Instituto de Física Manuel Sandoval Vallarta
Universidad Autónoma de San luis Potosí

Dynamic Arrest and Non-equilibrium


  • The Self-consistent Generalized Langevin Equation (SCGLE) theory of colloid dynamics

  • Colloidal dispersions are important for a wide variety of practical and fundamental reasons. Physicists find in colloidal systems beautiful mesoscopic analogs of atomic systems, in which the N micron-sized colloidal particles play the role of the N Angström-sized atoms of an atomic liquid. Thus, in spite of the huge disparity in space- and time-scales, one expects that there may be a correspondence not only regarding structure and phase behavior but also in the fundamental processes underlying other relevant physical phenomena, such as the dynamics of the glass transition.

    The dynamics of liquids is exhibited by the relaxation of the fluctuations δ n( r,t) of the local concentration n(r,t) of colloidal particles around its bulk equilibrium value n, whose average decay is best observed in the the van Hove function G(|r-r'| ,t)≈ ⟨ δ n(r,t)δ n(r',0)⟩. This property contains, in principle, all the relevant dynamic information of the equilibrium liquid, and is determined by techniques such as digital video microscopy. Dynamic light scattering measures its Fourier transform F(k,t), referred to as the intermediate scattering function.

    Based on general and exact expressions [1] for F(k,t) and for its self-diffusion counterpart FS(k,t), derived with the generalized Langevin equation (GLE) formalism [2,3], and complemented by a number of intuitively motivated approximations [4], our group proposed in 2001 a self-consistent theory of colloid dynamics [5,6]. In the absence of hydrodynamic interactions, this scheme allows the calculation of F(k,t) and FS(k,t), given the effective interaction pair potential u(r) between colloidal particles, and the corresponding equilibrium static structure, represented by the radial distribution function g(r) or the static structure factor S(k). This theory, referred to as the Self-consistent Generalized Langevin Equation (SCGLE) theory, was tested through its comparison with Brownian dynamics simulation data in specific idealized model systems [6]. In 2006 it was extended to colloidal mixtures [7,8], and this extension was later employed to describe the dynamics of colloidal liquids in random porous media [10].


  • > The SCGLE theory of dynamic arrest

  • The fundamental understanding of dynamically arrested states of matter is one of the most fascinating topics of condensed matter physics. Among the various approaches to the understanding of the transition from an ergodic to a dynamically-arrested state, the mode coupling theory (MCT) of the idel glass transition [11] provides perhaps the most comprehensive and coherent picture. The self-consistent character of the SCGLE theory also introduces a non-linear dynamic feedback, leading to the prediction of dynamic arrest, similar to that exhibited by the MCT. We refer to the application of the SCGLE theory of colloid dynamics to the description of this phenomenology, as the SCGLE theory of dynamic arrest.

    In 2007 our group put forward the initial proposal of the SCGLE theory of dynamic arrest, illustrating its use with the application to suspensions of neutral and electrically-charged hard-spheres [12,13]. Due to the fact that the detailed short-time properties of F(k,t) are completely overwhelmed by the non-linear long-time dynamics associated with the approach to the dynamic arrest transition, this initial proposal was later simplified in Ref. [14] to eliminate the irrelevant short-time details. The resulting simplified SCGLE theory of dynamic arrest was then applied to study the effects of the softness of purely repulsive interactions on the dynamic arrest behavior [15]. Regarding the effects of short-ranged attractive interactions on the dynamic arrest scenario, this theory was shown to describe the glass-fluid-glass reentrance involving repuslive and attractive glasses, originally discovered by MCT [16].


  • Dynamic arrest in multicomponent systems

  • The SCGLE theory of the dynamics of colloidal mixtures [8,9] was applied in 2008 to the description of dynamic arrest phenomena in multicomponent systems [17]. One of the most relevant results of this application refers to the format adopted by equations for the long-time value F(k,t) and FS(k,t) (the so-called non-ergodicity parameters), which involve an equation for the localization length γα of the particles of each species. This leads to the unambiguous prediction of the dynamic state of the multicomponent system (fully ergodic if all the parameters γα diverge, fully arrested if all these parameters have a finite value, and dynamically hybrid when some of these parameters diverge and others are finite).

    The simplest application of these general results involve the simplest model mixture, i.e., the binary mixture of hard-spheres of diameters σ1 and σ2 and volume fractions φ1 and φ2. Besides satisfying rather obviously-expected limits and conditions, for sufficient size asymmetry (σ1/ σ2 ≤ 0.4 ) the SCGLE theory predicts the existence of a region of dynamically mixed states, in which the larger particles become arrested but the smaller spheres continue to diffuse. As shown in Ref. [18], the topology of the resulting dynamic arrest diagram is preserved even if the interaction between the smaller spheres vanish, and the hard-sphere mixture becomes the Asakura-Oozawa mixture, which provides a simple model of colloid-polymer mixtures. This led to an alternative view of the dynamic arrest phenomena in these important experimental systems.

    In a still more interesting application, the dynamic arrest diagram of a mixture of electrically charged hard spheres (the so-called primitive model of ionic solutions and molten salts) was established in Ref. [19]. The resulting dynamic arrest scenario turned out to includes arrested phases corresponding to nonconducting ionic glasses, partially arrested states that represent solid electrolytes (or superionic conductors), low-density colloidal Wigner glasses, and low-density electrostatic gels associated with arrested spinodal decomposition [19].


  • Non-equilibrium SCGLE Theory of Liquid Dynamics

  • When a liquid system is driven to the region of dynamically arrested states, equilibrium theories such as MCT and our SCGLE theory predict that some relevant relaxation times will diverge. This implies the impossibility for the system to reach full thermodynamic equilibrium, thus invalidating the application of these equilibrium theories to the interpretation of the experimental measurements performed on real systems driven to such region of dynamically arrested states. Thus, any attempt to understand these measurements from a fundamental perspective requires the development of a genuine non-equilibrium theory of liquid dynamics.

    With this aim, in 2010 our group proposed the non-equilibrium extension of the SCGLE theory of colloid dynamics (referred to as the NE-SCGLE theory). For this, a non-equilibrium extension of Onsager's canonical theory of thermal fluctuations was employed to derive a self-consistent scheme for the description of the statistical properties of the instantaneous local concentration profile n(r,t) of a colloidal liquid in terms of the coupled time evolution equations of its mean value ñ(r,t) and of the covariance σ(r,r';t)≡ < δn(r,t)δ n (r',t) > of its fluctuations δ n(r,t) = n(r,t)-ñ(r,t). These two coarse-grained equations involve a local mobility function b(r,t) which, in its turn, is written in terms of the memory function of the two-time correlation function C(r,r',t,t') ≡ < δ n(r,t)δ n (r',t') >.

    For given effective interactions between colloidal particles and applied external fields, the resulting self-consistent theory is aimed at describing the evolution of a strongly correlated colloidal liquid from an initial state with arbitrary mean and covariance ñ0(r) and σ0(r,r') towards its equilibrium state characterized by the equilibrium local concentration profile ñeq(r) and equilibrium covariance σeq(r,r'). As explained in detail in Ref. \cite{nescgle1}, this theory also provides a general theoretical framework to describe irreversible processes associated with dynamic arrest transitions, such as aging, and the effects of spatial heterogeneities.


  • NE-SCGLE theory of aging

  • The non-stationary, slowly-evolving dynamics of deeply quenched fluids, referred to as aging, has been the subject of considerable attention over the last decade [22,23]. Concentrated emulsions, colloidal gels, and aqueous clay suspensions are some examples of aging systems. In spite of the apparent diversity of these structurally-disordered and out-of-equilibrium materials, the appearance of certain universal features in their non-equilibrium evolution suggests the existence of an underlying common source of the observed dynamic properties. Thus, the most immediate field of application of the non-equilibrium self-consistent generalized Langevin equation theory of colloid dynamics is, of course, the description of the non-stationary aging processes occurring in a suddenly quenched glass former.

    This is the most important ongoing project of our group. The first results in this direction were reported in Ref. [?], in which this nonequilibrium theory was applied to describe the irreversible evolution of the state of a model colloidal liquid with hard-sphere plus short-ranged attractive interactions, whose static structure factor and van Hove function evolve irreversibly from the initial conditions before the quench to a final, dynamically arrested state. The comparison of the numerical results of the theory with available simulation data turned out to be highly encouraging.


  • Equilibration of dense liquids

  • We report a systematic molecular dynamics study of the isochoric equilibration of hard-sphere fluids in their metastable regime close to the glass transition. The thermalization process starts with the system prepared in a non-equilibrium state with the desired final volume fraction φ for which we can obtain a well-defined non-equilibrium static structure factor S0(k;φ). The evolution of the α-relaxation time τα(k) and long-time self-diffusion coefficient DL as a function of the evolution time tw is then monitored for an array of volume fractions. For a given waiting time the plot of τα(k;φ,tw) as a function of φ exhibits two regimes corresponding to samples that have fully equilibrated within this waiting time (φ ≤ φ(c)(tw)), and to samples for which equilibration is not yet complete (φ≥ φ(c)(tw)). The crossover volume fraction φ(c)(tw) increases with tw but seems to saturate to a value φ(a) ≡ φ(c)(tw → ∞) ≈ 0.582. We also find that the waiting time tweqφ) required to equilibrate a system grows faster than the corresponding equilibrium relaxation time, tweq(φ) ≈ 0.27 \times[ταeq(k;φ)]1.43, and that both characteristic times increase strongly as φ approaches φ(a), thus suggesting that the measurement of equilibrium properties at and above φ(a) is experimentally impossible.


  • Density-Temperature-Softness Scaling of the Dynamics of Glass-forming Soft-sphere Liquids

  • We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density n, the temperature T, and the softness (characterized by a softness parameter ν-1) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard sphere fluid, constituted by the soft-sphere systems whose dynamic parameters, such as the α-relaxation time and the long-time self-diffusion coefficient, depend on n,t, and ν only through the reduced density n*≡ n σHS3(n,T,ν), where the effective hard-sphere diameter σHS(n,T,ν) is determined by the Andersen-Weeks-Chandler condition for soft-sphere-hard-sphere structural equivalence. A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short range interactions are found to be a direct manifestation of this general dynamic equivalence principle. The self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics is shown to accurately capture these scaling rules


  • Dynamic equivalence between atomic and colloidal liquids

  • We show that the kinetic-theoretical self-diffusion coefficient of an atomic fluid plays the same role as the short-time self-diffusion coefficient DS in a colloidal liquid, in the sense that the dynamic properties of the former, at times much longer than the mean free time, and properly scaled with DS, will be indistinguishable from those of a colloidal liquid with the same interaction potential. One important consequence of such dynamic equivalence is that the ratio DL/ DS of the long-time to the short-time self-diffusion coefficients must then be the same for both, an atomic and a colloidal system characterized by the same inter-particle interactions. This naturally extends to atomic fluids a well-known dynamic criterion for freezing of colloidal liquids [Phys. Rev. Lett. 70, 1557 (1993)]. We corroborate these predictions by comparing molecular and Brownian dynamics simulations on the hard-sphere system and on other soft-sphere model systems, representative of the "hard-sphere" dynamic universality class.



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