Survey of roughness by stochastic oscillations

0.26 MB·English

by  Eric Moreau

APS/123-QED
Survey of roughness by stochastic oscillations
E. Moreau
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0 Laboratoire des Sciences des Procédés Céramiques et Traitements de Surface
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2 ENSIL, 16 rue d’Atlantis 87068 Limoges, France

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a Abstract
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2 Inthispaper, connection between surface roughness and directed polymers inrandom medium arestud-

] ied, whenthesurface isconsidered asadirected line undergoing stochastic oscillations. Thisisperformed
h
c
e bystudyingtheinfluenceofastochasticelasticforcingterm κy+η(s),onaparticlemovingalongarough
m −
surface. Twomodels areproposed andanalysed inthis way: therandom-walk process (RW)initsdiscrete
-
t
a
andcontinuous form,andaMarkovian processviatheOrnstein-Uhlenbeck (O-U)process. Itisshownthat
t
s
.
t the continuous RW leads to an oscillator equation, via an effective action obeying a KPZ equation which
a
m
is solved analytically. The O-U process allows to obtain information on the profile of surface for a long
-
d
sizesubstrate. Theanalogy withtheroughness isachieved byintroducing aquantity suited todirected line
n
o
c formalism: theheightvelocityvariation ∂h/∂s.
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I. INTRODUCTION
Characterization of surface roughness or interface growth is of first importance in many ap-
plications: microelectronics, preparation of catalysts, magnetic materials (see e.g. [1]-[4]). And
numerous models based on numerical as well as analytical approaches have been proposed over
past decades to treat interface properties. These works concern in particular the KPZ equation
(scalelaws,roughness,growth)[5]-[15],andmodelstreatingconnectionsbetweensurfacegrowth
and directed polymers in random medium [16]-[24]. The present paper focus on the second class
of these models, considering the roughness of a surface as the result of specific stochastic pro-
cesses, and based on the directed line formalism. This formalism, much used in polymer physics
[25], allows to analyse the motion of a system by the way of transversal and curvilinear coordi-
nates [26]-[28] (respectively noted y and s in this paper). And we propose models showing the
analogies between the profile of a rough surface and the study of the transversal position equa-
tion. Note that, in this formalism, s plays the role of the time. For example dy/ds designs the
transversalvelocity. Moreover,thisformalismisalsolinkedtothestatisticalphysics: thepartition
functionofthesystemand its linkswith theKPZ ortheBurgers equation,which followfromthis
formalism, represent useful tools [29]-[31]. Concerning the roughness analysis, different works
have treated this problem [29], [32]-[37], but the roughness is nevertheless poor studied from a
purely stochastic point of view. Among recent works treating stochastic surface properties, we
can cite [38] and [39] which both have a Markovian approach, using a Langevin equation with a
stochasticnoise.
In the models presented here, we assumed that the profile of surface is described by the tra-
jectory of a particle undergoing a directing stochastic strength in addition to diffusion. We will
use for this purpose the elastic forcing term f(y,s) = κy + η(s) inducing stochastic oscilla-
−
tions, or equivalently, a potential Φ, such as f = ∂Φ/∂y; η having for example the properties
−
of a Langevin force. For deposition processes, we suppose that the surface start at the height h ,
0
and weare interested intheheight velocityvariation(hvv), ∂h/∂s, which giveslocal information
on the structure of the surface to analyse: for a fast variation of the height, we will have a high
value of the hhv, (Fig. (1)). Hence, this quantity is the reflect of local irregularities. But, the
hvv do not have to be confused with the quantity ∂h/∂x, where x stands for the horizontal coor-
dinate: locally, the slope ∂h/∂x may takes an infinite value (i.e parallel to the h axis), whereas
(h(s ) h(s ))/(s s ))is weak.
1 2 1 2
− −
2
FIG. 1: Profile of surface: result of the conflict between the random medium and a directing strength
∂Φ/∂y. Thelocalslope(a)(resp. (b))standsforaweak(resp. high)valueofthehvv.
−
Theoutlineofthepaperwillbeasfollow: wefirstderivepropertiesrelatedtoprofileofsurface
from pure and forced Random Walk (RW). In what follows, we will employed the term forced
RW to design a RW undergoing a forcing term in addition to the diffusion. Then, we justify the
useof an oscillatormodel comparing thedirected lineto astring vibrating. Thiswill allow to use
the restricted partition function to determine the transversal equation of motion, in presence of a
quadraticstochasticpotential. Inthelastpart,weshallconcentrateontheO-Uequationofmotion
forthetransversalposition,andparticularly ontheasymptoticsolutionsfors .
→ ∞
II. TOPOLOGYOFSURFACEVIEWASARANDOMWALKPROCESS
The models presented here concern the use of the directed line formalism associated to a RW
process. In its discrete form, we based our approach on the paper of Vilgis [25], treating directed
polymers in random media. Our derivation is somewhat different of Vilgis, but leads to the well-
knownpartitionfunctionofpolymerphysicsas showedlateron.
A. Discreterandomwalkprocess
We start from a set of vectors b b e ; i = 1,...,N , which are successively connected,
i i s
{ ≡ }
andrecoveringthewholeofthesurface(Fig. (2)). Wesupposetheyarestatisticallyindependent.
Theprobabilitytofind sucha setisgivenby:
N
P( b ;i = 1,...N ) = p(b ), (1)
i i
{ }
i=1
Y
where the probability p(b ) corresponds to a step of the random walk of b , with the mean value
i i
3
FIG. 2: Random Walk: N steps of length b are required in order to cover the surface on a linear size
substrate L.
properties:
b = 0 b b = b2δ . (2)
i i j ij
h i h i
δ is the Kronecker symbol and b is a characteristic length of the shift of position of the surface
ij
from oneto theother. Forthestepb , anormalizedgaussianprobabilitylaw istaken:
i
p(b ) = (1/2πb2)1/2exp( b2/2b2), (3)
i − i
and theassociated distributionofthesurfacelengthreads
N N
P(R) = db ... db δ R b p(b ); (4)
1 N i i
−
!
Z Z i=1 i=1
X Y
the vectorR = N b characterizing the line. Then, using the Fourier transform representation
i=1 i
oftheDiracdeltPa,aGaussianlaw forthelengthdistributionarises quitenaturally
P(R) = (1/2πb2N)1/2 exp( R2/2b2N). (5)
−
Andfrom thisexpression,thequadraticmean“length”ofthesurface can beevaluated:
R2 = R2 P(R2)dR = b2N. (6)
h i
Z
Note that the quantity R2 preserves information on the length of the trajectory, contrary to the
h i
geometrical quantity R. However, this result is a limiting case of a pure RW trajectory since it
impliesthat theentanglements,andso thelengthofthesurface, couldescapeto infinity. Thereby,
forthiscasewelabel N such as
R2 = b2N , (7)
rw rw
h i
4
with the meaning that N steps are necessary in average for a pure RW to describe the profile of
rw
surface. Theotherlimitingcase, i.e. thestraightline(sl),is givenby
R2 = b2N2 L2, (8)
h isl sl ≃
in sucha way thatthegeneral case (g), representing aforced RW, and includedbetween thecases
(7)and (8), maybewritten
R2 = b2N2γ; N N N , (9)
h ig g rw ≤ g ≤ sl
where γ is a critical exponent representative of a strength acting on the RW. In addition, we have
theconditions
b2N2γ b2N2 for N N ; (10)
g → sl g → sl
whichimpliesγ = 1, and
b2N2γ b2N for N N ; (11)
g → rw g → rw
implyingγ = 1/2. Thus,1/2 < γ < 1, availableforalinemovingintheplane(h,x). Thisresult
mayalso beexpressedinterm ofthelengthl R2 ofthesurface:
g
≃ h i
p
l = b N γ. (12)
g
Inthecaseofatwo-dimensionalRW,γ maytakestheformγ = 1/d ,whered designsthefractal
f f
dimension of the directed line. As a matter of fact, d describes the degree of irregularity of the
f
surface, and for γ = 1 and γ = 1/2, d becomes identical to the topological dimension of the
f
system. Clearly, the exponent γ is linked to the number of steps N . A relation between the
g
two quantities may be obtained from the probability of the general case, P (R), when the line
g
undergoesbotheffectsofdiffusionandofanexternalforce. Anticipatingthecontinuumapproach
described afterwards, wewritethisprobability
P (R) = exp R2/2b2N βΦ(R) , (13)
g g
N − −
(cid:0) (cid:1)
where Φ(R) stands for an external potential. For example, in the case of a harmonic potential,
Φ = (κ/2)R2, thequadraticmean lengthis
b2N
R2 = R2 P (R2)dR = g (14)
h ig g 1+b2βκN
g
Z
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whichleads, withEq. (9), to theexpression
ln(N /(1+b2βκN ))
γ = g g . (15)
2ln(N )
g
One can then obtain information on the topology of a surface, in the sense that the ratio l/L is
the reflection of its irregularities and roughness. The fractal dimension may be evaluated experi-
mentally,forexample,bythebox-countingmethodwithboxesoflengthb(inthesamewayasfor
electric discharges [40]). In simulations, the varying parameter would be the number of steps of
theRW.
B. Equationsofevolution
Ifweexpressnowthestepasb = R R ,andwritingshortlyP = P( b ;i = 1,...N ),
i i i 1 i
− − { }
theprobability(Eq. (1)) mayberewritten
N/2 N
1 1
P = exp (R R )2 . (16)
2πb2 "−2b2 i − i−1 #
(cid:18) (cid:19) i=1
X
Onecan then associatedtotheexponentialargument asymbolicHamiltonian
N
1
βH = (R R )2. (17)
0 2b2 i − i−1
i=1
X
This is the energy of a mass-spring chain of a one dimensional lattice. This suggest to take the
continuous limit of this discrete chain: R R ∂R/∂s, and N N, where s is the
i − i−1 → i=1 → 0
dimensionlesscurvilinearcoordinate. In thislimit,theprobabilityofPthepureRRWreads
1 N ∂R 2
P exp ds , (18)
→ N −2b2 ∂s
" Z0 (cid:18) (cid:19) #
where is a normalization constant. From this result, one can construct the partition function
N
as a sum over all the possible paths starting from s = 0 and reaching the linear size substrate at
s = N:
1 N ∂R 2
Z = exp ds . (19)
N −2b2 ∂s
AllpathsR(s) " Z0 (cid:18) (cid:19) #
X
The argument of the exponential is equivalent to the transversal kinetic energy, or more precisely
to the effective action of the system since we integrate over the “time” s. Furthermore, this path
integralobeystheclassical diffusionequationwhicharises naturallyfrom apureRWprocess:
∂ b2 ∂2
Z(R,s) = Z(R,s), (20)
∂s 2 ∂R2
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with the boundary condition Z(R,0) = δ(R). For a forced RW, we introduce a potential term
Φ(R(s)) standing for an external (virtual) strength field, in such a way that the total effective
actiontakes theform
1 N ∂R(s) 2 N
β = ds+ Φ(R(s))ds. (21)
S 2b2 ∂s
Z0 (cid:18) (cid:19) Z0
b
We note S to emphasize the fact that this quantity represents a pseudo action (the integration is
madeoveraHamiltonianand notoveraLagrangian). Thisyieldsto thetotal partitionfunctionof
b
thesystem,as theFeynman-Kacpathintegral:
Z(R,s) = exp β R(s). (22)
N − S D
Z h i
It is thenconvenientto introducethetransversal coordinabtey R/band so towritetheaction in
≡
theform[26]
s 1 dy 2 s
= ds +Φ(y,s) = dsH(y,s). (23)
′ ′ ′ ′
S 2 ds
Z0 (cid:18) (cid:16) ′(cid:17) (cid:19) Z0
H standing for the Habmiltonian of the system. By this way, we obtain the well-know expression
ofthepartitionfunctionofdirected polymers
y(s) s 1 dy 2
Z(y,s) = y exp β ds +Φ(y,s) ; (24)
′ ′
N D − 2 ds
Zy(0) (cid:20) Z0 (cid:18) (cid:16) ′(cid:17) (cid:19)(cid:21)
expression directly issued from a RW process with a directed line formalism. For a one dimen-
sionalsystem,Z(y,s)obeysthefollowingPDE [41]:
∂ ∂2Z(y,s)
Z(y,s) = ν βΦ(y,s)Z(y,s), (25)
∂s ∂y2 −
whereν = 1/2β isaviscositycoefficient.
Let us underline that there exists several versions of the Hamiltonian depending on system to
study[16]-[18], [20]-[24].
In view of these results, several analogies may be drawn between directed lines and vibrating
strings. In its discrete form, the directed line is composed of a succession of rigid chains and
the whole evolving with a resulting motion depending on the form of the noisy potential. For
the continuous form, the term of inertia in polymer physics is due to the thermal agitation of the
medium,andthepotentialtoanexternalfield(e.g. electricfield). Inourcase,theexternalpotential
isnecessarybecauseofthestrengthwhichhavetobeimposedtotheline,inordertoobtainrealistic
profiles of surface. So, the noisy potential of our model will be composed of a harmonicpart and
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of astochasticpart which shall maketheoscillationsstochastic: Φ(y,s) = 1ky2 +η(y,s),where
2
κ is the elastic constant of the line. The next step is to determine the equation of motion with the
frequency ofvibrationoftheline. In ordertodo this,letus proceed as follow:
In one hand, starting from Eq. (25), it is well-known that we may obtain successively a KPZ
and aBurgers equation(approach (A)):
2
∂S ∂2S 1 ∂S
= ν + Φ; (26)
∂s ∂y2 2 ∂y −
!
b b b
and
∂u ∂u ∂2u ∂Φ
+u = ν + . (27)
∂s ∂y ∂y2 ∂y
In thiscasethetransformationsare
∂S
u = and Z = exp S/2ν . (28)
−∂y
b (cid:16) (cid:17)
Usually one introduce the free energy. But the free energy tebnds to the total energy of the system
S whentheentropytends tozero.
In the other hand, starting from a general Burgers equation for a velocity field u, we have
b
(approach (B))
∂u ∂u ∂2u ∂V
+u = ν , (29)
∂s ∂y ∂y2 − ∂y
and
∂F ∂2F 1 ∂F 2
= ν + +V, (30)
∂s ∂y2 2 ∂y
(cid:18) (cid:19)
and finally,withZ = exp(F/2ν),
∂ V
b Z(x,s) = ν 2Z(x,s)+ . (31)
∂s ∇ 2ν
Whetherurepresents thetransversalbvelocity,wehabvein thiscase
dy ∂F
u = = . (32)
ds −∂y
From these two approaches, a question arises: “May the quantity u of Eq. (27) be interpreted as
thetransversalvelocityofEq. (29)?” In otherwords,is thereidentitybetween S and F ?
To remove this ambiguity, let us start from the Burgers equation (29), u given by (32), and let
b
ususetherelation
∂F ∂F
dF(y,s) = ds+ dy; (33)
∂s ∂y
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whichyieldsto
∂2F 1 ∂F 2 ∂F
dF = ν ds+ ds+V ds+ dy; (34)
∂y2 2 ∂y ∂y
(cid:18) (cid:19)
and consequentlyto theexpression
∂2F 1 ∂F 2 ∂F
F = ν ds+ ds+ V ds+ dy. (35)
∂y2 2 ∂y ∂y
Z Z (cid:18) (cid:19) Z Z
Thefirst and lastintegralsmay berewritten
∂2F ∂ ∂F ∂ dy
ds = ds = ds = 1. (36)
∂y2 ∂y ∂y −∂y ds −
Z Z Z
and
2
∂F dy dy
dy = dy = ds, (37)
∂y − ds − ds
Z Z Z (cid:18) (cid:19)
In sucha waythat the“free energy” willread
s 1 dy 2
F(y,s) = ν V(y,s) ds . (38)
′ ′
− − 2 ds −
Z0 (cid:18) ′(cid:19)
h i
Finally,theHopf-ColetransformationZ = exp(F/2ν), with1/2ν = β,yieldsexplicitlyto
b s 1 dy 2
Z = e 1/2 exp β ds V(y,s) . (39)
− ′ ′
× − 2 ds −
( Z0 (cid:18) ′(cid:19) !)
b
The heat equation (25) being invariant under the shift Z Z = c Z, c R, we wiil have
→ × ∈
equivalence between the two approaches if Φ = V. And, therefore, one obtain the classical
− b
Hamiltonequationofmotion:
dy ∂F ∂S d2y ∂H
= = = (40)
ds −∂y −∂y ⇐⇒ ds2 − ∂y
b
Thedeterminationofan equation ofmotionfory ishenceforth equivalentto solvetheKPZ equa-
tion(26). Moreover,wehavearelationbetweentheheightofthegrowingsurfaceandthetransver-
sal position: h(s) h y(s) y(0) ,and so, byextension,
0
| − | ≡ | − |
dy ∂h
. (41)
ds ≃ ∂s
Forexample∂h/∂s = 0 dy/ds = 0. Thereby, theaction mayas well bewritten[42, 43]
⇔
s 1 ∂h 2
= +Φ(h,s) ds, (42)
′ ′
S 2 ∂s
Z0 (cid:18) ′(cid:19) !
b
WeshallnowconcentrateonthesolutionoftheKPZ equation(26).
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AnalyticsolutionsofnoisyKPZequationshavealreadybeenstudiedintheliteratureinspecific
cases [44]-[46]. But the non-linearity of this equation added to the presence of a stochastic noise
make the resolution difficult. Nevertheless, it is known that this equation may be transformed
either into a Burgers equation or a Schr¨odinger equation with an imaginary time. And, among
works which treat exact solutions of theses kind of equations, we can cite [47]-[51]. This is our
concern hereto treat thecase ofa lineundergoing thestochasticelasticforcing term κy +η(s),
−
wherethefunctionηcontainsallthenoisethataffectstheoscillations(e.g. aLangevinforce). This
amountstosayingthatwesolvetheKPZequation(26)withapotentialΦgivenbyΦ(y,s) = κy2
2 −
yη(s). Moreover,weassumeaconstantinitialconditionforthetransversalvelocity: u(y,0) = u .
0
Following methods presented in Ref. [51], it is shown in appendix that the solution is of the
form:
ξ(s)
S(y,s) = y2 yψ(s)+Θ(s), (43)
2 −
where ξ(s), ψ(s) and Θ(s) arebfunctions of s and of η(s) (see the appendix). But, we have the
relation
dy ∂S
= = yξ(s)+ψ(s). (44)
ds −∂y −
b
Consequently, we obtain an equation standing for the equation of a forced oscillator with a term
offriction:
d2y dy
+ξ(s) +ω2(s) y = F(s), (45)
ds2 ds
where the pulsation ω2 and the forcing term F are respectively given by dξ/ds and dψ/ds. We
can notice from the expressions of ξ and ψ that, while the frequency f dξ/ds depends only
∝
ons,thetermofforcedependsalsoonthestochasticnoiseη. Thus,weexppectprofilesoftheform
drawnin Fig. (3).
Oncey determined,wecan deducethehvvfrom Eq. (44):
∂h s s t
ξ(s)exp ξ(t)dt 1+ ψ(t)exp ξ(t)dt dt +ψ(s). (46)
′ ′
∂s ≃ − −
(cid:18) Z0 (cid:19)(cid:26) Z0 (cid:18)Z0 (cid:19) (cid:27)
Then, the degree of irregularity of a surface, and so its global roughness, may be estimated by
adding up all the local slopes ∂h/∂s , in such a way that the interesting quantity to determine is
| |
∂h/∂s , where themean istaken overall thelengthofthesurface.
h| |i
Note that this problem is different of the one of a stochastic oscillator, which is a specific
problem[52, 53]
10