Slip effects in polymer thin films

TOPICAL REVIEW arXiv:0909.1946v1 [cond-mat.soft] 10 Sep 2009 Slip effects in polymer thin films O Bäumchen and K Jacobs Experimental Physics, Saarland University, Campus, D-66123 Saarbrücken, Germany E-mail: k.jacobs@physik.uni-saarland.de Abstract. Probing the fluid dynamics of thin films is an excellent tool to study the solid/liquid boundary condition. There is no need for external stimulation or pumping of the liquid due to the fact that the dewetting process, an internal mechanism, acts as a driving force for liquid flow. Viscous dissipation within the liquid and slippage balance interfacial forces. Thereby, friction at the solid/liquid interface plays a key role towards the flow dynamics of the liquid. Probing the temporal and spatial evolution of growing holes or retracting straight fronts gives, in combination with theoretical models, information of the liquid flow field and especially the boundary condition at the interface. We review the basic models and experimental results obtained during the last years with exclusive regard to polymers as ideal model liquids for fluid flow. Moreover, concepts that aim on explaining slippage on the molecular scale are summarized and discussed. PACS numbers: 68.15.+e, 83.50.Lh, 83.80.Sg, 47.15.gm Submitted to: J. Phys.: Condens. Matter Slip effects in polymer thin films 2 1. Introduction Understanding liquid flow in confined geometries plays a huge role in the field of microand nanofluidics [1]. Nowadays, microfluidic or so-called lab-on-chip devices are utilized in a wide range of applications. Pure chemical reactions as well as biological analysis performed on such a microfluidic chip allow a high performance while solely small amounts of chemicals are needed. Thereby, analogies to electronic large-scale integrated circuits are evident. Thorsen et al fabricated a microfluidic chip with a high density of micromechanical valves and hundreds of individually addressable chambers [2]. Recent developments tend to avoid huge external features such as pumps to control the flow by designing analogues to capacitors, resistors or diodes that are capable to control currents in electronic circuits [3]. By reducing the spacial dimensions of liquid volume in confined geometries, slippage can have a huge impact on flow dynamics. Especially the problem of driving small amounts of liquid volume through narrow channels has drawn the attention of many researchers on slip effects at the solid/liquid interface. The aim is to reduce the pressure that is needed to induce and to maintain the flow. Hence, the liquid throughput is increased and, what is important in case of polydisperse liquids or mixtures, a low dispersity according to lower velocity gradients perpendicular to the flow direction is generated. Confined geometries are realized in various types of experiments: The physics and chemistry of the imbibition of liquids by porous media is of fundamental interest and enormous technological relevance [4]. Channel-like three-dimensional structures can be used to artificially model the situation of fluids in confinement. Moreover, the small gap between a colloidal probe and a surface filled with a liquid (e.g realized in surface force apparatus or colloidal probe atomic force microscopy experiments) is a common tool to study liquid flow properties. Besides the aforementioned experimental systems, knowledge in preparation of thin polymer films has been extensively gained due to its enormous relevance in coating and semiconductor processing technology. Such a homogeneous nanometric polymer film supported by a very smooth substrate, as for example a piece of a silicon (Si) wafer, exhibits two relevant interfaces, the liquid/air and the substrate/liquid interface. Si wafers are often used according to their very low roughness and controllable oxide layer thickness. Yet, also highly viscous and elastically deformable supports, such as e.g. polydimethylsiloxane (PDMS) layers, are versatile substrates. The stability of a thin film is governed by the effective interface potential φ as function of film thickness h. In case of dielectric systems, φ(h) is composed of an attractive van-der-Waals part and a repulsive part [5, 6, 7, 8]. For the description of van-der-Waals forces of a composite substrate, the layer thicknesses and their respective polarisation properties have to be taken into account [9]. Thereby, three major situations of a thin liquid film have to be distinguished: stable, unstable and metastable films. As illustrated in Fig. 1 by the typical curves of φ(h), a stable liquid film is obtained if the effective interface potential is positive and monotonically decaying (cf. curve (1) in Slip effects in polymer thin films 3 Figure 1. Different shapes of the effective interface potential φ(h) associated with different wetting conditions. Curve (1) characterizes a stable liquid film. Curve (2) represents a metastable, curve (3) and (4) an unstable situation. Fig. 1). The equilibrium film thickness heq is infinite and the liquid perfectly wets the substrate. In case of a global minimum of φ(h), curves (2) and (3) in Fig. 1, the system can minimize its energy and a finite value for heq results. The metastable situation is furthermore characterized by a potential barrier that the system has to overcome to reduce its potential energy (cf. curve (2) in Fig. 1). Curve (3) and (4) characterize unstable conditions, since every slight fluctuation in film height will drive the system towards the global minimum. The wettability of the substrate by the liquid is correlated to the depth of the minimum at φ(heq ). The deeper the global minimum of φ, the larger is the equilibrium contact angle of the liquid on the surface. A nearly 180 ◦ situation is depicted in curve (4) of Fig. 1. For a 100 nm polystyrene (PS) film on a hydrophobized Si wafer with a native oxide layer, dewetting starts after heating the sample above the glass transition temperature of the polymer. Holes nucleate according to thermal activation or nucleation spots (dust particles, inhomogeneities of the substrate or of the polymer film) and grow with time, cf. Fig. 2. The subsequent stages of dewetting are characterized by the formation of liquid ridges by coalescence of growing holes and traveling fronts. Very thin films in the range of several nanometers may become unstable and can dewet according to thermally induced capillary modes, that are amplified by forces contributing to the effective interface potential. This phenomenon is characterized by the occurrence of a preferred wavelength and is called spinodal dewetting. To study thin film flow with regard to the influence of slippage, especially nucleated holes enable an easy experimental access for temporal and spatial observation. While fronts retract from the substrate and holes grow, a liquid rim is formed at the three-phase contact line due to conservation of liquid volume. A common phenomenon Slip effects in polymer thin films 4 Figure 2. Dewetting of a 80 nm PS(65 kg/mol) film at T = 135 ◦C from a hydrophobized Si substrate captured by optical microscopy (adapted from [10]). is the formation of liquid bulges and so-called ”fingers” due to the fact that the liquid rim becomes unstable, similar to the instability of a cylindrical liquid jet that beads up into droplets (e.g. in case of a water tap). This so-called Rayleigh-Plateau instability is based on the fact that certain modes of fluctuations become amplified and surface corrugations of a characteristic wavelength become visible. If two holes coalesce, a common ridge builds up that in the end decays into single droplets due to the same mechanism. The final stage is given by an equilibrium configuration of liquid droplets arranged on the substrate exhibiting a static contact angle. Actually, the final state would be one single droplet, since the Laplace pressure in droplets of different size is varying. Yet then, a substantial material transport must take place, either over the gas phase or via an equilibrium film between the droplets. This phenomenon is also called Ostwald ripening. In polymeric liquids, these transport pathways are usually extremely slow so that a network-like pattern of liquid droplets is already termed ”final stage”. Besides the dynamics and stability of thin liquid films driven by intermolecular forces, a recent article by Craster and Matar reviews further aspects such as e.g thermally or surfactant driven flows [11]. In the first part of this topical review, basic concepts from hydrodynamic theories in the bulk situation to corresponding models with regard to confinement are introduced. Polymers are regarded as ideal model liquids due to their low vapor pressure, the available chemical pureness and furthermore the fact that their viscosity can be controlled in a very reliable manner. By setting the viscosity via temperature, the experimental conditions can be tuned so that dewetting dynamics can be easily captured. Mass conservation can safely be assumed, which consequently simplifies the theoretical description. The dewetting dynamics governed by the driving forces and the mechanisms of energy dissipation will be discussed also with regard to the shape of liquid ridges. The second part summarizes experimental studies concerning the dynamics of two different dewetting geometries: the straight front geometry and the growth of holes. Especially the influence of parameters such as dewetting temperature, viscosity and molecular weight of the polymer will be discussed in detail. Moreover, we focus on various scenarios at the solid/liquid interface on the molecular level. Simulations such as for example molecular-dynamic (MD) studies help to obtain more information and are supportive to gain insight into the molecular mechanism of slippage. Slip effects in polymer thin films 5 2. Basic theoretical concepts In this section we aim to describe the main concepts of fluid dynamics especially in a confined geometry. Depending on the type of liquid, viscous or even viscoelastic effects have to be considered and deviations from Newtonian behavior might become non-negligible. Since we concentrate in this article on polymer melts, viscosity and viscoelasticity can be varied by chain length and branching of the polymer. An important aspect of a moving liquid is the velocity profile. 2.1. Polymer properties For a comprehensive understanding of slippage, some important polymer properties like glass transition temperature, viscosity and viscoelasticity have to be taken into account. Especially in geometries like in a liquid film, confinement effects are a concern. A detailed description can be found in textbooks [12, 13]. 2.1.1. Polymer physics Polymers are synthesized by polymerization of monomers of molar mass Mmono . To characterize the polydispersity of polymer chains in a solution or in a melt, the polydispersity index Mw /Mn is calculated as the ratio of mean values given by the weight Mw and number averaged Mn molecular weight. Polymers are able to change their conformations. The radius of gyration characterizes the spatial dimension of the polymer and is given as the mean square displacement between monomers and the polymer’s center of mass. In an isotropic configuration, the shape of the polymer chain can be approximated as a spherical entity. Polystyrene, abbreviated PS, which is commonly used as a melt in dewetting experiments, is a linear homopolymer with Mmono = 104 g/mol. Besides other properties concerning the micro-structure of polymer chains such as the tacticity and the architecture (linear, branched, ring-shaped), physical properties are of special interest. If a melt is heated above its glass transition temperature Tg , a phase transition from the glassy phase to the liquid phase occurs and the polymer becomes liquid. Randomly structured macromolecules such as atactic polymers avoid the formation of semi-crystalline domains below Tg and exhibit a pure amorphous phase. The glass transition of a bulk polymer or of a polymeric thin film can be observed e.g. via probing its linear expansion coefficient. Although, the glass transition is based on a kinematic effect and does not occur due to a rearrangement process of polymer chains. Therefore, the glass transition usually takes place on a specific temperature range and not at a exactly allocable temperature. According to the increased mobility of shorter chains, their glass transition temperature is decreased significantly. For PS with a sufficiently large chain-length, Tg =100 ◦ C. The viscosity η of a polymer melt measures the inner friction of polymer chains and governs the time scale of flow processes. Due to the fact that the mobility of chains increases while the temperature increases, the viscosity decreases. Internal stresses relax and dynamical processes proceed faster. The most common description of the functional Slip effects in polymer thin films 6 dependency of the viscosity η of a polymer on temperature T was developed by Williams, Landel and Ferry: η = ηg exp B(Tg − T ) . fg (T − T∞ ) (2.1) In this so-called WLF-equation, ηg denotes the viscosity at Tg , B an empirically obtained constant, T∞ the so-called Vogel temperature and fg the free liquid volume fraction. Besides the above discussed impact of temperature on the viscosity of a polymer melt, also the molecular weight Mw strongly influences η. While Mw increases, the chain mobility decreases and therefore the relaxation times and thus η increase. For sufficiently small Mw , the Rouse model predicts a linear increase of η for increasing Mw . At this point, another characteristic number of polymer physics has to be introduced: the critical chain length for entanglements Mc . Above Mc , the polymer chains form chain entanglements exhibiting a specific mean strand length called entanglement length Me . For PS, Mc =35 kg/mol and Me =17 kg/mol is found [12]. According to the reptation model of de Gennes [14], the viscosity increases stronger in the presence of chain entanglements and an algebraic behavior of η ∝ Mw3 is expected. Empirically, the linear regime below Mc is well reproduced, above Mc an exponent of 3.4 is found experimentally for different polymers. For a detailed description of chemical and physical properties of polymers as well as for the Rouse or the reptation theory we refer to the book Polymer physics by Rubinstein and Colby [12]. 2.1.2. Properties in confined geometries In contrast to the bulk situation, where volume properties of the polymers are measured, liquids in confined geometries such as a thin film often show deviations from the behavior in the volume due to additional interface effects. One of these properties is the aforementioned glass transition temperature. It has been shown in numerous studies, that Tg changes with film thickness h. On the one hand, in case of free-standing or supported films exhibiting no or repulsive interactions with the substrate, Tg (h) decreases with decreasing film thickness [15, 16, 17, 18]. On the other hand, Keddie and Jones have shown that an increase of the glass transition temperature with decreasing film thickness is possible for attractive interactions between substrate and polymer film [19]. The influence of the interfacial energy on the deviation of Tg from its bulk value has been studied and quantified by Fryer and co-workers for different polymers [20]. In case of PS on a solid support, a significant change of Tg is found for films thinner than 100 nm (see Fig. 3) [21]. For PS(2k) below 10 nm for instance, the glass transition temperature and the viscosity of the polymer film are affected such that these films are liquid at room temperature and may dewet spinodally. Several attempts have been made to explain the change of Tg according to the film thickness. Besides interface-related effects such as reorientation of polymer chains or accumulation of chain-ends at the interface, finite-size effects have been proposed to be responsible. Herminghaus et al discussed the strain relaxation behavior of thin Slip effects in polymer thin films glass transition temperature T g [K] 0 7 film thickness [nm] 50 100 150 200 320 10 300 280 100 320 300 280 260 260 Figure 3. Glass transition temperature Tg of polystyrene films of 2 kg/mol against film thickness (adapted from [21]). viscoelastic polymer films with regard to surface melting and the shift of the glass transition temperature [21]. Kawana and Jones studied the thermal expansivity of thin supported polymer films using ellipsometry and attributed their results concerning Tg to a liquid-like surface layer [22], a result that was also found by other authors [23, 24]. Besides confinement effects on Tg , further interface-related phenomena have been studied: Si et al have shown that polymers in thin films are less entangled than bulk polymers and that the effective entanglement molecular weight Me is significantly larger than the bulk value [25]. 2.1.3. Viscosity and viscoelasticity One of the major characteristics of a liquid in general or a polymer in particular is its viscosity η. Applying shear stress σ to a liquid, it usually reacts with a strain γ. If stress and strain rate γ̇ are proportional, the fluid is called Newtonian. The constant of proportionality is identified as the viscosity η of the liquid. σ = η γ̇. (2.2) Liquids such as long-chained polymers show a shear rate dependent viscosity η(γ̇) due to the fact that the liquid molecules are entangled. If the viscosity increases while shearing the liquid, we call this behavior shear thickening, whereas in case of lowered viscosity so-called shear thinning is responsible. In contrast to the elastic deformation of a solid, a deformation of a viscoelastic liquid might induce an additional flow and can relax on a specific time scale τ . On short time scales (t < τ ), the liquid behaves in an elastic, on long time scales (t > τ ) in a viscous manner. Thereby, strain γ is connected to stress σ via the elastic modulus G of the liquid: σ = Gγ. (2.3) Slip effects in polymer thin films 8 Figure 4. Maxwell model represented by a dashpot and a spring in a serial connection. To cover the stress relaxation dynamics of a polymer film, several modelling attempts have been proposed. Mostly, so-called Maxwell or Jeffreys models are applied. The simplest model is the Maxwell model (see Fig. 4), which assumes a serial connection of a perfectly elastic element (represented by a spring) and a perfectly viscous one (represented by a dashpot). Consequently, the total shear strain γ is given by the sum of the corresponding shear strains γe and γv of both mechanisms: γ = γe + γv . (2.4) With (2.3) and (2.2) we get σ = GM γe = ηM γ̇v (2.5) since both react on the same shear stress σ. Thereby, the ratio of viscosity of the viscous element to the elastic modulus of the elastic one can be identified with a specific time scale, the relaxation time τM : ηM . (2.6) τM = GM The relaxation of stress after a step strain γ leads to a time-dependent stress function σ(t) for a viscoelastic liquid. Due to the fact that the total strain γ is constant, a first order differential equation for the time-dependent strain γv (t) is obtained: τM γ̇v = γ − γv (t). (2.7) Solving this differential equation using the initial condition γv (t = 0) = 0 gives a simple exponential decay of σ(t) on the time scale of the stress relaxation time τM in the Maxwell model: γe (t) = γ exp (−t/τM ), σ(t) = GM γe (t) = GM γ exp (−t/τM ). (2.8) A situation of special interest is the linear response region: For sufficiently small values of γ, the stress/strain-relation (2.3) is valid and the stress relaxation modulus G(t) is independent of the strain γ. In this regime, a linear superposition of stresses resulting from an infinite number of strain steps can be used to model a steady simple shear flow of a viscoelastic liquid. For larger applied shear rates, linear response and the linear superposition fails. The viscosity is still defined as the ratio of stress and strain rate, but it has to be regarded as an apparent viscosity which differs from the above described Slip effects in polymer thin films 9 ”zero shear rate” viscosity. Polymers with shear-thinning or shear-thickening properties can be described by the function σ ∝ γ̇ n , (2.9) where the exponent n can be extracted from experimental data. These type of fluids are also called ”power law fluids”. Moreover, also other nontrivial stress-strain relations can be considered or alternatively non-linear extensions can be applied to the linear Maxwell models. In case of the linear Jeffreys model, the stress tensor σij relaxes according to the following constitutive relaxation equation: (1 + λ1 ∂t )σ = η(1 + λ2 ∂t )γ̇, (2.10) where the strain rate is given by the gradient of the velocity field γ̇ij = ∂j ui + ∂i uj . Hereby, λ1 governs the relaxation of stress, whereas λ2 (λ2 < λ1 ) describes the relaxation of the strain rate, respectively. This model accounts for the viscous and the elastic properties of a fluid and was used by Blossey, Rauscher, Wagner and Münch as basis for the development of a thin-film equation that incorporates viscoelastic effects [26, 27, 28]. For a more elaborate description of non-Newtonian flows we refer e.g. to the correspondent work of Te Nijenhuis et al [29]. 2.1.4. Reynolds and Weissenberg number The flow of a liquid can be characterized by specific numbers. One of these numbers is the so-called Reynolds number Re, which describes the ratio of inertia effects to viscous flow contributions. In case of thin liquid films, Re can be written as Re = ρuh , η (2.11) where ρ denotes the density of the liquid, u describes the flow velocity and h stands for the film thickness [30]. For thin dewetting polymer films, the Reynolds number is very small, i.e. Re ≪ 1, and a low-Re lubrication theory can be applied. To quantify and to judge the occurrence of viscoelastic effects versus pure viscous flow, the so-called Weissenberg number W i has been introduced as W i = τ γ̇. (2.12) Thereby, τ denotes the relaxation time and γ̇ the strain rate as introduced in the previous section. If W i ≪ 1, an impact of viscoelasticity on flow dynamics can be neglected and viscous flow dominates. 2.2. Navier-Stokes equations The Navier-Stokes equations for a Newtonian liquid mark the starting point for the discussion of fluid dynamics in confined geometries. According to conservation of mass, the equation of continuity can be formulated as ∂t ρ + ∇ · (ρu) = 0, (2.13) Slip effects in polymer thin films 10 where u = (ux , uy , uz ) is the velocity field of the fluid. For an incompressible liquid, which implies a temporally and spatially constant liquid density ρ, (2.13) can be simplified to ∇u = 0. (2.14) With the conservation of momentum, the Navier-Stokes equations for an incompressible liquid can be written as ρ(∂t + u · ∇)u = −∇p + η△u + f, (2.15) with the pressure gradient ∇p and the volume force f of external fields acting as driving forces for the liquid flow. We already stated that for small Reynolds numbers, i.e. Re ≪ 1, the terms on the left hand side of (2.15) can be neglected as compared to terms describing the pressure gradient, external volume forces and viscous flow. By that, we can simplify (2.15) to the so-called Stokes equation 0 = −∇p + η△u + f. (2.16) In section 2.5, we will demonstrate how these basic laws of bulk fluid dynamics can be applied to the flow geometry of a thin film supported by a solid substrate. 2.3. Free interface boundary condition At the free interface of a supported liquid film, i.e. at the liquid/gas or usually the liquid/air interface, no shear forces can be transferred to the gas phase due to the negligible viscosity of the gas. In general, the stress tensor σij∗ is given by the stress tension σij , see (2.2), and the pressure p: σij∗ = σij + pδij = η(∂j ui + ∂i uj ) + pδij . (2.17) The tangential t and normal n (perpendicular to the interface) components of the stress tensor are: (σ ∗ · n) · t = 0 (σ ∗ · n) · n = γlv κ, (2.18) where κ denotes the mean curvature and γlv the interfacial tension (i.e. the surface tension of the liquid) of the liquid/vapor interface. If the liquid is at rest, i.e. the stationary case u = 0, the latter boundary condition gives the equation for the Laplace pressure pL : pL = γlv κ = γlv ( 1 1 + ). R1 R2 (2.19) R1 and R2 are the principal radii of curvature of the free liquid/gas boundary; the appropriate signs of the radii are chosen according to the condition that convex boundaries give positive signs. Such convex liquid/gas boundaries lead to an additional pressure within the liquid due to its surface tension. In the next section, the solid/liquid boundary condition will be discussed, which yields a treatment of slip effects. Slip effects in polymer thin films partial slip no slip u u 11 full slip ”apparent” slip u u z0 b=0 b b=¥ b Figure 5. Different velocity profiles in the vicinity of the solid/liquid interface and illustration of the slip (extrapolation) length b. The situation of so-called ”apparent” slip is illustrated on the right: According to a thin liquid layer of thickness z0 that obtains a significantly reduced viscosity, the slip velocity ux |z=0 is zero, but a substantial slip length is measured. 2.4. Slip/no-slip boundary condition 2.4.1. Navier slip boundary condition In contrast to fluid dynamics in a bulk volume, where the assumption that the tangential velocity u|| at the solid/liquid interface vanishes (no-slip boundary condition), confined geometries require a more detailed investigation as slippage becomes important. In 1823, Navier [31] introduced a linear boundary condition: The tangential velocity u|| is proportional to the normal component of the strain rate tensor; the constant of proportionality is described as the so-called slip length b: u|| = b n · γ̇ (2.20) In case of simple shear flow in x -direction, the definition of the slip length can be alternatively written as ux η η ux |z=0 = = , (2.21) b= ∂z ux σ ξ where ξ = σ/ux denotes the friction coefficient at the solid/liquid interface. The xyplane thereby represents the substrate surface. According to these definitions, the slip length can be illustrated as the extrapolation length of the velocity profile ”inside” the substrate, cf. Fig. 5. Moreover, both limiting cases are included within this description: For b = 0, we obtain the no-slip situation, whereas b = ∞ characterizes a full-slip situation. The latter case corresponds to ”plug-flow”, where the liquid behaves like a solid that slips over the support. 2.4.2. How to measure the slip length? In recent years, numerous experimental studies were published using diverse methods to probe the slip length at the boundary of different simple or complex liquids and solid supports. For details concerning these experimental methods we refer to the review articles from Lauga et al [32], Neto et Slip effects in polymer thin films 12 al [33] and Bocquet and Barrat [34] (and references therein). To probe the boundary condition, scientists performed either drainage experiments or direct measurements of the local velocity profile using e.g. tracer particles. In case of drainage experiments, the liquid is squeezed between two objects, e.g. a flat surface and a colloidal probe at the tip of an AFM cantilever, and the corresponding force for dragging the probe is measured (colloidal probe AFM ). Alternatively, in an surface force apparatus (SFA), two cylinders arranged perpendicular to each other are brought in closer contact and force/distance measurements are performed to infer the slip length. The use of tracer particles as a probe of the local flow profile might bring some disadvantages. The chemistry of these particles is usually different from the liquid molecules and their influence on the results might not be negligible. A similar method is called fluorescence recovery after photo bleaching. Thereby, a distinct part of a fluorescent liquid is bleached by a laser pulse and the flow of non-bleached liquid into that part is measured. The disadvantage of this method is that diffusion might be a further parameter that is hard to control. Recently, Joly et al showed that also thermal motion of confined colloidal tracers in the vicinity of the solid/liquid interface can be used as a probe of slippage without relying on external driving forces [35]. 2.4.3. Which parameters influence slippage? Of course, many interesting aspects in the field of micro- and nanofluidics are related to intrinsic parameters that govern slippage of liquid molecules at the solid/liquid interface. For simple liquids on smooth surfaces, the contact angle is one of the main parameters influencing slippage [36, 37, 38, 39]. This originates from the effect of molecular interactions between liquid molecules and the solid surface: If the molecular attraction of liquid molecules and surface decreases (and thereby the contact angle increases), slippage is enlarged. Further studies aim to quantify the impact of roughness [37, 40, 41, 42] or topographic structure [43, 44, 45, 46, 47] of the surface on slippage. For different roughness length scales, a suppression (see e.g. [37, 40]) or an amplification (see e.g. [43]) of slippage can be observed. Moreover, the shape of molecular liquids itself has been experimentally shown to impact the boundary condition. Schmatko et al found significantly larger slip lengths for elongated linear compared to branched molecules [48]. This might be associated with molecular ordering effects [49] and the formation of layers of the fluid in case of the capability of these liquids to align in the vicinity of the interface [50]. Cho et al identified the dipole moment of Newtonian liquids at hydrophobic surfaces as a crucial parameter for slip [51]. De Gennes proposed a thin gas layer at the interface of solid surface and liquid as a possible source of large slip lengths [52]. Recently, MD studies for water on hydrophobic surfaces by Huang et al revealed a dependence of slippage on the amount of water depletion at the surface and a strong increase of slip with increasing contact angle [53]. Such depletion layers for water in the vicinity of smooth hydrophobic surfaces have also been experimentally observed using scattering techniques [54, 55, 56, 57]. Contamination by nanoscale air bubbles (so-called nanobubbles) and its influence on Slip effects in polymer thin films 13 z h(x,y,t) u||(ux,uy) y x Figure 6. Illustration of the nomenclature of the thin film length scales (x and y are parallel to the substrate) and the velocity contribution u|| = (ux , uy ). slippage has been controversially discussed in literature (see e.g. [58, 59, 60, 61]). In case of more complex liquids such as polymer melts further concepts come into play. They will be illustrated in section 3.4. 2.5. Thin-film equation for Newtonian liquids 2.5.1. Derivation Confining the flow of a liquid to the geometry to the one of a thin film, we can assume that the velocity contribution perpendicular to the substrate is much smaller than the parallel one. Furthermore, the lateral length scale of film thickness variations is much smaller than the film thickness itself. On the basis of these assumptions, Oron et al [62] developed a thin-film equation from the rather complex equations of motion, p (2.13) and (2.15). In case of film thicknesses smaller than the capillary length lc = γlv /ρg, (which is typically in the order of magnitude of 1 mm) (2.16) can be written as 0 = −∇(p + φ′ (h)) + η△u (2.22) Additional external fields such as gravitation can be neglected, but a secondary contribution φ′ (h), the disjoining pressure, has been added to the capillary pressure p. The disjoining pressure originates from molecular interactions of the fluid molecules with the substrate. The effective interface potential φ(h) summarizes the inter-molecular interactions and describes the energy that is required to bring two interfaces from infinity to the finite distance h. As already discussed in the introductory part, the stability of a thin liquid film is also governed by φ(h). For a further description of thin film stability, we refer to [8] and the references therein. The derivation of a thin-film equation for Newtonian liquids starts with the kinematic condition Z h ∂t h = −∇|| u|| dz, (2.23) 0 i.e. the coupling of the time derivative of h(x, y, t) to the flow field, where the index || in general denotes the components parallel to the substrate (∇|| = (∂x , ∂y ) and u|| = (ux , uy )) as illustrated in Fig. 6. Slip effects in polymer thin films 14 For thin liquid films, film thickness variations on lateral scale L are much larger than the length scale of the film thickness H. Introducing the parameter ǫ = H/L ≪ 1 yields the so-called lubrication approximation and is used in the following to re-scale the variables to dimensionless values. In a first approximation, linearized equations are obtained while neglecting all terms of the order O(ǫ2 ). For reasons of simplicity and due to translational invariance in the surface plane, a one-dimensional geometry is used: ∂x (p + φ′ ) = ∂z2 ux , ∂z (p + φ′ ) = 0, ∂x ux + ∂z uz = 0. (2.24) While the substrate is supposed to be impenetrable for the liquid, i.e. uz = 0 for z = 0, friction at the interface implies a velocity gradient ∂z ux = ux /b for z = 0. Moreover, the tangential and normal boundary condition at the free interface, i.e. z = h(x), can be simplified in the following manner: ∂z ux = 0, p + ∂x2 h = 0. (2.25) From (2.24) and the boundary conditions, the velocity profile ux (z) can be obtained. Using the kinematic condition (2.23), the equation of motion for thin films in three dimensions is derived: ∂t h = −∇[m(h)∇(γlv △h − φ′ (h))], where m(h) denotes the mobility given by 1 m(h) = (h3 + 3bh2 ). 3η (2.26) (2.27) 2.5.2. Lubrication models including slippage As discussed in the previous section, the derivation of the thin-film equation is based on the so-called lubrication approximation and the re-scaling of relevant values in ǫ. As a consequence, the slip length b is supposed to obtain values smaller than the film thickness h, i.e. b ≪ h. To extend this so-called weak-slip situation with regard to larger slip b ≫ h, Münch et al [30] and Kargupta et al [63] developed independently so-called strong-slip models. Thereby, the slip length is defined as b = β/ǫ2 . The corresponding equation of motion together with the kinematic condition in one dimension for a Newtonian thin liquid film read as: bh 2b u = ∂x (2ηh∂x u) + ∂x (γlv ∂x2 h − φ′ (h)) η η (2.28) ∂t h = −∂x (hu). In fact, a family of lubrications models, cf. Tab. 1, accounting for different slip situations have been derived. In the limit b → 0, i.e. the no-slip situation, the mobility is given by m(h) = h3 /3η. If the slip length is in the range of the film thickness b ∼ h, the mobility in the corresponding intermediate-slip model is m(h) = bh2 /η. Recently, Fetzer et al [64] derived a more generalized model based on the full Stokes equations, developed up to third order of a Taylor expansion. The authors were able to show that this model is in good agreement with numerical simulations of the full hydrodynamic equations and is not restricted to a certain slip regime as the aforementioned lubrication models. Slip effects in polymer thin films 15 Table 1. Summary of lubrication models for Newtonian flow and different slip situations. model validity equation limiting cases ref. weak-slip b≪h (2.26), (2.27) b → 0 (no-slip) b → ∞ (intermediate-slip) [62] strong-slip b≫h (2.28) β → 0 (intermediate-slip) β → ∞ (”free”-slip) [63, 30] 2.5.3. Lubrication models including viscoelasticity In the meanwhile, the derivation of a thin film equation for the weak-slip case including linear viscoelastic effects of Jeffreys type (such as described by equation (2.10) in section 2.1.3) has been achieved (see [26]). To cover relaxation dynamics of the stress tensor σ, an additional term ∇ · σ on the right hand side of (2.22) has to be included to the aforementioned model for Newtonian liquids. Furthermore, the treatment of linear viscoelastic effects was also achieved for the strong-slip situation by Blossey et al [27]. To summarize these extensions, the essential result is the fact that linear viscoelastic effects are absent in the weak-slip case and the Newtonian thin-film model is still valid. The strong-slip situation, however, is more complicated. Slippage and viscoelasticity are combined and strongly affect the corresponding equations. In the meanwhile, the authors were able to fully incorporate the non-linearities of the co-rotational Jeffreys model for viscoelastic relaxation into their thin-film model [28]. The extensions of the aforementioned thin-film models for different slip conditions with or without the presence of viscoelastic relaxation (Newtonian and non-Newtonian models) affect on the one hand the rupture conditions, but also on the other hand the shape of a liquid ridge. These two phenomena will be discussed in the next two subsections. A elaborate description of these theoretical aspects can be found in a recent review article by Blossey [65]. 2.5.4. Application I - Spinodal dewetting One of the main applications of the theoretical thin-film models is the dewetting of thin polymer films. As introduced in section 1 and illustrated by Fig. 1, the stability of a thin liquid film is governed by the effective interface potential. Basically, long-range attractive van der Waals forces add to short-range repulsive forces. Due to the planar geometry of two interfaces of distance d, the van der Waals contribution to the potential is φ(d)vdW ∝ −A/d2 , where A is the Hamaker constant. For the description of the explicit calculation of Hamaker constants from the dielectric functions of the involved materials we refer to [8] and to the book by Israelachvili [66]. Experimental systems often exhibit multi-layer situations, cf. Fig. 7. A hydrophobic film and/or an oxide layer of distinct thicknesses di exhibiting Hamaker constants Ai require a superposition of contributions to the potential: φ(d)vdw = − A2 − A1 A3 − A2 A1 − − 2 2 12πd 12π(d + d1 ) 12π(d + d1 + d2 )2 (2.29) Slip effects in polymer thin films 16 Figure 7. Two examples for polystyrene films prepared on multi-layer substrates. Left: Silicon wafer with native oxide layer covered with a hydrophobic layer (e.g. a self-assembled monolayer (SAM)). Right: Silicon wafer with an increased (compared to a native silicon oxide) oxide layer thickness. Figure 8. Effective interface potential φ(h), gained from experimental data, plotted against the thickness h of a thin PS film prepared on silicon wafers with different oxide layer thicknesses d1 (adapted from [7]). The cross-hatched rectangle marks the error for heq and the depth of the global minimum. Consequently, the shape of the effective interface potential (and thereby also the thin film stability) is governed by the set of Hamaker constants Ai and film thicknesses di . E.g. for a thin PS film of thickness h on a Si substrate with a native oxide layer of 2.4 nm, φ(h) shows a global minimum at an equilibrium film thickness heq (c.f. Fig. 8). Moreover, a local maximum at h > heq is found. If the PS film is sufficiently thin (φ′′ (h) < 0) , it may become unstable due to the amplification of thermally induced capillary waves. To track the evolution of small fluctuations of the film thickness h, i.e. f (x, t) = h(x, t) − h with f (x, t) ≪ h, a Fourier transform of the linearized thin film equation (2.26) has to be performed. The amplitudes of the capillary waves grow exponentially in time. Their growth rate α can be calculated as a function of the wavenumber q and depends on the sign of the local curvature of the interface potential. If the second derivative of the effective interface potential φ′′ at the film thickness h is positive, α is negative for all q and the amplitudes Slip effects in polymer thin films 17 Figure 9. In situ (at elevated temperature T = 53 ◦ C) atomic force microscopy (AFM) images (corresponding annealing times given in pictures) of a spinodally dewetting 3.9(2) nm PS(2 kg/mol) film on a Si wafer with a thick (191 nm) oxide layer (adapted from [67]). of the capillary waves are damped. If φ′′ < 0, the growth rate α is positive for a certain range of wavenumbers up to a critical wavenumber qc and capillary waves are amplified. The wavenumber that corresponds to the maximum value of α and therefore exhibits the fastest amplification, is called preferred wavenumber q0 and is connected to the preferred wavelength λ0 = 2π/q0 . The latter is also called spinodal wavelength and can be written as s 8π 2 γlv . (2.30) λ0 = −φ′′ (h) The spinodal dewetting process can be monitored e.g. by atomic force microscopy (AFM) as shown in Fig. 9. By measuring λ0 as a function of film thickness, φ′′ (h) can be inferred and conclusions can be drawn with regard to the effective interface potential φ(h) [7]. For further details concerning stability of thin films we refer to [8] and references therein. Using the strong-slip model while taking slip into account (2.28), Rauscher and coworkers could theoretically show that slippage is supposed to influence the capillary wave spectrum due to a different mobility at the solid/liquid interface [68]. The position of the maximum q0 shifts to smaller wavenumbers and larger wavelengths for increasing slip length. As for today, to the best of our knowledge experimental studies concerning the impact of slippage on the spinodal wavelength are not available. The description of the influence of thermal noise on the temporal and spatial dynamics of spinodally dewetting thin polymer films has been recently achieved by Fetzer and coworkers [69]. A stochastic Navier-Stokes equation with an additional random stress fluctuation tensor that accounts for thermal molecular motion is utilized to model the flow while assuming a no-slip boundary condition at the solid/liquid interface. The stochastic model matches the experimentally observed spectrum of capillary waves and thermal fluctuations cause the coarsening of typical length scales. Slip effects in polymer thin films 18 Figure 10. Left: AFM image (scan size 10 µm) of a liquid rim formed during hole growth in a PS film dewetting from a hydrophobic substrate (Si wafer covered with a 21 nm hydrophobic Teflonr coating (AF 1600)). Right: Different rim morphologies for PS on AF 1600 (AFM cross-sections): Profile exhibiting a trough (depicted in the inset) for PS(35.6 kg/mol) at 120 ◦C and a monotonically decaying rim shape for PS(125 kg/mol) at 130 ◦ C. 2.5.5. Application II - Shape of a dewetting rim Besides the implications on spinodal dewetting, the one-dimensional thin-film model has been successfully applied to the shape of the rim along the perimeter of e.g. nucleated holes. Experimentally, researchers have studied and observed different types of rim profiles [70, 71, 72]. As shown in Fig. 10, profiles either decay monotonically into the undisturbed film or they show a more symmetrical profile exhibiting a trough (termed ”oscillatory shape”). If the depth of this through is in the range of the film thickness, a ring of so-called satellite holes can be generated [73, 74]. The shape of a dewetting rim can be understood by the aforementioned thin film theory for Newtonian liquids: Introducing a small perturbation δh(x, t) ≪ h of the film thickness h(x, t) and small velocities u(x, t) leads to linearized thin-film equations that describe the temporal and spatial evolution of δh. Thereby, the disjoining pressure φ′ (h) can be neglected due to the fact that films thicker than 10 nm are considered. To obtain stationary solutions of the linearized equations, a frame that is co-moving with the rim ζ(x, t) = x − s(t) is introduced. Thereby, s(t) denotes the position of the three-phase contact line; ṡ stands for the dewetting velocity V as described in the next sections. Fetzer et al used a normal modes ansatz δh(ζ) = δh0 exp kζ and u(ζ) = u0 exp kζ in the linear stability analysis, which leads to a characteristic polynomial of third order. Depending on the ratio of slip length to film thickness b/h and on the capillary number Ca = η ṡ/γlv , the parameter k obtains complex or real solutions. Fetzer and coworkers successfully identified the morphological transition from oscillatory to monotonically decaying rims and were able to extract slip lengths and capillary numbers from diverse experiments on dewetting surfaces [75, 76, 64, 77]. Slip effects in polymer thin films 19 Figure 11. Temporal series of optical micrographs showing the growth of a hole in a 120 nm PS(13.7 kg/mol) film at T = 120 ◦ C prepared on a Si wafer covered with a 21 nm hydrophobic Teflonr coating (AF 1600). In the last stage of hole growth (right image) perturbations of the three-phase contact line (according to the liquid rim instability) become visible. 3. Flow dynamics of thin polymer films - Experimental studies and theoretical models One of the main aspects of experimental studies concerning the flow dynamics of thin polymer films is to obtain a comprehensive view on the molecular mechanisms of slippage and on the responsible parameters. Although these insights are rather indirect, several models have been proposed to explain diverse experimental results. In this section, we will focus on these studies, with special regard to the proposed mechanisms of slippage at the solid/polymer interface. In general, we have to distinguish two different dewetting geometries: growth of holes (cf. Fig. 11) and receding straight fronts. On the one hand driving forces and on the other hand dissipation mechanisms have to be considered. 3.1. Dewetting dynamics - Driving forces According to the description of the effective interface potential in section 2.5.4, a global minimum of φ(h) occurs at heq in case of unstable or metastable films. This means that the film will thin until a thickness of heq is reached. In other words, a thin wetting layer remains on top of the substrate, if heq > 0 and if heq has a size that is not below the size of the molecules ‡. After dewetting has taken place (the dynamics of which is not covered by φ(h) but depends on viscosity, viscoelasticity, contact angle and the solid/liquid boundary condition), single droplets remain on top of the wetting layer. The droplets - in equilibrium - exhibit the Young’s contact angle θY . Parallel to the Young’s equation that characterizes the contact angle via the involved surface tensions, another characteristic parameter can be defined to specify the wettability of a surface by a liquid. This is the so-called spreading parameter S given by S = γsv − (γsl + γlv + φ(heq )), (3.1) ‡ Note that the concept of the effective interface potential is a continuum approach. It may fail if molecular sized phenomena are to be captured, such as extremely thin films close to the size of the molecules. Slip effects in polymer thin films 20 where γij denotes the interface tension of the solid (s), liquid (l ) and the vapor (v ) phase. S describes the energetic difference per unit area between a dry surface and a surface that is covered by a liquid layer. Consequently, if S < 0 the system can gain energy by reducing the contact area with the substrate. Thereby, the liquid forms a √ dynamic contact angle θd = θY / 2 at the three-phase contact line [81]. The link between the contact angle in equilibrium and the depth of the minimum of the effective interface potential becomes obvious, if the equation for the Young’s contact angle is inserted in (3.1): φ(heq ) = γlv (cos θY − 1) = S (3.2) Driving forces for wetting and dewetting are capillary forces (see [78]). Thereby, the capillary force per unit length of the contact line is in general given by Fc = γlv (cos θY − cos θd ). (3.3) l As long as θd < θY remains valid, the capillary force is negative and the three-phase contact line will recede; the liquid film dewets. Dewetting ends as soon as droplets exhibit their Young’s contact angle θY on the surface. During dewetting, a rim which is formed from accumulated material and retracts from the substrate while further growing. De Gennes developed a theoretical description of the resulting driving force for dewetting [78]: On the ”dry” side of the rim, which denotes the side where the three-phase contact line is located, a negative capillary force pulls at the contact line: Fc = S + γlv (1 − cosθd ). (3.4) l On the other side, where the rim decays into the liquid film (the ”wet” side of the rim), a positive capillary force occurs: Fc = γlv (1 − cosθd ). (3.5) l We have to remark that in experimental systems, the Laplace pressure of course avoids edges and therefore forces the dry side of the rim to decay smoothly into the film. Summing up both forces per unit length gives the effective driving force for retracting contact lines in dewetting systems: Fc 1 = |S| = γlv (1 − cos θY ) ≃ γlv θY . (3.6) l 2 Thereby, it is assumed that the system is in a quasi-stationary state, which means that changes in the shape of the rim occur much slower that the dewetting velocity. All in all, this result implies that the driving force for dewetting is given by the absolute value of the spreading parameter, which solely depends on the surface tension γlv of the liquid (a property which is purely inherent to the liquid) and the Young’s contact angle of the liquid on the surface. The same conclusion is drawn if considering energies instead of driving forces. The depth of the minimum of the effective interface potential φ(h) is equal to S and gives the energy per unit area that is set free during dewetting [79]. Thereby, S also stands for the force per unit length of the contact line (see (3.6)). Slip effects in polymer thin films 21 3.2. Dewetting dynamics - Dissipation mechanisms In the previous section, we focused on the driving forces for dewetting. The experimentally observed dynamics represents a force balance of driving forces and friction forces. The resulting dewetting velocity V is connected to the spreading parameter S and the occurring energy dissipation mechanisms (Fi gives the friction force per unit length of the contact line) and their corresponding velocity contributions vi via the power balance X |S|V = Fi vi . (3.7) i In case of dissipation due to viscous friction within the liquid (Fv ) and dissipation due to friction of liquid molecules at the solid/liquid interface (Fs ), i.e. slippage with a finite velocity vs = u|z=0, we end up with the following balance (the index v stands for ’viscous’, s denotes ’slip’): |S|V = Fv vv + Fs vs . (3.8) 3.2.1. Viscous friction According to the work of Brochard-Wyart et al [80], the no-slip boundary condition for fluid flow implies a friction force that is proportional to the liquid viscosity and the dewetting velocity v and means that the dissipation is solely due to viscous friction within the liquid. The largest strain rates occur in the direct vicinity of the three-phase contact line. The dissipation is therefore mainly independent from the size of the dewetting rim. However, the flow geometry (given by the dynamic contact angle θd ) at the contact line influences viscous dissipation. In case of pure viscous flow V = vv , the following expression for the velocity contribution is obtained for viscous flow [80]: vv = Cv (θd ) |S| . η (3.9) Cv (θd ) denotes the constant of proportionality and displays a measure for the flow field in the vicinity of the contact line. According to the description of Redon et al [81], the velocity in case of viscous dissipation strongly depends on θY and can also be written as γlv 3 vv ∝ θ , (3.10) η Y where the constant of proportionality accounts for the flow singularity near the contact line. In detail, this constant represents a logarithmic factor that includes i.a. a shortdistance cutoff and accounts for the fact that viscous dissipation diverges near the contact line. Moreover, a small impact of the slip length b on this logarithmic factor has been experimentally found [81] by variation of molecular weight Mw (cf. (3.12)) by Redon [82]. Consequently, only in the case of a no-slip boundary condition, the constant of proportionality in (3.10) and also Cv (θd ) in (3.9) is purely independent of slip. Slip effects in polymer thin films 22 H0 w/2 R or D w/2 x0 h Figure 12. Schematic representation of a rim cross-section illustrating the rim width w (usually obtained as the distance between the three-phase contact line and the position where the rim height is dropped to 1.1 h), the distance x0 for a given film thickness h and hole radius R (or dewetted distance D of a straight front). (Height scale in sketch is exaggerated as compared to the lateral length scale.) For growing holes of radius R and dewetting velocity V = dR/dt, integration of (3.9) gives a linear proportionality of the radius versus dewetting time t: R∝t (3.11) 3.2.2. Slippage - Friction at the solid/liquid interface Besides viscous dissipation within the rim, de Gennes expected a long-chained polymer film to exhibit an exceptional amount of slippage. For entangled polymer melts on a non-adsorbing surface, de Gennes stated that the slip length should strongly increase with the chain length of the polymer [83]: N3 b = a 2, Ne (3.12) where a is the size of the monomer, N the polymerization index and Ne the entanglement length. In case of dominating slippage, a minor contribution of viscous dissipation is expected due to very small stresses within the rim. The corresponding analytical model (see [80, 82]) is based on the linear friction force per unit length of the contact line given by Fs ∝ ξvs . Dissipation occurs along the distance of the solid/liquid interface, where liquid molecules are moved over the substrate. In general, this distance is identified as the width w of the rim (cf. Fig. 12). Consequently, Fs ∝ w can be assumed. If the slip length is introduced according to Navier’s model by b = η/ξ (see (2.21) in section 2.4), the slip velocity contribution vs is given by: vs = 1 |S| b . 3 η w (3.13) Let’s consider the conservation of mass for a growing hole of radius R, π(R + x0 )2 h = 2πQ(R + w/2), (3.14) Slip effects in polymer thin films 23 and for a straight dewetting front of dewetted distance D, (D + x0 )lh = Ql, (3.15) where x0 stands for the distance as depicted in Fig. 12, Q denotes the cross sectional area of a rim (approximated as a half-circle §, i.e. Q = π(x0 /2)2 ≈ π(w/2)2), and l is the length of a straight front, which cancels out. Based on the assumption of selfsimilar growing √ √ rims and the fact that x0 /R ≪ 1 and x0 /D ≪ 1, the width of the rim w ∝ h R for both geometries. The constant of proportionality, however, which we name Cs in the following, depends on the shape of the rim and furthermore on the dewetting geometry itself: √ √ w = Cs h R. (3.16) Values for Cs can be obtained collecting a temporal series of AFM snapshots of rim profiles and fitting (3.16) to the measured rim width w plotted versus the hole radius R. The initial film thickness h can also be measured by AFM. Of course, the validity of this simplified model and especially the approximation w ≈ x0 becomes questionable for asymmetric rims and should lead to systematic differences in Cs . We point out that, as described in section 2.5.5, the degree of asymmetry of the cross-section of the rim is strongly correlated to the capillary number and the ratio of slip length to film thickness. Replacing the above derived relation of rim width w and hole radius R in (3.13) gives the slip velocity contribution in terms of the hole radius R for purely slipping liquids: vs = 1 1 |S| b √ √ . 3 η Cs h R (3.17) Via integration, a characteristic growth law for the radius R of a dewetting hole with time t is found (separation of variables), R ∝ t2/3 . (3.18) Using the lubrication models described in section 2.5, Münch compared numerical simulations of polymer melts dewetting from hydrophobized substrates to the above described dewetting dynamics obtained from scaling arguments (based on energy balances) [84]: In the no-slip situation (mobility m(h) ∝ h3 ), an exponent α = 0.913 is reported instead of 1. The deviation is traced back to the fact that the logarithmic factor in the constant of proportionality of equation (3.10) also depends on the width of the rim (which evolves with time t). Numerical simulations of the slip-dominated case (m(h) ∝ h2 ) give α = 0.661, which captures α = 2/3 very well. § Note that usually a dynamic contact angle θd < 90 ◦ at the three-phase contact line is obtained in experiments. The difference between a segment of a circle and the half-circle approximation is marginal and contributes to the constant of proportionality Cs . Slip effects in polymer thin films 24 Figure 13. Left: Optical measurement of the hole radius versus time in 130 nm PS(13.7 kg/mol) films on different substrates. Right: Plot of the dewetting velocity V √ versus 1/ R. The y-axis intercept is identical for all substrates. The difference in the slope K indicates substantial differences in slip for dewetting PS (adapted from [86] and [87]). 3.2.3. Models based on the superposition of dissipation mechanisms Considering the models derived before for pure viscous flow and for pure slippage, a combination of these models seems to be reasonable for situations, where viscous dissipation as well as dissipation at the solid/liquid interface act together. Jacobs et al proposed an additive superposition of the corresponding velocity contributions, i.e. V = vs +vv [85], according to the fact that both mechanisms counteract the same driving force |S|. Separation of variables leads to the implicit function: √ Ks R Kv 2 Kv √ Kv )) (3.19) R + 2( ) ln (1 + (R − 2 t − t0 = |S| Ks Ks Kv with Kv = η , Cv (θd ) Ks = 3η w √ , b R √ Kv ∝ b R. Ks (3.20) In that √ notation, the velocity contributions are given by vv = |S|/Kv and vs = |S|/( RKv ). One of the preconditions of this superposition was the successful inclusion of both border cases: For b → 0, R ∝ t is obtained, whereas for b → ∞, R ∝ t2/3 follows. As a more practicable alternative for fitting this relation between radius R and time t to experimental data (typical data is shown in Fig. 13), Fetzer and Jacobs recently proposed a more simple way of visualizing slip effects [86]: According to the additive superposition, the dewetting velocity can be written in terms of K V = vv + √ , R K= 1 |S| b √ . 3 η Cs h (3.21) √ which leads to a linear relationship when plotting the dewetting velocity v versus 1/ R (c.f. Fig. 13). Then, the y-axis intercept of the straight line can be identified as the viscous velocity contribution vv , whereas the the slope K is connected to the slip Slip effects in polymer thin films 25 length as illustrated in (3.21). The validity of this model, which assumes the linear superposition of velocity contributions, was checked by plotting the viscous velocity contribution vv versus the reciprocal viscosity η. An excellent agreement has been demonstrated in view of melt viscosity data obtained from independent measurements. Moreover, the slope has been used to calculate the slip length b. In the experiments, substantial differences in slip lengths for different substrates (cf. Fig. 13) at identical liquid properties could be detected [86]. 3.2.4. Molecular-kinetic theory and further approaches Besides theoretical models based on continuum hydrodynamics (see 3.2.1), contact line dynamics such as the spreading of a liquid droplet on a planar surface has been analyzed according to the so-called molecular-kinetic theory (MKT) [88, 89, 90]. The movement of a contact line is described as an activated rate process, where the liquid molecules close to the substrate jump from one potential well to another. The resulting contact line friction coefficient is proportional i.a. to the viscosity η of the liquid and to the exponential of the reversible work of adhesion γlv (1 + cos θY )/kB T . As a consequence, increasing the temperature reduces the friction coefficient and increases the velocity. In case of a squalane droplet spreading on gold surfaces covered with different self-assembled monolayers (SAM) of thiols, the corresponding contact line friction coefficient turns out to increase linearly with the chain length of the SAM [91]. SAM surfaces have also been target of friction experiments using AFM tips. Barrena et al explained discrete changes in the frictional behavior with discrete molecular tilts of the chains of the SAM [92]. Viscoelastic deformation of the substrate at the contact line due to the normal component of the liquid surface tension has also attracted attention with regard to fluid dynamics on the nanoscale and energy dissipation. The phenomenon of a reduced velocity of contact-line displacements compared to a corresponding non-deformable substrate (termed ”viscoelastic braking”) was extensively studied and described by Shanahan and Carré [93, 94, 95, 96, 97]. In this context, Long et al theoretically examined the static and dynamic wetting properties of liquids on thin rubber films [98] and grafted polymer layers [99]. To the best of our knowledge, there is to-date no experimental evidence of deformations and tilting on the molecular scale in case of SAM surfaces due to retracting contact-lines of a dewetting polymer film. However, dewetting on soft deformable rubber surfaces has recently been shown to be influenced by viscoelastic deformations of the substrate [100]. In case of non-volatile liquids such as polymers, evaporation into and condensation from the vapor phase can be excluded as relevant mechanism of energy dissipation. 3.3. Dynamics of growing holes Besides very early theoretical and experimental studies of thin film rupture [5, 101, 102, 103, 104, 105, 106], one of the first studies concerning dewetting experiments had been published by Redon et al [81]. Films of alkanes and polydimethylsiloxane (PDMS) Slip effects in polymer thin films 26 have been prepared on different hydrophobized silicon wafers. The authors observed a constant dewetting velocity which was inversely proportional to the viscosity and very sensitive to the equilibrium contact angle of the liquid on a corresponding surface. In 1994, Redon et al investigated the dewetting of PDMS films of different thicknesses on silanized hydrophobic surfaces [82]. They found that for films larger than 10 µm a constant dewetting velocity is found, whereas R(t) ∝ t2/3 in films thinner than 1 µm is observed. These results corroborate the models represented by (3.11) and (3.18) and indicate that for sufficiently thick films (h ≫ b) viscous dissipation dominates whereas slippage becomes relevant in case of thin films (h < b) [82]. 3.3.1. Stages of Dewetting While studying the growth of holes, Brochard-Wyart et al proposed a set of subsequent stages that can be attributed to distinct growth laws [107]: Starting with the birth of the hole, the radius of the hole is supposed to grow exponentially √ with time, i.e. R(t) ∝ exp (t/τ ), as long as the radius R is smaller than Rc ≈ bh. Experimentally, Masson et al [108] found relaxation times orders of magnitude larger than the largest translational reptation times expected. For Rc < R < Rc′ ≈ b, the rim is formed and viscous dissipation √ dominates the hole growth dynamics, i.e. R ∝ t. The special case of R ≫ Rc = bh moreover gives an analytic expression for the hole growth: |S| b 1/2 ( ) t (3.22) R(t) ≈ η h √ During this linear regime, a dynamic (receding) contact angle θd = θY / 2 is formed at the three-phase contact line. In the subsequent stage (for hole radii approximately larger than the slip length b), the rim is fully established and grows in a self-similar manner. As discussed in section 3.2.1, this results in a characteristic R ∝ t2/3 growth law if large slip is present. That stage of self-similar growing rims is often called ”mature” regime. These distinct regimes have been experimentally observed by Masson et al [108] and Damman et al [72]. Moreover, a dissipation dominated by slippage has been shown to be restricted to sufficiently small hole radii: If the rim has accumulated a very large amount of liquid (so that the height of the rim H0 is much larger than the slip length b, i.e. H0 ≫ b), viscous dissipation dominates again. This consequently results in a transition to a linear growth law for the radius with respect to the dewetting time, i.e. R ∝ t. In this article, we mainly focus on dewetting phenomena in terms of growing holes or retracting straight fronts. Additionally, also the subsequent regime influenced by the fingering instability has attracted the interest of researchers [109]. Thereby, facets such as the onset of the instability and the morphology of liquid profiles have also shown to be sensitive to rheological properties of thin films and the hydrodynamic boundary condition at the solid/liquid interface [110, 111, 112, 113]. 3.3.2. ”Mature” holes Recently, Fetzer et al experimentally studied the dewetting dynamics of ”mature” holes in thin PS films on two different types of hydrophobized Slip effects in polymer thin films 27 surfaces. The aforementioned assumption of linear superposition of viscous dissipation and slippage (see 3.2.3) was used to gain information about the slip contribution [86]. Hydrophobization was achieved by the preparation of a dense, self-assembled monolayer of silane molecules (silanization). In that case two different silanes have been utilized, octadecyltrichlorosilane (OTS) and dodecyltrichlorosilane (DTS). The slip length turns out to depend strongly on temperature (and thereby on the melt viscosity); in fact, the slip length decreased with increasing temperature. Furthermore, slippage was about one order of magnitude larger on DTS compared to OTS. The results are in good agreement with the results for the slip length gained by the analysis of the shape of rim profiles of the corresponding holes (see 2.5.5). The molecular mechanisms of slippage are widely discussed and several models have been proposed with regard to different experimental conditions (shear rates, polymers, surfaces, ...). This will be discussed in the next section. 3.4. Molecular mechanisms of slippage For small flow velocities, slip lengths lower than expected have been found (see e.g. [114]). This has been explained by the adsorption of polymer chains at the solid surface. These adsorbed chains are able to inhibit slippage due to entanglements with chains in the melt. At larger shear strains, disentanglement of adsorbed and melt chains occurs, the friction coefficient decreases rapidly and slippage is ”switched on” [115, 116, 117, 118]. The authors interpret the results as follows: Strongly adsorbed chains stay at the solid/liquid interface followed by a coil-stretch transition of these tethered chains. Thereby, the interaction of anchored chains (which are stretched under high flow velocities) with the chains in the melt are reduced and slippage is enhanced. This has been experimentally observed by Migler et al [114] and Hervet et al [119]. Thereby, the critical shear strain depends on the molecular weight of chains attached to the surface and on their density [39]. This situation is also called ”stick-slip transition” and has moreover been reported for pressure-controlled capillary flow of polyethylene resins [120]. The authors find a characteristic molecular weight dependence of the slip length b ∝ Mw3.4 , whereas the critical stress σc for the transition described before scales as σc ∝ Mw−0.5 [121]. Also de-bonding of adsorbed chains might occur. Molecular dynamics (MD) studies aiming to investigate the slip length in thin, short-chained polymer films subject to planar shear revealed a dynamic behavior of the slip length b(γ̇) upon the shear rate according to b = b∗ (1 − γ̇/γ̇c )−1/2 , where γc represents a critical shear rate [122]. Slippage strongly increases as γ̇/γ̇c → 1. This relation was also found for simple liquids. Additionally, the authors studied the border case γ̇ → 0 and especially the molecular parameters affecting the slip length. Moreover, a special situation has been discussed in the literature: A thin polymer film prepared on a surface decorated with end-grafted polymer chains of the same species. Due to entropic reasons, an interfacial tension between identical molecules occurs and dewetting can take place. This phenomenon is called autophobicity or Slip effects in polymer thin films 28 autophobic dewetting [123, 124]. Reiter and Khanna observed that PDMS molecules slipped over grafted PDMS brushes. They determined a slip length on the order of 10 µm for film thicknesses between 20 and 850 nm of the dewetting PDMS film (high Mw of 308 kg/mol in most cases). For lower grafting densities, slippage is reduced, indicating a deeper interpenetration of free melt chains [123]. 3.5. Impact of viscoelasticity and stress relaxation As already introduced in 2.1.3 and furthermore theoretically described in section 2.5, viscoelasticity can be included in the description of thin film dynamics. In case of thin film flow with strong slippage, we already mentioned that the impact of viscoelasticity might be not distinguishable from slippage effects. Numerous experimental studies have been published by Reiter, Damman and coworkers describing the influence of viscoelastic effects on the flow dynamics of thin polymer films. These studies have been supported with theories by Raphaël and coworkers. In the following, a snap-shot of recent results in chronological order is presented. It reflects the progress from simple to more and more sophisticated models and experiments. Some of the results, however, may have tentative character, as brought up by a very recent manuscript by Coppée et al (cf. section 3.7). 3.5.1. Thin film rupture Reiter and de Gennes [125] pointed out that the usual spin-casting preparation (based on fast solvent evaporation) of thin, long-chained polymer films may induce a cascade of distinct states. Annealing the samples to high temperatures for times larger than the reptation time of the polymer (depending on Mw and the temperature T ) might induce a complete healing of the preparation effects. Non-equilibrated conformational states and residual stresses have shown to be capable to cause rupture of thin films [126]. Thereby, the areal density of holes appearing at a certain temperature above Tg has been measured: After storage at elevated temperatures below Tg , an exponential reduction of the hole density can be observed. Vilmin and Raphaël were able to show that lateral stress reduces the critical value for surface fluctuations initiated by an anisotropic diffusion of polymer molecules to induce the formation of holes [127]. 3.5.2. Stages of distinct dewetting dynamics - Experimental results Reiter and coworkers furthermore studied the temporal and spatial evolution of dewetting PS fronts on Silicon wafers covered with a PDMS monolayer at times shorter than the relaxation (reptation) time τrept in equilibrated bulk samples [126]. At the beginning of their experiments, the dewetted distance and the rim width w increased in a logarithmic way with respect to the dewetting time t, consequently V ∝ t−1 is found. They correlated a maximum of the rim width w occurring at a distinct time t1 (nearly independent on Mw and significantly shorter than τrept ) of the experiment to a change in dewetting √ dynamics, namely V ∝ t−1/2 (i.e. R ∝ t). Slip effects in polymer thin films 29 Even before, Damman, Baudelet and Reiter compared the growth of a hole and the dynamics of a dewetting straight front [72]. Thereby, strong influence of the dewetting geometry on the dynamics of early stages became obvious. Moreover, the authors correlated their findings to the shape of the corresponding rim and clearly identified distinct dewetting stages: At the beginning of hole growth, capillary forces dominate the dynamics and exponential growth is observed. In contrast to that, dewetting of a straight front at the beginning starts almost instantaneously at a high velocity and decreases as V ∝ t−1 . As already observed by Redon et al [81], V = const and R ∝ t due to viscous dissipation is found for growing holes. For both dewetting geometries, Damman et al find a maximum of the rim width w which they interpret in terms of a transition of the shape of the rim from very asymmetric profiles towards more symmetric rims, while the volume still increases during dewetting. In the following dissipationdominated regime, V ∝ t−1/3 and R ∝ t2/3 (slip boundary condition) or V = const and R ∝ t (no-slip boundary condition) is found, depending on the boundary condition for the solid/liquid interface. Afterwards, again a constant dewetting velocity (until the ”mature rim” regime is reached and slippage dominates) is obtained. Recently, Damman and Reiter determined the strain γ from rim shapes of dewetting fronts, which can be written according to (2.3) by γ = (S/h + σ0 )/G, where σ0 denotes residual stresses [128]. These values were one order of magnitude larger than for equilibrated PS films (γ = S/(hGbulk )) and increased with increasing Mw . Consequently, larger residual stresses σ0 or smaller elastic moduli G can be responsible for this effect. Indeed, the elastic modulus of a thin film is supposed to be smaller than its bulk value according to the fact that larger numbers for the entanglement length Me (for larger Mw ) are expected (see section 2.1.2). Additionally, instead of one transition time for both, two different transitions times tw for the maximum of the rim width and tv for the power law of the dewetting velocity (to V ∝ t−1/3 ) are found for larger molecular weights. Time tv is comparable to the reptation time trept (also for larger Mw ). The authors interpret this as follows: After tv ∼ trept, the equilibrium entanglement conformation has been reached via interdiffusion and re-entangling of chains. Relaxation of stress, which is correlated to the rim shape transition and tw , occurs at shorter times. 3.5.3. Stages of distinct dewetting dynamics - Theoretical models These observations have motivated a theoretical model by Vilmin and Raphaël [129], that takes viscoelastic properties of the liquid and slippage into account. The logarithmic increase of the dewetted distance with time can be explained if residual stress is regarded as an extra driving force for dewetting that has to be added to the capillary forces. Subsequent relaxation of the stress reduces this additional contribution. For times larger than the reptation time τrept , Vilmin and Raphaël predict a constant dewetting velocity and V ∝ t−1/3 as soon as the ”mature regime” is reached (and slippage becomes important). They consider a simple equation for a viscoelastic film which includes one relaxation time τ1 (and an elastic modulus G), but two distinct viscosities: η0 is a short-time viscosity which describes the friction between monomers, and η1 is the melt viscosity Slip effects in polymer thin films 30 (η1 ≪ η0 ) governed by disentanglements of polymer chains. Theoretically, three regimes of distinct time-response are expected: For t < τ0 = η0 /G, Newtonian flow accompanied with a small viscosity η0 . Afterwards (τ0 < t < τ1 = η1 /G), the elastic modulus governs the dynamics. For longer times, i.e. t > τ1 , Newtonian flow is again obtained, but with a much larger viscosity η1 . During the first and the last regime, asymmetric rims and constant velocities are predicted (the latter velocity of course is much smaller than the initial velocity). The intermediate regime, governed by viscoelastic behavior, is expected to show V ∝ t−1/2 according to the fact that the width of the rim w will increase proportional to the dewetted distance D, i.e. w ∝ D (in contrast to the squareroot dependence in case of viscous flow). Including residual stresses into their model, Vilmin and Raphaël obtained the aforementioned experimentally observed V ∝ t−1 law instead of V ∝ t−1/2 . Recently, Yang and co-workers experimentally quantified the molecular recoiling force stemming from non-equilibrium chain conformations. They obtained values that are at least comparable to (or even larger than) the dispersive driving forces [130]. 3.5.4. Further remarks Concerning studies and models including viscoelastic effects, one has to bear in mind that stress relaxation dynamics and relaxation times in thin films have been shown to be different from bulk polymer reptation values. As mentioned in section 2.1.2 and indicated by the characteristic relaxation times τ1 of elastic constraints being significantly shorter than bulk values, the entanglement length has shown to be significantly increased in case of thin films. Recently, Reiter and coworkers extended their studies with regard to viscoelastic dewetting on soft, deformable substrates [100]. The essential result was that transient residual stresses can cause large elastic deformations in the substrate which almost stop dewetting for times shorter than the relaxation time τrept of the polymer film. For times longer than τrept , the elastic behavior and the elastic trench in the deformable substrate vanishes. Vilmin and Raphaël applied their model for viscoelastic liquids and residual stresses to the hole growth geometry [131] with regard to the early stage (exponential growth for Newtonian liquids). They discovered a very fast opening regime followed by a slow exponential growth of the radius of the holes. 3.6. Non-linear friction Up to now, concerning energy dissipation at the solid/liquid interface, exclusively linear friction has been considered. As described in section 3.4, often not only smooth and passive (non-adsorbing) surfaces have been experimentally considered, but also grafted or adsorbed polymer layers are used as a support for dewetting experiments. These systems motivate the theoretical treatment of non-linear friction, as described in the following subsection. Slip effects in polymer thin films 31 3.6.1. Theoretical model To cover non-linear friction, Vilmin and Raphaël introduced a friction force per unit area that is linear below a certain transition velocity vr and non-linear above [131]: Fs = ξvr (v/vr )1−r , v > vr , (3.23) where r denotes a so-called friction exponent. Consequently, the effective (velocity-dependent) friction coefficient ξef f can be written in terms of ξef f (v) = ξ(vr /v)r (3.24) and a velocity-dependent slip length b(v) can be defined as b(v) = η/ξef f (v). (3.25) An increasing velocity leads to a decreasing friction coefficient and to pronounced slippage. Thereby, the friction at the solid/liquid interface influences the power law of the velocity decrease. Assuming non-linear friction in the intermediate regime (which is governed by the viscoelastic behavior) gives (according to w ∝ D) V ∝ t−1/(2−r) . Including residual stress σ0 to this model leads to a formula for the maximum width of the rim (depending on r and σ0 ). Experimentally, values for r between 0 (for low Mw ) and 1 (for large Mw ) have been found [128]. The dependence of the nonlinearity of friction upon molecular weight Mw could be explained by the influence of chain length on slippage (see (3.12)). 3.6.2. Variation of substrate properties Hamieh et al focused on the frictional behavior of dewetting viscoelastic PS films on PDMS-coated (irreversibly adsorbed) silicon wafers [132]. Thereby, thickness (via the PDMS chain-length) and preparation of the PDMS support (via the annealing temperature) were varied. In summary, the observations are consistent with the aforementioned (section 3.5) experiments by Reiter, showing a characteristic time t1 for the change in dewetting dynamics and rim shape (from highly asymmetric towards a more symmetric equilibrated shape). Again, this transition is interpreted by the Laplace pressure that overcomes elastic effects. Probing the maximum rim width enables the authors to identify the impact of the friction coefficient or the friction exponent r: If they prepare thicker PDMS layers, the result is a larger maximum rim width. This result can be explained in terms of a small increase of r. Consequently, the velocity-dependent slip length b(v) increases (see (3.25)): Thicker PDMS-layers lead to more slippage. Concerning the characteristic time for stress relaxation t1 , no significant influence of the preparation procedure (annealing temperature of the PDMS coating) and thickness of the PDMS layer have been found. Slip effects in polymer thin films 32 3.7. Temporal evolution of the slip length Most of the aforementioned dewetting experiments of thin (viscoelastic) PS films are based on supports consisting of a PDMS layer prepared onto a Si wafer. These PDMS surfaces, which were assumed to be impenetrable by PS chains, turned out to be less ideal. Recently, the aforementioned fast decay of the dewetting velocity and the maximum of the rim width upon dewetting time (ascribed to relaxation of residual stresses, see section 3.5), has also been observed in case of low molecular weight PS [133]. Furthermore, neutron reflectometry experiments revealed an interdiffusion of polymer chains at the PS-PDMS interface below the PS bulk glass transition temperature. The dewetting velocity accelerated as the brush thickness is increased. In view of energy dissipation due to brush deformation (pronounced by increasing thickness), this is contradictory to the slower dewetting velocities expected in that case. Ziebert and Raphaël recently investigated the temporal evolution of the energy balance (viscous dissipation and sliding friction) of thin film dewetting by numerical treatment, especially concerning non-linear friction [134]. They point out that both mechanisms have different time dependencies and propose that simple scaling arguments such as the mass conservation w ∝ D should be revisited. In case of non-linear friction, viscous dissipation is even more important than sliding friction for times larger than τ1 , whereas for linear friction both mechanisms are approx. equally important. The stage of ”mature” rims afterwards is again dominated by friction at the solid/liquid interface. Using scaling arguments and numerical solutions of the thin film model for viscoelastic liquids [129], they moreover showed that the aforementioned dewetting characteristics (fast decay of the dewetting velocity, maximum of the rim width in course of dewetting time) can be explained by the temporal decrease of the slip length during the experiment instead of stress relaxation [135]. Therefore, roughening of the PS-PDMS interface (as detected by neutron reflectometry) or potentially also an attachment of very few PS chains to the silicon substrate might be responsible, if very low PDMS grafting densities are prepared. The attachment of melt chains to the substrate has been recently considered by Reiter et al [136]. They pointed out that the driving force is reduced by a certain pullout force for molecules attached to the surface as they get stretched while resisting to be pulled out. Consequently, this force (per unit length of the contact line) can be written in terms of Fp = νLf ∗ , where ν represents the number of surface-connected molecules, L the length according to the stretching and f ∗ the pull-out force per chain. 4. Conclusions and outlook To conclude, dewetting experiments can be regarded as a very powerful tool to probe rheological thin film properties and frictional mechanisms. The validity of the no-slip boundary condition at the solid/liquid interface, for a long period of time accepted as a standard approach in fluid dynamics, fails. Even for Newtonian liquids such as polymers Slip effects in polymer thin films 33 below their critical length for entanglements substantial amount of slippage has been found. For larger molecular weights, slippage can be even more pronounced if chain entanglements come into play. We have reviewed in this article theories and experiments characterizing the statics and dynamics in thin liquid polymer films, focussing on energy dissipation mechanisms occurring during dewetting. Experiments are relatively easy to perform, since usually no very low or very high temperatures are needed, neither are high-speed cameras necessary. However, careful preparation, preferably in a clean-room environment, of thin films is inevitable. Concerning the experimental data, diligent interpretation and consideration of all relevant parameters are of essential importance: Molecular weight, film thickness (also with regard to the molecular dimensions of a polymer coil in the respective melt), dewetting temperature, melt viscosity, dewetting velocity and capillary number, residual stresses, relaxation times and ageing time represent parameters that are sometimes coupled and not easy to disentangle. Their unique impact on the friction coefficient or the slip length on a specific, ideally non-adsorbing and non-penetrable substrate (to reduce the number of parameters of the system) is not always easy to identify. In particular, the specific stage of dewetting (early or mature stage) and the dewetting geometry (holes or straight fronts) represent further facets and playgrounds for experimentalists and theoreticians that, under careful consideration, enable to gain insight into rheological or frictional mechanisms. The dynamics and the morphology of the fingering instability can provide additional access to the solid/liquid boundary condition and the rheology. To obtain a deeper understanding of the molecular mechanisms at the solid/liquid interface is one of the main tasks in micro- and nanofluidics. In this context, we like to highlight two essential questions and possible pathways to answer them: a) The liquid: What is the impact of the polydispersity of the liquid on dewetting? Dewetting studies of polymer melts by adding a second chemical component have shown to influence slippage [137]. However, also the driving force is changed due to the difference in chemical composition for different species. To overcome this problem, the influence of chain length distribution on dewetting can be probed by studying polymer mixtures instead of monodisperse polymer melts [138]. Concerning a theoretical approach to this question, dissipative particle dynamics (DPD) simulations enable to locate energy loss in the rim and reveal interesting results in case of two immiscible fluids of different viscosity: In case of a low-viscosity layer at the solid/liquid interface, faster dewetting dynamics is found that is attributed to a lubrication effect, i.e. the sliding of the upper high-viscosity layer [139]. Finally, these aspects lead to the fundamental discussion, whether ”apparent” slip, possibly induced by the formation of a shortchained layer of low viscosity, is present. b) The substrate: What is the impact of the molecular structure of the substrate? As indicated before, the set of parameters concerning the topographical and/or chemical structure of the support is large. Besides parameters such as surface roughness and surface energy, experiments can be performed on substrates e.g. decorated with Slip effects in polymer thin films 34 an amorphous coating, a self-assembled monolayer or ever grafted polymer brushes of the same or different species. Scattering techniques (using neutrons or X-rays) provide access to the solid/liquid interface, complementing dewetting experiments and confirming proposed mechanisms of slippage. Simulations based on molecular dynamics (MD) of near-surface flows can help to compare experimental results from dewetting studies to molecular parameters, easily tunable in theoretical models, and structural changes [140, 141]. Moreover, the evolution of coarse-grained polymer brush/melt interfaces under flow has also been identified as a potential application of MD studies [142, 143]. In the end, all these facets help to obtain a universal picture of sliding friction which can potentially lead to surfaces precisely tailored for special microfluidic applications. Acknowledgments The authors acknowledge financial support from the German Science Foundation (DFG) under grant JA905/3 within the priority program 1165 ”Micro- and nanofluidics”. References [1] Squires T M and Quake S R 2005 Microfluidics: Fluid physics at the nanoliter scale Rev. Mod. Phys. 77 977 [2] Thorsen T, Maerkl S J and Quake S R 2002 Microfluidic large-scale integration Science 298 580 [3] Leslie D C, Easley C J, Seker E, Karlinsey J M, Utz M, Begley M R and Landers J P 2009 Frequency-specific flow control in microfluidic circuits with passive elastomeric features Nature Physics 5 231 [4] Bear J 1972 Dynamics of fluids in porous media (New York: Elsevier) [5] Vrij A 1966 Possible mechanism for the spontaneous rupture of thin, free liquid films Discuss. Faraday Soc. 42 14 [6] Herminghaus S, Jacobs K, Mecke K, Bischof J, Fery A, Ibn-Elhaj M, Schlagowski S 1998 Spinodal dewetting in liquid crystal and liquid metal films Science 282 916 [7] Seemann R, Herminghaus S and Jacobs K 2001 Dewetting patterns and molecular forces: A reconciliation Phys. Rev. Lett. 86 5534 [8] Jacobs K, Seemann R and Herminghaus S 2008 Stability and dewetting of thin liquid films Polymer Thin Films ed O K C Tsui and T P Russell (Singapore: World Scientific) [9] Seemann R, Herminghaus S and Jacobs K 2001 Gaining control of pattern formation of dewetting liquid films J. Phys.: Condens. Matter 13 4925 [10] Jacobs K, Seemann R and Mecke K 2000 Dynamics of structure formation in thin films: A special spatial analysis Lecture Notes in Physics (Statistical Physics and Spatial Statistics) ed D Stoyan and K Mecke (Heidelberg: Springer) [11] Craster R V and Matar O K 2009 Dynamics and stability of thin liquid films Rev. Mod. Phys. 81 1131 [12] Rubinstein M and Colby R H 2003 Polymer Physics (New York: Oxford University Press) [13] Jones R A L and Richards R W 1999 Polymers at surfaces and interfaces (Cambridge University Press) [14] De Gennes 1971 Reptation of a polymer chain in the presence of fixed obstacles J. Chem. Phys. 55 572 [15] Keddie J L, Jones R A L and Cory R A 1994 Size-dependent depression of the glass transition temperature in polymer films Europhys. Lett. 27 59 Slip effects in polymer thin films 35 [16] Mattsson J, Forrest J A and Börgesson L 2000 Quantifying glass transition behavior in ultrathin free-standing polymer films Phys. Rev. E 62 5187 [17] Dalnoki-Veress K, Forrest J A, de Gennes P-G and Dutcher J R 2000 Glass transition reductions in thin freely-standing polymer films : A scaling analysis of chain confinement effects J. Phys. IV France 10 221 [18] Herminghaus S, Jacobs K and Seemann R 2001 The glass transition of thin polymer films: Some questions, and a possible answer Europ. Phys. J. E 5 531 [19] Keddie J L and Jones R A L 1995 Glass transition behavior in ultrathin polystyrene films J. Isr. Chem. Soc. 35 21 [20] Fryer D S, Peters R D, Kim E J, Tomaszewski J E, De Pablo J J and Nealey P F 2001 Dependence of the glass transition temperature of polymer films on interfacial energy and thickness Macromolecules 34 5627 [21] Herminghaus S, Jacobs K and Seemann R 2003 Viscoelastic dynamics of polymer thin films and surfaces Eur. Phys. J. E 12 101 [22] Kawana S and Jones R A L 2001 Character of the glass transition in thin supported polymer films Phys. Rev. E 63 021501 [23] Herminghaus S, Seemann R, and Landfester K 2004 Polymer surface melting mediated by capillary waves Phys. Rev. Lett. 93 017801 [24] Seemann R, Jacobs K, Landfester K, and Herminghaus S 2006 Freezing of polymer thin films and surfaces: The small molecular weight puzzle J. polym. Sci. B 44 2968 [25] Si L, Massa M V, Dalnoki-Veress K, Brown H R and Jones R A L 2005 Chain entanglement in thin freestanding polymer films Phys. Rev. Lett. 94 127801 [26] Rauscher M, Münch A, Wagner B and Blossey R 2005 A thin-film equation for viscoelastic liquids of Jeffreys type Eur. Phys. J. E 17 373 [27] Blossey R, Münch A, Rauscher M and Wagner B 2006 Slip vs. viscoelasticity in dewetting thin films Eur. Phys. J. E 20 267 [28] Münch A, Wagner B, Rauscher M and Blossey R 2006 A thin-film model for corotational Jeffreys fluids under strong slip Eur. Phys. J. E 20 365 [29] Te Nijenhuis K, McKinley G H, Spiegelberg S, Barnes H A, Aksel N, Heymann L, Odell J A 2007 Non-Newtonian flows Springer Handbook of experimental fluid mechanics ed C Tropea, A L Yarin and J F Foss (Berlin, Heidelberg, New York: Springer) [30] Münch A, Wagner B A and Witelski T P 2005 Lubrication models with small to large slip lengths J. Eng. Math. 53 359 [31] Navier C L M H 1823 Mémoire sur les lois du mouvement des fluids Mem. Acad. Sci. Inst. Fr. 6, 389 and 432 [32] Lauga E, Brenner M P and Stone H A 2007 Microfluidics: The no-slip boundary condition Springer Handbook of experimental fluid mechanics ed C Tropea, A L Yarin and J F Foss (Berlin, Heidelberg, New York: Springer) [33] Neto C, Evans D R, Bonaccurso E, Butt H-J and Craig V S J 2005 Boundary slip in Newtonian liquids: A review of experimental studies Rep. Prog. Phys. 68 2859 [34] Bocquet L and Barrat J-L 2007 Flow boundary conditions from nano- to micro-scales Soft Matter 3 685 [35] Joly L, Ybert C and Bocquet L 2006 Probing the nanohydrodynamics at liquid/solid interfaces using thermal motion Phys. Rev. Lett. 96 046101 [36] Barrat J-L and Bocquet L 1999 Large slip effect at a nonwetting fluid-solid interface Phys. Rev. Lett. 82 4671 [37] Pit R, Hervet H and Léger L 2000 Direct experimental evidence of slip in hexadecane: Solid interfaces Phys. Rev. Lett. 85 980 [38] Cottin-Bizonne C, Jurine S, Baudry J, Crassous J, Restagno F and Charlaix E 2002 Nanorheology: An investigation of the boundary condition at hydrophobic and hydrophilic interfaces Eur. Phys. J. E 9 47 Slip effects in polymer thin films 36 [39] Leger L 2003 Friction mechanisms and interfacial slip at fluid-solid interfaces J.Phys.: Condens. Matter 15 S19 [40] Zhu Y and Granick S 2002 Limits of the hydrodynamic no-slip boundary condition Phys. Rev. Lett. 88 106102 [41] Schmatko T, Hervet H and Léger L 2006 Effect of nano-scale roughness on slip at the wall of simple fluids Langmuir 22 6843 [42] Kunert C and Harting J 2007 Roughness induced boundary slip in microchannel flows Phys. Rev. Lett. 99 176001 [43] Cottin-Bizonne C, Barrat J-L, Bocquet L and Charlaix E 2003 Low-friction flows of liquid at nanopatterned interfaces Nat. Mater. 2 237 [44] Priezjev N V and Troian S M 2006 Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: Molecular-scale simulations versus continuum predictions J. Fluid Mech. 554 25 [45] Joseph P, Cottin-Bizonne C, Benoı̂t J-M, Ybert C, Journet C, Tabeling P and Bocquet L 2006 Slippage of water past superhydrophobic carbon nanotube forests in microchannels Phys. Rev. lett. 97 156104 [46] Ybert C, Barentin C, Cottin-Bizonne C, Joseph P and Bocquet L 2007 Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries Phys. Fluids 19 123601 [47] Steinberger A, Cottin-Bizonne C, Kleimann P and Charlaix E 2007 High friction on a bubble mattress Nature Materials 6 665 [48] Schmatko T, Hervet H and Léger L 2005 Friction and slip at simple fluid/solid interfaces: The roles of the molecular shape and the solid/liquid interaction Phys. Rev. Lett. 94 244501 [49] Priezjev N V, Darhuber A A and Troian S M 2005 Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamics simulations Phys. Rev. E 71 041608 [50] Heidenreich S, Ilg P and Hess S 2007 Boundary conditions for fluids with internal orientational degrees of freedom: Apparent velocity slip associated with the molecular alignment Phys. Rev. E 75 066302 [51] Cho J-H J, Law B M and Rieutord F 2004 Dipole-dependent slip of Newtonian liquids at smooth solid hydrophobic surfaces Phys. Rev. Lett. 92 166102 [52] De Gennes P G 2002 On fluid/wall slippage Langmuir 18 3413 [53] Huang D M, Sendner C, Horinek D, Netz R R and Bocquet L 2008 Water slippage versus contact angle: A quasiuniversial relationship Phys. Rev. Lett. 101 226101 [54] Steitz R, Gutberlet T, Hauss T, Klösgen B, Krastev R, Schemmel S, Simonsen A C, Findenegg G H 2003 Nanobubbles and their precursor layer at the interface of water against a hydrophobic substrate Langmuir 19 2409 [55] Doshi D A, Watkins E B, Israelachvili J N and Majewski J 2005 Reduced water density at hydrophobic surfaces: Effect of dissolved gases PNAS 102 9458 [56] Mezger M, Reichert H, Schröder S, Okasinski J, Schröder H, Dosch H, Palms D, Ralston J and Honkimäki V 2006 High-resolution in situ x-ray study of the hydrophobic gap at the wateroctadecyl-trichlorosilane interface PNAS 103 18401 [57] Maccarini M, Steitz R, Himmelhaus M, Fick J, Tatur S, Wolff M, Grunze M, Janeček, Netz R R 2007 Density depletion at solid-liquid interfaces: A neutron reflectivity study Langmuir 23 598 [58] Tyrrell J W G and Attard P 2001 Images of nanobubbles on hydrophobic surfaces and their interactions Phys. Rev. Lett. 87 176104 [59] Tretheway D C and Meinhart C D 2004 A generating mechanism for apparent fluid slip in hydrophobic microchannels Phys. Fluids 16 1509 [60] Poynor A, Hong L, Robinson I K, Granick S, Zhang Z and Fenter P A 2006 How water meets a hydrophobic surface Phys. Rev. Lett. 97 266101 [61] Hendy S C and Lund N J 2009 Effective slip length for flows over surfaces with nanobubbles: Slip effects in polymer thin films [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] 37 The effect of finite slip J. Phys.: Condens. Matter 21 144202 Oron A, Davis S H and Bankoff S G 1997 Long-scale evolution of thin liquid films Rev. Mod. Phys. 69 931 Kargupta K, Sharma A and Khanna R 2004 Instability, dynamics, and morphology of thin slipping films Langmuir 20 244 Fetzer R, Münch A, Wagner B, Rauscher M and Jacobs K 2007 Quantifying hydrodynamic slip: A comprehensive analysis of dewetting profiles Langmuir 23 10559 Blossey R 2008 Thin film rupture and polymer flow Phys. Chem. Chem. Phys. 10 5177 Israelachvili J 1992 Intermolecular and surface forces 2nd edition (New York: Academic Press) Becker J, Grün G, Seemann R, Mantz H, Jacobs K, Mecke K R and Blossey R 2003 Complex dewetting scenarios captured by thin-film models Nature Materials 2 59 Rauscher M, Blossey R, Münch A and Wagner B 2008 Spinodal dewetting of thin films with large interfacial slip: Implications from the dispersion relation Langmuir 24 12290 Fetzer R, Rauscher M, Seemann R, Jacobs K and Mecke K 2007 Thermal noise influences fluid flow in thin films during spinodal dewetting Phys. Rev. Lett. 99 114503 Seemann R, Herminghaus S and Jacobs K 2001 Shape of a liquid front upon dewetting Phys. Rev. Lett. 87 196101 Reiter G 2001 Dewetting of highly elastic thin polymer films Phys. Rev. Lett. 87 186101 Damman P, Baudelet N and Reiter G 2003 Dewetting near the glass transition: Transition from a capillary force dominated to a dissipation dominated regime Phys. Rev. Lett. 91 216101 Neto C, Jacobs K, Seemann R, Blossey R, Becker J and Grün G 2003 Correlated dewetting patterns in thin polystyrene films J. Phys.: Condens. Matter 15 421 Neto C, Jacobs K, Seemann R, Blossey R, Becker J and Grün G 2003 Satellite hole formation during dewetting: Experiment and simulation J. Phys.: Condens. Matter 15 3355 Fetzer R, Jacobs K, Münch A, Wagner B and Witelski T P 2005 New slip regimes and the shape of dewetting thin liquid films Phys. Rev. Lett. 95 127801 Fetzer R, Rauscher M, Münch A, Wagner B A and Jacobs K 2006 Slip-controlled thin film dynamics Europhys. Lett. 75 638 Bäumchen O, Fetzer R, Münch A, Wagner B and Jacobs K 2009 Comprehensive analysis of dewetting profiles to quantify hydrodynamic slip IUTAM Symp. on Adv. in Micro- and Nanofluidics ed M Ellero, X Hu, J Fröhlich and N Adams (Springer) De Gennes P G 1985 Wetting: Statics and dynamics Rev. Mod. Phys. 57 827 Frumkin A N 1938 On the wetting phenomena and attachment of bubbles I J. Phys. Chem. USSR 12 337 Brochard-Wyart F, De Gennes P-G, Hervet H and Redon C 1994 Wetting and slippage of polymer melts on semi-ideal surfaces Langmuir 10 1566 Redon C, Brochard-Wyart F and Rondelez F 1991 Dynamics of dewetting Phys. Rev. Lett. 66 715 Redon C, Brzoska J B and Brochard-Wyart F Dewetting and slippage of microscopic polymer films 1994 Macromolecules 27 468 De Gennes P G 1979 Ecoulements viscométriques de polymères enchevêtrés C. R. Acad. Sci. B 288 219 Münch A 2005 Dewetting rates of thin liquid films J. Phys.: Condens. Matter 17 S309 Jacobs K, Seemann R, Schatz G and Herminghaus S 1998 Growth of holes in liquid films with partial slippage Langmuir 14 4961 Fetzer R and Jacobs K 2007 Slippage of newtonian liquids: Influence on the dynamics of dewetting thin films Langmuir 23 11617 Bäumchen O, Jacobs K and Fetzer R 2008 Probing slippage and flow dynamics of thin dewetting polymer films Proc. Eur. Conf. on Microfluidics (Bologna) Dec 10-12, 2008 Blake T D and Haynes J M 1969 Kinetics of liquid/liquid displacement J. Colloid Interface Sci. 30 421 Slip effects in polymer thin films 38 [89] De Ruijter M J, Blake T D and De Coninck J 1999 Dynamic wetting studied by molecular modeling simulations of droplet spreading Langmuir 15 7836 [90] Blake T D and De Coninck J 2002 The influence of solid/liquid interactions on dynamic wetting J. Adv. Colloid Interface Sci. 96 21 [91] Voué M, Rioboo R, Adao M H, Conti J, Bondar A I, Ivanov D A, Blake T D and De Coninck J 2007 Contact-line friction of liquid drops on self-assembled monolayers: Chain-length effects Langmuir 23 4695 [92] Barrena E, Kopta S, Ogletree D F, Charych D H and Salmeron M 1999 Relationship between friction and molecular structure: Alkylsilane lubricant films under pressure Phys. Rev. Lett. 82 2880 [93] Shanahan M E R and Carré A 1994 Anomalous spreading of liquid drops on an elastomeric surface Langmuir 10 1647 [94] Shanahan M E R and Carré A 1995 Viscoelastic dissipation in wetting and adhesion phenomena Langmuir 11 1396 [95] Carré A and Shanahan M E R 1995 Influence of the ”wetting ridge” in dry patch formation Langmuir 11 3572 [96] Carré A, Gastel J-C and Shanahan M E R 1996 viscoelastic effects in the spreading of liquids Nature 379 432 [97] Shanahan M E R and Carré A 2002 Spreading and dynamics of liquid drops involving nanometric deformations on soft substrates Colloids Surf. A 206 115 [98] Long D, Ajdari A and Leibler L 1996 Static and dynamic wetting properties of thin rubber films Langmuir 12 5221 [99] Long D, Ajdari A and Leibler L 1996 How do grafted polymer layers alter the dynamics of wetting Langmuir 12 1675 [100] Al Akhrass S, Reiter G, Hou S Y, Yang M H, Chang Y L, Chang F C, Wang C F and Yang A C-M 2008 Viscoelastic thin polymer films under transient residual stresses: Two-stage dewetting on soft substrates Phys. Rev. Lett. 100 178301 [101] Ruckenstein E and Jain R K 1974 Spontaneous rupture of thin liquid films J. Chem. Soc. Faraday Trans. II 132 [102] Brochard-Wyart F, Di Meglio J-M and Quéré D 1987 Dewetting C. R. Acad. Sc. Paris Série II 304 553 [103] Sharma A and Ruckenstein E 1989 Dewetting of solids by the formation of holes in macroscopic liquid films J. Coll. Int. Sci. 133 358 [104] Brochard-Wyart F and Daillant J 1990 Drying of solids wetted by thin liquid films Can. J. Phys. 68 1084 [105] Brochard-Wyart F, Redon C and Skykes C 1992 Dewetting of ultrathin liquid films C.R. Acad. Sci. Paris Série II 314 19 [106] Reiter G 1992 Dewetting of thin polymer films Phys. Rev. Lett. 68 75 [107] Brochard-Wyart F, Debregeas G, Fondecave R and Martin P 1997 Dewetting of supported viscoelastic polymer films: Birth of rims Macromolecules 30 1211 [108] Masson J-L and Green P F 2002 Hole formation in thin polymer films: A two-stage process Phys. Rev. Lett. 88 205504 [109] Brochard-Wyart F and Redon C 1992 Dynamics of rim instabilities Langmuir 8 2324 [110] Reiter G and Sharma A 2001 Auto-optimization of dewetting rates by rim instabilities in slipping polymer films Phys. Rev. Lett. 87 166103 [111] Münch A and Wagner B 2005 Contact-line instability of dewetting thin films Physica D 209 178 [112] Gabriele S, Sclavons S, Reiter G and Damman P 2006 Disentanglement time of polymers determines the onset of rim instabilities in dewetting Phys. Rev. Lett. 96 156105 [113] King J R, Münch A and Wagner B 2009 Linear stability analysis of a sharp-interface model for dewetting thin films J. Eng. Math. 63 177 [114] Migler K B, Hervet H and Leger L 1993 Slip transition of a polymer melt under shear stress Phys. Slip effects in polymer thin films [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] 39 Rev. Lett. 70 287 Brochard F and de Gennes P G 1992 Shear-dependent slippage at a polymer/solid interface Langmuir 8 3033 Ajdari A, Brochard-Wyart F, De Gennes P-G, Leibler L, Viovy J-L and Rubinstein M 1994 Slippage of an entangled polymer melt on a grafted surface Physica A 204 17 Gay C 1999 New concepts for the slippage of an entangled polymer melt at a grafted solid interface Eur. Phys. J. B 7 251 Tchesnokov M A, Molenaar J, Slot J J M and Stepanyan R 2005 A molecular model for cohesive slip at polymer melt/solid interfaces J. Chem. Phys. 122 214711 Hervet H and Leger L 2003 Flow with slip at the wall: From simple to complex fluids C. R. Physique 4 241 Drda PP and Wang S-Q 1995 Stick-slip transition at polymer melt/solid interfaces Phys. Rev. Lett. 75 2698 Wang S-Q and Drda P A 1996 Stick-slip transition in capillary flow of polyethylene. 2. Molecularweight dependence and low-temperature anomaly Macromolecules 29 4115 Priezjev N V and Troian S M 2004 Molecular origin and dynamic behavior of slip in sheared polymer films Phys. Rev. Lett. 92 018302 Reiter G and Khanna R 2000 Kinetics of autophobic dewetting of polymer films Langmuir 16 6351 Reiter G and Khanna R 2000 Negative excess interfacial entropy between free and end-grafted chemically identical polymers Phys. Rev. Lett. 85 5599 Reiter G and De Gennes P-G 2001 Spin-cast, thin, glassy polymer films: Highly metastable forms of matter Eur. Phys. J. E 6 25 Reiter G, Hamieh M, Damman P, Sclavons S, Gabriele S, Vilmin T and Raphaël E 2005 Residual stresses in thin polymer films cause rupture and dominate early stages of dewetting Nature Materials 4 754 Vilmin T and Raphaël E 2006 Dynamic instability of thin viscoelstic films under lateral stress Phys. Rev. Lett. 97 036105 Damman P, Gabriele S, Coppée S, Desprez S, Villers D, Vilmin T, Raphaël E, Hamieh M, Al Akhrass S and Reiter G 2007 Relaxation of residual stress and reentanglement of polymers in spin-coated films Phys. Rev. Lett. 99 036101 Vilmin T and Raphaël E 2005 Dewetting of thin viscoelastic polymer films on slippery substrates Europhys. Lett. 72 781 Yang M H, Hou S Y, Chang Y L and Yang A C-M 2006 Molecular recoiling in polymer thin film dewetting Phys. Rev. Lett. 96 066105 Vilmin T and Raphaël 2006 Dewetting of thin polymer films Eur. Phys. J. E 21 161 Hamieh M, Al Akhrass S, Hamieh T, Damman P, Gabriele S, Vilmin T, Raphaël E and Reiter G 2007 Influence of substrate properties on the dewetting dynamics of viscoelastic polymer films J. of Adhes. 83 367 Coppée S, Gabriele S, Jonas A M, Jestin J, Damman P 2009 Influence of chain interdiffusion between immiscible polymers on dewetting dynamics arXiv:0904.1675v1 Ziebert F and Raphaël E 2009 Dewetting dynamics of stressed viscoelastic thin polymer films Phys. Rev. E 79 031605 Ziebert F and Raphaël E 2009 Dewetting of thin polymer films: Influence of interface evolution Europhys. Lett. 86 46001 Reiter G, Al Akhrass S, Hamieh M, Damman P, Gabriele S, Vilmin T and Raphaël E 2009 Dewetting as an investigative tool for studying properties of thin polymer films Eur. Phys. J. Special Topics 166 165 Besancon B M and Green P F 2007 Dewetting dynamics in miscible polymer-polymer thin films mixtures J. Chem. Phys. 126 224903 Fetzer R 2006 Einfluss von Grenzflächen auf die Fluidik dünner Polymerfilme (Berlin: Logos) Slip effects in polymer thin films 40 [139] Merabia S and Avalos J B 2008 Dewetting of a stratified two-component liquid film on a solid substrate Phys. Rev. Lett. 101 208304 [140] Servantie J and Müller M 2008 Temperature dependence of the slip length in polymer melts at attractive surfaces Phys. Rev. Lett. 101 026101 [141] Müller M, Pastorino C and Servantie J 2008 Flow, slippage and a hydrodynamic boundary condition of polymers at surfaces J. Phys.: Condens. Matter 20 494225 [142] Pastorino C, Binder K, Kreer T and Müller M 2006 Static and dynamic properties of the interface between a polymer brush and a melt of identical chains J. Chem. Phys. 124 064902 [143] Pastorino C, Binder K and Müller M 2009 Coarse-grained description of a brush/melt interface in equilibrium and under flow Macromolecules 42 401