The Interplay between the Theories of Mode Coupling and of Percolation Transition in Attractive Colloidal Systems

Abstract

In the recent years a considerable effort has been devoted to foster the understanding of the basic mechanisms underlying the dynamical arrest that is involved in glass forming in supercooled liquids and in the sol-gel transition. The elucidation of the nature of such processes represents one of the most challenging unsolved problems in the field of material science. In this context, two important theories have contributed significantly to the interpretation of these phenomena: the Mode-Coupling theory (MCT) and the Percolation theory (PT). These theories are rooted on the two pillars of statistical physics, universality and scale laws, and their original formulations have been subsequently modified to account for the fundamental concepts of Energy Landscape (EL) and of the universality of the fragile to strong dynamical crossover (FSC). In this review, we discuss experimental and theoretical results, including Molecular Dynamics (MD) simulations, reported in the literature for colloidal and polymer systems displaying both glass and sol-gel transitions. Special focus is dedicated to the analysis of the interferences between these transitions and on the possible interplay between MCT and PT. By reviewing recent theoretical developments, we show that such interplay between sol-gel and glass transitions may be interpreted in terms of the extended F13 MCT model that describes these processes based on the presence of a glass-glass transition line terminating in an A3 cusp-like singularity (near which the logarithmic decay of the density correlator is observed). This transition line originates from the presence of two different amorphous structures, one generated by the inter-particle attraction and the other by the pure repulsion characteristic of hard spheres. We show here, combining literature results with some new results, that such a situation can be generated, and therefore experimentally studied, by considering colloidal-like particles interacting via a hard core plus an attractive square well potential. In the final part of this review, scaling laws associated both to MCT and PT are applied to describe, by means of these two theories, the specific viscoelastic properties of some systems.

Keywords: dynamical arrest; fragile-strong crossover; sol-gel transition; viscoelasticity.

Figure 1

Schematic representation of the potential-energy profiles for strong Arrhenius and fragile Super-Arrhenius liquids [17].

Figure 2

This figure illustrates the fragile-to-strong dynamical process by considering the viscosity and diffusion data of more than 80 glass forming liquids [21]. Figure adapted from Ref. [21], Copyright (2010) National Academy of Sciences.

Figure 3

The phase diagram correspond to the F12 ($v_1,v_2$-plane) and F13 ($v_1,v_3$-plane) MCT models. In the latter case, the presence of cusp-like singularity A${}_3$ and the re-entrant behavior attractive repulsive glass are highlighted.

Figure 4

The logarithmic decays observed the measured self Intermediate Scattering Function of an adhesive colloidal system [37].

Figure 5

The decays of the density correlators in a supercooled liquids (colloidal suspension of a dendrimer in methanol) close to the dynamical arrest. Highlighted in particular are the different MCT regions. The initial non-universal intra-cage motion is followed by a region characterized by a power law caging dynamics. After than, there is the onset of the slow inter-cage dynamics (described by the stretched exponential).

Figure 6

The von Schweidler universal plot measured in the previous colloidal PAMAM dendrimers suspensions.

Figure 7

The experimental phase diagram (Temperature–Volume fraction) of the Pluronic L64 in D${}_2$O characterized by a critical point (cloud point line, CP), the critical micellar concentration line (CMC), and a percolation line PT [35,37,48,49]. The inset reports an expanded region of the corresponding glass phase showing the attractive-glass–repulsive-glass transition line and the cusp-like singularity A${}_3$.

Figure 8

The phase diagram corresponding to a square well system described in the text in the plane $T^{*}$-$\phi$. Reported are curves for some different fractional attractive width $\Theta$. Can be observed here are the glass lines’ re-entrance and that the glass–glass line appears for $\Theta <0.04$, and increases as $\Theta$ decreases. A${}_4$ identifies the place where the two characteristic lines (attractive and repulsive) meet continuously, while A${}_3$ is the end point of the attractive line. It is shown that this last line tends, at the low $\phi$ values, with continuity towards the PT line [34]. Copyright 2000 by the American Physical Society.

Figure 9

The specific heat data, Cp(J/gK) versus T, in the temperature interval 273–348 K, for our AHS system at the volume fraction $\phi$ = 0.535 (top panel), $\phi$ = 0.544 (central panel) and $\phi$ = 0.55 (bottom panel).

Figure 10

The scaling plots of the SANS intensities for $\phi =0.495$ measured in the PluronicL64-D${}_2$O, at different temperatures in the range 310–327.4 K.

Figure 11

The shear viscosity $\eta \left(T\right)$ measured at different volume franction reported in a log-log plot as a function of $|T-T_p|$.

Figure 12

The density correlators ${\varphi }_q\left(t\right)$ calculated at the peak position ($q=7.3$) of the static structure factor $S\left(q\right)$ for different T (top). The dashed curves deal with the ideal MCT and shows the bifurcation (in the long t limit above and below $T_c$). The system self-diffusion D versus $T_c/T$ (bottom). The solid curves deal with the extended MCT while the dashed one refers to $D^{id}$ from the ideal MCT. The dash-dotted curve represents $D^{hop}$ originated to the hopping processes [67]. Adapted with permission from Ref. [67]. Copyright IOP Publishing.

Figure 13

Fit of the propylene carbonate DS data (T = 157 K) by means of the extended schematic-MCT model incorporating KWW-like hopping rates (solid line). The hopping contributions stemming from the collective correlator $\varphi \left(t\right)$ and from the probe variable correlator ${\varphi }_A^s\left(t\right)$ are indicated separately (dashed and dash-dotted lines) [72]. Reprinted with permission from Domschke, M. et al. Phys. Rev. E 84, 627 031506 (2011). Copyright (2011) by the American Physical Society.

Figure 14

The data collapse in the $\beta$ regime $t\ll t_{\alpha }$ of the dynamical susceptibility, ${\chi }_4\left(t\right)$ (for the FA model on the Bethe lattice). In particular, the scaling relation are shown: $\gamma =1$, ${\tau }_{\beta }\sim {\epsilon }^{-1/2a}$. Straight lines show the power law behaviors in the early $t^{2a}$ and late $\beta$ regime ($t^{2b}$), with $a=0.29$ and $b=0.50$ [47]. Figure adapted under the terms of the Creative Commons CC BY license from Ref. [47].

Figure 15

A log-log representation of the self-diffusion D versus $\alpha -$relaxation time ${\tau }_{q^{*}}$ of the coherent ${\varphi }_q\left(t\right)$ at the $S\left(q\right)$ peak $q^{*}$ (solid curve). The arrow indicates $T_c$. The dotted line refers to the SER prediction ($D_s\sim {\left({\tau }_q^{*}\right)}^{-1}$), while the dashed (${\left({\tau }_q^{*}\right)}^{-0.73}$) and dash-dotted (${\left({\tau }_q^{*}\right)}^{-0.95}$) curves to fractional relations (see the text). Inset: D versus ${\tau }_{q^{*}}$ from the main panel (solid curve) is compared with the fractional relation (${\left({\tau }_q^{*}\right)}^{-0.85}$ ) (dashed curve) [25]. Adapted with permission from Ref. [67]. Copyright IOP Publishing.

Figure 16

Propylene carbonate data (DS and light spectroscopy) and the related fits obtained by means of memory kernel schematic model are shown in a log-log plot. Dashed lines are fits with the ideal MCT model, i.e., without hopping term for comparison. Solid lines represent instead the fits with the extended schematic-MCT model [72]. Reprinted with permission from Domschke, M. et al. Phys. Rev. E 84, 627 031506 (2011). Copyright (2011) by the American Physical Society.

Figure 17

The dynamic response function $\chi (q,t)$ evaluated for a Lennard–Jones system at peak position of $S\left(q\right)$ ($q=7.3$). As it can be observed, the peak height of $\chi (q,t)$, denoted as ${\chi }^{*}(q,t)$ increases as one approaches $T_c$, but below $T_c$ it decreases. The latter behavior is due to the dynamical crossover [121]. Adapted with permission from Ref. [68]. Copyright IOP Publishing.

Figure 18

The real and imaginary part of the shear moduli for the volume franction 0.49 and 0.52. The G′ and G″ in the temperature range $291<T<305$ K are reported as a function of the frequency range $0.0147<\omega <9.613$ s${}^{-1}$.

Figure 19

The Arrhenius plot of the shear viscosity L64/D${}_2$O at twelve different volume fractions as a function of $1000$/T. Starting from low T, $\eta$ increases steeply, first going through a PT, then to a liquid–glass transition, and finally for $\phi >0.53$ to a glass–glass transition. Note that, for $\phi =0.3$, only the percolation is present, denoted by crosses and straight line.

Figure 20

The power-law dependence of $t^{\prime }={\left({\omega }_{min}\right)}^{-1}$ and $G_{min}^{″}$ on $|T-T_c|$ for some different micellar volume fractions.

Figure 21

The left panel shows the $G^{{}^{″}}\left(\omega \right)$ data measured at different temperatures for $\phi =0.50$. From these data, the minima and their frequencies, on approaching $T_c$, are well evident. On the right side, the corresponding data fitting according to the MCT interpolation form (IF) are shown.

Figure 22

The rescaled loss moduli $G\left(\omega \right)/G_{min}$ versus $\omega /{\omega }_{min}$, for the volume fractions $0.505$ and $0.53$. The line is the interpolation MCT form for $\lambda =0.75$ and $\gamma =3.1$.

Figure 23

The $\omega$-power law dependence of both $G^{{}^{\prime }}\left(\omega \right)$ and $G^{{}^{″}}\left(\omega \right)$, near the the percolation threshold, for $\phi =0.355, 0.42, 0.46$ and $0.51$.

Figure 24

The normalized viscosity with the volume fraction as control parameter: $\eta \left(\phi \right)/\eta \left({\phi }_c\right)$ vs. ${\phi }_c/\phi$. Data for $T=305, 309, 311, 313, 315$, and 318 K are reported.

Figure 25

The elastic loss modulus $E^{″}$ measured in SBR-CNT samples at the phrCNT of 1 and 40 (for different temperatures from 233 to 303 K (Symbols)). Curves are the results of the fitting procedure according to the MCT.

Figure 26

The time–temperature superposition (TTS) of the data illustrated in Figure 25. The curves report data (fully symbols) obtained according to the described procedure. The lines (proposed by the empty symbols) are obtained by a background subtraction. Empty symbols lines and the superimposed straight lines have an ${\omega }^{-1/2}$ decay, which confirms the suggested generality of TTS.