Slip effects in polymer thin films.
ABSTRACT Probing the fluid dynamics of thin films is an excellent tool for studying 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. Thus, 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 on the liquid flow field and, especially, the boundary condition at the interface. We review the basic models and experimental results obtained during the last several years with exclusive regard to polymers as ideal model liquids for fluid flow. Moreover, concepts that aim to explain slippage on the molecular scale are summarized and discussed.

Article: Solid surface structure affects liquid order at the polystyreneselfassembledmonolayer interface.
Philipp Gutfreund, Oliver Bäumchen, Renate Fetzer, Dorothee van der Grinten, Marco Maccarini, Karin Jacobs, Hartmut Zabel, Max Wolff[Show abstract] [Hide abstract]
ABSTRACT: We present a combined xray and neutron reflectivity study characterizing the interface between polystyrene (PS) and silanized surfaces. Motivated by the large difference in slip velocity of PS on top of dodecyltrichlorosilane (DTS) and octadecyltrichlorosilane (OTS) found in previous studies, these two systems were chosen for the present investigation. The results reveal the molecular conformation of PS on silanized silicon. Differences in the molecular tilt of OTS and DTS are replicated by the adjacent phenyl rings of the PS. We discuss our findings in terms of a potential link between the microscopic interfacial structure and dynamic properties of polymeric liquids at interfaces.Physical Review E 01/2013; 87(11):012306. · 2.31 Impact Factor  SourceAvailable from: Colm O'Dwyer[Show abstract] [Hide abstract]
ABSTRACT: The stability of thin poly(methylmethacrylate) (PMMA) films of low molecular weight on a solid substrate is controlled by the areal coverage of gold nanoparticles (NPs) present at the airpolymer interface. As the polymer becomes liquid the Au NPs are free to diffuse, coalesce, and aggregate while the polymer film can change its morphology through viscous flow. These processes lead at the same time to the formation of a fractal network of Au NPs and to the development of spinodal instabilities of the free surface of the polymer films. For thinner films a single wavelength is observed, while for thicker films two wavelengths compete. With continued heating the aggregation process results in a decrease in coverage, the networks evolve into disordered particle assemblies, while the polymer films flatten again. The disordering occurs first on the smallest scales and coincides (in thicker films) with the disappearance of the smaller wavelength. The subsequent disordering on larger scales causes the films to flatten.Langmuir 05/2013; · 4.19 Impact Factor  SourceAvailable from: Colm O'Dwyer[Show abstract] [Hide abstract]
ABSTRACT: The stability of polystyrene thin films of low molecular weight on a solid substrate is shown to be controlled by the presence of uniformly distributed gold sputtered at the air–polymer interface. Continuous gold coverage causes the formation of wrinkles. High coverage and Au nanoparticle (NP) density leads to the development of a spinodal instability while low coverage and NP density retards the nucleation dewetting mechanism that beads up the thin polymer film into drops when no coverage is present. Heating at temperature larger than the polymer glass transition temperature for extended periods allows the gold NPs to coalesce and rearrange. The area of polymer surface covered by NPs decreases as a result and this drives the films from unstable to metastable states. When the gold NPs are interconnected by polymer chains a theoretically predicted spinodal instability that patterns the film surface is experimentally observed. Suppression of the instability and a return to a flat film occurs when the polymer interconnections between particles are broken. While the polymer films maintain their physical continuity changes in their chemical surface composition and thickness are observed. The observed film metastability is nevertheless in agreement with theoretical prediction that includes these surface changes.Soft Matter 05/2013; 9:2695–2702. · 3.91 Impact Factor
Page 1
arXiv:0909.1946v1 [condmat.soft] 10 Sep 2009
TOPICAL REVIEW
Slip effects in polymer thin films
O B¨ aumchen and K Jacobs
Experimental Physics, Saarland University, Campus, D66123 Saarbr¨ ucken, Germany
Email: k.jacobs@physik.unisaarland.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
Page 2
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 socalled labonchip 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 largescale 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]. Channellike threedimensional 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 vanderWaals part and a repulsive part [5, 6, 7, 8]. For the description of
vanderWaals 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
threephase contact line due to conservation of liquid volume. A common phenomenon
Page 4
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 socalled ”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 socalled RayleighPlateau 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 networklike 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
moleculardynamic (MD) studies help to obtain more information and are supportive
to gain insight into the molecular mechanism of slippage.
Page 5
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
nonnegligible. 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 microstructure of
polymer chains such as the tacticity and the architecture (linear, branched, ringshaped),
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 semicrystalline 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 chainlength, 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
Page 6
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 socalled WLFequation, ηgdenotes the viscosity at Tg, B an empirically obtained
constant, T∞the socalled 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 freestanding 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 coworkers 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 interfacerelated effects such as reorientation of polymer chains
or accumulation of chainends at the interface, finitesize 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
Page 7
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 liquidlike surface layer [22], a result that was also found by other authors
[23, 24]. Besides confinement effects on Tg, further interfacerelated 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 longchained 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 socalled 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)
Page 8
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, socalled 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 timedependent 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 timedependent 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/strainrelation (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
Page 9
Slip effects in polymer thin films9
”zero shear rate” viscosity. Polymers with shearthinning or shearthickening 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 stressstrain relations can
be considered or alternatively nonlinear 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 thinfilm equation that incorporates viscoelastic effects
[26, 27, 28]. For a more elaborate description of nonNewtonian 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 socalled 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 lowRe lubrication theory can be applied. To quantify
and to judge the occurrence of viscoelastic effects versus pure viscous flow, the socalled
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. NavierStokes equations
The NavierStokes 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)
Page 10
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 NavierStokes 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 socalled 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.
Page 11
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 socalled ”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 uxz=0 is zero, but a
substantial slip length is measured.
2.4. Slip/noslip boundary condition
2.4.1. Navier slip boundary condition
where the assumption that the tangential velocity u at the solid/liquid interface
vanishes (noslip 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 uis proportional to the normal component
of the strain rate tensor; the constant of proportionality is described as the socalled
slip length b:
In contrast to fluid dynamics in a bulk volume,
u= bn · ˙ γ(2.20)
In case of simple shear flow in xdirection, the definition of the slip length can be
alternatively written as
ux
∂zuxz=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 noslip situation, whereas b = ∞ characterizes a fullslip
situation. The latter case corresponds to ”plugflow”, 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
Page 12
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 nonbleached 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 (socalled nanobubbles) and its influence on
Of course, many interesting aspects in the
Further
Page 13
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. Thinfilm 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 thinfilm 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 intermolecular
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 thinfilm 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
udz,(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 socalled lubrication approximation and is used in the following to rescale
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 onedimensional 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 thinfilm equation is based on the socalled lubrication approximation
and the rescaling 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 socalled
weakslip situation with regard to larger slip b ≫ h, M¨ unch et al [30] and Kargupta et
al [63] developed independently socalled strongslip 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 noslip 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 intermediateslip 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.
weakslipb ≪ h(2.26), (2.27)
b → 0 (noslip)
b → ∞ (intermediateslip)
β → 0 (intermediateslip)
β → ∞ (”free”slip)
[62]
strongslipb ≫ h(2.28)[63, 30]
2.5.3. Lubrication models including viscoelasticity
a thin film equation for the weakslip 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 strongslip situation by Blossey et al [27]. To summarize these extensions, the
essential result is the fact that linear viscoelastic effects are absent in the weakslip case
and the Newtonian thinfilm model is still valid. The strongslip 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 nonlinearities of the corotational Jeffreys model for viscoelastic relaxation into
their thinfilm model [28].
The extensions of the aforementioned thinfilm models for different slip conditions
with or without the presence of viscoelastic relaxation (Newtonian and nonNewtonian
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 thinfilm 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, longrange attractive van der Waals forces add
to shortrange 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 multilayer 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)
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