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How Do Grafted Polymer Layers Alter the Dynamics of Wetting? Didier Long,* Armand Ajdari, and Ludwik Leibler Laboratoire de Physico-Chimie The´ orique, URA CNRS 1382, ESPCI, 10, rue Vauquelin, 75231 Paris Cedex 05, France Received August 21, 1995. In Final Form: December 4, 1995X We model the static and dynamic wetting properties of nonsoluble liquids on surfaces protected by end-grafted molten polymers. We show that in many situations the spreading dynamics should be controlled by viscoelastic phenomena in the polymer films. Temperature changes can cause the spreading velocities to be varied spectacularly by many orders of magnitude. Conversely, studying wetting on grafted polymer layers should provide a method to characterize their viscoelastic properties.
I. Introduction In many useful systems such as paints, adhesives, and textiles, solid surfaces are protected by soft adsorbed polymer or surfactant layers. Protective films enable one to control the wetting properties of the surfaces as their presence modifies the surface tension of the solid. The aim of the present paper is to discuss how the kinetics of spreading of liquids on polymer-protected surfaces can be profoundly modified by dissipative processes that occur in the thin surface layers. We focus on surfaces covered by terminally anchored polymer chains (so-called polymer brushes)1,2 and propose a model which describes both the statics and the dynamics of partial wetting of nonsoluble liquids on such surfaces. We predict that the presence of a molten polymer layer can spectacularly slow down the spreading process. The dynamics can therefore be controlled by temperature changes or by an appropriate choice of polymer chemical structure. On a rigid surface the wetting dynamics is controlled by the viscous dissipation of the excess surface energy in the wedge of the spreading droplet.3,4 The spreading velocity is thus determined by the viscosity of the liquid. For the spreading on a soft solid (an elastomer, for example), the situation may be quite different. In an elegant experiment, Carre´ and Shanahan5-7 have demonstrated that the spreading of a liquid on such substrates can be much slower than that on a rigid solid with a similar surface tension. They introduced the concept of viscoelastic braking through the dissipation in the substrate and proposed a phenomenological description of this mechanism. Here, we extend this concept to polymercoated surfaces (brushes) and develop a microscopic model which provides quantitative predictions. In this context of wetting phenomena, polymer brushes are particularly interesting systems. Due to the grafting of chain ends, the molten brushes behave as very thin elastic films even when the chains are not entangled or cross-linked.8 Yet, many local viscous dissipative phenomena characteristic of polymers are still present and can be easily controlled X
Abstract published in Advance ACS Abstracts, March 1, 1996.
(1) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 31. (2) Milner, S. Science 1991, 251, 845. (3) De Gennes, P.-G. Rev. Mod. Phys. 1985, 57, 827. (4) Le´ger, L.; Joanny, J.-F. Rep. Prog. Phys. 1992, 431. (5) Carre´, A.; Shanahan, M. E. R. C. R. Acad. Sci., Ser. II 1993, 317, 1153. (6) Carre´, A.; Shanahan, M. E. R. Langmuir 1995, 11, 24. (7) Carre´, A.; Shanahan, M. E. R. Langmuir 1995, 11, 1396. (8) Fredrickson, G. H.; Ajdari, A.; Leibler, L.; Carton, J.-P. Macromolecules 1992, 25, 2882.
by standard methods such as the introduction of plasticizers or the modification of temperature. In section II we discuss the viscoelastic properties of polymer brushes, in the nonentangled case. This will constitute the basis for our consideration of static and dynamic wetting. A liquid drop partially wetting a soft substrate induces a deformation of the latter. This is a well-known phenomenon for both liquid and elastomeric subtrates.9,10 In section III we calculate the corresponding static deformation for a molten brush, characterized by a length ξ0 which depends on the surface tension of the polymer, its molecular weight, and its grafting density. The deformation of the brush is determined by a balance between elasticity, which dominates at distances larger than ξ0, and the brush surface tension, which dominates at shorter distances. During the spreading of a droplet the deformed region moves. The resulting dissipation and other factors that control the dynamics are analyzed in section IV. In particular, we exhibit a characteristic velocity Vc ) ξ0/2πτb, τb being the relaxation time of brush deformations. For spreading velocities V smaller than Vc, the dissipation in the brush can be described in terms of a constant viscosity and is proportional to V2. In many cases the velocity is actually of the same order as Vc and can be higher than Vc only when the instantaneous contact angle is very far from its equilibrium value. In the latter case spreading is no longer controlled by the sole dissipation in the substrate. In section V we present some detailed predictions and discuss in particular the temperature dependence of the spreading velocity. This allows us to propose wetting analysis as a way to gain insight into viscoelastic properties of polymer coatings. II. Polymer Brushes Are Viscoelastic Materials We consider a layer of flexible polymer chains endanchored to a rigid surface. We assume that the grafting density σ is sufficiently high for the chains to be stretched in the direction normal to the surface.1,2 Such polymer brushes can be produced by covalent bonding or selective adsorption of end-functionalized polymers. A self-assembled surface of block copolymers with a molten and a glassy block also provides an example of a molten polymer brush. The height of the brush is h0 ) σNv where N denotes the polymerization index and v the monomeric volume. For the sake of simplicity we assume that v = (9) Shanahan, M. E. R. J. Phys. D: Appl. Phys. 1988, 21, 981. (10) Shanahan, M. E. R.; De Gennes, P.-G. C. R. Acad. Sci., Ser. II 1986, 302, 517.
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Figure 2. Liquid droplet on a soft substrate: picture of the vicinity of the contact line. A vertical force f0 applied at point A induces the deformation of the substrate on a scale ξ0.
deformation ∆h(x) ) cos(qx). The surface tension contribution is Figure 1. Schematic picture of a molten brush. The chains are elongated in the direction normal to the surface and can be viewed as stretched sequences of blobs.
∆FS ) (1/4)γSq22
(3)
whereas the elastic contribution reads b3, where b is the Kuhn length of the polymer. Thus the height of the brush is given as
h0 ) σNb3
(1)
When the grafting density is high, i.e., σ > N-1/2b-2, the chains are indeed stretched as h0 > N1/2b (N1/2b is the typical end-to-end distance of the chains in their unperturbed Gaussian conformation). The stretched chains can be viewed as a sequence of N/g Gaussian blobs of g ) σ-2b-4 monomers each and of end-to-end distance g1/2b (Figure 1). The stretching energy per chain is equal to 3/ (h 2/Nb2)k T ) 3/ (N/g)k T, where k denotes the Boltz2 0 B 2 B B mann constant and T the temperature. Further deformation of the brush implies a free energy penalty of two different origins: surface tension and elasticity. When the brush is deformed, its free surface increases and so does its surface energy. Let us note γS the brush surface tension. Because the chains are anchored by one end to a rigid surface and as incompressibility is assumed, molten brushes behave like incompressible elastic films.8,11,12 Within the Alexanderde Gennes approximation,13,14 their shear modulus is simply8 µ0 ) 3kBT(vσ2/b2), which we write as
µ0 ) 3kBTbσ2 ) 3kBT/(gb3)
(2)
It is interesting to note that the modulus µ0 is equal to the modulus of a rubber with g monomers between crosslinks, even though the chains are not cross-linked (but attached to the surface). Typically, the grafting density is about σb2 ) 10-1, b = 5 Å, and the polymerization index N ) 500, so that the brush height is h0 ) σNb3 = 250 Å and the modulus µ0 = 106 Pa. Brushes can thus be quite soft compared to rigid substrates, for which µ0 = 109 Pa. We suppose that the deformations undergone by the brush are small; therefore, its response to a perturbationse.g., an applied force on the surfacesis linear and the corresponding deformation free energy is the sum of the contributions of each Fourier mode of the deformation. Following ref 8, let us then consider the (11) Williams, D. R. M. Macromolecules 1993, 26, 5096. (12) Williams, D. R. M. Macromolecules 1993, 26, 6667. (13) Alexander, S. J. Phys. (Paris) 1977, 36, 983. (14) De Gennes, P.-G. Macromolecules 1980, 13, 1069.
∆FEL ) (1/2)µ0q2
(4)
for short-wavelength deformations (qh0 . 1) and
µ02 ∆FEL ) (3/4) 2 3 q h0
(5)
for long-wavelength deformations (qh0 , 1). Let us compare the relative importance of the surface tension and elasticity terms at short distances (qh0 > 1). In this regime, we deduce from eqs 3 and 4 that the ratio of the surface energy to the elastic energy 1/2(γSq2)/(µ0q) is everywhere larger than its value at the crossover q = h0-1:
γS 2µ0h0
R)
(6)
For γS ) 10-1 N m-1 and the previously quoted typical values one has R ) 2. More generally, with this value of γS, R > 1 when
σb2
250. As in general R > 1, for wave numbers q > h0-1 the surface tension term dominates the elastic one (actually by a factor Rqh0). Thus the contribution of eq 4 will be neglected. The free energy is then the sum of eqs 3 and 5, so that elasticity dominates surface tension at distances larger than the characteristic length:
( )
ξ0 ) h0
γS 3µ0h0
1/4
(8)
In an exact description, the deformation of the substrate is a function of the distance F from the center of the droplet. However, as we will verify in section III, the deformation induced by a localized force is strongly damped beyond ξ0 (=100-1000 Å), which is much smaller than the typical radius R of the droplet. We will thus consider that the
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deformation ∆h is localized in the vicinity of the contact line. This region is then well approximated by a translationally invariant geometry. In this 2D picture, a Cartesian variable x is used to denote the distance from the contact line. The local thickness of the brush in the perpendicular direction y is h(x) ) h0 + ∆h(x) (Figure 2). We are thus now interested in the deformation free energy per unit length of the triplet line, F. It is a functional of the deformation ∆h(x) which we write
F[∆h] )
1 P(q) hˆ (q) hˆ (-q) dq ∫-∞∞2π
1 2
(9)
hierarchy of long relaxation times such as those found, e.g., in lightly cross-linked rubbers19 due to long dangling chains,20 which would complicate the description of wetting dynamics. In what follows, however, we neglect the effects of the entanglements. Hence our model of viscoelastic properties of brushes is analogous to the Rouse model19 valid for a perfectly regular rubber: at low frequencies, dissipation is correspondingly described by the bulk viscosity
η0 ) µ0τb
(14)
In terms of complex elastic moduli, we can write with, according to eqs 3 and 5,
P(q) ) h0-2[γS(qh0)2 + 3µ0h0(qh0)-2]
(10)
We use here the following definition of the Fourier transform:
hˆ (q) )
∫-∞∞ ∆h(x) exp(iqx) dx
(11)
The inverse Fourier transform is then given by
1 ∆h(x) ) 2π
∫-∞ hˆ (q) exp(-iqx) dq ∞
(12)
2
(13)
where ζ denotes the monomeric friction coefficient. At finite time scales, another contribution to the modulus could arise from chain entanglements inside the brush.16 We consider here brushes for which this contribution is small compared to the blobs contribution, i.e.,17
kBT 3
Neb
kBT
100. Entanglements could also lead to a very long relaxation time as the only allowed motion of the chains is retraction as in star-polymer systems.18 However, in most experimental situations, chains in a brush are hardly entangled. There may be typically 1 or 2 entanglements per chain. For such a small number of entanglements, their effect on the dynamics should not be essential. For example, viscoelastic measurements for polystyrene melts of molecular weight 59 000,19 which corresponds roughly to twice the critical molecular weight of entanglement, show that both storage and loss moduli are correctly described by the Rouse model. Of course, if the brush chains are very long, we might expect a continuous
These relations (eqs 15-17) hold up to a crossover frequency ωc ) τb-1. For higher frequencies, the elasticity of the brush is given no longer by the elasticity of static blobs, but by that of smaller blobs, namely, the largest ones which have time to relax during one cycle. This leads to 15,19
µ′(ω) ) µ0(ωτb)1/2
(18)
µ′′(ω) ) µ0(ωτb)1/2
(19)
The viscosity η(ω) and tan δ(ω) are then
η(ω) ) µ′′(ω) ω-1 ) µ0τb(ωτb)-1/2
(20)
tan δ(ω) ) 1
(21)
Within the present Rouse model, the bulk viscoelastic properties of the brush are in scaling form the same as those of an ideal, regularly reticulated rubber. This picture is valid only when the surface to which the chains are attached is not deformed. The value of τb depends on the friction coefficient ζ, which depends strongly on the chemical structure and on the temperature. Typically, for the polyisobutylene at room temperature ζ = 4 × 10-8 N s m-1,19 which leads to τb = 10-3 s (with a number of monomers per blob g ) 100). Then the low-frequency viscosity is η0 = 103 Pa s. For such polymer films, viscous dissipation in the substrate can play an important role. III. Deformation of Polymer Brushes by Capillary Forces When a droplet is at equilibrium on a solid substrate, the forces on the triple line are balanced. Projecting this condition on the horizontal plane, one obtains Young’s equation:3
cos θe ) (15) Doi, M.; Edwards, S.-F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (16) Witten, T.; Leibler, L.; Pincus, P. Macromolecules 1990, 23, 824. (17) Semenov, A. N. Langmuir 1995, 11, 3560. (18) De Gennes, P.-G. J. Phys. (Paris) 1975, 36, 1199. (19) Ferry, J. D. Viscoelastic Properties of Polymers; Wiley: New York, 1980.
(17)
γS - γSL γL
(22)
γS, γL, and γSL are the surface tensions respectively of the air/substrate, air/liquid, and substrate/liquid interfaces, (20) Curro, J. G.; Pincus, P. Macromolecules 1983, 16, 559.
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Long et al. ∞ Then ∆h(x) ) ∫-∞ (1/2π)(e-iqx/P(q))f0 dq, and we find
∆h(x) ) Figure 3. Liquid droplet at equilibrium on a rigid substrate. The equilibrium is characterized by the contact angle θe.
and θe is the equilibrium contact angle (Figure 3). The vertical force due to the liquid surface tension, f0 ) γL sin θe, is equilibrated by that resulting from the deformation of the substrate. This deformation ∆h is determined by the substrate elasticity and is of order (γL sin θe)/µ0.9,10 Typically, one has γL sin θe ) 2 × 10-2 N m-1 and for rigid substrates µ0 = 109 Pa. Thus ∆h = 10-11 m, which is negligible, and vertical deformations are usually altogether forgotten. However, for polymer brushes the elastic modulus is much lower and the brush is deformed by capillary forces. To estimate this deformation, we will make in this section the simplifying assumption that γSL = γS. Figure 2 shows a typical shape of a brush deformed by a droplet. Near the triple line, the brush is raised by the vertical force f0. At distances smaller than ξ0, the elasticity is negligible and the vertical force is equilibrated by the brush surface tension. Then, by writing the equilibrium of vertical forces at the triple line, one finds γS|dh(x)0)/dx| ) 1/2f0. The term due to the elasticity, proportional to q-2, dominates at distances larger than ξ0, where it induces a fast decay of the deformation. Note that as the brush is incompressible, there must be a trough at a distance of order ξ0 to ensure the conservation of the volume. As shown below, the deformation is actually a damped oscillation at long distances, a point difficult to infer a priori. There is inside the liquid droplet an excess pressure π1 due to the curvature of the air/liquid interface:
π1 )
2γL sin θ R
(23)
This Laplace pressure21 exerts an additional stress on the liquid/substrate interface. For droplets of radius R much larger than ξ0, the corresponding deformation of the brush is anyway negligible. Effectively, the total pressure force has the same intensity as the total contribution of the vertical force f0 exerted along the triple line (it has the opposite sign). However, it is uniformly spread on the liquid/substrate interface. Incompressibility then prevents deformation except in the vicinity of the triple line, so that the deformation due to the pressure is typically smaller than that due to the vertical force f0 by a factor of ξ0/R and can be neglected. Let us now calculate the shape of the brush. The deformation due to the vertical force f0 is obtained by minimizing the functional:
∫
1 F[∆h] ) 2
1 P(q) hˆ (q) hˆ (-q) dq - f0∆h(0) -∞2π
∞ (1/2π)hˆ (q) dq, we obtain Using ∆h(0) ) ∫-∞
f0 P(q)
(24)
(21) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: New York, 1959.
) (
)
(25)
In particular, as mentioned previously, the deformation at long distances is a damped oscillation (Figure 2). The amplitude of the deformation measured by its maximum value is
∆h(0) )
f0 2x2γS
ξ0
(26)
With the data quoted above and using eq 9, we obtain ξ0 = 350 Å and ∆h(0) = 30 Å. In the same way, the deformation due to the sole Laplace pressure can be estimated:
∆hp(x) )
{ ( ) ( ) ( ) ( )}
-ξ0 f0ξ0 x2x -x2|x| sin exp + R γS 2ξ0 2ξ0 sin
x2(2R-x) -x2|2R-x| exp 2ξ0 2ξ0
(27)
Thus ∆hp = (ξ0/R)∆h as anticipated, and this effect is therefore negligible in most practical cases. Note that incompressibility imposes conservation of the substrate volume, i.e., ∫∆h(x) dx ) 0, as ensured by the divergence at q ) 0 of P(q): one has indeed hˆ (q)0) ) f0/P(q)0) ) 0. In conclusion, we find that for a small contact angle the deformation is relatively small, about 30 Å. Yet its very existence will have considerable consequences on spreading dynamics. IV. Viscoelastic Braking Consider now a droplet deposited on a rigid substrate with an initial contact angle θ > θe: it is thus out of equilibrium and there is a nonzero total horizontal force on the contact line
fh ) -γL cos θ - γSL + γS ) γL (cos θe - cos θ)
(28)
This pulling force induces spreading. The dissipation in the droplet is3
Pdrop )
3ηLV2 ln(r) θ
(29)
where ηL is the viscosity of the liquid and r a ratio between a macroscopic length (the radius of the droplet) and a microscopic cutoff. Typically ln(r) is of order 10. There is here no dissipation in the rigid substrate, and the velocity of the triple line is obtained by writing Pdrop ) fhV:
V)
∞
hˆ (q) )
(
f0 x2|x| π -x2|x| ξ cos + exp 2γS 0 2ξ0 4 2ξ0
θγL(cos θe - cos θ) 3ηL ln(r)
(30)
In contrast, if the substrate is a brush, it will be deformed and contribute to the dissipation. To estimate this contribution, we consider the following simple model: the point A where the punctual vertical force f0 is applied moves at a velocity V. The deformation thus also moves and induces a flow inside the brush, leading to a viscous dissipation that we aim to calculate. We do not take into account the deformation due to the horizontal force γL(cos θe - cos θ). This term is at most of the same order
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Grafted Polymer Layers and the Dynamics of Wetting
as γL sin θ for π/2 g θ g θe and becomes negligible as a source of dissipation when θ approaches θe. First, let us note that the inertial effects are negligible. Indeed, for typical velocities V < 10-2 m s-1, and typical flow distances ξ0, the Reynolds number Re ) VFξ0/η0 e 10-8 is very small, whereas the sound velocity c ) (µ0/F)1/2 = 30 m s-1 is high. Even in dynamical conditions, the stress is integrally and instantaneously transmitted throughout the brush, at least over the relevant scales h0 and ξ0. In particular, the brush is deformed on its entire depth. Near A, we still expect a region where the elasticity is negligible and where the deformation is dominated by the brush surface tension. Consider first a velocity V smaller than Vc ) ξ0/(2πτb). Then the frequency ω ) 2πV/ξ0 involved in the motion is smaller than τb-1 and the storage modulus µ′(ω) is constant and equal to µ0. The elasticity still dominates beyond ξ0 as in the static case. The dynamical shape has essentially the same features as the static one, with a characteristic width ξ0 and an amplitude given by eq 24. For higher velocities (V > Vc), the deformation near the triple line must still be controlled by the surface tension of the brush. However, the characteristic frequency is now higher than τb-1. Thus the storage modulus is higher than µ0. According to eq 8 the characteristic length depends on the storage modulus and therefore on ω. Thus the frequency ω and the characteristic length ξ must be determined self-consistently by requiring that ω ) 2πV/ ξ(ω), where ξ(ω) ) h0(γS/3µ′(ω)h0)).14 This leads to
()
(V > Vc)
ξ ) ξ0
V Vc
-1/7
()
ω ) τb-1
V Vc
8/7
(31) (32)
The amplitude of the deformation is still given by the same expression as in the static case, ξ0 being replaced by ξ, i.e.,
(V > Vc)
∆h(0) =
f0 ξ γS
(33)
We can define a characteristic frequency ω both at low and high velocities. The dissipation in the substrate per unit length of the triple line is, from a classical result of viscoelasticity,
Pbrush ) E(ω) ω tan δ(ω)
(34)
where E is the elastic energy per unit length of the triple line and tan δ(ω) ) µ′′(ω)/µ′(ω). From the expression of the deformation energy (eq 9), we get
1 1 E= 2 2π
∫ξ
∞ -1
2
γSq hˆ (q) hˆ (-q) dq
(35)
and 2
E)
1 f0 ξ 4π γS
(36)
For low velocities the characteristic frequency ω ) 2π(V/ξ0) is lower than τb-1 and tan δ(ω) ) ωτb, whereas for
Langmuir, Vol. 12, No. 6, 1996 1679
higher velocities, one has ωτb > 1. From eqs 21 and 34 we obtain 2
()
1 f0 V for V < Vc, Pbrush = V 2 γS c Vc
2
(37)
2
for V > Vc, Pbrush =
1 f0 V 2 γS
(38)
It is interesting to stress that this result can be derived from a hydrodynamic picture. Indeed, the dissipated power can be expressed in terms of the velocity field inside the brush, u(x,y), namely, Pbrush ) η∫[∇u|2 dx dy. Since the deformation is characterized by a long-wavelength ξ, and because of the incompressibility of the brush, the chains are mostly elongated in the x direction. Then we find the typical velocity u to scale as
∆h(0) ω ξ ) V u = ∆h(0) 2π h0 h0
(39)
On the other hand, the velocity at the grafting surface is 0. Thus the typical square velocity gradient is
∂ux 2 ∆h(0) 2 V| | =| ∂y h2
|∇u|2 = |
(40)
0
We can neglect the contribution from |∂ux/∂x|, which is smaller than |∂ux/∂y| by a factor h0/ξ < 1. By integrating over the height h0 in the y direction and the length ξ in the x direction we get the dissipated power
()
Pbrush ∝ η
f 0 2 ξ3 2 V γS h 3 0
(41)
In the case V e Vc, one has η ) µ0τb. Then, using eq 8 we recover the same result as eq 37. In the case V g Vc, by writing η ) µ0τb(ωτb)-1/2, we recover Pbrush ∝ (f02/γS)V as in eq 38 above. V. Discussion Spreading Velocity. We have found two different regimes for the dissipated power Pbrush in the brush: at low velocity, Pbrush is proportional to V2, while at velocities larger than Vc, Pbrush is proportional to V. Hence, the braking force Pbrush/V due to the viscoelastic dissipation in the brush reaches a maximum value 1/2(f02/γS) at V ) Vc and stays at this constant value for V > Vc. This dissipative mechanism can control the spreading dynamics only if the horizontal force fh is smaller than this maximal value 1/2(f02/γS). From eq 28, this condition is equivalent to
cos θe - cos θ γL >2 γS sin2 θ
(42)
When the surface tension of the brush, γS, is much larger than that of the liquid, γL, this condition is verified only when θ is very close to θe. Then the spreading dynamics is controlled by the viscoelastic dissipation in the brush only during the last stages of the spreading. For γS of the same order of magnitude as γL, which is the most frequent case, condition 42 is not very restrictive. Then, if dissipation in the brush indeed controls the spreading, the velocity V is smaller than Vc and the viscoelastic dissipation Pbrush is proportional to V2. The force balance
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Pbrush(V) ) fhV determines the velocity. The dissipation Pdrop in the liquid has to be smaller than the dissipation in the brush Pbrush for this picture to hold, which reads 2 2 1 f0 V2 3ηLV > ln(r) 2 γS Vc θ
(43)
which for a small angle θ is equivalent to
Vc
