Lubrication Models with Small to Large Slip Lengths

Münch, Andreas; Wagner, Barbara; Witelski, Thomas P.

doi:10.1007/s10665-005-9020-3

Cited by 131 publications

(212 citation statements)

References 37 publications

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“…While a well-defined wave speed is maintained, in the frame of reference moving with that speed, the solution exhibits a self-similar growing form for t → ∞; similar behavior has been observed in other thin film problems [8,24]. We have been able to obtain expressions for the accumulation of surfactant and leading order asymptotic forms for the film and surfactant profiles subject to relatively few assumptions on the dynamics.…”

Section: Resultssupporting

confidence: 61%

Growing surfactant waves in thin liquid films driven by gravity

Witelski¹,

Shearer²,

Levy³

2006

Applied Mathematics Research eXpress

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The dynamics of a gravity-driven thin film flow with insoluble surfactant are described in the lubrication approximation by a coupled system of nonlinear PDEs. When the total quantity of surfactant is fixed, a traveling wave solution exists. For the case of constant flux of surfactant from an upstream reservoir, global traveling waves no longer exist as the surfactant accumulates at the leading edge of the thin film profile. The dynamics can be described using matched asymptotic expansions for t → ∞. The solution is constructed from quasi-statically evolving traveling waves. The rate of growth of the surfactant profile is shown to be O( √ t) and is supported by numerical simulations.

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Section: Resultssupporting

confidence: 61%

Growing surfactant waves in thin liquid films driven by gravity

Witelski¹,

Shearer²,

Levy³

2006

Applied Mathematics Research eXpress

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“…The boundary condition for the velocity component parallel to the substrate is u = b∂u/∂z. Recently, a thin film model has been developed for the case where the slip length is much larger than the film thickness scale H [18][19][20]. Neglecting inertia, the lateral velocity in the film for this model is given by…”

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Slip-controlled thin-film dynamics

et al. 2006

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PACS. 83.80.Sg -Polymer melts. PACS. 47.15.gm -Thin film flows. PACS. 83.50.Lh -Slip boundary effects (interfacial and free surface flows).Abstract. -In this study, we present a novel method to assess the slip length and the viscosity of thin films of highly viscous Newtonian liquids. We quantitatively analyse dewetting fronts of low molecular weight polystyrene melts on Octadecyl-(OTS) and Dodecyltrichlorosilane (DTS) polymer brushes. Using a thin film (lubrication) model derived in the limit of large slip lengths, we can extract slip length and viscosity. We study polymer films with thicknesses between 50 nm and 230 nm and various temperatures above the glass transition. We find slip lengths from 100 nm up to 1 µm on OTS and between 300 nm and 10 µm on DTS covered silicon wafers. The slip length decreases with temperature. The obtained values for the viscosity are consistent with independent measurements.

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“…In [2] a family of thin film models ranging from weak to strong slip regimes could be derived depending on the order of magnitude of the slip length (see also [3]). Modelling the viscoelastic properties of such polymers generalized Maxwell and Jeffreys models have been widely used.…”

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“…We employ the strong-slip scaling, as in [2]. In this limit, the friction between the liquid and the substrate is too weak to maintain a non-zero xz-shear stress to lowest order, and lateral pressure gradients are balanced by the xx-component of the stress tensor.…”

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“…At the substrate, z = 0, we have as usual the impermeability condition w = 0 and the slip condition u = bτ xz . In the strong-slip regime we have b = β s /ε 2 , β s being the slip length parameter of order O(ε 0 ), see [2] for details. For the derivation of the lubrication approximation we assume that the dynamic variables can be given in form of the asymptotic expansions…”

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A thin-film model for corotational Jeffreys fluids under strong slip

et al. 2006

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Abstract. We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.PACS. 83.60.Bc Viscoelasticity -47.50.+d Non-newtonian fluid flows -68.15.+e Liquid thin films Dewetting liquid polymer films on nonwetting substrates, such as silicone wafers grafted with a monolayer of brushes, play a prominent role in many nanotechnological applications. It is known that for these situations, polymer films on the scale of a few hundred nanometers typically show large slippage [1]. Furthermore, for highly entangled polymers, the assumption of a Newtonian fluid will seize to be valid. To understand the interplay of viscous and viscoelastic properties of liquid polymers on hydrophobically coated substrates, there is a need for refined theoretical methods that are able to capture and evolve the emerging morphologies and their longtime dynamics. Dimension-reduced thin film models have shown in the past to be extremly successful to enable quantitative predictions that are hardly being attained simply via the underlying free-boundary problem.In this paper we make an important step in that direction by developing a new thin film model that combines large slippage with viscoelastic properties. In [2] a family of thin film models ranging from weak to strong slip regimes could be derived depending on the order of magnitude of the slip length (see also [3]). Modelling the viscoelastic properties of such polymers generalized Maxwell and Jeffreys models have been widely used. In [4] a weak slip model could be combined with the linearized Jeffreys model to discuss effects of viscoelastic relaxation. More recently [5] we have shown that the strong slip limit can also be recovered for the linear Jeffreys model. In this Rapid Note we show that, for the strong slip regime, we are able to fully incorporate the general corotational Jeffreys model into our thin film model.We begin by presenting the underlying free boundary problem for incompressible, viscous flow with velocity u,where we assume that the traceless part of the symmetric stress tensor τ obeys the corotational Jeffreys model [6]Here, D/Dt denotes the Jaumann derivative which for arbitrary tensor fields Λ is given bywhereγ and ω denote the rate of strain tensor and the vorticity tensor, given bẏrespectively; d/dt is the material derivative ∂ t + u · ∇. We assume the viscosity µ as well as the relaxation parameters λ 1 , λ 2 to be constant material parameters.

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Lubrication Models with Small to Large Slip Lengths

Cited by 131 publications

References 37 publications

Growing surfactant waves in thin liquid films driven by gravity

Growing surfactant waves in thin liquid films driven by gravity

Slip-controlled thin-film dynamics

A thin-film model for corotational Jeffreys fluids under strong slip

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