PhD Thesis Defence: Arkaprava Pal

Mesoscale Modelling of sheared lamellar mesophases.

Oil, water, and surfactants self-assemble into various phases like micellar, lamellar, hexagonal, and onion. These phases exhibit nonlinear, history dependent rheological behaviour due to their anisotropic structure. The lamellar phase or smectic lyotropic liquid crystalline medium, which is made up of alternating flat sheets of water and surfactant bilayers shearing past each other in an imposed flow. The viscosity of this is expected to be a few times that of water. In practical applications, the viscosity is two to three orders of magnitude larger than that of water. This is because the stacking of the bilayers is no perfect, and there is disorder and different kinds of defects like edge, screw, and focal conic defects.  Moreover, it is observed that the defect density and viscosity increase with system size. In a sheared system, these defects are created, annihilated, and modified, which again influences the rheology, resulting in a two-way coupling between structure and rheology. In addition, flow-induced instabilities also play a major role in rheology. Though the lamellar phase is ubiquitous in food and health care industries, where products such as hair conditioners and butter substitutes, are made of lamellar mesophases, there is still no fundamental systemic model to predict rheology variation with respect to material and system parameters. 

 Molecular and Brownian dynamics simulations are computationally infeasible to capture the macroscopic rheology of tens of layers. So, a mesoscopic continuum model is developed in which only relevant parameters can be determined from molecular dynamics simulations. In this model, the evolution of the concentration order parameter, which is an indicator of the hydrophobic and hydrophilic phases, is studied. This is accomplished by simultaneous solution of the coupled advection-diffusion and momentum equations. These equations are modified versions of the model H equation where the chemical potential responsible for diffusion is determined from augmented Landau-Ginzberg free energy functional. This contains an additional term which is a minimum when there is a sinusoidal modulation of the concentration field of a particular wavelength (λ = 2 × π/k). The regions with positive concentration are hydrophobic regions, those with negative concentration are hydrophilic and surfaces of zero concentration are interfaces between hydrophobic and hydrophilic parts. There is two-way coupling between the concentration and momentum conservation equations due to the presence of the convective term in the concentration equation, and a reciprocal term in the momentum equation which results from the Poisson bracket relationship. The Lattice Boltzmann method is used to solve coupled concentration and momentum equations for different system sizes in three dimensions. 

 In this simulation study, the coupling between the structure and rheology of a sheared lamellar phase in a cubic simulation box. The system is initially prepared in a random state, the shear is initiated, and the evolution of the structure and rheology is examined. The dimensionless groups are the Schmidt number Sc which is the ratio of momentum and mass diffusion, the parameter Σ which is a dimensionless ratio of inertia and viscosity at the layer scale, the macroscopic Reynolds number Re and the Ericksen number Er which is the ratio of the viscous stress and the stress due to concentration variations. The structure-rheology relationship can be understood on the basis of the relative magnitude of coarsening time (λ² /D), the time for momentum diffusion across the sample (L² /ν), the inverse of strain rate (γ̇) and Ericksen Number (Er) which is the ratio of viscous and layer bending modulus. Here, D is the diffusion coefficient, ν is the kinematic viscosity and L is the system size. At low Er, when diffusion time is smaller compared to the inverse of strain rate, there is the formation of local misaligned layer-like domains, which are stretched and rotated by imposed shear. There is near-perfect ordering with titled layers and isolated defects; this is in contrast to the evolution in two dimensions where there is a persistent disorder. The tilt and defects cause high excess viscosity and second normal stress difference due to substantial layer stiffness. At high Er, the formation of layers is disrupted by shear. Cylindrical structures aligned along the flow are formed after tens of strain units. These cylinders evolve into layers at long time because of diffusion perpendicular to flow. The system does not reach a perfectly aligned state even after hundreds of strains as defects persist at steady state. The excess viscosity shows an order of magnitude decrease as layer stiffness is small compared to layer stiffness. At intermediate Er, evolution shows a strong dependence on ScΣ. At low ScΣ, there is perfect ordering at a long time where, but at high ScΣ, the creation of highly deformed structure perpendicular to flow direction at high ScΣ, which shows no sign of relaxation with shear.  However, this configuration does not have very high viscosity as defects are perpendicular to the flow plane. Another significant finding is that the alignment direction is a stochastic process between parallel (layer normal along velocity gradient) and perpendicular (layer normal along vorticity direction) at low Er, whereas the layers are always aligned in the perpendicular direction at high Er. The direction of alignment is correlated to the second normal stress difference, which is positive for parallel configuration and negative for perpendicular configuration. The first normal stress difference is always negative irrespective of the direction of alignment. 

 The effect of viscosity contrast between the hydrophobic and hydrophilic parts has also been studied three dimensions. It is found that at high Er, the viscosity contrast has no effect on the concentration evolution.  The formation of a cylindrical structures takes place when the concentration amplitude is low, and therefore this process is independent of the viscosity contrast. However, the cylindrical structures with high local viscosity cause effective viscosity to decrease as the low-viscosity matrix transmits most of the stress. For lower Er, the initial concentration evolution depends on diffusivity, but the later evolution is affected by viscosity contrast. The alignment is not perfect, and the excess viscosity increases when viscosity contrast is introduced at moderate Er at low ScΣ. The excess viscosity also increases for high ScΣ, though there is a decrease in the extent of deformation. The layer alignment shifts from perpendicular to parallel direction with a tilt at low Er and low ScΣ, which alters the second normal stress difference from negative to positive. However, the alignment is in the perpendicular direction for high ScΣ. The excess viscosity and second normal stress difference exceed the shear stress by an order of magnitude at intermediate time during shear. 

 Earlier studies on the structure-rheology relationship in two dimensions reveal that there is high shear localization near walls and no shearing at low Er. This is because the bulk consists of defect, and there is a plug with zero strain rate between two defects due to defect pinning. To better understand the defect dynamics, the shearing of a configuration with two edge defects of opposite signs at a finite cross-stream separation is studied in both compressional and extensional shear. The defect dynamics can be classified into four categories when defects are sheared in such a way that the portion between them is compressed. The regimes are finite separation of defects with a solid plug in between, defect cancellation, defect crossing to the extensional axis, and defect creation outside the pinned portion in order of increasing Er. The defects are created due to the buckling of highly deformed layers close to the original defect outside the pinned portion. A statistical steady state is attained due to creation and annihilation at the intermediate time, but at long time limit, a well-aligned system is formed, with cancellation the defects. However, there is finite defect density for larger system sizes. Though there is high defect density in this regime, the viscosity is relatively low because there is a shearing of the zones between the defects due to lower stiffness. The defect dynamics are qualitatively different when the defects are sheared so that there is an extension of the region between the defects. The steady-state separation regime in this case consists of the highly deformed layers, which break and reforms into well-aligned layers upon the slight increase of Er.  When two defects are sheared along the extensional direction, two or more additional defects of opposite signs are created in the plug region due to the buckling of layers. These pairs could either reach a steady state or cancel depending on the Er. This mechanism of cancellation is observed at high system sizes. The Ericksen number regimes and the time scale for defect evolution along the extensional axis are much lower than those along the compression axis. The approaching velocity of defects and apparent viscosity of the sample decreases faster in case of compression than extension at any ScΣ number in all regimes expect defect cancellation regime in extensional axis.