Dynamic glass transition in two dimensions

The question of the existence of a structural glass transition in two dimensions is studied using mode coupling theory (MCT). We determine the explicit $d$ dependence of the memory functional of mode coupling for one-component systems. Applied to two dimensions we solve the MCT equations numerically for monodisperse hard disks. A dynamic glass transition is found at a critical packing fraction ${\phi }_c^{d=2}\cong 0.697$ which is above ${\phi }_c^{d=3}\cong 0.516$ by about 35%. ${\phi }_c^d$ scales approximately with ${\phi }_{\mathrm{rcp}}^d$, the value for random close packing, at least for $d=2$, 3. Quantities characterizing the local, cooperative “cage motion” do not differ much for $d=2$ and $d=3$, and we, e.g., find the Lindemann criterion for the localization length at the glass transition. The final relaxation obeys the superposition principle, collapsing remarkably well onto a Kohlrausch law. The $d=2$ MCT results are in qualitative agreement with existing results from Monte Carlo and molecular dynamics simulations. The mean-squared displacements measured experimentally for a quasi-two-dimensional binary system of dipolar hard spheres can be described satisfactorily by MCT for monodisperse hard disks over four decades in time provided the experimental control parameter $\Gamma$ (which measures the strength of dipolar interactions) and the packing fraction $\phi$ are properly related to each other.

Figure 1

Comparison between theoretical MHNC structure factors and simulation for hard disks at packing fractions $\varphi =0.5(+)$, 0.628 (엯), and 0.68 $\left(\Delta \right)$; $q$ is given in units of $1∕\left(2R\right)$, the inverse diameter. Inset (a) concentrates on the vicinity of the main peak and gives additional comparison with Baus-Colot and PY theories for packing fraction $\varphi =0.628$. Inset (b) demonstrates how the MHNC theory correctly captures the asymptotic behavior for large $q$ values for packing fraction $\varphi =0.628$.

Figure 2

(Color online) Comparison of 2D MHNC structure factors (black/dark) to 3D PY (magenta/light). The packing fractions correspond to $\epsilon =0$ (solid lines), $\epsilon =-{10}^{-4∕3}$ (dotted lines), and $\epsilon =+{10}^{-4∕3}$ (dashed lines). The critical packing fractions are 0.697 for 2D and 0.516 for 3D.

Figure 3

Nonergodicity parameter of coherent (2D diamond, 3D dashed line) and incoherent (2D circles, 3D solid line) correlators at critical packing fraction ${\phi }_c^{2D}=0.697$ and ${\phi }_c^{3D}=0.516$. The $q$ values for $q<1$ are not included since they can not be determined accurately for numerical reasons. The $q=0$ results are from the analytic expansions, Eqs. (11, 12), and are included as triangles.

Figure 4

(a) Structure factor $S_q$ as function of wave vector $q$ for $\phi ={\phi }_c\approx 0.697$ (solid line), $\phi \approx 0.729$ (dotted line), and $\phi \approx 0.664$ (dashed line); the latter correspond to $\epsilon =\pm {10}^{-4∕3}$. The arrows mark the wave vectors $q_1=6.46$, $q_2=10.06$, and $q_3=18.26$. (b) Critical NEP $f_q^c$ (diamonds) and critical amplitude $h_q$ (squares). (c) The amplitudes $K_q$ (triangles) and ${\overline{K}}_q$ (circles).

Figure 5

Nonergodicity parameter $f_q$ for $q_1=6.46$, $q_2=10.06$, and $q_3=18.26$ (solid lines). The leading asymptotes (dashed line) describe $f_q-f_q^c$ within 10% up to the values marked by diamonds (0.005, 0.075, 0.0015). The next-to-leading asymptotes (dotted line) are within 10% for $\epsilon$ up to the values marked by circles (0.03, 0.033, 0.05). The values for the correction amplitudes are $C=2.08$, $\kappa =1.18$, $h_1=0.337$, $h_2=0.654$, $h_3=0.508$, ${\overline{K}}_1=-1.86$, ${\overline{K}}_2=-0.456$, and ${\overline{K}}_3=0.844$. The inset shows the separation parameter $\sigma$ as a function of $\epsilon$. The dashed line is the linear asymptote $\sigma =C\epsilon$. The range where the asymptote deviates less than 10% from $\sigma$ is between the diamonds that are at $\phi =0.676$ and $\phi =0.714$.

Figure 6

Coherent correlators ${\varphi }_1\left(t\right)$ (top) and ${\varphi }_2\left(t\right)$ (bottom) for different $\epsilon =\pm {10}^{-n∕3}$. Curves are labeled by $n$.

Figure 7

Timescales ${\tau }^{\pm }$ for the $\beta$ process and ${\tau }^{\prime }$ for the $\alpha$ process in the liquid for wave vector $q_1$ (in double-logarithmic presentation). The lines are the asymptotic power laws. $\lambda =0.7167$, $a=0.320$, $b=0.613$, $\delta =\frac{1}{2a}=1.56$, $c_{+}=0.044$, $c_{-}=0.009$, $\gamma =\frac{1}{2a}+\frac{1}{2b}=2.38$, and $c^{\prime }=0.0173$. (For the definitions of $c_{\pm }$ and $c^{\prime }$ see Ref. 22.)

Figure 8

Determination of $t_0$. $[{\varphi }_1\left(t\right)-f_1^c]t^a$ is plotted over $\log t$, for glass and liquid curves close to the transition. The position where the plateau is best reached by both curves is marked by arrows. The value found is 0.114. $t_0$ is then given by ${\left(\frac{0.114}{h_1}\right)}^{1∕a}=0.034$, with $h_1=0.337$ and $a=0.320$.

Figure 9

The critical laws. The leading asymptotes (dashed line) describe the solution within 10% up to the points marked by diamonds (41, 0.52) for $(q_1,q_2)$. The next-to-leading results (dotted) are within 10% up to the circles (0.73, 0.17) for $(q_1,q_2)$. Correction amplitudes are $\kappa \left(a\right)=-0.021$, $K_1=-1.007$, and $K_2=-0.120$.

Figure 10

The von Schweidler law. The leading asymptotes (dashed line) describe the solution within 10% up to the points marked by diamonds (0.29, 0.053) for $(q_1,q_2)$. The next-to-leading results (dotted line) are within 10% up to the circles (0.37, 0.47) for $(q_1,q_2)$ ($b=0.613$ and $t_{\sigma }^{\prime }=7.45\times {10}^9$). Correction amplitudes are $\kappa \left(b\right)=0.496$, $K_1=-1.007$, and $K_2=-0.120$.

Figure 11

(Color online) Functions ${\widehat{\varphi }}_q\left(t\right)=[{\varphi }_q\left(t\right)-f_q^c]∕\left(h_q\sqrt{C}\right)$ for $q_1=6.46$, $q_2=10.06$, and $q_3=18.26$ for $\epsilon =\pm {10}^{-n∕3}$ for 3 values of $n$ (solid lines). The scaling asymptotes $G\left(t\right)∕\sqrt{C}$ are shown as dashed lines. The region where the functions for different $q$ collapse onto a master function ($\beta$ region) increases with decreasing $\epsilon$. $C=2.08$; the curves for $\epsilon >0$ are shifted down by 1.

Figure 12

(Color online) Correlators ${\varphi }_q\left(\tilde{t}\right)$ for different $\epsilon =-{10}^{-n∕3}$, $n=4,6,8,10$, as function of rescaled time $\tilde{t}=t∕t_{\sigma }^{\prime }$ (solid lines). The thick solid line is the $\alpha$-master function ${\tilde{\varphi }}_q\left(\tilde{t}\right)$. The dotted lines are the short time parts of leading-plus-next-to-leading approximation for the $\alpha$ correlators ${\tilde{\varphi }}_q\left(\tilde{t}\right)+h_q{B}_1\sigma {\tilde{t}}^{-b}$ according to Eq. (22) with $B_1=0.5∕[\Gamma (1-b)\Gamma (1+b)-\lambda ]=0.374$. Light (magenta) curves give the Kohlrausch laws fitted to the ${\tilde{\varphi }}_q\left(\tilde{t}\right)$ in the range ${\log }_{10}\left(\tilde{t}\right)=[-3.86,2.14]$; the parameters are given in Fig. 13.

Figure 13

Kohlrausch fit parameters of a least-squares fit of Eq. (31) to the MCT $\alpha$-master functions ${\tilde{\varphi }}_q\left(\tilde{t}\right)$ on the logarithmic time axis in the interval ${\log }_{10}\left(\tilde{t}\right)=[-3.86,2.14]$. $A_q$ (circles) is quite close to $f_q^c$ (thick dashed). The Kohlrausch exponent ${\beta }_q$ (diamonds) converges to the von Schweidler exponent $b=0.613$ (thick line) for high $q$ values. The times ${\tilde{\tau }}_q$ are shown by a dotted line.

Figure 14

(Color online) Scaled mean-squared displacement $\Delta r^2\left(t\right)=\delta r^2\left(t\right)∕\left(2d\right)$ in units of ${\left(2R\right)}^2$ for disks in $d=2$ (black, solid line) and spheres in $d=3$ (magenta, dashed line) for different $\epsilon =\pm {10}^{-n∕3}$ as labeled. The thick lines are the critical curves.

Figure 15

Comparison of experimental data (dotted line) from [36] with theoretical curves (solid). Since the experimental system is a binary mixture three MSD curves are measured. Curve $b$ is measured considering only big particles, $s$ only small particles and $b+s$ takes all particles into account without discerning big or small. The top panel is at $\Gamma =592$, $\epsilon =-{10}^{-3}$, the bottom one at $\Gamma =474$, $\epsilon =-{10}^{-2}$. The data at $\Gamma =592$ can in the given region also be fitted by a critical curve and could thus correspond to a state in the glass. At $\Gamma =592$, the unit of time is fitted as ${\tau }^{\mathrm{BD}}=1∕470\mathrm{s}$, and the unit of length is fitted as $\left(\mathrm{420,700,500}\right)\mu {\mathrm{m}}^2$ for the big (curve $b$), the small (curve $s$), and all particles (curve $b+s$). This leads to a short time diffusion coefficient $D_0\approx 0.9e^{-13}{\mathrm{m}}^2∕\mathrm{s}$ and an effective diameter of $2R_{\mathrm{eff}}^{\text{big}}\approx 20.5\mu \mathrm{m}$ for the big particles. The lower inset shows the fitted separation parameter $\epsilon$ versus $\Gamma$. The upper inset shows a comparison of the (partial) structure factor of the big particles at $\Gamma =592$ with the one of the hard disk system at $\epsilon =0$; fortuitously, deviations only appear at larger $q$.