Mode Coupling Theory (MCT) is a microscopic theory to describe the glass transi-tion and was rst introduced by Bengtzelius, Götze, and Sjölander in 1984 [9]. It makes numerous predictions that were largely successfully checked by experiments and simulations [52, 53]. The main success is that it predicts a dynamical bifurcation as a function of the thermodynamic control parameter: the long-time dynamics changes from ergodic to non-ergodic, while thermodynamic or structural quantities do not be-come singular. This describes the characteristic behavior of experimental glass formers:
The viscosity shows a drastic increase while structural or thermodynamic quantities are almost not aected (e.g static structure factors or isothermal compressibility). MCT therefore states that the glass transition is purely dynamical in nature, an important point where it diers from other interpretations of the glass transition, e.g. ideas where crystallization plays a key role for the dynamic arrest [16]. Although in MCT struc-tural quantities do not become singular at the glass transition, dynamics and structure are formally closely related. In fact, the static structure factor is the only quantity that enters the MCT equations that generate the dynamical density correlation func-tions. Thus, the information if a system is ergodic or non-ergodic is already uniquely contained in the static structure factor.
MCT has indeed some signicant predictive power for the physics of the glass transi-tion as discussed in [10] and [54]. Some of the successes are: MCT predicts distinct values like packing fractions, where dynamical arrest occurs, power law exponents of the α- andβ-decays and their direct relation, the divergence of the α-relaxation time, and non-ergodicity parameters fq. It also predicts the time-temperature superposition of the α-process: the correlators collapse onto a non exponential master function.
3In this denition ofΓit is not considered that the underlying structure might change when altering the magnetic eld, e.g. when the system is undergoing a phase transition.
A well known failure of MCT is the predicted value of the critical packing fractionφd=3c in a 3D hard sphere system: the experimental value is determined as φd=3c ≈ 0.58 [6]
whereas MCT predicts a value of φM CT ≈ 0.51 [55]. However, the predicted critical packing fraction sensitively depends on certain features of the structure factors used as the input for the MCT equations, especially on the height of the main peak. Dierent methods can be used to determine the input structure factors like simulation, Liquid integral equation theory, or even experiments. However, dierent approximate methods lead to dierent results of the critical packing fraction. Thus, the failure to predict the exact value of φd=3c may partially lie in the approximative methods used to determine the input structure factors.
2.2.1 Basic formalism
As the mathematical description of MCT is very extended and complex, it is not intended to give a sucient introduction here. For a deeper understanding see review articles and introductions [10, 26, 54, 56, 57] and the references therein. Here, only the basic equations, ideas and consequences are presented. Emphasis is put on the parts that concern the comparison with the experimental system at hand.
In the following the MCT equations for a one-component liquid are shown dening the collective particle correlator Φq(τ)of density uctuations with wave vector q. The elaborate description of this is given in [26, 58].
MCT provides equations of motion for the collective and the tagged particle correlators of density uctuations. These quantities are analogous to the intermediate scattering function or its self part respectively (see appendix B.7). For a system with Brownian dynamics the equation of motion is
γq˙Φq(τ) + Φq(τ) +
Z ∞
0 dτ0mq(τ −τ0) ˙Φq(τ0) = 0 (2.8) with the memory kernel mq(τ) and a characteristic microscopic timescale γq. These equations of motion are derived from the Langevin- or Smoluchowski-equation respec-tively, by projection onto the slowly varying variables of the system, i.e. the density uctuations [10, 54]. The memory kernel can be interpreted as a generalized friction coecient that describes the uctuating stresses of the system. If the memory kernel is set to zero, a simple dierential equation is left that has an single exponential de-cay as solution for Φq(τ), the solution of free diusion. Thus, the information about interaction of particles and retardation of the dynamics is contained in this integral.
To solve equation 2.8, in MCT it is assumed that the main contribution at long times are given by density pair uctuations. With this approximation the memory kernel is
then written as
mq(τ) =
Z ddk
(2π)dV(q,k,p)Φk(τ)Φp(τ) (2.9) where the vertices V(q,k,p) express the overlap of uctuating stresses with the pair density modes. They are determined by the equilibrium structure of the system, by
V(q,k,p) = n 2
SqSkSp
q4 [q·kck+q·kcp]2δ(q−k−p) (2.10) where Sq is the static structure factor and n the number of particles per unit d-dimensional volume. The direct correlation function cq is uniquely related to the pair correlation function via the Ornstein-Zernike equation
h(r) = c(r) +n
Z
ddr0c(r0)h(r−r0) (2.11)
where h(r) ≡ g(r) −1 [23]. As the pair correlation function g(r) is connected via Fourier transform to the static structure factor Sq, equation 2.10 is only dependent on the latter. Thus, the solution of Φq(τ) in equation 2.8 only depends on the static structure factor.
The vertices V(q,k,p) are nonzero only for values that fulll the relation q = k+p which expresses the assumption that only density pair uctuations contribute that are coupled like this. The name 'Mode Coupling Theory' originates from this approximation. Static three-point correlations do not explicitly enter the memory kernel of MCT.
Here, only the equation of motion for the collective density correlatorΦq(τ)was shown.
The equations of the self part Φ(s)q (τ) are almost analogous. There, the integral over the memory function is not only a functional of the self part Φ(s)q (τ) but additionally depends on the collective part Φq(τ). Furthermore, the mean square displacement h∆r(τ)2ican be calculated directly from the MCT equations in the small wave vector limit q →0.
Note, that the interaction potential does not explicitly enter the MCT equations, only indirectly via the static structure factor.
The only direct dependence of MCT on the dimensionality is found in the integration element of the memory kernel in equation (2.9) which is (2π)−dddk ∝ (2π)−dkd−1dk. Further dependence on dimensionality only enters indirectly via the input static structure factors.
2.2.2 MCT in 2D for hard discs
The system used by Bayer et al. in [26] consists of two-dimensional hard discs with packing fraction φ and will be used for comparison with the experimental system at hand in chapter 4.
The structure factors are calculated using the Ornstein-Zernike equation with a modied hypernetted chain approximation as closure relation. Unlike in real 2D monodiperse systems, crystallization is prohibited by the method itself. It is assumed that this is a reasonable model for polydisperse hard discs, at least on a qualitative level.
The MCT equations reveal a bifurcation in dynamics where the system has an ergodic-nonergidic transition at a critical packing fractionφd=2c ≈0.697. The density correlators do not decay to zero for all times analogous to a hard sphere system in 3D, where MCT predicts an arrest at φd=3c ≈0.51 [55]. The predicted packing fraction in 2D is higher but they only dier by 5% if both critical packing fractions are scaled with the corre-sponding fraction of random close packing (φd=2rcp ≈0.84, φd=3rcp ≈0.64) [26, 59, 60].
2.2.3 Von Schweidler law
As mentioned above, MCT predicts power law behavior for the long-timeα-relaxation before the system nally relaxes via a Kohlrausch decay Φq(τ) ∝ exp(−τ /τ0)β. The scale free decay of the correlation function
Φq(τ) =fq−aτb (2.12)
is known as the von Schweidler law, and the exponentbis the von Schweidler exponent being independent of temperature or packing fraction respectively (constant a > 0, non-ergodicity parameter fq). This decay behavior is a general observation in glass formers [4].
Figure 2.2 shows MCT data of the hard disc system introduced above [26]. The data is close to, but below the glass transition with ²= φcφ−φc =−10−2. To show the powerlaw decay, the data is displayed in the form of the normalized self-intermediate scattering function P(q, τ) in Gaussian approximation for q=π. This function is dened as
P(q, τ) = log10
where FS(q, τ) is the conventional self-intermediate scattering function, and fq = FS(q, τmax) is the plateau height with τmax being the time in the point of inection of the mean square displacement (for details see appendix B.7). The second part clearly follows a straight line revealing the von Schweidler-law of equation 2.12 with an exponent of b= 0.73.
Figure 2.2: Normalized intermediate scattering functions of a hard disc system cal-culated from MCT in 'Gaussian approximation' (see appendix B.7, raw data ob-tained from [58]). The packing fraction is close to, but below the glass transition (² = φcφ−φ
c = −10−2). The curve is divided in a fast and slow part at the time of maximum stretching marked by the discontinuity (dened by the time of inection in the mean square displacement). The fast part follows a stretched decay, whereas the slow part shows power law behavior before converging into an stretched exponential de-cay. A power law with exponent 0.73 is plotted as the dashed line in the slow part. This scale free relaxation behavior is known as the 'von Schweidler' law.