Rheometry

Rheometry has been employed in order to measure the elastic modulus of our rubber substrates and confront it with the results of other independent experimental techniques.

Rheological measurements were carried out using a Haake Mars Rheometer from Thermo Scientific Karlsruhe that was operated in temperatures up to 80^◦, kindly provided by Prof. C. Wagner, at the Department of Technical Physics of the University of Saarland.

(a) (b) (c) (d)

Figure 2.12: Sketch of shear rheometers of different geometries: (a) capillary (b) rota-tional cylinder, (c) cone-plate, (d) plate-plate.

There are two distinctively different types of rheometers, depending on the geometry of the applied stress. When the applied stress is tensional, the rheometer is called extensional rheometer, whereas, when shear stress is applied, the rheometer is called shear rheometer. In this section, we will deal with the latter.

As seen in Figure 2.12, the shear rheometers can be categorised into three main groups:

• Capillary, in which the liquid is forced through a tube of constant cross-section and precisely known dimensions under conditions of laminar flow (Figure 2.12.(a)).

Either the flow-rate or the pressure drop is fixed and the other is measured. Know-ing the dimensions, the flow-rate can be converted into a value for the shear rate and the pressure drop into a value for the shear stress. Varying the pressure or flow allows for a flow curve to be determined. The capillary rheometer is a common device for measuring viscosities.

• Rotational cylinder, in which the liquid is placed within the annulus formed by one cylinder being inside the other (Figure 2.12.(b)). One of the cylinders is rotated at a set speed. This determines the shear rate inside the annulus. The liquid tends to drag the other cylinder around, and the force it exerts on that cylinder (torque) is measured and converted to a shear stress. This type of rheometer is used widely for determining the flow character of drilling fluids.

• Cone and plate, The cone and plate geometry consists of an inverted cone in near contact with a lower plate (Figure 2.12.(c)). The cone is usually designed with an angle of less than 4^◦. Either the upper or lower surface may rotate de-pending on the instrument design. The parallel plate geometry can be considered a simplified version of the cone and plate, having an angle of 0^◦ (Figure 2.12.(d)).

The test fluid is constrained in the narrow gap between the two surfaces. Cone and plate measurement tools are most often used for highly viscous pastes, gels, and concentrated suspensions.

Given the rheological properties of the PDMS before cross-linking, a shear rheometer in plate - plate geometry was chosen for our measurements. PDMS was prepared as explained on page 41 and was cured in situ, while the change of its rheological properties during curing was monitored and measured.

Chapter 3

Polystyrene nanodroplets on rubber elastic substrates

In this chapter we analyse the complete shape of PS nanodroplets in equilibrium on rubber elastic substrates. Even though nanodroplets in equilibrium is the last stage of dewetting, we start our analysis from here, because it will give us information that is necessary in order to analyse and comprehend the more complicated phenomena involved in the dewetting dynamics, which we will discuss in Chapter 4.

3.1 From rigid to deformable substrates

When a liquid nanodroplet is deposited on a solid surface, gravity effects can be neglected and its shape can be approximated to a spherical cap. The equilibrium con-tact angle θ_e of the droplet with the substrate is a result of the balance between the solid/liquid, solid/vapour and liquid/vapour interfacial free energies. This balance is represented by Young’s equation [17]

cosθe= γ_SV −γ_SL

γ_LV (3.1)

Equation 3.1 was derived considering a perfectly smooth, homogeneous and rigid solid surface. If now the droplet is placed on an elastic surface, it causes a deformation, whose

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shape depends on the properties of the materials involved and their surface interactions.

Specifically, the sessile drop pulls upwards the surface at the three-phase contact line (TPCL), while it pushes downwards at its contact area due to the Laplace pressure inside the liquid (Figure 3.1).

Pericet-C`amara et al. [63] were the first ones to image experimentally in-situ the effect of capillary pressure on the deformation of an elastic surface caused by a sessile microdrop. Their experiments involved imaging by laser scanning confocal microscopy the deformation of a silicone elastomeric polymer below a microdrop of an ionic liquid, as well as the rim of the external profile of the deformation in the vicinity of the TPCL using a white-light confocal profilometer. The resolution of the techniques used, as the authors sustain, was not high enough for the specific system, such that no precise and detailed imaging could be achieved, but, even so, there was a satisfactory agreement between their experimental results and the theoretical curves extrapolated from the Linear Elasticity Theory for the specific system.

e

Figure 3.1: Sketch of a droplet on a viscoelastic substrate. The dotted line represents the solid without the deformation caused by the vertical component of the liquid surface tension, γ_LV ·sinθ_e. The system has not been sketched to scale, as the wetting ridge near the three-phase contact line of the solid (S)/liquid (L)/vapour (V) phases has been emphasized in order to allow for a better viewing.

In the following paragraphs, we analyse the shape of nanometre-sized polystyrene droplets generated by the complete dewetting of PS films on PDMS substrates of different elasticities, taking into account both the air/PS interface and the PS/PDMS interface.

Our interest is focused mainly on the shape of the deformation at the outer part of the droplet in the vicinity of the three-phase contact line, as well as on the shape of the deformed substrate below the droplet.