Interfacial Instability of Charged End-Group Polymer Brushes

0.19 MB·English

by  Yoav Tsori

epl draft
Interfacial Instability of Charged End-Group Polymer Brushes
Yoav Tsori1, David Andelman2 and Jean-Franc¸ois Joanny3
1 Departmentof Chemical Engineering, Ben-Gurion Universityof theNegev, P.O.Box 653, 84105 Beer-Sheva, Israel.
8 2 Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978,
0 Israel.
0 3 Institut Curie, UMR 168, 26 rue d’Ulm, F-75248, Paris Cedex 05, France.
2

n
a
J PACS 61.25.Hq–Macromolecular and polymer solutions; polymer melts; swelling

2 PACS 41.20.Cv–Electrostatics; Poisson and Laplace equations, boundary-valueproblems
2
Abstract. - We consider a polymer brush grafted to a surface (acting as an electrode) and

] bearing a charged group at its free end. Usinga second distant electrode, thebrush is subject to
t
f a constant electric field. Based on a coarse-grained continuum model, we calculate the average
o brush height and find that thebrush can stretch or compress depending on theapplied field and
s
chargeend-group. Wefurtherlookatanundulationmodeoftheflatpolymerbrushandfindthat
.
t the electrostatic energy scales linearly with the undulation wavenumber, q. Competition with
a
m surface tension, scaling as q2, tends to stabilize a lateral q-mode of the polymer brush with a
well-defined wavelength. This wavelength depends on the brush height, surface separation, and
-
d several system parameters.
n
o
c
[

1
Introduction. – There are different ways to bind chargedend-group is that we can control the layer height
v
polymers to surfaces. Either by adsorption from solu- andother properties by applying an externalelectric field
8
1 tion or grafting them onto the surface with a terminal and varying it continuously. This field stretches (or com-
3 grouporhavinganadheringblockincaseofblockcopoly- presses)the chains and is in directcompetition with their
3 mers. Such coated surfaces have many important appli- elastic energy and entropy. Even in the absence of any
.
1 cations in colloidal and interfacial science. The polymer external field, we expect the brush to be affected by the
0 layer can change the hydrophobicity of the surface, pre- chargeend-groups,becauseoftheirrepulsiveinteractions.
8
vent absorption of other molecules from solution and, in Indeed, the height profile depends on the charged group
0
general, plays an important role in colloidal suspensions andaninstability of the flatbrushtowardsanundulating
:
v by preventing flocculation and aggregation of coated col- one may occur.
i
X loidal particles [1,2].
A densely grafted polymer layer is called a polymer
r
a brush. The layer is grafted irreversibly on a solid sur-
face by an end–group. Both neutral and charged poly- Flat end-charged polymer brush. – Letusbriefly
mer brushes have been studied extensively in the last recalltheequilibriumpropertiesofaneutralgraftedlayer.
few decades [3–11]. If there is no strong interaction be- The conditionof highly dense layers(the brushregime)is
tween the monomers and the surface, the brush proper- ℓ≪R , where R is the chainradius ofgyration,and ℓ is
g g
ties are mainly determined by the chain entropy. Neu- the averagedistance between chains (Fig. 1a). The brush
tral brushes, to a large extent, are characterized by their height, defined as the averagedistance of chain ends from
height that scales linearly with N, the polymerization in- thesubstrate,isdenotedbyh. Inthe1970s,asimplefree-
dex [4–7]. Charged polymer brushes depend in addition energywasproposedbyAlexanderandde Gennes[4,5]to
on the charge density of the chain as well as the solution determinethelayerequilibriumheight. IntheAlexander–
ionic strength [8–13]. deGennesmodel,thebrushheightistakentobethesame
In this Letter we aim at understanding another variant forallchains;namely,the heightdistributionis step-wise.
of polymer brushes having a terminal charge group, Ze, Later,inmorerefinedtheories,itwasfoundthatthe free-
at their free end, where e is the electronic charge and Z enddistributionisparabolic[6,7]. Inthe presentworkwe
the valency (see Fig. 1). The main advantage of having a remain within the step-wise distribution approximation,
p-1
Y. Tsori et al.
(a) (b) the following way. The entire gap 0<y <L between the
y =V two electrodes is assumed to contain the same dielectric
y=L
mediumwithdielectricconstantε. Thediscreteend-group
e
charges are replaced by a two-dimensional layer having a
continuous surface charge density. We assume that the
Ze Ze Ze Ze s = Ze/ℓ2 continuum limit is adequate enough and provides a good
description of the system electrostatics. Finally, we treat
e
h h the charged brush without taking into account the pres-
y=0 y =0 ence of counter-ions. This assumption and the possible
x
ℓ
influence of counter-ions is discussed further below for a
few relevant limits.
Fig. 1: (a) Schematic illustration of a polymer brush with a The electric field has a jump at y = h, ∆E(h) ≃ σ/ε,
terminalcharge group. The polymerisgrafted onto aflat and whereσ =Ze/ℓ2 isthe layerchargedensityperunitarea.
conductingsurfaceaty=0,whiletheotherelectrodeaty=L
For a typical grafting density of ℓ ≃ 100nm, dielectric
provides an electrostatic potential difference V, with an aver-
age electric field, E = −V/L. Each chain end-group carries a constant ε ≃ 10ε0 with ε0 being the vacuum permittiv-
charge Ze, the grafting chain density is ℓ−2 and the average ity and Z ≃ 5−10, the electric field jump is of order
∆E ≃ 106V/m = 1V/µm. Because the charged layer
brushheightish. (b)Acontinuummodelofthebrushusedto
calculatetheelectrostaticproperties. Thedielectricconstantε at y =h creates a discontinuity in the electric field E(y),
has the same value throughout the gap between the two elec- the electrostatic problem is solved separately in two re-
trodes. Thechainend–chargesareboundtoatwodimensional gions: the potential is marked as ψ in the region below
a
2
layer, with a charge density per unit area, σ=Ze/ℓ . the charged layer, 0 < y < h, and as ψ for the region
b
above it, h <y < L. Solving the Laplace equation in the
gap 0<y <L we get
whichisadequateenoughaslongasthewavelengthofthe
predicted instability (see discussion below) is larger than ψ = by 0<y <h
a
the width of the chain-end distribution [6,7].
ψ = c + dy h<y <L (3)
b
TheAlexander–deGennesexpressionforthebrushfree-
energy is: where the coefficients b, c, and d are determined from the
boundary conditions: ψ | = 0, and ψ | = V. At
a y=0 b y=L
F = 1Kh2+ kBTv ℓ−2N2h−1 (1) the charged layer itself y=h, the potential is continuous,
brush 0
6 2 ψ | =ψ | , and the jump in its electric field is pro-
a y=h b y=h
portional to the charge density σ:
wheretheentropic“spring”constantisK =9k T/(Na2),
B
a is the Kuhn statistical length, N the polymerization V σ σh V σh
b= + (1−h/L) ; c= ; d= − (4)
index, k T the thermal energy, and v = a3(1−2χ) is L ε ε L εL
B 0
the excluded volume parameter depending on the Flory-
The total free-energy can be written as the sum [15]
Huggins parameter χ. For a neutral brush, minimization
1
of Fbrush with respect to the profile height h gives the F =F − ε(∇ψ)2 d3r+ ρψ d3r (5)
brush
well-known Alexander–de Gennes height of the brush at 2Z Z
equilibrium h . It scales linearly with N:
0 where F is the brush free energy, and the last two
brush
termsrepresenttheelectrostaticenergy(inSIunits). The
1/3
1
h =N v a2ℓ−2 (2) volume charge density ρ is related to the surface one via
0 0
(cid:18)6 (cid:19) the Dirac delta–function, ρ=σδ(y−h).
We first consider the case where the electrostatic inter-
Next, the end-charged brush is considered. As before,
actionshaveasmalleffectonthethicknessandweexpand
the chains are grafted onto the surface located at y =
F around its value at h to second order in h−h .
brush 0 0
0, but the surface is taken as conducting and is held at
The resulting total free-energy per grafting site is
potential ψ =0 (see Fig. 1) [14]. At the other (free) end,
thechainscarryachargeofZe. Withoutlossofgenerality 1 1 V2
F = K(h−h )2 − ℓ2ε
we will take this charge to be positive, Ze>0. A second 2 0 2 L
conductingandflatsurfaceaty =Lis heldata potential σV 1σ2 1ℓ2σ2
+ ℓ2 + h − h2 + const.(6)
ψ = V with respect to the surface at y = 0. Hence, the (cid:18) L 2 ε (cid:19) 2 εL
polymerbrushissubjecttoaverticalaverageelectricfield
Minimization of F with respect to the brush height h
E =−V/L. ForV >0,thefieldiscompressingthechains,
yields the equilibrium brush height h for the charged
while for V <0, it is stretching them. el
case:
Althoughthechainelasticdeformationisconsideredex-
plicitly, the electrostatic properties are calculated within σ2ℓ2 1 1 σℓ2V
h ≃h 1− − − (7)
acontinuummodelwherethechainsarecoarsegrainedin el 0(cid:18) Kε (cid:20)2h L(cid:21) KLh (cid:19)
0 0
p-2
Interfacial Instability of Charged End-Group Polymer Brushes
This expression is valid for low enough σ, i.e. when σ2 ≪ ψ=V
y=L
Kεh /ℓ2 and σV ≪KLh /ℓ2. ψ
0 0 b
Note that the brush height is compressed, h < h ,
el 0
even when the external potential gap between the two
electrodes vanishes, V = 0. In this case the charges at
2π/q
the brush end–groups are attracted to the induced image
σ
eff
charges on the two grounded electrodes, and the interac- h
q
tion with the closer electrode (at the origin) is stronger.
In the opposite case of strong charge-charge interac-
tions,thebrushheightcanbemuchsmallerthanh0,which hel ψa
makes eq (6) invalid. In this case the brush free energy is
y=0 ψ=0
dominated by excluded volume interactions [last term in x
eq (1)]. In the limit h≪L the brush height is:
Fig. 2: A brush with an undulating height profile given by
1 1
h2el = 2kBTv0ℓ−4N2 σV + 1σ2 (8) hfla(xt)su=rfahceels+lohcqatceodsqaxt yco=nfi0naedndbyet=weLen. two conducting and
L 2 ε
(cid:0) (cid:1)
Undulating charged brush. – Because neighbor-
ing chains carry identical charges they repel each other. Laplace equation is satisfied separately for each order n
The electrostatic energy can be further reduced if the in the expansion: ∇2ψ(n) =0 and ∇2ψ(n) =0. Note also
a b
chains compress or stretch alternatively, thereby increas- thatthezerothorderterms,ψ(0) andψ(0),arethesolution
a b
ing the distance between chain ends. In order to investi- of the flat chargedlayer[eqs. (3) and (4)]. It then follows
gate whether such an interfacial instability exists, we as- that for n>0
sume that the brush height has a single undulation mode
along one of the lateral surface dimensions, x: ψ(n) = a(n)exp(ky)+b(n)exp(−ky) coskx
a k k
kX6=0(cid:16) (cid:17)
h(x)=h +h cos(qx) (9)
el q ψ(n) = d(n)exp(ky)+e(n)exp(−ky) coskx (11)
b k k
where h is the equilibrium location of a flat brush [eq. kX6=0(cid:16) (cid:17)
el
(7)], q =2π/λ is the modulation q-mode, and h the am-
q The leading contributions in h , the layer undulation
q
plitude. As was done above for the flat layer, the gap is
amplitude, can be examined by assuming that h ≪ h
q el
divided into two regions and the potential is ψ = ψ for
a and expanding the electrostatic free energy up to order
y < h(x) and ψ = ψ for h(x) < y < L. The poten-
b ∼(h )2. We,therefore,limitourselvestothefirstorderin
tialψ(x,y)satisfiesLaplace’sequation∇2ψ =0,withthe h : ψq =ψ(0)+ψ(1)h . Furthermore,we focus on the long
q q
following four boundary conditions:
wavelength limit, qh ≪ 1, relevant to small amplitude
q
modulations.
ψ | =0 ; ψ | =V
a y=0 b y=L For linear order in h , only the first Fourier component
q
ψ =ψ | ,
a b y=h(x) k=q does not vanish, and we find
εnˆ·∇(ψ − ψ )| =σ (x) . (10)
a b y=h(x) eff σ coshq(L−h )
a(1) = −b(1) =− el
q q 2ε sinhqL
Here nˆ =(qh sinqx,1)/ 1+(qh sinqx)2 is a unit vec-
q q
σ coshqh
tornormaltotheundulatping interfacegivenbyh(x). The d(1) = −e(1) =− el e−qL (12)
densityσ (x)appearingintheaboveboundarycondition q q 2ε sinhqL
eff
is defined as: σ = σ/ 1+(qh sinqx)2 and is related
eff q Expanding to second order in h , both the k = 0 mode
q
to the constant densitypσ on the projected area, as de-
and the second harmonics k =2q do not vanish.
fined in the previous section for the flat layer. In our
The electrostatic energy difference ∆f (per unit area)
el
simplified treatment, the media dielectric constant ε has
between the undulating (h 6=0) and the flat layer (h =
q q
the same value below and above the brush surface. Even
0) is given to second order in h by
q
within the uniform dielectric media assumption, we are
able to show that the flat interface can be unstable. This L L ∆f =
x z el
demonstratesthattheinstability,duetocharge-chargein- ε y=h
itnerRacetfiso.n[1s,6–is18d]i,ffaenrdenrtelfartoemdtoothheerteirnosgteanbeiloituises(ucnocnhsaidrgereedd) − 2Z Z dxdzZy=0 dy h2hq∇ψa(0)·∇ψa(1)
2
dielectric materials placed in external electric fields. +h2 ∇ψ(1)
The potential within the polymer layer, ψa, and above q(cid:16) a (cid:17) (cid:21)
it,∞n=ψ0b,ψa(anr)(ehqw)nri,ttaenndaψsb =a pow∞n=e0rψsb(enr)i(ehsq)inn. hCql:earψlyath=e − 2ε Z Z dxdzZy=y=hL dyh2hq∇ψb(0)·∇ψb(1)
P P
p-3
Y. Tsori et al.
2
+h2 ∇ψ(1) 0 (a)
q(cid:16) b (cid:17) (cid:21)
−0.04
+ σh dx dz a(1)exp(qh)+b(1)exp(−qh) cosqx
qZ Z h q q i f −0.08
(13) D
−0.12
where L and L are the two lateral dimensions and h
x z −0.16
is h(x) from eq. (9). Straightforward algebraic manip-
0.1 0.2 0.3 0.4 0.5 0.6
ulations give the final answer for the electrostatic energy
differenceperunitareaofthebrushinthelongwavelength q
limit (qh ≪1):
q
σ2cosh(qh )cosh[q(L−h )]
∆f =− el el qh2 (14) (b)
el ε sinh(qL) q
Thescalingofthelastexpressioncouldhavebeenguessed 0.4
*
from the outset. The electrostatic energy is symmetric in q
h → −h and, to lowest orders, is quadratic in h . In
q q q 0.3
addition, the prefactor σ2/ε has dimensions of dielectric
constant times electric field squared, and thus ∆f must
el
be linear in q. The derivation above gives us in addition
0 0.5 1 1.5 2
the numerical factors and an extra dependence contain-
h
ing trigonometric functions. These are symmetric with el
respect to the transformation h →L−h .
el el
Note that the externally imposed potential V does Fig. 3: (a) The brush free-energy ∆f from eq. (15) in dimen-
not appear explicitly in ∆f . The external field simply sionless units [rescaled by (σ2/ε)2(h2/γ)], as a function of the
el q
stretchesthe brushuniformly,therebyincreasingh . The dimensionlesswavenumberq[rescaledbyσ2/εγ]. Solid,dashed
el
interfacial instability is solely due to charge-charge inter- andcircle-decoratedcurvescorrespondtorescaledbrushheight
2
actions, as exemplified by the σ2 prefactor. The simple hel(σ /εγ) = 0.5, 1 and 1.5, respectively, and rescaled film
thickness L(σ2/εγ) = 10. The location of the minimum in-
case of a thin isolated charged layer embedded in an infi-
nite medium of uniform dielectric constant is obtained in creases as hel decreases. (b) The most stable dimensionless
∗
the symmetric limit h = 1L, and L → ∞. In this case wavenumberq as a function of rescaled height hel.
el 2
one finds ∆f =−(σ2/2ε)·qh2.
el q
Brush surface instability. – The brush instability by taking the brush charge to be Z =5, chain separation
ℓ=20nm, dielectric constant ε=5ε andsurface tension
mentionedabovecausesadeformationoftheflatlayerand 0
of the brush layer γ = 1mN/m. We use h = γε/σ2 ≃
costs interfacial and elastic energy. We consider first the el
10nmandL=10h ,andfindq∗ ≃2.8·106m−1,resulting
effect of surface tensionand then comment onthe elastic- el
in an undulation wavelength λ∗ = 2π/q∗ ≃ 2.2µm. This
ity. For a single q mode, the interfacial energy per unit
area is ∆f = 1γq2h2. The total free energy difference is indeed the long wavelengthlimit which agrees with the
γ 4 q
variousapproximationswemade. Byfurtherchangingthe
∆f =∆f +∆f is:
γ el
system parameters: ℓ, Ze and γ it is possible to tune q∗
σ2cosh(qh )cosh[q(L−h )] 1 and adapt its value (in the micrometer range) in specific
∆f/h2 ≃− el el q+ γq2 (15)
q ε sinh(qL) 4 experimental setups.
∆f has a minimum at a finite wavenumber q∗ as can be Discussion and Conclusions. – In this Letter we
seen in Fig. 3(a), where ∆f is plotted with a choice of revisit the problem of polymer brushes. The new feature
typicalparametervalues. Thevalueofq∗ canbe obtained consideredwastoattachachargeZetothechainterminal
by solving the transcendental equation free-end. The brushcanbe graftedontoanelectrode(flat
surface), while a second distant electrode is placed above
2 ∂∆f the brush and provides a voltage gap V. The net effect is
q∗ = el (16)
γh2 ∂q (cid:12) to have a controlled way to compress or expand the layer
q (cid:12)q∗
(cid:12) height, h.
(cid:12)
In Fig. 3(b) we show the dependence of q∗ on h /L. The The charged brush creates an effective surface charge
el
moststablewavenumberq∗ decreasesmonotonicallyash density that is localized at the y=h interface. These
el
increases. charges repel each other and also interact with the ex-
In order to check whether the predicted instability can ternal electric field. The equilibrium height h depends
el
beseeninexperiments,weestimateitsorderofmagnitude on the competition between all electrostatic interactions
p-4
Interfacial Instability of Charged End-Group Polymer Brushes
andthe elasticity andentropyofthe neutralchain,ascan More recently [21], a q-mode instability was found for
be seen from eqs. (7) or (8). It can lead to a compres- an electric double layer where a charge density bound to
sion or expansion of the layer height with respect to the a surface was allowed to fluctuate laterally. The model is
neutralbrushcase. Butmaybethe moreinterestingeffect motivated by an experimental setup where charged am-
is the onset of an instability in the layer height. Em- phiphiles coat heterogeneously a mica surface. In the ex-
ploying a linear stability analysis, the conditions leading perimentthesurfacecontainspatchesofpositiveandneg-
to an instability of the uniform layer are analyzed, and a ativechargesbutthe overallsurfacecharge(summedover
preferred wavenumber q∗ stabilized by surface tension is all patches) remains zero. As the surface was placed in
found. When elasticity of the polymer brush is included, contact with a salt solution, a local electric double layer
it contributes a q4 term in eq. (15) [19]in addition to the is formed. Positively and negatively charged counterions
q2 term originating from surface tension. The qualitative areattracted,respectively,tonegativeandpositivecharge
system behavior is similar, with a modified expressionfor domains,resembling our system as well as the undulating
the preferred wavenumber q∗. dipolar one [20,21].
The full system behavior in the presence of counterions Wehope thatthe simple considerationsmentionedhere
is quite complicated and should be explored in a sepa- willmotivateexperimentalstudiesofend-chargedbrushes
rate work, especially the limit of highly charged brushes. where some of our predictions can be tested.
Here, we comment briefly on two extreme limits. In the
firstlimit, the brushchargeandtheexternalpotentialare ∗∗∗
taken to be small enough so that the counterions are uni-
formly distributed throughout the available volume, and We would like to thank Terry Cosgrove for suggesting
behave like an ideal gas. It then follows that the electro- us this problem and Dan Ben-Yaakov for helpful discus-
static potential depends quadratically on the direction y. sions. YT acknowledges support from the Israel Science
In this approximation, the counterions do not contribute Foundation (ISF) under grant no. 284/05, and German-
to the pressure difference ∆P across the brush end. The Israeli Foundation (GIF) grant no. 2144-1636.10/2006.
only source of this pressure difference is electrostatic and DA acknowledges support from the Israel Science Foun-
is due to the difference in 1εE2 between the two sides of dation (ISF) under grant no. 160/05 and the US-Israel
2
the brush. We find ∆P =σV/L+σ2/ε−3σ2h/εL. Binational Foundation (BSF) under grant no. 2006055.
In a second scenario,the brush charge is small, but the
electrostaticenergyofcounterionsincontactwiththeelec-
REFERENCES
trode, eV, is much larger than the thermal energy k T.
B
Here we find that all counterions migrate to one of the
[1] Evans D. F. and Wennerstro¨m H., The Colloidal Do-
electrodes. However, since the above calculation assumes
main: where Physics, Chemistry, Biology and Technology
a fixed voltage gap, V, the electrodes will accumulate ex- meet (Wiley-VCH,New-York) 1999.
tra charge to balance exactly the counterions. Therefore, [2] Fleer G. J., Cohen Stuart M. .A., Scheutjens, J.
the results in eqs. (6), (7) and (8) stay valid. Lastly, M. H. M., Cosgrove T. and Vincent B. Polymers at
we point out that in a more physically feasible setup, the Interfaces (Chapman & Hall, London) 1993,
system may contain added salt [12,13]. In this case, the [3] For a recent review see: Netz R. R. and Andelman D.,
electric field is screened and the brush ends do not “feel” Phys. Rep., 380 (2003) 1, and references therein.
[4] Alexander S.,J. Phys. (France), 38 (1977) 983.
the electrodes as long as the brush length is larger than
[5] de Gennes P.-G., Macromolecules, 13 (1980) 1069.
the Debye-Hu¨ckel screening length.
[6] Milner S. T., Witten T. A. and Cates M. E., Macro-
It is worthwhile to mention some similarities between
molecules, 21 (1988) 2610.
our charged brush and other two-dimensional systems of [7] Semenov A. N., Sov. Phys. JEPT, 61 (1985) 733.
charges or dipoles. A two-dimensional layer of electric [8] Pincus P. A., Macromolecules, 24 (1991) 2912.
dipoles pointing in the perpendicular direction was inves- [9] Borisov O. V., Birshtein T. M.andZhulina E. B.,J.
tigated [20]in relationwith dipolar Langmuir monolayers Phys. II, 1(1991) 521.
at the water/air interface [20]. When the dipoles have a [10] Israels R., Leermakers F. A. M., Fleer G. J. and
fixed out-of-plane moment but their inplane density can Zhulina E. B.,Macromolecules, 27 (1994) 3249.
vary,amodulatedphaseintheinplanedensitycanbesta- [11] Miklavic S. J. and Marcelja S., J. Phys. Chem., 92
bilized with a preferred wavenumber q∗. In addition, the (1988) 6718.
[12] Carignano M. A. and Szleifer I., Mol. Phys., 100
dipolar free energy also scales linearly in q [20]. The sim-
(2002) 2993.
ilarity between the two systems can be understood in the
[13] Gong P.andSzleiferI.,Ind. and Eng. Chem. Res.,45
following way. The charge displacement from their aver-
(2006) 5466.
agepositionaty =h inourcaseissimilartoaneffective
el [14] Thepresenceofathininsulatinglayertowhichthechains
dipole whose moment points ‘up’ or ‘down’ with respect
are grafted is not expected to change our results.
to this reference plane. Increase in the external E field in [15] Lifshitz E. M., Landau L. D. and Pitaevskii L.
our system translates into an increase proportionalto the P., Electrodynamics of Continuous Media (Butterworth-
averagedipole strength in the dipolar system. Heinemann,2nd ed., Boston) 1984.
p-5
Y. Tsori et al.
[16] Scha¨fferE.,Thurn-AlbrechtT.,RussellT.P.and
Steiner U.,Nature, 403 (2000) 874.
[17] Scha¨fferE.,Thurn-AlbrechtT.,RussellT.P.and
Steiner U.,Europhys. Lett., 53 (2001) 518.
[18] Lin Z, Kerle T., Russell T. P., Scha¨ffer E. and
Steiner U.,Macromolecules, 35 (2002) 3971.
[19] Lifshitz E. M. and Landau L. D., Theory of Elasticity
(Butterworth-Heinemann,3rd ed., Boston) 1986.
[20] Andelman D., Brochard F. and Joanny J.F., J.
Chem. Phys., 86 (1987) 3673.
[21] Naydenov A., Pincus P. A. and Safran S. A., Lang-
muir, 23 (2007) 12016.
p-6