Rheology of Brownian Discs

In this thesis the properties of binary mixtures of hard discs, undergoing Brownian motion, have been studied. Two major cases have been considered: First the approach of the glass transition for a binary mixture along with the effect of changes in the composition within that mixture. For one selected mixture a detailed discussion of its transition followed. Second the rheological properties of this selected mixture were investigated by focusing on the distorted microstructure and the dynamical correlation functions of mainly tagged quantities. Finally time dependent shear flows were investigated. Performing simulations for different binary mixtures at the glass transition was motivated by recent results of MCT for two dimensions. With the simulations it was possible to confirm four mixing effects, predicted by MCT for these binary mixtures, by David Hajnal. It could be shown that for big radius ratios when we keep the total packing fraction constant, the increase in the concentration of the small particles leads to a melting of the glass, the plasticization. For small radius ratios the increase in the concentration of small particles at constant volume fraction leads to an even stronger glass. Both remaining effects, the increase of non-ergodicity parameters, and the slowing down of the relaxation towards the non-ergodicity parameters on increasing the small particles concentrations could be confirmed. These qualitative effects, especially the first two ones, are not only of theoretical interest, as industrial applications of so called plasticizers show. Going into detail for a selected mixture made it possible to determine the ideal MCT glass transition point. Along with the verification of the factorization close to the critical point, the α-scaling and even the increase of the plateaus according to a square root law could be validated. The simulation shows additional decay processes, leading to a decay of the correlation functions even above the ideal glass transition. This is not in accordance with MCT, however, the rise of the plateau values can be explained. This analysis closes a gap in the field of the glass transition as: First, a quantitative test for exactly the model system MCT uses was performed, and second to the knowledge of the author, no such analysis for a this two dimensional glass former has been performed before. With the characterization of the considered system the foundation for the rheological part was laid. That the simulation algorithm yields Brownian motion for the quiescent system was known before from the work of Erik Lange. This thesis goes one step further and shows, with the same arguments, that even for the sheared case the algorithm is still in accordance with short time expansions of the Smoluchowski equation for the shear modulus. The significance here is, that even on the two particle level used in the theoretical derivation, the simulation agrees in its short time asymptote with the theoretical findings. Having ensured that the simulation actually solves the Smoluchowski equation for Brownian motion under shear, and having characterized the glass transition for one special mixture in the preceding chapters, the rheological properties of exactly that mixture could be probed. In the framework of an extended MCT, MCT-ITT, which was developed by Matthias Fuchs and Michael Cates, the microscopic structure was investigated. The appearance of a yield stress, shear thinning and the distortion of the structure factor in the simulation can be qualitatively explained by MCT-ITT and the interplay of different relaxation time scales involved in the glass and in the liquid. Quantitatively we found that MCT-ITT overestimates the distortion of the structure and the resulting shear stresses and viscosities by about a factor of 10. However, it should be remarked that a simplified monodisperse MCT-ITT calculation was used, which could explain a part of the deviation. Future works should consider using a multi-component approach, as used for the analysis of the mixture effects in this thesis. Scrutinizing the tagged particle correlation functions, the interplay between purely structural relaxation and shear induced relaxation could be observed in detail in the simulation: For correlators in the liquid, the shear induced decay competes with the intrinsic decay, whereas in the glassy system the time scale is always set by the shear field. The relaxation in the shear melted glass thus follows a master function which could be found for very low shear rates in the simulation. The famous Taylor dispersion, expressing itself, for example, through a cubic growth in the mean squared displacement in the shear direction could be found in the simulations. The MCT-ITT prediction, that actually the shear and the gradient direction are connected via one single long time diffusion coefficient, could be confirmed. Furthermore the MCT-ITT scaling properties of these coefficients, stating that they scale linearly with the shear rate in the limit of vanishing shear, could be worked out with the simulation. Yet, the Taylor dispersion still leaves a riddle, as next-to-leading order asymptotes in MCT-ITT and the simulation seem to differ: A term proportional to the square of time couldn’t be found as proposed by MCT-ITT in the simulations, which propose a next-to-leading order dependence that is linear in time. Along with the discussion of the mean squared displacement, a connection between the superdiffusivity and the overshoot in the transient stress could be pointed out. Also the stress overshoot is connected to the anisotropy in the correlation functions for certain directions. Coherent correlators for the stationary state have been shown. In principle they show that simulating this quantities, although it requires a lot more numerical tricks and computational power, is possible. Calculating the transient correlators, which are the main quantity in MCT-ITT, will be a task for the near future, and will make further tests possible. Finally in the last chapter with a method, developed by Matthias Krüger, it was possible to close the gap between stationary and transient correlation functions. The simulation showed that for small waiting times the connection given between transient, stationary and equilibrium correlator is perfectly reproduced by Krüger’s relation. Furthermore, it could be shown, that even for large waiting times reasonable results can be achieved, although the shear modulus entered as a fitting parameter. Extending the simulation to oscillatory shear a connection between the yield stress of a glassy system and its non-linear reaction upon oscillatory shear for low frequencies could be inferred. The results support the theoretical results from a schematic MCT model invented by Joseph Brader. For time dependent stresses below the yield stress the system shows the elastic behavior of a solid state body. Above the yield stress the system shows the dissipative properties of a liquid accompanied by higher harmonics in the stress response. An upper bound for the stress can be estimated from the flow curves