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Register a new userGlass and jamming transition of simple liquids: static and dynamic theory
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Hugo Jacquin
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We study the glass and jamming transition of finite-dimensional models of simple liquids: hard- spheres, harmonic spheres and more generally bounded pair potentials that modelize frictionless spheres in interaction. At finite temperature, we study their glassy dynamics via field-theoretic methods by resorting to a mapping towards an effective quantum mechanical evolution, and show that such an approach resolves several technical problems encountered with previous attempts. We then study the static, mean-field version of their glass transition via replica theory, and set up an expansion in terms of the corresponding static order parameter. Thanks to this expansion, we are able to make a direct and exact comparison between historical Mode-Coupling results and replica theory. Finally we study these models at zero temperature within the hypotheses of the random-first-order-transition theory, and are able to derive a quantitative mean-field theory of the jamming transition. The theoretic methods of field theory and liquid theory used in this work are presented in a mostly self-contained, and hopefully pedagogical, way. This manuscript is a corrected version of my PhD thesis, defended in June, 29th, under the advisorship of Frederic van Wijland, and also contains the result of collaborations with Ludovic Berthier and Francesco Zamponi. The PhD work was funded by a CFM-JP Aguilar grant, and conducted in the Laboratory MSC at Universite Denis Diderot--Paris 7, France.
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We develop a full microscopic replica field theory of the dynamical transition in glasses. By studying the soft modes that appear at the dynamical temperature we obtain an effective theory for the critical fluctuations. This analysis leads to several results: we give expressions for the mean field critical exponents, and we study analytically the critical behavior of a set of four-points correlation functions from which we can extract the dynamical correlation length. Finally, we can obtain a Ginzburg criterion that states the range of validity of our analysis. We compute all these quantities within the Hypernetted Chain Approximation (HNC) for the Gibbs free energy and we find results that are consistent with numerical simulations.
We show in numerical simulations that a system of two coupled replicas of a binary mixture of hard spheres undergoes a phase transition in equilibrium at a density slightly smaller than the glass transition density for an unreplicated system. This result is in agreement with the theories that predict that such a transition is a precursor of the standard ideal glass transition. The critical properties are compatible with those of an Ising system. The relations of this approach to the conventional approach based on configurational entropy are briefly discussed.
The complex behavior of confined fluids arising due to a competition between layering and local packing can be disentangled by considering quasi-confined liquids, where periodic boundary conditions along the confining direction restore translational invariance. This system provides a means to investigate the interplay of the relevant length scales of the confinement and the local order. We provide a mode-coupling theory of the glass transition (MCT) for quasi-confined liquids and elaborate an efficient method for the numerical implementation. The nonergodicity parameters in MCT are compared to computer-simulation results for a hard-sphere fluid. We evaluate the nonequilibrium-state diagram and investigate the collective intermediate scattering function. For both methods, nonmonotonic behavior depending on the confinement length is observed.
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This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k times k$ squares (E-problem) or a mixture of $k times k$ and $m times m$ ($m leqslant k$) squares (M-problem). The larger $k times k$ squares were assumed to be active (conductive) and the smaller $m times m$ squares were assumed to be blocked (non-conductive). For equal size $k times k$ squares (E-problem) the value of $p_j = 0.638 pm 0.001$ was obtained for the jamming concentration in the limit of $krightarrowinfty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j = 0.564 pm 0.002$. It was observed that the value of percolation threshold $p_{mathrm{c}}$ (i.e., the ratio of the area of active $k times k$ squares and the total area of $k times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k times k$ and $m times m$ squares (M-problem), the value of $p_{mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $kgeqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.
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