Coarsening of stripe patterns: Variations with quench depth and scaling
Ashwani K. Tripathi and Deepak Kumar*
School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India(Received 30 May 2014; published 26 February 2015)
The coarsening of stripe patterns when the system is evolved from random initial states is studied by varyingthe quench depth ε, which is a measure of distance from the transition point of the stripe phase. The dynamicsof the growth of stripe order, which is characterized by two length scales, depends on the quench depth. Thegrowth exponents of the two length scales vary continuously with ε. The decay exponents for free energy, stripecurvature, and densities of defects like grain boundaries and dislocations also show similar variation. This impliesa breakdown of the standard picture of nonequilibrium dynamical scaling. In order to understand the variationswith ε we propose an additional scaling with a length scale dependent on ε. The main contribution to this lengthscale comes from the “pinning potential,” which is unique to systems where the order parameter is spatiallyperiodic. The periodic order parameter gives rise to an ε-dependent potential, which can pin defects like grainboundaries, dislocations, etc. This additional scaling provides a compact description of variations of growthexponents with quench depth in terms of just one exponent for each of the length scales. The relaxation of freeenergy, stripe curvature, and the defect densities have also been related to these length scales. The study is doneat zero temperature using Swift-Hohenberg equation in two dimensions.
The understanding of growth dynamics of phases wherethe order parameter has spatial periodicity as in crystals, is animportant and long-standing problem. The stripe phase, whichis the simplest of such examples, has been studied extensivelydue to its occurrence in a number of physical, chemical,and biological sytems in equilibrium and nonequilibriumsituations [1,2]. Results of these studies [3–15] have beensomewhat ambiguous, when interpreted in the frameworkof nonequilibrium dynamical scaling, which has been quitesuccessful for a number of systems [16–18].
The dynamical scaling principle for the growth of spa-tially homogeneous order as in a ferromagnet or a nematicliquid crystal is based on the observation that the systemat different times looks statistically similar if it is scaledby a time-dependent length scale. The length scale l(t) hastypically a power-law dependence, i.e., l(t) = l0t
1/z. Thedynamical exponent z is fairly universal and just dependson its universality class. There are few universality classesdetermined by broad factors like nature of the order parameter,viz scalar, vector, etc., and the nature of dynamics, conservingor not conserving the order parameter. Defects, particularlythe topological defects, which are essentially determinedby the manifold of the order parameter, play a significant rolein the dynamics [16–18].
There are two levels of problems when these ideas areapplied to stripe order, which is composite with a rathercomplex manifold [19]. This order allows two length scalesassociated with (i) the full stripe order and (ii) the orientationalorder of the stripe director [20]. Further, three kinds oftopological defects [19] can occur, which are grain boundaries,dislocations, and disclinations. The stripe order is degraded byall types of defects, but the orientational order is not muchaffected by dislocations and in two dimensions orientational
order can persist even when the full stripe order is absent. Thegrowth exponents of the length scales associated with theseorders may also differ.
The second level of problem, as shown by the literaturesurvey below, is that the growth exponents and relaxationof quantities like free energy seem to depend on the quenchdepth and noise (temperature), which is quite a departurefrom the above universality paradigm. To resolve these issues,in this paper we present a systematic study of the variationof growth dynamics with quench depth ε [see Eq. (2) for thedefinition of ε]. We find that the dynamical exponents dependcontinuously on the quench depth over a range where thestripe order is well defined.
We provide a compact description of the variation withquench depth by using a scaling with respect to a length de-pendent on the quench depth. An important contribution to thislength is due to the nonadiabatic or pinning effects, which ariseas the stripe periodicity provides an oscillatory potential to alllarge-scale structures, such as grain boundaries, dislocations,etc., as was pointed out by Pomeau [21]. A compact descriptionis reached in terms of just two growth exponents related to fullstripe order and the orientational order.
The stripe patterns are locally described in the followingtypical form:
ψ(�r,t) = A(�r,t) cos(�q · �r + φ), (1)
where the amplitude A, wave vector �q, and phase φ varyover a length scale much bigger than q−1. These parameterstogether define the stripe order. A simple model that has beenextensively studied to understand the general properties ofstripe patterns is the Swift-Hohenberg model [22]. Here thescalar order parameter ψ(�r,t) satisfies the dynamical equation
∂ψ(�r,t)∂t
= [ε − (∇2 + q2
0
)2]ψ(�r,t) − ψ3(�r,t) + η(�r,t). (2)
ε is termed as the quench depth and η(�r,t) is the random noiserepresenting thermal fluctuations. The present study is at zero
ASHWANI K. TRIPATHI AND DEEPAK KUMAR PHYSICAL REVIEW E 91, 022923 (2015)
noise and for small ε up to a value of 0.25. The linear termfor ε > 0 leads to the growth of stripe order at wavevectors ofmagnitude around q0, which is saturated by the cubic term.
When we study the evolution from random initial conditionssmall domains of stripes with different orientations and phasesform along with a number of defects. With time the patterncoarsens through annihilation of defects, growth of domains,and straightening of stripes. The amplitude A reaches itssaturation value of order
√ε early, in time of order ε−1,
except in defected regions. In defected regions, such as grainboundaries and dislocations, the amplitude is much smaller.It builds up as the defects annihilate, which occurs on amuch longer time scale. Similarly, for small ε, the wave-vectormagnitude relaxes to values close to q0 in each domain. Forthe subsequent slower evolution, a description in terms of asimpler order-parameter consisting of direction �n of �q andphase φ is quite physical. The order-parameter manifold is akind of two-dimensional torus whose defect structure has beenanalyzed by Chen et al. [19].
This manifold allows for topological defects, dislocations,and disclinations, but since �q and φ are part of an overall phase�(�r) = φ + �r · ∇�|�r=0, there are restrictions on standardtopological analysis. First, the Goldstone mode correspondingto the rotation of �q does not exist, which allows for sharpdomain walls. Second, disclinations with winding numbergreater than +1 are not allowed [19].
The stripe order is characterized by the structure factorassociated with the field ψ(�r,t), which is
Sψ (q,t) = 〈|ψ(�q,t)|2〉q , (3)
where ψ(�q,t) denotes the Fourier transform of ψ(�r,t) andbracket, the angular average for a fixed magnitude of q.Even in the restricted description mentioned above, the stripeorder requires spatial coherence of phase φ and wave-vectordirection �n. In light of the work of Toner and Nelson [20],which shows that in two dimensions the loss of coherence in φ
need not imply a similar loss of coherence in �n, it is importantto study the orientational order. This is done using the directorcorrelation function defined as
Cnn(�r,t) = 2
N
∑�x
〈[�n(�x + �r,t) · �n(�x,t)]2〉 − 1, (4)
where N is the total number of lattice points and for each latticepoint �n = �∇ψ/| �∇ψ | (see Sec. III for details of computation).Cnn is a measure of nematic order as �n and −�n are equivalentfor stripes. We further perform an angular average for a fixed r ,Cnn(r,t) = 〈Cnn(�r,t)〉r . For the scaling to hold these functionsshould have the form
Sψ (q,t) = lp(t)FS[(q − q0)lp(t)], (5)
Cnn(r,t) = GS[r/ ln(t)], (6)
with lp(t) = a1txp and ln(t) = a2t
xn .Swift-Hohenberg equation (SHE) has potential dynamics
described by the free-energy functional of the following form:
F (t) =∫
dr
{1
2
[(q2
0 + ∇2)ψ
]2 − 1
2εψ2 + 1
4ψ4
}. (7)
One can monitor the evolution of the excess free energy�F (t) = F (t) − Feq. Here Feq = −ε2/6, which is the min-
imum energy for a single domain pattern of the form ψ =√4ε/3 cos(q0x).In general, stripes are curved and the growth of stripe order
requires relaxation of this curvature. Accordingly, we will alsostudy the relaxation of the stripe curvature. The stripe curvatureis defined as
κ(�r,t) = 2π | �∇ · �n|/q0. (8)
We have also tracked the densities of defects in grainboundaries, disclinations, and dislocations with time.
II. REVIEW OF THE PREVIOUS LITERATURE
We now review briefly the earlier numerical work on thecoarsening of stripe patterns using SHE. The first quantitativestudy was reported by Elder, Vinals, and Grant [3], who workedat ε = 0.25. They included noise and showed the existence ofa transition with noise between a phase exhibiting power lawrelaxation to a phase having exponential relaxation toward aliquid-like state. They also studied the dynamical scaling of thestructure factor [Eq. (5)] and found good scaling with exponentxp = 1/5 at zero noise. This is a departure from commonvalues of 1/2 and 1/3 seen in systems with homogeneousorder. Further they found that the exponent changes to xp =1/4 in the presence of noise.
This paper was followed by several studies [6–8,10,13],which extended this work and confirmed some findings for ε =0.1 and 0.25. The orientational correlation function was stud-ied and the growth exponent for these correlations was foundto be 1/3 for ε = 0.1 [13] and 1/4 for ε = 0.25 [6,8] withoutnoise, showing the presence of two length scales. The excessfree energy was found to decay as power law with exponents1/4 for ε = 0.25 [7] and 1/3 for ε = 0.1 at zero noise [13].An exhaustive study of defect densities at zero noise [13]finds that defects in grain boundaries and dislocations decaywith exponent 1/3, whereas disclinations decay faster withexponent 0.57. Both the growth and relaxation exponentsbecome larger with noise. Simulations at a higher ε = 0.75 [7]find that the coarsening process slows down considerably, andthe growth of lp is possibly logarithmic. This account showsthat no unique value for the growth exponents has been settledupon. There are indications that the exponents depend on ε andnoise, but the matter has not been investigated systematically.
In Refs. [3,8] arguments were proposed for the exponentof 1/4. These arguments assume that the relaxation of thestripe curvature or that of the interface parallel to such stripesis the rate-limiting process. This relaxation was then shownto be proportional to the second derivative of the curvaturewith respect to coordinate along the stripes. The dimensionalargument then yields the exponent 1/4.
Boyer and Vinals [10] attempted to resolve the issuesregarding two length scales and noise dependence by arguing,that close to the threshold coarsening is dominated bygrain boundary motion and relaxation. They argued that asingle exponent, 1/3 dominates the domain growth and otherrelaxational processes as ε → 0. In particular, at ε = 0.04 thegrowth exponent for lp and relaxation exponents for grainboundary and curvature are equal. They further claimed thatthe scaling exponents do not change with noise at small ε.They provided a rationale for the exponent value 1/3 by
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studying an example of the grain boundary motion betweentwo semi-infinite domains such that stripes on one side areparallel to the boundary, while on the other side they areperpendicular to the boundary [9]. They showed that theboundary moves only if it has a nonzero curvature κ , andits speed is vGB = ε−1/2κ2. Then the dimensional argumentssuggest that the length scale grows as t1/3.
A significant point made by Boyer and Vinals [11] isregarding nonadiabatic or pinning effects. Such effects ariseas the stripe periodicity provides an oscillatory potential to alllarge scale structures, such as grain boundaries, dislocations,etc. [21]. Clearly this becomes important when the largelength scale ∝ε−1/2 and λ0 = 2πq−1
0 are of similar order.The strength of this potential was calculated to be of theform ε2 exp(−2α/
√ε) by Malomed et al. [23] and Boyer and
Vinals [11]. Accounting for this Boyer and Vinals showed thatthe grain boundary discussed above gets trapped if its curvatureis less than a critical value R−1
g ,
Rg = 1√ε
exp
(α√ε
), (9)
where α is of the order of unity.
III. COARSENING AND VARIATION OF EXPONENTS
We now present our numerical results. We have solvedSHE using finite difference scheme described in Ref. [8].All of our simulation have been done on 512 × 512 latticewith lattice spacing �x = π/4 and time interval �t = 0.03with periodic boundary condition. We have used two types ofrandom initial conditions, Gaussian and uniform, with zeromean and a variance of 0.1. Ensemble average has beendone on 60 independent runs. We report results on orderparameter structure factor Sψ (q,t), director correlator Cnn(r,t),free energy, relaxation of stripe curvature, and populations ofdefects in grain boundaries and dislocations for eight differentvalues of ε ranging from 0.02 to 0.25.
A. Coarsening of Sψ (q,t)
We begin with results on Sψ (q,t). Figure 1(a) shows thestructure factor Sψ (q,t) for ε = 0.1 at different times, whileFig. 1(b) shows its plots for different values of ε at t = 30 000.These curves show that the angle-averaged structure factorhas peaks at q ≈ 1.02 for all times and all values of ε studiedhere. To find the time dependence of the correlation lengthlp we calculate the width δq at which Sψ (q,t) reduces tohalf of its maximum value (lp is taken as 1/δq). For this
t 6000t 24000t 42000t 60000
0.90 0.95 1.00 1.05 1.100
5
10
15
20
25
30
q
S ψq
(a)
0.040.080.150.25
0.90 0.95 1.00 1.05 1.100
10
20
30
40
q
S ψq
(b)
t 6000t 24000t 42000t 60000
10 5 0 5 100.00
0.05
0.10
0.15
0.20
0.25
0.30
q q0 lp t
S ψq,
tl p
t
(c)
5000 1 104 2 104 5 104
0.010
0.020
0.030
0.015
t
δq
0.04
0.08
0.15
0.25
(d)
FIG. 1. (Color online) (a) Sψ (q,t) versus q at different times for ε = 0.1. (b) Sψ (q,t) versus q at different values of ε at a time t = 30,000.(c) Data collapse of Sψ (q,t) according to Eq. (5) for ε = 0.1. (d) Variations of δq with time on a log-log plot for four values of ε.
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TABLE I. The Table shows variations with ε of effectivedynamical exponents of lp (column 2) and ln (column 3).
purpose we fit the structure factor data to square Lorentzianfit, a2/[(q2 − b)2 + c2]2. Then δq is given by 0.322c/
√b.
The peak height of Sψ (q) is proportional to 1/δq. The datacollapse on scaling with lp is shown in Fig. 1(c). This justifiesthe scaling in Eq. (5). Figure 1(d) shows plots of δq with timeon a log-log plot for different values of ε. Straight lines provegood power law fits of the form at−xp .
The dynamical exponents xp for different values of ε
are shown in the second column of Table I. As can beseen, the exponent decreases continuously with increasing ε
showing the slow down of coarsening. We regard ε-dependentexponents as effective exponents, as one expects that a betterargument should provide a more compact description (seeSec. V). Note that the fits to power law show deviations atlarge t , which become larger as ε decreases. These are due tofinite-size effect, which comes into play when lp becomescomparable to system size. For this reason exponents areextracted from the data within a time range that is dependenton ε. The lower limit here is the time (∼3000) by which stripesare well formed, while the upper limit is the time (∼30 000)until finite-size effects are insignificant. While the correlationcurves show a clear trend with ε, the values of exponents atclose intervals have some fluctuation.
B. Scaling of Cnn(r,t)
Next we present the results for the correlation in stripedirector field Cnn(r,t). In order to compute Cnn(r,t), we havefirst evaluated the director field �B,
Bx = n2x − n2
y ; By = 2nxny, (10)
t 6000t 24000
t 42000t 60000
0 50 100 150 2000.0
0.2
0.4
0.6
0.8
1.0
r
Cnn
r
(a)
0.040.08
0.150.25
0 50 100 150 2000.0
0.2
0.4
0.6
0.8
1.0
r
Cnn
r
(b)
5000 1 10 42 10 4 5 10 4
100
50
20
30
15
70
t
Rα
t 0.4
0.5
0.6
0.7
(c)
t 12000
t 24000
t 36000
t 48000
t 60000
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
r ln t
Cr,
t
(d)
FIG. 2. (Color online) Plots of Cnn(r,t) with r , (a) for ε = 0.1 at different times, (b) at t = 30 000 for different values of ε. (c) Rα(t) witht on a log-log plot for different values of α, (d) Data collapse for Cnn(r,t) according to Eq. (6) for ε = 0.1.
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at each lattice point [13]. Then,
Cnn(r,t) = 1
N
∑�x
〈 �B(�x + �r,t) · �B(�x,t)〉r . (11)
The contribution from points where the stripe order is notpresent is negligible.
In Fig. 2(a) we have shown Cnn(r,t) as a function of r
at four different times for ε = 0.1, while in Fig. 2(b) thesame is plotted for four values of ε at t = 30 000. Clearly,this correlation grows with time and the rate of growth slowswith increasing ε. In order to establish scaling and extract thecorrelation length ln associated with Cnn we find the positionRα(t) where C(r,t) = α at five different values of α from 0.3to 0.7 for a given time. Rα(t) for different values of α areshown in Fig. 2(c) on a log-log plot. These parallel straightlines establish scaling as well as the power law growth of ln.Using ln = R0.5(t), we find very good collapse of the data tothe form, Cnn(r,t) = GS[r/ ln(t)] as shown in Fig. 2(d). Herealso deviations due to finite-size effects are seen for largertimes. These are larger than those of Sψ as ln > lp. The growthexponent xn for ln at all values of ε are shown in the thirdcolumn of Table I. The dynamical exponent decreases steadilywith ε, but the values are larger compared to the exponents of
lp. These results show that the orientational order grows fasterthan the full stripe order and has a larger length scale.
C. Relaxation of stripe curvature, free energy, and defects
We now discuss relaxation of stripe curvature and determinethe decay exponent associated with its relaxation. In orderto calculate the stripe curvature, we need to isolate the fullyformed stripe region from the defect sites, where stripe orienta-tions are well defined. To identify these regions we use a filter,which is based on the amplitude of the stripes. We calculatethe stripe amplitude A(�r,t) using the following definition:
A2(�r,t) = ψ2(�r,t) + (−→∇ ψ)2. (12)
This expression can be obtained using Eq. (1) with theassumption that the phase φ is varying slowly and wave vector�q has a very small variation around the critical value q0 = 1.0.Now a site belongs to stripe region if 0.95 < (A/A0)2 < 1.05,where A0 = 2
√ε/3 is the amplitude of the ideal stripe
patterns. After identifying the defect-free stripe regions, stripecurvature can be calculated using Eq. (8). However, we needto incorporate the symmetry of the director �n during thecalculation of the derivatives of �n. For example, suppose thatthe stripe director has the value (nx,ny) and (−nx,−ny) at
two adjacent sites. The calculation of the derivative−→∇ · �n
t 15000t 30000t 45000t 60000
0 0.3 0.6 0.90
0.04
κ(a)
0.040.080.150.25
0 0.3 0.6 0.90
0.04
κ
Pκ
Pκ
Pκ
(b)
t 15000
t 30000
t 45000
t 60000
0 5 10 15 20 25 30 350.0000
0.0005
0.0010
0.0015
0.0020
κtφc
tφc
(c)
5000 1 104 2 104 5 104
4.
5.
6.
7.
8.
9.
t
κt
0.25
0.15
0.08
0.04
(d)
FIG. 3. (Color online) (a) P (κ,t) versus κ at different times for ε = 0.1. (b) P (κ,t) versus κ at different values of ε at time t = 30 000.(c) Scaling of P (κ,t) with time at ε = 0.1. Best data collapse is obtained for φc = 0.33. (d) Relaxation of the average stripe curvature κ withtime for four values of ε.
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TABLE II. Variations with ε of effective dynamical exponentsof �F (column 2), P (C,t) (column 3), NGB (column 4), and NDis
Figure 3(a) shows the probability distribution of stripecurvature P (κ,t) at four different times for ε = 0.1. It is seenthat with time the curvature distribution shifts toward lowervalues of κ . In Fig. 3(b) variation of the distribution P (κ,t)with ε is shown for time t = 30 000. Here with increasingε the distribution shifts generally toward larger κ , but thedistributions are overlapping and the trend is not very clearcut.
We have also looked for scaling of the distribution withrespect to time by using the following scaling ansatz:
P (κ,t) = tφcS(κtφc ). (14)
As shown in Fig. 3(c) a good collapse of data for ε = 0.1at various times is obtained for exponent φc = 0.33. Similarquality collapse is obtained at other values of ε. Exponents fordifferent ε are shown in the column 2 of Table II. Here againwe observe the continuous decrease in φc with ε. We have also
calculated the average stripe curvature κ(t) defined as
κ(t) =⟨
1
Nc
∑�r
κ(�r,t)⟩
. (15)
Nc is the total number of sites where stripe order is defined and〈〉 denotes the average over initial conditions. The relaxationof κ with time is shown in Fig. 3(d) on a log-log plot. A power-law relaxation is seen with exponents decreasing with ε.
We next report results on the decay of the excess free energy�F (t,ε) with time. This is shown on a log-log plot in Fig. 4(a)for four values of ε. The straight-line plots establish the decayas a power law with exponents φf dependent on ε. Theseexponents are listed in the column 3 of Table II. One againfinds a decrease of φf with ε.
Finally, we come to the study of defects and their relaxation.We follow the work of Qian and Mazenko [13] for identifyingdefects and their nature. The total defect density increaseswith ε. The largest proportion of defect sites is in grainboundaries. However, with ε the proportion of defects in grainboundaries decreases while that of isolated defects increases.The isolated defects are mostly dislocations whose numberincreases rapidly with ε. The number of disclinations alsoincreases with ε, but it remains too small (10 percent of thenumber of dislocations) to be analyzed statistically in theε-range studied here.
In Figs. 4(b) and 4(c) the time variations of defect numbersin grain boundaries, NGB(t,ε) and in dislocations, NDis(t,ε)are shown on log-log plots for some values of ε. These plots,particularly for dislocations, show more fluctuations as defectnumbers tend to vary more over the runs and with time. Boththese quantities are seen to have power-law decays. The expo-nents φGB and φDis, extracted from best fits over time rangesdiscussed above, are shown in columns 4 and 5 of Table II.
The grain boundary exponents show the earlier trend ofdecreasing continuously with ε, but surprisingly the exponentsfor dislocations have no significant variation with ε.
IV. PINNING POTENTIAL
These results imply that the standard notion of the dynami-cal scaling does not apply here, since growth exponents for the
103 104 105100
101
102
103
t
FE
t
0.04
0.08
0.15
0.25
(a)
5000 1 104 2 104 5 104
2000
3000
1500
t
NG
Bt
0.040.08
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0.25
(b)
5000 1 104 2 104 5 104
500
300
700
t
ND
ist
0.04
0.08
0.15
0.25
(c)
FIG. 4. (Color online) (a) Excess free energy with time at four different values of ε. (b) NGB vs. t on a log-log plot. (c) NDis vs. t on alog-log plot.
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0 10 000 20 000 30 000100
400
600
800
τ
Vpτ
0.25
0.2
0.15
0.1
(a)
0.10.150.20.25
0 1000 2000 27000
1000
2000
3000
4000
5000
τ
fτ
(b)
0.10.150.20.25
0 10 000 20 000 30 000
6000
7000
8000
τ
fτ
(c)
FIG. 5. (Color online) (a) Plot of �Vp(τ ) = Vp(t) − Vp(t0) vs. τ = t − t0 for four different values of ε. (b) Data collapse of �Vp(τ ) forε with a scaling of the form �Vp(τ ) = ε1.55f (τ ), for the time range 300 < t < 3 000. (c) Data collapse of �Vp(τ ) for ε for the time range3 000 < t < 30 000. Black line is the fit to the collapsed data using the fitting function f (τ ) = ae−b/
√τ .
Sψ (q,t) and Cnn(r,t) are different and decrease continuouslywith increasing ε. One important factor that surely contributesto the slowing down of dynamics with ε is the “pinningpotential.”
This is a unique feature of systems with spatially periodicorder. This potential represent a coupling between the large-scale variation of order parameter such as due to variousdefect structures and the underlying periodicity of stripe pat-terns [21,23]. The order parameter variations occur typicallyon a length scale of the order of ε−1/2. For small ε, whereamplitude equation [1,2] provides a good account of spatialvariations, this scale is much larger than the stripe wavelength.In this regime the coupling is weak because the large scaleof the defect structure makes it immune to underlying stripes.The amplitude-equation formalism ignores these couplings,termed as nonadiabatic effects. However, with the increase inε this separation of length scales becomes less and the stripepotential increasingly affects the dynamics. Clearly when thewidth of the grain boundary or distortions associated withdislocations are comparable to stripe wavelength, the periodicpotential can pin them.
The pinning effects have been studied in the amplitudeequation formalism. In the multiscale perturbation theoryused to derive the amplitude equation only terms with basicperiodicity of the form ei �q0·�r are retained. For larger values ofε, this approximation naturally becomes less and less accurateand one needs to include the nonadiabatic corrections in theform of terms having spatial dependence eim�q0·�r with m � 2.Incorporating nonadiabatic corrections in the amplitude equa-tion, Malomed et al. [23] showed that the potential has the form
Vp(ε) = Bε2 exp(−2α/√
ε). (16)
Boyer and Vinals [11] evaluated the pinning effect on thecurved grain boundary mentioned above, Their calculationshowed that the grain boundary can move only if it has aminimum curvature R−1
g given by Eq. (9).More generally, this leads to the notion of a length scale
associated with the pinning potential, R2g ∝ ε/Vp(ε). An
analysis similar to the multiple-scale pertubation theory [23]shows that the most significant term in the nonadiabaticcorrection to the amplitude equation is of the form A3e2i �q0·�r ,where A is the complex stripe amplitude. The lyapunov
functional corresponding to this term is given by
Vp(t ; ε) =∫
d2rA4e2i �q0·�r + c.c. (17)
We extend the above considerations to the general evolvingstate. For this purpose it is useful to evaluate Vp(t ; ε), whichwe term as the average pinning potential, by adding a positiveterm so as to make its minimum value zero:
Vp(t ; ε) =∫
d2rA4 cos2(�q0 · �r + φ), (18)
where A is given by Eq. (12). We have studied time evolutionof the average pinning potential numerically for four differentvalues of ε ranging from ε = 0.1 to 0.25 as shown in Fig. 5.A(�r,t) for the evolving configurations are evaluated usingEq. (12). Figure 5(a) shows the time evolution of the pinningpotential from the time t0 = 300, when stripe domains arewell formed. This time is fixed by monitoring the stripeconfigurations. As expected, the average potential increaseswith time as the number of defects go down and the stripe orderenhances. Its strength increase by a factor of 5 as ε increasesfrom 0.1 to 0.25. To extract the ε dependence of the potential,we have done a scaling analysis using the following equation:
�Vp(τ ) = Vp(t) − Vp(t0) = εαf (τ ), (19)
where τ = t − t0. A very good data collapse is found forα = 1.55 as shown in Figs. 5(b) and 5(c), in small andlarge time ranges. A fit of the collapsed data gives the timedependence of the pinning potential, f (τ ). At large times,
f (τ ) = ae−b/√
τ , (20)
with the fitting parameters a = 9748 and b = 32.04 for3 000 < t < 30 000.
While the above numerical evaluation of Vp has correcttrends with respect to ε, it does not match the theoreticalprediction given in Eq. (16), which due to the exponentialfactor decreases far more rapidly at small ε in comparisonto numerical results. Two factors may be responsible forthis disagreement. The exponential factor arises due to theassumption that the amplitude varies at a length scale of1/
√ε. The spacing of defects and amplitude variations in
simulations occur at a shorter length scale, which is possibly
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ASHWANI K. TRIPATHI AND DEEPAK KUMAR PHYSICAL REVIEW E 91, 022923 (2015)
due to boundary effects. The values of Rg using Eq. (9) aremuch larger than the system size for ε smaller than 0.1.
As far as we are aware, ours is the first numerical workaimed to test the analytical form of the pinning potential.Boyer and Vinals [11] studied the curvature distribution P (κ,ε)numerically and found that P (0,ε) is proportional to Rg
for 0.3 � ε � 0.5. Though the relation between these twoquantities is not clear to us, it is possible that the discrepancybetween the numerical and analytical results for Vp is larger forsmaller values of ε. Nevertheless, we find this study serves auseful purpose. It suggests a form for Rg that is quite successfulfor scaling of the numerical data with respect to ε as shown inthe next section.
V. SCALING DESCRIPTION FOR VARIATIONSWITH QUENCH DEPTH
In order to understand variations of numerical results withquench depth ε, we seek a scaling description that can providea compact account of these variations. We first note that Rg
derived from the numerical results on Vp does not providescaling to the numerical data for lp or ln. We take it to indicatethat while pinning is a very significant factor for the slow-downwith ε, other nonlinear factors also play a role. Accordingly wedetermine this length scale empirically guided by numericaldata and above ideas. We propose the following scaling ansatzfor the length scale lp(t,ε), which enters the order parametercorrelation function
lp(t,ε) = RgG(tνp/Rg), (21)
where G(x) is a universal function such that it is linear at smallx and Rg is given by
Rg = 1√ε
exp
(α√ε
+ b
2√
t
). (22)
Note that Rg(ε,t) is a combination of Eqs. (9) and (20). Thetime-dependence is needed as the pinning potential increaseswith time and we take it to be that of numerically determinedVp. We have chosen α to be π/2, as this gives the best collapseof data for different ε with maximal overlap on a universal
scaling curve. In Fig. 6(a) we plot lp/Rg against tνp/Rg fordata sets of lp at 8 values of ε. A very good data collapse isobtained for νp = 0.31 in the time range 300 � t � 60 000.The function G(x) is found to be of the form βx/(1 + γ x).The concave nature of the scaling function G accounts for thedecrease of effective growth exponents xp with ε.
Similarly, for the variation of growth exponents for ori-entational correlation, we again try the scaling ansatz ln =RgH (tνn/Rg). In Fig. 6(b) we plot ln(t)/Rg against tνn/Rg fordata sets of ln(t) at 8 values of ε. Now the best fit is obtained forνn = 0.45 in the time range 300 � t � 60 000. The functionH (y) has the similar concave variation as that of G(x). Theeffective growth exponents xn for ln decrease from the value0.45 with ε, which is clearly distinct from those of lp.
In order to understand the dependence of relaxationexponents for free energy within the above scaling scenario,one notes that �F should decrease with the increase inthe order-parameter correlation. Accordingly, we looked forthe relationship between �F (t,ε) and [lp(t,ε)]−1. Plots of�F with (lp)−1 exhibited in Fig. 7(a) show an excellentproportionality between the two, with slopes K(ε) dependenton ε. This dependence is shown in Fig. 7(b) and can be fittedto a form a + bε + cε2.
Next we explore quantitative relationships between defectdensities and the lengths lp and ln. The orientational correla-tions are clearly affected by grain boundaries, but dislocationsaffect orientational order minimally at long range [20].Disclinations also disrupt this order, but their number is toosmall. So one expects a relationship between ln and NGB. Theplots of NGB against [ln(t,ε)]−1 exhibited in Fig. 7(c) show anice linear relation between the two with slopes dependent onε. It correlates the faster growth of ln to the faster decay of NGB.The length lp on the other hand is affected by all defects asit represents the additional phase order. It is therefore shorterand its slower growth is possibly controlled by dislocationsthat annihilate with a smaller exponent. We earlier observedthat the decay exponent for the dislocation density does notshow much dependence on ε. This can be rationalized in thepinning scenario by noting that dislocations are pinned evenby a very weak potential as they necessarily have to be in the
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.2
0.4
0.6
0.8
1.0
1.2
x
l pR
g
(a)
0.0 0.5 1.0 1.5 2.0 2.50.0
0.1
0.2
0.3
0.4
0.5
0.6
y
l nR
g
(b)
FIG. 6. (Color online) (a) The plot of lp/Rg against x = t0.31/Rg establishing scaling of lp with ε as in Eq. (21). (b) The plot of ln/Rg
against y = t0.45/Rg establishing scaling of ln with ε.
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COARSENING OF STRIPE PATTERNS: VARIATIONS . . . PHYSICAL REVIEW E 91, 022923 (2015)
0.040.080.150.25
0.01 0.02 0.030
30
60
90
120
l p1
F
(a)
0.00 0.05 0.10 0.15 0.20 0.250
1000
2000
3000
4000
K
(b)
0.040.080.150.25
0.007 0.02 0.04 0.05
1200
1800
2400
ln1
NG
B
(c)
0.040.080.150.25
0.01 0.015 0.02 0.025
5
6
7
8
9
10
l p1
k
(d)
FIG. 7. (Color online) (a) �F (t) vs. (lp)−1. (b) Variation of K with ε. (c) NGB vs. (ln)−1. (d) Average stripe curvature κ(t) as a functionof (lp)−1.
middle of two stripes. At zero temperature their pinning haslittle dependence on the strength of the potential. The totaldefect density also controls the free energy, which rationalizesthe simple dependence of the free energy on (lp)−1.
A similar argument can also be given for the relaxationof stripe curvature with ε. Here we search for a functionalrelationship between (lp)−1 and κ(t), as we expect therelaxation of stripe curvature is controlled by all types ofdefects. Figure 7(d) shows a nice linear relation between thesetwo quantities for four different values of ε. The slope of thefits depends on ε.
VI. SUMMARY AND CONCLUSION
To summarize, we have studied the coarsening of stripepatterns that are generated by Swift-Hohenberg equation ateight different values of quench parameter ε from 0.02 to0.25. We find that the dynamics slows down with ε as reflectedin continuous variation in growth and relaxation exponents.The stripe order is characterized by correlation functions oforder parameter and stripe orientations. The growth exponentsfor the length scales associated with these correlations aredifferent and decrease with ε. We have additionally studied
the relaxation of free energy, stripe curvature and defectdensities of grain boundaries, and dislocations. These alsoshow power-law behaviors with exponents decreasing with ε.An important factor that gives rise to unusual dependence ofdynamics on ε is the pinning potential, which arises due tothe periodic potential exerted by stripes on various large-scaledefects. We have done a numerical study of the pinningpotential and tested analytical results. We find a discrepancythat we attribute to boundary effects and system size. In order tounderstand the variation with quench depth, we have proposeda scaling with respect to a length dependent on ε. This lengthscale is determined empirically but provides a very goodcollapse of data at different values of ε. Thus, a comprehensivedescription of ε variation is obtained in terms of just twoexponents.
The variation of relaxation exponents quantities like freeenergy, defect densities, and stripe curvature are also includedin this description, as these are shown to be related to thegrowth length scales lp and ln. Since pinning is a commonfeature of crystalline order, this characterization should beuseful for understanding growth of other crystalline phasesand patterns.
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ASHWANI K. TRIPATHI AND DEEPAK KUMAR PHYSICAL REVIEW E 91, 022923 (2015)
ACKNOWLEDGMENTS
A.K.T. acknowledges gratefully a fellowship from theCouncil of Scientific and Industrial Research, India. D.K.
thanks the Department of Atomic Energy, Government ofIndia, for support through the award of Raja RamannaFellowship.
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