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arXiv:0909.1946v1 [cond-mat.soft] 10 Sep 2009

TOPICAL REVIEW

Slip effects in polymer thin films

O B¨ aumchen and K Jacobs

Experimental Physics, Saarland University, Campus, D-66123 Saarbr¨ ucken, Germany

E-mail: k.jacobs@physik.uni-saarland.de

Abstract.

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.

Probing the fluid dynamics of thin films is an excellent tool to study the

PACS numbers: 68.15.+e, 83.50.Lh, 83.80.Sg, 47.15.gm

Submitted to: J. Phys.: Condens. Matter

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Slip effects in polymer thin films2

1. Introduction

Understanding liquid flow in confined geometries plays a huge role in the field of micro-

and 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

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Slip effects in polymer thin films3

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 heqis 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 heqresults. 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 100nm 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

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Slip effects in polymer thin films4

Figure 2.

hydrophobized Si substrate captured by optical microscopy (adapted from [10]).

Dewetting of a 80nm PS(65kg/mol) film at T = 135◦C from a

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.

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Slip effects in polymer thin films5

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.

important aspect of a moving liquid is the velocity profile.

An

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/Mnis 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 = 104g/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 Tgand 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

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Slip effects in polymer thin films6

dependency of the viscosity η of a polymer on temperature T was developed by Williams,

Landel and Ferry:

η = ηgexpB(Tg− T)

fg(T − T∞).(2.1)

In this so-called WLF-equation, ηgdenotes the viscosity at Tg, B an empirically obtained

constant, T∞the so-called Vogel temperature and fgthe 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=35kg/mol and Me=17kg/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 ∝ M3

regime below Mcis well reproduced, above Mcan 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].

wis expected. Empirically, the linear

2.1.2.

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 Tgchanges 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 Tgfrom 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 100nm (see Fig. 3) [21]. For

PS(2k) below 10nm 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

Properties in confined geometries In contrast to the bulk situation, where

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Slip effects in polymer thin films7

?

?

?

???

?

?

?

?

Figure 3. Glass transition temperature Tg of polystyrene films of 2kg/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 Meis significantly

larger than the bulk value [25].

2.1.3.

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 ˙ γ are proportional, the fluid is

called Newtonian. The constant of proportionality is identified as the viscosity η of the

liquid.

Viscosity and viscoelasticity One of the major characteristics of a liquid in

σ = η˙ γ.(2.2)

Liquids such as long-chained polymers show a shear rate dependent viscosity η(˙ γ)

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)

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Slip effects in polymer thin films8

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 γeand γvof both mechanisms:

γ = γe+ γv.(2.4)

With (2.3) and (2.2) we get

σ = GMγe= ηM˙ γv

(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:

τM=ηM

GM.(2.6)

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˙ γv= 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) = GMγe(t) = GMγ 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

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Slip effects in polymer thin films9

”zero shear rate” viscosity. Polymers with shear-thinning or shear-thickening properties

can be described by the function

σ ∝ ˙ γn,(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 σijrelaxes according to

the following constitutive relaxation equation:

(1 + λ1∂t)σ = η(1 + λ2∂t)˙ γ,(2.10)

where the strain rate is given by the gradient of the velocity field ˙ γij = ∂jui+ ∂iuj.

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 M¨ unch

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

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

The flow of a liquid can be characterized by

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 Wi has been introduced as

Wi = τ ˙ γ.(2.12)

Thereby, τ denotes the relaxation time and ˙ γ the strain rate as introduced in the previous

section. If Wi ≪ 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

∂tρ + ∇ · (ρu) = 0,(2.13)

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Slip effects in polymer thin films10

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,

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

(2.15)

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 σ∗

tension σij, see (2.2), and the pressure p:

ijis given by the stress

σ∗

ij= σij+ pδij= η(∂jui+ ∂iuj) + 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 = γlvκ, (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= γlvκ = γlv(1

R1

+

1

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.

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Slip effects in polymer thin films 11

no?slip

partial?slipfull?slip

”apparent”?slip

bb

b=0

b=?

z0

u

u

u

u

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

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:

In contrast to fluid dynamics in a bulk volume,

u||= bn · ˙ γ(2.20)

In case of simple shear flow in x-direction, the definition of the slip length can be

alternatively written as

ux

∂zux|z=0=uxη

σ

b =

=η

ξ,

(2.21)

where ξ = σ/uxdenotes the friction coefficient at the solid/liquid interface. The xy-

plane 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?

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

In recent years, numerous experimental studies

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Slip effects in polymer thin films12

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.

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].

Thereby, a distinct part of a

2.4.3. Which parameters influence slippage?

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.

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

Of course, many interesting aspects in the

Further

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Slip effects in polymer thin films13

x

y

z

h(x,y,t)

u (u ,u )

||xy

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.

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, (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 1mm)

0 = −∇(p + φ′(h)) + η△u

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

Derivation Confining the flow of a liquid to the geometry to the one of a

(2.16) can be written as

(2.22)

∂th = −∇||

?h

0

u||dz,(2.23)

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.

Page 14

Slip effects in polymer thin films14

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′) = ∂2

zux,∂z(p + φ′) = 0,∂xux+ ∂zuz= 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 ∂zux= 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:

∂zux= 0,p + ∂2

xh = 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:

∂th = −∇[m(h)∇(γlv△h − φ′(h))],

where m(h) denotes the mobility given by

1

3η(h3+ 3bh2).

(2.26)

m(h) =

(2.27)

2.5.2. Lubrication models including slippage

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, M¨ unch 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:

u =2b

η∂x(γlv∂2

∂th = −∂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.

As discussed in the previous section, the

η∂x(2ηh∂xu) +bh

xh − φ′(h))

(2.28)

Page 15

Slip effects in polymer thin films15

Table 1.

situations.

Summary of lubrication models for Newtonian flow and different slip

modelvalidity equationlimiting casesref.

weak-slipb ≪ h(2.26), (2.27)

b → 0 (no-slip)

b → ∞ (intermediate-slip)

β → 0 (intermediate-slip)

β → ∞ (”free”-slip)

[62]

strong-slipb ≫ h(2.28)[63, 30]

2.5.3. Lubrication models including viscoelasticity

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].

In the meanwhile, the derivation of

2.5.4.

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 diexhibiting Hamaker

constants Airequire a superposition of contributions to the potential:

Application I - Spinodal dewetting One of the main applications of the

φ(d)vdw= −

A1

12πd2−

A2− A1

12π(d + d1)2−

A3− A2

12π(d + d1+ d2)2

(2.29)