Contact line dynamics of a liquid meniscus advancing in a microchannel with chemical heterogeneities C. Wylock1, M. Pradas2, B. Haut1, P. Colinet1 and S. Kalliadasis2 1 Université Libre de Bruxelles – Transfers, Interfaces and Processes 2 Imperial College London – Chemical Engineering Department International Conference on Multiscale Complex Fluid Flows and Interfacial Phenomena MULTIFLOW 2010 Brussels, Belgium In this work, we aim to study the motion of a liquid meniscus and associated contact lines advancing into a microchannel with chemically heterogeneous inner walls.

Goal Gas-liquid meniscus moving in a "Hele-Shaw cell like " microchannel Surface chemically heterogeneous  spatial distribution of wetting properties 2 configurations Effect of chemical heterogeneities on meniscus dynamics ? 3D configuration 2D configuration Especially, we are interested in the the contact line motion of a gas-liquid meniscus when the liquid forced to flow between two parallel flat plates separated by a narrow distance as it is typically the case with a Hele-Shaw cell. The surface of the plates is taken to be chemically heterogeneous inducing a spatial distribution of the wetting properties 2 configurations are investigated, as depicted on these 2 figures. You can see the walls of the channel and the liquid moving, advancing or receding, in the channel Firstly, a 2-dimensionnal configuration is considered, where the wetting properties are uniform in the transverse direction of the flow. Secondly, a 3-dimensional configuration, presented on the right is considered, where the wetting properties are random in both directions of the flow. Of particular interest is the influence of the chemical disorder strength under constant forced flow conditions on the meniscus dynamic

Modelling Phase field approach f represents the 2 phases Interface at f=0 A Cahn-Hilliard phase field model is used to describe the interface separating the two phases of the system. Here is a schematic representation of the phase field profile, phi, in one dimension. In this approach, the interface is not sharp but diffuse, and the thickness of this interface is controlled by the parameter epsilon. The equilibrium values phi_e = +1 and phi_e = -1 represent the liquid and the air phases, respectively, and the interface position is then located at phi = 0.

Modelling Phase field approach f represents the 2 phases Interface at f=0 Equilibrium given by Ginzburg-Landau model Free energy formulation Double-well potential This model is based on a Ginzburg-Landau formulation of the free energy of the system for a locally conserved field phi, which is given by this functional equation. This term represents the free energy density function, where V is a potential, usually chosen to have a double-well form as presented in this figure in equilibrium situation, and it is combined with the square gradient term to ensure the existence of a well defined interface with a width of the order of epsilon. The chemical potential can also be defined as the change in this functional for a local change in concentration. Chemical potential

Modelling Phase field approach f represents the 2 phases Interface at f=0 Equilibrium given by Ginzburg-Landau model Free energy formulation Double-well potential When this potential is destabilized, in one of the two phases as presented in this figure, by applying a pressure in the liquid phase for instance, it forces the interface to advance at the expense of the air phase. Therefore, this approach enables contact line motion by diffusive interfacial fluxes. Chemical potential

Modelling Wetting boundary condition Conserved dynamic equation with Standard deviation s = disorder strength [1] with When the phases spread on a solid surface, the interactions/capillary effects can be included in the model by adding such a terme in the free energy, following the Cahn formulation , which destabilize one of the 2 phases and therefore favorize one of the phase From this formulation can be derived this equation for the phase field time evolution and this wetting boundary condition on the wall surfaces. The form of this f function is chosen like that, where A represents a stochastic variable that plays the role of the chemical disorder on the wall surfaces. The A values follow a normal distribution and therefore is characterised by its average and its standard deviation sigma, which characterises its strength. For a constant value of A, the corresponding equilibrium contact angle can be obtained by this relation. The dynamical evolution of the phase field is therefore given by this conserved dynamic equation, where M is a mobility parameter that can depend on phi (it is chosen as a constant in the liquid phase and equals to zero in the gasesous phase) [1] Cahn, J. Chem. Phys. 66 (1977), 3667

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations Now, I will present you a typical simulation results for the 2D configuration. In this short movie, you see the time evolution of a gas-liquid meniscus when the liquid is forced to advance in the microchannel. For this configuration, we are especially interested in the behaviour of the contact angle of this meniscus on the solid surface when the liquid move in this chemically heterogeneous channel. We have a statistical analysis of our numerical results by generating a sufficiently large number of the chemical disorder realisations and our analysis shows that the apparent contact angle suffers of a hysteresis behaviour induced by the disorder of the walls. In this figure is presented the contact angle, averaged on several realisations, as a function of the mean interface velocity. Positive values correspond to advancing motion and negative value to receding motion. The black points and line correspond to a homogeneous case, without disorder

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations With chemical disorder, we observe a hysteresis jump

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations Moreover we observe the contact angle hysteresis is enhanced by the disorder strength. This jump is increased with an increase of the disorder strength, as you can observe on the graph on the right, where are presented the widths of the contact angle gap at the zero velocity as a function of the strength Chemical disorder  contact angle hysteresis enhanced by disorder strength

Results and discussion 3D configuration Typical simulation results Contact line dynamics: preliminary analysis interface width follows fractal dynamics ( scale-invariant growth) pinning-depinning effects avalanche dynamics induced by the chemical disorder  Statistical analysis to perform for various disorder configurations Now for the 3D configuration, here is a short movie presenting a typical simulation results. For this configuration, we are interested in the behaviour on the contact line of the meniscus on the wall surface. In this movie, the two inner walls are presented. The blue area is the wetted part of the walls. The wetting properties of the dry part are presented in gray level, the white zones correspond to hydrophilic zone and the black zone to hydrophobic zone. Our preliminary numerical results show several effects induced by the chemical disorders. The mean interface width follows fractal dynamics when the liquid invades the channel. Moreover, pinning and depinning events and associated avalanche dynamics are observed under certain conditions. A statiscal analysis has now to be performed for various disorder configurations in order to characterize these effects.

Conclusion and future plans Phase field  contact line dynamics in chemically heterogeneous microchannel Chemical disorder induces 2D: hysteresis of contact angle  hysteresis “jump” function of disorder strength 3D: kinetic roughening process of contact line motion, pinning-depinning effects Future plans Statistical analysis for 3D configuration: Characterization of the scaling growth factors Avalanche dynamics In conclusion, the phase field approach enables to study the contact line dynamics of a gas-liquid meniscus in motion in a chemically heterogeneous microchannel. We observe several phenomena induced by the chemical disorder: in the 2D configuration, a statistical analysis reveals that the contact angle suffers of hysteresis behaviour which is enhanced by the disorder strength for the 3D configuration, we observe a kinetic roughening process of the contact line motion : the mean interface width follows fractal dynamics. We observe also pinning-depinning effect and associated avalanche dynamics. Our future plan is to perform a detailed statistical analysis in the 3D configuration in order reach a good characterization of the fractal scaling growth factors and the statistical behaviour of the avalanche events.

Modelling Boundary conditions for 2D configuration

Modelling Boundary conditions for 3D configuration

Results and discussion 2D configuration Typical simulation result Statistical analysis on several disorder realisations Chemical disorder  contact angle hysteresis enhanced by disorder strength

Results and discussion 3D configuration Typical simulation results Contact line dynamic: preliminary analysis interface width growth follows fractal dynamic

Results and discussion 3D configuration Typical simulation results Contact line dynamic: preliminary analysis interface width growth follows fractal dynamic ( size-invariant scaling factors) pinning-depinning effect s avalanche dynamic Pinning site Avalanche site