is highly non-linear, m(x+r,t)m(x,t) is obtained from (11) as
h i
however, due to the implicit dependence of n on m
through (3). The OJK approximation involves lineariz- 1
ing the m equation by replacing the product nanb by its G(r,t)=G(0,t) exp−2 ra(M−1)abrb , (14)
spatial average, a,b
X
 
D (t)= n (x,t)n (x,t) , (5) where the precise expression for G(0,t) is not relevant
ab a b
h i in what follows. All the information concerning domain
leading to the following equations for m and D: growthin this system is contained in the matrix M (t).
ab
2
To obtain a closed set of equations we have to express Using these limiting forms, the integrals (17) can be
the elements of the matrix D in terms of the elements evaluated asymptotically. After some algebra one finds
of M. To to this we first rewrite (7) using an integral
representation for the denominator: 1 1 D∞ 1/2 −1
D∞ = 1+ − 22 (20)
22 2 D∞
1 ∞ u (cid:18) 22 (cid:19) !
D = du ∂ m∂ m exp ( m)2 . (15)
ab 2Z0 D a b (cid:16)−2 ∇ (cid:17)E w(2h0i)lehDas3∞3th=e n1o−n-tDri2∞v2iaalnsdolDut1i3on=DD∞23==4/05.,Eimqupalytiinong
22
Since m is a Gaussian field, the required average can D∞ =1/5 and demonstrating the self-consistency of the
33
be computed using the probability distribution P(v) of initial ansatz. Finally we find D11 3ln(γt)/(γt)2 and
the vector v = m. This distribution is determined →
∇ D12 6/(5γt),consistent with our initial assumption.
bmy2th(eMco−r1r)ela,tofrrohmvawvhbiic=h o−n∂eai∂nbfhemrs(trh,at)tm(0,t)i|r=0 = Th→ec−haracteristiclengthscalesinthesystemaregiven,
h i ab from (14), by the square roots of the eigenvalues of the
matrix M. Using (19), with D∞ =4/5, we find
22
1
P(v) exp v M v . (16) 2 1 4
∝ −2hm2i Xa,b a ab b Lk = √15γt3/2, L⊥ = √5t1/2, L3 = √5t1/2, (21)
 
Carrying out the average in (15) gives for t . The corresponding eigenvectors are
→∞
Dab = 12 ∞du ddeettMN 1/2 (N−1)ab (17) ek = 213γt , e⊥ =−123γt , e3 =00 , (22)
Z0 (cid:18) (cid:19) 0 0 1
Nab =Mab+uδab . (18)      
implying that the principal axes in the xy plane are ro-
We now proceed to outline the solution of this closed tated, relative to the x and y axes, through an angle
setofequations. Fulldetails willbe givenelsewhere[12]. 3/2γt, which we can interpret as the angle between the
We are interested in the large-t asymptotics. This limit averageorientationofthe domainstructure and the flow
simplifies the analysis which is still, however, quite sub- direction.
tle. The results are very different in three and two di- Thethreelengthscalesarealldistinct,thoughL⊥ and
mensions, so we discuss these cases separately. L3growinthesameway,andcoarseningcontinuesindef-
We begin with some general remarks on the expected initely–thesystemdoesnotapproachastationarystate.
form of Dab. The effect of the shear is to produce elon- The matrix elements Mab grow in the way expected if a
gateddomainstructuresaligned,atlatetimes,atasmall naive form of scaling holds: M11 = L2k, M12 ∼ LkL⊥,
angle(∼1/γt- see below)to the flowdirection. As a re- M22 ∼ L2⊥, and M33 = L23. This means that dynamical
sult, the component n1 of the normal to the interface scaling holds, and the scaling variables can be taken to
is very small almost everywhere at late times, implying be x/Lk, y/L⊥, and z/L3. The same simple structure
D11 0 for t . The sum rule, TrD = 1, is there- does not, however,hold in d=2.
→ → ∞
fore exhaustedby the remainingdiagonalcomponents of The case d=2. For d=2 the self-consistency prob-
D for large t. In particular, for d = 2 we have D22 1 lem is more tricky, because the quantity 1 D22(t) =
→ −
fort ,andΩ11 0in(8). Thiscaserequiresspecial D11(t) tends to zero as t . For d = 2 the inte-
→∞ → → ∞
care. For d=3, on the other hand, D22+D33 1, and grals(17)canbe evaluatedexactly. Making the assump-
→
it turns out that both D22 and D33 approach non-zero tions, to be verified subsequently, that asymptotically
limits. This case is, therefore, simpler to analyse. M11 M12 M22, that TrM √detM M22,
≫ ≫ ≫ ≫
The case d=3. With the assumption that D11 0 and that D12 1/t, one can derive the following self-
→ ∼
and D12 1/t, while D22 and D33 remain non-zero for consistent equation for D11(t):
∼
t , the asymptotics of the matrix elements M are
ab
re→adi∞ly obtainedMfro11m=(94),γ(21t30)(,1(12D)2∞a2n)d (13): D11(t)= γ1t2 R0t0tdtd′tt′′D2D111(1t(′t)′)!1/2 , (23)
3 − with asymptotic solutionR
M12 =2γt2(1 D2∞2)
− 1
M22 =4t(1 D2∞2) D11(t)= . (24)
− 2γt√lnγt
M33 =4tD2∞2 , (19)
Using this result, the asymptotic results for the matrix
whereD2∞2 isthelarge-tlimitofD22,whileM13 =M23 = elements, and the determinant, of the correlationmatrix
0 by symmetry. M are obtained as
3
M11(t)=4γt2 lnγt treatment is based on the ‘thin wall’ limit, in which do-
mainwallsaretreatedashavingzerowidth, itwillbreak
M12(t)=4t lpnγt
down when L becomes comparable with the width, ξ,
⊥
M22(t)=(4/pγ) lnγt of the walls, at which point we conjecture that domains
detM(t)=4t2 , p (25) willbreak,possiblyarrestingthecoarsening. Thiscanbe
testedbyexperimentsontwistednematicliquidcrystals.
while D12 = −1/γt. These results confirm, a posteriori, Ind=3,thepresentworkprovidesstrongevidencethat,
the assumptions made in their derivation, i.e. the solu-
atleastforthenonconservedscalarfieldconsideredhere,
tionisself-consistent. TheasymptoticresultsfortheM
ab thecoarseningstateproceedsindefinitely. Inthisrespect
seem to imply detM = 0, in contradiction to (25). To
itisinterestingthat,intheird=3simulations(including
obtain (25) one has to keep subdominant contributions
both conservation of the order parameter and hydrody-
to the M . [Note that detM =detR from (12)].
ab namics), Cates et al. [6] found no evidence for a steady
The characteristic length scales are given, as before,
state structure emerging that is independent of the sys-
by the square roots of the eigenvalues of M. The eigen-
tem size, i.e. observed steady states could be attributed
values of any 2 2 matrix can be expressed as λ =
× ± to finite size effects.
[TrM (TrM)2 4detM]/2. UsingTrM √detM
± − ≫
we obtaipn λ+ =TrM and λ− =detM/(TrM), whence We thank Peter Sollich for a useful discussion. This
1 work was supported by EPSRC (UK) under grant
L =2√γt(lnγt)1/4, L = . (26)
k ⊥ √γ(lnγt)1/4 GR/L97698.
The corresponding eigenvectors are
1 1
ek = 1 , e⊥ = −1γt , (27)
(cid:18) γt (cid:19) (cid:18) (cid:19)
giving a tilt angle 1/γt between the domain orientation
[1] A. J. Bray, Adv. Phys. 43, 357 (1994), and references
and the flow direction.
therein.
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→∞
4