2.5 Phase Transition
2.5.4 Shear-Induced Phase Transition
lamellar phases under steady shear flow present different orientations. Shear force induces a hydrodynamic instability; a lamellar phase exposed under certain shear stress is of kinetically stable at its steady state, whereas is not of thermo-dynamically stable at rest. The lamellar orientations were drawn in the dynamic phase diagram (Figure 1.2) by Diat et. al, who developed the theoretical description of the shear-induced phase transition.
At very low shear rates and high surfactant concentrations, the membranes are mainly parallel to the flow with the smectic director parallel to the velocity gradient direction (region I in Figure 1.2).
In this state a lot of defects or dislocations would persist in the two directions perpendicular to the director,78 which is presumably similar to the thermotropic systems.
At higher shear rates or for more dilute systems, a new state appears where the smectic layers form multilayer spherical droplets of well-defined size, controlled by the shear rate, ranging typically from 10 μm to less than 1 μm (region II). At even higher shear rates, the membranes orient parallel to the flow with the smectic director parallel to the gradient of the velocity direction. This state has some resemble patterns with the first orientation but no defects remain in the direction of the flow (region III).
The bilayer structures are stabilized dominantly by the electrostatic interaction between charged membranes and the undulation introduced by Helflich.79 This undulation force seems to be destabilized by shear flow. At very low shear rate, the system flows by moving dislocations and is very viscous but Newtonian. For the intermediate shear rates, the sample is forced to move faster and the dislocation cannot follow anymore. The fact that the plate movements are faster than the rate of displacement of the dislocations creates a pressure perpendicular to the layer and the lamellar bilayers develop undulation instability. These dislocations forbid the system to flow and instead the system bifurcates to another orientation which consists of small spheres rolling on each other in order to allow the flow to proceed. Therefore the shear stress would be correlated with the sphere size. Diet et al. have observed a relation between the size R, of lamellar droplets in an onion phase and the applied shear rate,γ&:
v z
∇v
Figure 2.12. Orientation of lamellar sheets in shear flow. v and ∇v represent the directions of the flow and the velocity gradient, respectively.
R~1/√γ&.10,80 They calculated the characteristic size of the onions by balancing two forces: an elastic one, fel, and the viscous force fvis as follows81:
( )
The elastic force corresponds to one required to maintain a lamellar phase with a size R, and the viscous force is one experienced in a flow. d is the repeating distance of bilayer, η the viscosity, γ& the applied shear rate. According to the balance of the equations [2.21] and [2.22], the equilibrium size for the steady state can be calculated,
( )
This theoretical view agrees fairly with the empirical results.
van der Linden et. al, furthermore, addressed the volume fraction dependence on the onion size.82 They used another balance between the Laplace pressure and the local shear stress during shear. The difference in curvature of the droplet surface with respect to its surroundings would imply, during shear, a surface stress, equal to the Laplace pressure, 4σeff/R, where σeff refers to the effective surface tension of the droplet. The surface stress due to the curvature of the surface, created at a certain shear rateγ&, balances the shear stress, ηγ&. One then finds for the onion diameter
γ
Essentially σeff can be described by the product of the bending elasticity and compression modulus. For the case of non-charged or highly charge-screened system, the compression modulus B is given by83
( )
respectively. One thus can formulate the effective surface tension using Eq. [2.25] and a geometric relationship, dw = δ(1-φ)/φ.Hoffmann and Rehage, on the other hand, found out an analogous phase transition, L3→Lαh, under shear,84 and their group had experimentally explored it.85 Cates and Milner, furthermore, theoretically studied the L3→Lαh transition in the presence of shearing on the basis of a physical picture that the transition in a lyotropic system also belongs to the category of the fluctuation-induced first-order transition.86 They predicted the decrease in the first-order nature of the transition with an increase in the shear rate, based on the idea that symmetry-breaking shear field suppresses the origin of the first-order nature, i.e. large fluctuation effects associated with the degeneracy of the possible orientations of the ordered state. It is well-known that the sponge phase becomes birefringent once the shear field is exposed,87 and recently Yamamoto et. al showed experimentally the sponge-lamellar phase transition under shear.88 In the sponge phase, shear flow tears off selectively the passage of the
membrane that is not along the flow direction, if it is strong enough. And the sponge phase starts to have anisotropy with an increase in the shear rate. This may lead to the shortening of the characteristic length of the system.89 When the shear rate becomes sufficiently strong to erase all the passages, the characteristic length would be equal to that of the lamellar phase, and thus the shear-induced phase transition takes place. The sponge-lamellar phase transition should be first-order thermodynamically because of the symmetry difference between these phases, while the order of phase transition increases upon addition of shear flow. This could be due to the contribution of the flow mechanics instead of the thermodynamics.
Chapter 3 EXPERIMENT
3.1 INGREDIENTS