Generic morphologies of viscoelastic dewetting fronts

6 Pages·0.1 MB·English

by  Stephan Herminghaus

Generic morphologies of viscoelastic dewetting fronts
Stephan Herminghaus, Ralf Seemann, Karin Jacobs
Applied Physics Lab., University of Ulm, D-89069 Ulm, Germany
(February 6, 2008)
lefttoright). Theprofilesareobtainedbyscanningforce
2
A simple model is put forward which accounts for the
0 microscopy from the rims of circular dry patches nucle-
occurrence of certain generic dewetting morphologies in thin
0 ated in the films [16–18]. Most commonly, one observes
liquid coatings. It demonstrates that by taking into account
2 profiles as those shown in fig. 1a, with a simple decay
the elastic properties of the coating, a morphological phase
n from the crest into the undisturbed film to the right.
diagram may be derived which describes the observed struc-
a However, when the molecular weight of the polymer is
tures of dewetting fronts. It is demonstrated that dewetting
J
morphologies may also serveto determinenanoscale rheolog- small enough [14], the shape may be qualitatively differ-
2
ical properties of liquids. ent. This is shown in fig. 1b, which has been obtained
1
with polystyrene films with a molecular weight of only 2

kg/mol. Aditchisclearlyvisibleinfrontofthecrest,and
]
t even another small elevation to the right of the ditch is
f
o presentas well,such thatthe frontappears as a damped
s When a thin liquid film beads off a solid substrate,
oscillation. The only difference between the film materi-
t. it is eventually transformed into an ensemble of individ-
a als used for fig. 1a and 1b is their molecular weight, and
ual liquid droplets, the arrangement of which may vary
m thus their viscoelastic properties. If the film is not too
strongly according to the basic mechanisms involved in
- thick, the ditch may reach the substrate and pinch off
the dewetting process[1–7]. In by farmostcases,dewet-
d the crestfromthe film, forming a new contactline. This
n ting is initiated by heterogeneous nucleation of individ-
happens repeatedly, such that a series of isolated crests
o ual holes in the initial film, thus forming contact lines
formsinacascadeofpinch-offevents,asshowninfig.1c.
c between the film surface and the substrate. The surface
[ We shall now create a suitable mathematical descrip-
forces acting upon these contact lines give rise to mov-
tion of these findings. An incompressible viscoelastic
1 ing dewetting fronts,the motion,shape, andinterplayof
fluid may be described by the force balance [19–21]
v which largely determine the final dewetting morphology.
3
9 It would be very desirable to have a theory describ- η∆j = p E∆φ (1)
ing the dynamics of these fronts precisely enough for ex- ∇ −
1
1 tracting information on the dewetting mechanisms from where η is the viscosity, j is the material current in the
0 the characteristics of their shape and the final droplet film, p is the pressure, E is Young’s modulus, and φ is
2 structure. However, the task of setting up such a theory a local displacement field describing the strain [21,22].
0
has proveduntractable so far. Solving the Navier-Stokes In the present treatment, we will adopt the lubrication
/
t equationwiththemovingcontactlineasaboundarycon- approximation [23], thereby neglecting the normal (z-)
a
m dition is particularly complicated and entails ad hoc as- componentsofthecurrentandthepressuregradient,and
sumptions on the dynamics at the contact line [8–10]. consider quantities averaged over the film thickness, h.
-
d Furthermore, it has meanwhile become clear that vis- Wethencanrewriteeq.(1)asJ(x)= C∂x(p αE∂xφ),
− −
n coelastic effects, which are not described by the Navier- where J is the total current in the film, and α is a nu-
o
Stokes equation, may be decisive for the evolving mor- merical factor which characterizes the flow profile. The
c
: phologyinboth advancingandrecedingfronts[4,11–14]. latterdepends,e.g.,onthefrictionofthefilmatthesub-
v In particuilar, it has recently been shown [13,14] that strate,anditisclearthatinthe caseoffullslippage,the
i
X ‘elastic dewetting fronts may strongly differ, as to their coupling of the flow to the strain will not be the same
r shape and dynamics, from Newtonian ones [15], and it as in the absence of slip. These effects are difficult to
a has been pointed out that these findings are not ac- treat explicitly, and thus will be absorbed here into α.
counted for by theory so far [13]. Consequently, what Similarly, C depends upon the viscosity of the film, η,
is needed is a handy theoretical model, which is just as and its friction coefficient at the substrate surface, κ. If
detailed as necessary to describe the main physics in- the latter is infinite (Poiseuille flow with no slip at the
3
volvedinthe dewettingdynamicsofviscoelasticfilms. It substrate), we have C = h /3η, while for κ 0 (plug
is the purpose of this paper to propose such a model. flow), C =h2/κ [15,23]. →
Letusfirstrecallthemainfeaturestobeexplained,by Since the derivative of the displacement is the strain,
summarizing the generic front morphologies observed so σ(x,t) = ∂xφ, and excess pressure in the film comes
far. Fig. 1 shows dewetting fronts in liquid polystyrene about from the curvature of its surface, we have
filmsbeadingoffsiliconsubstrates(the frontsmovefrom
J(x,t)=C∂ γ∂ ζ(x,t)+αEσ(x,t) (2)
x xx
{ }
1
where γ is the surface tension, and ζ(x,t) is the ver- not depend on the friction of the film at the substrate
tical displacement of the film surface. We explicitely [11,12]. A profile like that is shown in fig. 1b, and it is
neglect long range wetting forces acting through the obtained only for small molecular weight indeed, where
film, such as van der Waals forces. In sufficiently thick elastic effects are particularly small. The real and imag-
films,whichdewetbynucleation[17,24](suchasassumed inary part of Q can be determined from the profile, and
above), these do not play a significant role. in the case of the one displayed in fig. 1b, it turns out
Since the strain in the film may decay by internal re- that they are not precisely as expected for a Newtonian
laxations of the material according to ∂tσ = ω0σ, its fluid [14]. Since the complex roots of eq. (6) satisfy the
−
couplingtothematerialcurrentcanbewrittenas[21,22] relation
3 (Q)2 (Q)2 =E˜ (7)
(∂t+ω0)σ =∂xJ/h (3) ℜ −ℑ
the elasticity of the film can be inferred from the mea-
where we have neglected the nonlinear convective term
surements, and we get αE = 8.7 kPa. We would like
J∂ φ. This restricts our discussion to small excursions
x to point out that this is a non-invasive method of deter-
ζ, but enables a linear treatment. By combination of
mining rheological properties of liquids on a sub-micron
the above equations with the continuity equation, ∂ ζ +
t scale.
∂ J =0,itiseasytoobtaintheequationofmotionforσ
x As the height of the crest, H, increases while more
and ζ. A precondition for elasticity to play a significant
material is collected in the front, the amplitude of the
roleisthatω0 <<ω,inwhichcasethedispersionrelation
oscillation increases, and so does the depth of the ditch
reads
[14]. When the latter reaches the substrate, the crest is
αE pinched off and a cascade is formed as shown in fig. 1c.
2 2
ω =ω0+Cq h +γq (4) However, this works only as long as the width, W, of
(cid:18) (cid:19)
the crest is less than half the wavelength of the damped
where q is the wave number of the perturbation. oscillation. As H increases, so does W [17,23], with the
In order to determine the shape of the moving dewet- ratioH/W =:G(Θ)beingonlyafunctionofthedynamic
ting front, we now look for travelling-wave solutions of contact angle at the substrate, Θ [17]. For a cylindrical
theformζ(x,t)=ζ(x vt),wherev isthevelocityofthe rim [23], we have simply G(Θ) = (1 cosΘ)/2sinΘ.
− −
dewetting front. We remain in Fourier space,i.e., we de- For more asymmetric shapes, the form of G is different,
compose the front profile into modes ζ exp(iqx ωt). but similar as to the overall behavior and the order of
q
∝ −
Writing Q := iqh, it follows that ω = Qv/h, and from magnitude. IfW islargerthanW =λ/2=π/ (Q),the
c
ℑ
eq. (4) we then get depth of the ditch is rather determined by the contact
angle of the rim at the leading edge. Consequently, it
4 αEh 2 vh3 ω0h4 will reach a final maximal depth [14], and a dry spot
Q Q Q+ =0. (5)
− γ − γC γC forms only if this depth exceeds the film thickness. The
condition for a cascading front to form is readily seen to
Asitwillbecomeclearbelow,thetermω0h4/γCisrather be
small. If we neglect it for the moment, we get the cubic
(Q)
equation πG(Θ)> (Q) exp π|ℜ | (8)
|ℑ |· (Q)
Q3 E˜Q v˜=0 (6) (cid:18) |ℑ |(cid:19)
− − As the film thickness is decreased, so is the right hand
sideofeq.(8)(hcancelsoutintheexponential),suchthat
where we have introduced the dimensionless quantities
v˜ := vh3 and E˜ := αEh. This can be easily solved, and a cascade is expected at sufficiently small h. Eq. (8) can
Cγ γ be used to determine the boundary line for the appear-
thesolutionsofthequarticequation(5)turnouttodevi- ance of cascade structures in the (v˜,E˜)-plane. In fig. 2a,
atesignificantlyfromthoseofthecubicequationonlyfor −1
this is indicated by the lower curve for G = π . For
parameters which are not relevant in most experimental
differentvaluesofG,themaximumoftheboundaryline,
situations. We will thus primarily discuss eq. (6), and
indicatedby the dotin the figure,is quite accuratelyde-
refer to eq. (5) only occasionally, where appropriate. scribed by E˜ = 0.060 G2. For very thin films, where
If viscoelastic effects are absent, i.e. E = 0, we have ·
van der Waals forces play a role, the cascade region will
Q= √3v˜. Weareinterestedonlyinmodeswith (Q)<0
[25], which are Q = 1√3v˜(1 i√3). The shaℜpe of the slightly extend or shrink, depending on the sign of the
−2 ± Hamaker constant.
real surface of the leading front is thus e−xcos(√3x) Let us now turn to the influence of stronger elastic
∝
[14,26], which clearly exhibits not only a ditch, but a effects. It is interesting to investigate at which system
damped oscillation. For Newtonian liquids, it can be parameters the damped oscillation, as shown in fig. 1b,
seenfromnumericalsolutionsofthe NavierStokesequa- vanishes. Thesearegivenbythezeroofthediscriminant
tion for the moving front problem that this result does of the cubic equation (6), which is
2
v˜ 2 E˜ 3 quantitative by determining the numerical constant α,
D = (9) which we leave to further work.
(cid:18)2(cid:19) − 3! The authors are indebted to Klaus Mecke and Ralf
The critical modulus above which there is no oscillation Blossey for many fruitful discussion. We also thank
isthusgivenbyE˜ =3 v˜ 2/3. Thismaybeviewedasan- Gu¨nter Reiter for stimulating remarks. Funding from
2
othermorphological‘phase boundary’inthe(E˜,v˜)-plane, theDeutscheForschungsgemeinschaftwithinthePriority
(cid:0) (cid:1)
andisshowninfig.2aastheuppersolidcurve. Theexact Program‘WettingandStructureFormationatInterfaces’
boundary, according to eq. (5), deviates noticeably only is gratefully acknowledged.
for very smallvalues of E˜ and v˜. To quantify this devia-
tion,itisusefultointroducethedimensionlessparameter
Ω := ω0h/v, which represents the importance of intrin-
sic relaxation in the material with respect to the shear
flow exerted by the dewetting process itself. In fig. 2b,
the‘exact’phaseboundary,whichwasdetermiedbysolv- [1] D. J. Srolovitz, S. A. Safran, J. Appl. Phys. 60 (1986)
ing eq.(5) graphically,is shownas the solidcurve, along
247.
withtheapproximationobtainedfromthecubicequation [2] C. Redon, F. Brochard-Wyart, F. Rondelez, Phys. Rev.
(dashedline). ForΩ 0,the solidcurveapproachesthe Lett. 66 (1991) 715.
→
dashed line asymptotically. The diagram presented in [3] G. Reiter, Phys. Rev. Lett. 68 (1992) 75.
fig.2a,whichcorrespondstoΩ=0,shallthusbeconsid- [4] K. Jacobs, Thesis, Konstanz 1996; ISBN 3-930803-10-0.
ered quite reliable within the range displayed if Ω < 0.1 [5] P.Lambooy,K.C.Phelan,O.Haugg,G.Krausch,Phys.
(in the experiments in ref. [13], Ω 10−5). Rev. Lett. 76 (1996) 1110.
≈ [6] T.G.Stange,D.F.Evans,W.A.Hendrickson,Langmuir
The twocomplexconjugaterootsgivingrisetothe os-
13 (1997) 4459.
cillations for D > 0 merge into a degenerate real root
[7] S. Herminghaus et al., Science 282 (1998) 916.
at a critical elastic modulus (D = 0). As the latter in-
[8] J. Moriarty, L. Schwartz, E. Tuck, Phys. Fluids A 3
creases further, they bifurcate again into two real roots,
(1991) 733.
Qlong and Qshort. In fact, ‘elastic’ fronts can be well fit- [9] H. Greenspan, J. Fluid Mech. 84 (1978)125.
ted by a superposition of two exponentials, as shown in [10] S. Troian, E. Herbolzheimer, S. Safran, J. F. Joanny,
fig.1abythedottedcurves. Themodelpredictstheratio Europhys. Lett. 10 (1989) 25.
Qshort/Qlong to increase with E˜, and thus the front to [11] M.Spaid,G.Homsy,J.Non-Newt.FluidMech.55(1994)
become more and more asymmetric, in accordance with 249.
observation [4,13,14]. We can again determine E˜ from [12] M. A.Spaid, G. M. Homsy, Phys. Fluids8 (1996) 460.
the front profile via the relation [13] G. Reiter, Phys. Rev. Lett. 87 (2001) 186101.
[14] R.Seemann,S.Herminghaus,K.Jacobs,Phys.Rev.Lett.
Qlong = sin{π6 − 31arccosB} (10) 87 (2001) 196101.
Q sin π + 1arccosB [15] K. Jacobs, R. Seemann, G. Schatz, S. Herminghaus,
short {6 3 } Langmuir 14 (1998) 4961.
withB := 3√3v˜E˜−3/2,oncetheconstantC isknownfor [16] G. Reiter, P. Auroy, L. Auvray, Macromolecules 29
2
the system. (1996) 2150.
The long range shape of the front is determined by [17] K. Jacobs, S. Herminghaus, K. Mecke, Langmuir
14(1998) 965.
thesmallerroot,Q ,whichasymptoticallyapproaches
long
[18] D. Podzimek et al., cond-mat/0105065 (2001).
the simple law Q= v˜/E˜. Thus the width of an elastic
− [19] L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Vol.
frontshouldremainconstantduringdewetting, inagree- VII), Butterworth, London 1995.
ment with recent observations [13], but in contrast to [20] L. Landau, Lifshitz, Hydrodynamics (Vol. VI), Butter-
quasi-Newtonianfronts[15,23]. Sinceourmodelpredicts worth, London 1995.
Q to scale as E˜ for sufficiently large E˜, we have [21] S. Herminghaus, K. Jacobs, R. Seemann, Eur. Phys. J.
short
Q Q2 if all other parameters in the system are E 5 (2001) 531.
long ∝ short p [22] S. Herminghaus, Eur. Phys. J. E(2002)in print.
keptconstant. Infact,forthetwocurvesshowninfig.1a,
[23] F.Brochard-Wyart,P.-G.deGennes,H.Hervert,C.Re-
theratiooftheQ is1.40 0.07,whiletheratioofthe
Q2 is 1.34 0l.o1n3g. This is±clear agreement. don, Langmuir 10 (1994) 1566.
short ± [24] R.Seemann,S.Herminghaus,K.Jacobs,Phys.Rev.Lett.
We thus have put forward a tractable model which 86 (2001) 5534.
describes the generic morphologies of liquid dewetting [25] ℜ(·)andℑ(·)denotetherealandimaginarypartoftheir
fronts observed so far. It correctly accounts for the im- argument, respectively.
pact of the fluid properties upon their occurrence and [26] Thesame result isobtained forsurface diffusion kinetics
structure. Itistherebycapableofextractinginformation (D. J. Srolovitz, S. A. Safran, J. Appl. Phys. 60 (1986)
on the sub-micron scale rheologicalproperties of the liq- 255).
uidfilmfromtheobservedprofiles. Thismayberendered
3
FIG.1. Dewettingmorphologies observedinvariousliquid
polystyrene films beading off a silicon substrate. The height
has been normalized with respect to the film thickness. a)
Profiles of the rims of holes nucleated in 40 nm thick films.
The steep profile was obtained with a molecular weight of
MW =101k,theflatterprofilewithMW =600k. Eachprofile
is superimposed with a double exponential fit as a dotted
line, the agreement is almost perfect (see text). b) Profile
obtained with MW = 2 k. c) Same as b), but with smaller
film thickness.
FIG. 2. a) Morphological phase diagram for viscoelastic
dewetting fronts in the (E˜,v˜)-plane, according to eq. (6). b)
Deviation of the phase boundary according to eq. (5) (solid
curve)fromtheapproximationgivenbyeq.(6)(dashedline).
4
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