Short-range stationary patterns and long-range disorder in an evolution equation for one-dimensional interfaces

0.43 MB·English

by  Javier Muñoz-García

Short range stationary patterns and long range disorder in an evolution
equation for one-dimensional interfaces
Javier Mun˜oz-Garc´ıa,1 Rodolfo Cuerno,1 and Mario Castro2
1Departamento de Matem´aticas and Grupo Interdisciplinar de Sistemas Complejos (GISC),
7 Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911 Legan´es, Spain
0 2GISC and Grupo de Din´amica No Lineal (DNL), Escuela T´ec. Sup. de Ingenier´ıa (ICAI),
0 Universidad Pontificia Comillas, E-28015 Madrid, Spain
2
(Dated: February 6, 2008)

n
Anovellocalevolutionequationforone-dimensionalinterfacesisderivedinthecontextoferosion
a
byion beam sputtering. Wepresentnumerical simulations ofthisequationwhich showinterrupted
J
coarsening in which an ordered cell pattern develops with constant wavelength and amplitude at
3
intermediate distances, while the profile is disordered and rough at larger distances. Moreover, for
2
a wide range of parameters the lateral extent of ordered domains ranges up to tens of cells. This

behaviorisnewinthecontextofdynamicsofsurfacesorinterfaceswithmorphologicalinstabilities.
]
h Wealsoprovideanalyticalestimatesforthestationarypatternwavelengthandmeangrowthvelocity.
c
e PACSnumbers: 47.54.-r,68.35.Ct,05.45.-a,79.20.Rf,surfaces
m
-
t Patternformationisubiquitousinnatureandone sulting interface equation was studied numerically
a
of the most fascinating features of nonequilibrium formoderatesystemsizessuggestingthe occurrence
t
s systems[1]. Typicalexamplescanbefoundinmany of a stationary ordered pattern and interrupted
.
t startlingly similar interfaces which emerge in very coarsening [11]. Here, we derive the 1d counterpart
a
m different processes,suchas growthof amorphous[2] of the height equation in [10] from a physical model
and epitaxial thin films [3], or erosion by ion beam of IBS and perform a systematic numerical analysis
-
d sputtering (IBS) [4]. These pattern forming sur- of its coarsening properties in relation with Krug’s
n faces can be classified into different categories ac- conjecture.
o cording to the stationary or time-dependent (coars- Following the standard assumption made in hy-
c
ening) behavior of the typical pattern length scale drodynamic models of aeolian sand dunes [12], we
[
l. Actually, a large effort has been devoted recently will consider that ripples formed under IBS are

1 to describe these systems through height equations translationallyinvariantintheydirection;addition-
v [5], since these provide compact and efficient ana- ally, we assume symmetry under x x, as occurs
8 →−
lytical/numerical descriptions that successfully de- undernormalincidenceconditionsforthe bombard-
6
scribe global morphological properties such as ki- ing ions [13]. As proposed in [10], the evolution of
5
neticroughening,andsurfacepatternformationand the thickness of the mobile surface adatoms layer R
1
0 coarsening. Specifically, in order to assess the pre- and the height of the bombarded surface h is pro-
7 dictive power of (1d) height equations, it would be vided by a pair of coupled equations, namely,
0 importanttoproducecriteriaforthepresenceorab-
2
t/ sence of coarsening. In Ref. [6] up to four different ∂tR=(1−φ)Γex−Γad+D∂xR, (1)
a scenarios have been proposed depending on the be- ∂th= Γex+Γad, (2)
m −
haviorofl asafunctionofthepatternamplitudeA.
where Γ and Γ are, respectively, rates of atom
- The conclusionis thatcoarseningstops (interrupted ex ad
d excavationfrom and addition to the immobile bulk,
coarsening) ifthe functionl(A)attains amaximum,
n (1 φ) measures the fraction of eroded atoms that
o afterwhichtheamplitudeincreasesindefinitelywith bec−ome mobile, and the third term in Eq. (1) de-
c time. AsimilarconclusionwasreachedatbyKrugin
scribes thermal diffusion of mobile adatoms.
: Ref.[7],whereitisactuallyconjecturedthatno(1d)
v The rate at which material is sputtered from the
Xi local height equation can “describe the emergence bulkisdescribedbymicroscopicderivations[14]and
and evolution of patterns with constant wavelength
depends on the local morphology of the surface
r and amplitude”. Hence, a 1d counterexample of an
a interface equation [5] leading to a stationary pat- Γex =α0 1+α2∂x2h+α3(∂xh)2 , (3)
ternwithconstantwavelengthandamplitude would
be interesting to improve our understanding of in- where α0 is the sp(cid:2)uttering rate for a plan(cid:3)ar surface.
The rate of nucleation is also related to the local
terruptedcoarseningand, specifically,of the type of
shape of the surface, and is given by
nonlinearities that induce it [8].
duRceedcewnthliyc,hadceoscnrtiibneusuimnte2rdesmtiondge(lsuhbas)mbiecernominettrroic- Γad =γ0 R(1+γ2∂x2h)−Req , (4)
features of surfaces eroded by IBS [9, 10]. The re- where Req is the th(cid:2)ickness due to the m(cid:3)obile atoms
that are thermally generated even in the absence of
2
−1
bombardment,and γ is the averagetime between
0
nucleation events. After a multiple scale expansion
of (1)-(4), R can be adiabatically eliminated from
Eqs. (1) and (2) [9, 10], obtaining, to lowest order
near threshold of the morphological instability, the
following1dequationfortheevolutionofthesurface
height,
2 4 2 2 2
∂th(x,t)=−ν∂xh−K∂xh+λ1(∂xh) −λ2∂x(∂xh) ,
(5)
whereparametersarerelatedtothosein(1)-(4)and
depend onthe experimentalconditions. To the best
of our knowledge, the deterministic Eq. (5) has not
been systematically studied. We will restrict our-
selves to positive values of ν and K, which are re-
quired in order to produce a long-wavelength insta-
bility. Moreover,λ1 andλ2 arerequiredto have the
samesignformathematicalwell-posedness,asshown
in [15, 16]. If the signs of the non-linear terms are
simultaneously changed, Eq. (5) remains invariant
after h h. Thus, we will only consider positive
→ −
values of these parameters. For λ2 = 0, Eq. (5) re-
FIG. 1: Height profiles for Eq. (6) with r = 50 on a
duces to the celebratedKuramoto-Sivashinsky(KS)
systemsizeL=128attimes(a)t=2,6,11;(b)t=11,
equation [17, 18], which is a paradigm of spatio-
24, 41; (c) t=41, 70, 103; (d) t=103, 153, 205; (e) t=
temporal chaos. Its nonlinear term stabilizes the
205, 302, 425; (f) t=425, 764, 1024. Height profiles at
system and a (disordered) pattern develops that is
different times evolve by increasing the maximum value
characterizedbyawavelengththatdoesnotcoarsen, of h with time. Allunits are arbitrary.
andbychaoticcelldynamics. Onlargelengthscales,
the KS system can be effectively described by the
stochasticKardar-Parisi-Zhang(KPZ)equation[19],
250randominitialconditions. The standardsystem
paradigmatic of kinetic roughening. In particular,
size of our simulations was L = 512 (1024 nodes),
thesurfaceroughness(globalrmswidth,W)[20]for
except when other values are indicated, and the pa-
a KPZ interface scales as a power law with the lat-
rameters fixed to ν = 1, K = 1, λ1 = 0.1, varying
eral system size L. On the other hand, for λ2 = 0
6 λ2 in order to check for the different values of r.
and λ1 = 0, Eq. (5) reduces to the “conserved” KS
In Fig. 1 the evolution of the height profile is
equation. Thisequationhasbeenstudiedinthecon-
depicted for r = 50. Starting from an initial ran-
text of amorphous thin film growth [15] and step
dom distribution, a periodic surface structure with
dynamics on vicinal surfaces [21]; in this case, the
a wavelength of about the maximum of the linear
linear instability evolves into an ordered pattern of
dispersionrelation[1],namelyl =2√2π,arises
paraboloids with uninterrupted coarsening. linear
and the amplitude of h increases (Fig. 1a). At later
In order to reduce the number of parameters and stages, the conserved KPZ nonlinearity ∂2(∂ h)2
simplify the analysis of (5), we rescale x, t and h x x
by (K/ν)1/2,K/ν2 and ν/λ1, respectively, resulting (cKPZ) induces coarsening of the ordered cell-like
structure,whereinthecellsgrowinwidthandheight
into a single-parameter equation, namely,
and the number of cells decreases. This coarsening
∂ h(t,x)= ∂2h ∂4h+(∂ h)2 r∂2(∂ h)2, (6) issuchthatsmallercellsare“eaten”bylargerneigh-
t − x − x x − x x bors(Figs.1b-e). WeshowinFig.2athetimeevolu-
where r = (νλ2)/(Kλ1) is the (squared) ratio of a tion of the mean height h¯L(t)=1/L xh(x,t), the
linearcrossoverlengthscaletoanon-linearcrossover wavelength l(t), and the amplitude A(t) of the pat-
P
lengthscale. We have performed a numerical in- tern defined as the mean lateral distance between
tegration of (6) using a fourth-order Runge-Kutta two consecutive local minima and the mean verti-
method and the improved spatial discretization in- cal distance from a local minimum to the next lo-
troduced by Lam and Shin [22] for the nonlinear cal maximum, respectively. As seen in Fig. 2a, for
terms. We have used periodic boundary conditions, t&70,thenon-conservedKPZterm(∂ h)2becomes
x
lattice constant ∆x = 0.5 and time step ∆t = 0.01, relevantandthemeanheightofthesurfaceh¯ starts
L
checking that results do not differ significantly for increasing to reacha constant velocity (Fig. 1f). At
smallerspaceandtime steps. The initialheightval- the same time, the coarsening process slows down
ueswerechosenuniformlydistributedbetween0and until stopping completely in the stationary state.
1andstatisticaldatawereobtainedasaveragesover This behavior suggests that the cKPZ term, which
3
102 hlL 106 360 h336505
A 350
101 W S100 350 8250 x 8400
q h
340
100
10-6
330
101 10t2 103 104 0,01 0,1q 1 0 2000 4000x6000 800010000
(a) (b) (a)
(b)
FIG. 2: (a) Time evolution of mean surface height av-
FIG.3: (a) Heightprofileat t=15000 for asystem size
erageh¯L(t),wavelength l(t),amplitudeA(t),andglobal L = 10000 and r = 50; (b) local width vs window size
widthW(t),forr=50;(b)correspondingsurfacestruc-
att=15000 forseveralvaluesofr andL=8192. Inset:
ture factor as a function of wave number q at times samedataplottedvsx0/(2√6r). Allunitsarearbitrary.
t=4,15,30,60,125,250,1129,and10653,bottomtotop
(curvesare offset vertically). All units are arbitrary.
towardsits KSvalue (1/2)for decreasingr. Indeed,
the profile is more disordered for small r, as can be
actsatsmallscales,inducestheorderandthecoars- observedfromFig.4a,totheextentthatforr 0.2,
ening process until the slopes and the characteristic and independently of L, the secondary peaks≈in the
wavelengthof the pattern are large enough to make structurefactorvanishcompletely (not shown),and
the KPZ term no longer negligible. At this time, only a weak peak about the linear instability per-
the KPZ term interrupts the coarsening process, as sists, as in the KS equation. Thus, the KPZ term is
claimed in [8, 10], and a constant average velocity seentoactatlargerscalesandisresponsibleforthe
value is achieved as a consequence of a constant av- disorder of the profile, while the cKPZ terms domi-
erage of slopes across the interface. Thus, the fi- natesatsmallerscaleswithatrendtoorderthecells
nal wavelength of the pattern depends on the inter- vertically.
playbetweenthe cKPZandthe KPZterms. Fig.2a Asindicatedabove,forlargervaluesandafterthe
alsoshowstheglobalsurfacermswidthorroughness initial times, the cell-like structure is well ordered
W(t) [20], and Fig. 2b shows the surface structure and the length scales of the profile are large enough
factor, defined as S(q,t)= hˆ(q,t)hˆ( q,t) , where so that the linear fourth order derivative can be ef-
− fectivelyneglected. Thus,wecanrescalex r1/2x,
hˆ(q,t) is the Fourier transfoDrm of the fieldEh(x,t). →
t rt and h h to obtain an effective parameter-
Thefirstpeakinthestructurefactorindicatesthe → →
free equation. This means that, for large values of
dominantwavelengthofthepattern. Wecanseehow
r, the solution of Eq. (6) remains unchanged if we
this peak moves to larger wavelengths (coarsening) rescale lengths by r1/2 and times by r. In order to
until a fixed mode is reached which corresponds to
check this hypothesis we present in Fig. 4b the fi-
the stationary value l 36 in Fig. 2a. At this time
≈ nal amplitude and wavelength of the structure for
coarseninginterruptsandtheamplitudesaturatesto
different values of r. We can observe that, for large
a constant value. Nevertheless, for large enough L
values of r, the height amplitude does not change
andatlongdistances,theprofiledisordersinheights with r and the wavelength l is proportional to r1/2
(Fig. 3a), although the lateral cell-like order is still
as predicted above. This behavior can be also seen
preserved for intermediate distances (Fig. 3a, inset;
intheinsetofFig.3b,wherewerescalethehorizon-
note the difference in scales between the x and y
tal axis by l = 2√6r (see below), obtaining almost
axes). This disorder reflects in the power law be-
perfect collapse of the curves for large r.
havior of S(q,t) for q much smaller than 2π/l and
Further progress can be made by recalling that
long enough times (Fig. 2b) or, equivalently, in the
the solution of (5) with λ1 = 0 is a periodic juxta-
behaviorofthelocalwidthw(x0)[20]atlongtimes, position of parabolas of the form [15, 21]
displayed in Fig. 3b as a function of window size,
x0. Duetotheparabolicshapeofthecells,thelocal 4A
width scales as w ∼ x20 for distances smaller than h(x)=A− l2 x2. (7)
thecellsize,reachingaplateauforintermediatedis-
tances,andfinallyincreasingasa(smaller)powerof Wecanconsiderthisfunctionasanapproximateso-
x0 for large enough distances (kinetic roughening). lution of Eq. (6) for large values of r. In Fig. 5a
Theplateauinw(x0)isrelatedtothelateralorderof we compare the numerical profile with the function
the pattern, reaching up to several tens of cells (see (7) for r = 50. Using this solution and assuming
e.g. inFig. 4aanordereddomaincontainingover30 as a condition for the final (interrupted) structure
cells for r = 10). The effective exponent character- that the averagednonlinear contributions along one
izing the long distance behavior of w(x0) increases period must be equal, we obtain a relation between
4
300 100 r-independent if we rescale t rt, as already indi-
l →
A cated. Furthermore,atlongtimes,thegrowthveloc-
200
itygivenby(8)agreesaccuratelywiththenumerical
h
observations for a wide range in r.
100
10 In conclusion, we have derived a novel determin-
0 istic1dequationwithshiftsymmetry inthe context
0 100 200x300 400 500 1 1r0 100 of ion-beam sputtering and performed a numerical
(a) (b) analysis to obtain some relevant information about
itssolutions. Forlargevaluesofrwehaveestimated
FIG. 4: (a) Height profiles at t = 15000 for (top to the final pattern wavelength assuming a parabolic
bottom) r =0.1,0.5,1,5,10,50, and 100; (b) stationary
solution analogous to that of the “conserved” KS
wavelengthlandamplitudeAasfunctionsofr(statisti-
equation and checked our assumption by compar-
calerrorsaresmallerthanthesymbolsizes). Thedashed
ing with the numerical mean height evolution. The
line corresponds tol=2√6r. All unitsare arbitrary.
resulting single-parameter Eq. (6) interpolates be-
tween the KS equation for r = 0, which presents a
chaotic solution and no coarsening, and the “con-
8 104 r=2.5
r=5 served” KS equation for r (upon rescaling
6 rr==1300 h h/r), which displays u→nbo∞unded coarsening.
h4 h102 rr==5705 Th→is behavior is similar to the 1d convective Cahn-
L r=100
Hilliard (cCH) equation studied in [23]. However,
2
100 as reported in Ref. [23], coarsening does not in-
0
-30 -20 -10 x0 10 20 30 10-2 100t/r 102 terrupt in the cCH system for a whole range of
parameter values but, rather, proceeds logarithmi-
(a) (b) cally for long times, whereas Eq. (6) does present
interrupted coarsening with an ordered pattern of
FIG.5: (a)Stationaryheightprofileatt=15000forr=
constantwavelengthandamplitudefor intermediate
50. The solid line represents the approximate solution
given by (7) for l = 2√6r and A = 7; (b) mean height distances. Thisbehaviorprovidesanewscenariofor
fordifferentvaluesofr asafunctionoft/r. Thedashed the classification of surface coarsening phenomena
line is given by (8). All unitsare arbitrary. inRef.[6],whereallthestudiedevolutionequations
display perpetual coarsening, or else develop a pat-
tern with a frozen wavelength while the amplitude
continues growing without bound in the course of
the final wavelength and r, namely l = 2√6r. This time. Regarding Krug’s conjecture [7], the present
function is represented in Fig. 4b, fitting accurately example reinforces its validity for long range order
the stationarywavelengthsobtainednumericallyfor properties, since the pattern produced by Eq. (6)
large values of r. The net mean growth velocity of disordersatlargelengthscales. However,ourresults
the heightaverageis only due to the KPZnonlinear suggest that it is still possible to stabilize a well or-
termandisgivenbyv = λ1[∂xh(x)]2L .Assuming dered pattern over (intermediate) distances ranging
even up to several tens of cell sizes in contrast with
(7) as an approximate solDution and inteEgrating over
other equations in this context.
oneperiod,weobtainthenetmeanprofileevolution.
It reads
h¯L(t)=vt λ1 1 l/2 [∂xh(x)]2dx t=λ111t,
≈ (l Z−l/2 ) r Acknowledgments
(8)
where we have substituted l = 2√6r, and A 7 This work has been partially supported by MEC
≈
is obtained from Fig. 4b. As seen from Fig. 5b, (Spain), throughGrants Nos. BFM2003-07749-C05,
for large values of r the evolution of h¯ becomes -01, -05, and the FPU programme (J.M.-G.).
L
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