Ideal MCT Glass transition lines

Transition lines from ergodic to non-ergodic for several values of the volume fractionΦ are shown in Figure2.1. Once in the usual dimensionless units of the hard sphere diameter𝜎and once in the units of the mean interparticle distance𝑛^−1∕3, where the potential is given as:

𝛽 ̃𝑢(𝑥) =̃ ̃𝛾𝑒^{−̃𝑘 ̃𝑥}

̃

𝑥 =𝛽 𝑢(𝑥𝑛̃ ^−1∕3) =𝛾𝑛^1∕3 𝑒^{−𝑘 ̃𝑥𝑛−1∕3}

̃

𝑥 , so that ̃𝑘=𝑘𝑛^−1∕3 and̃𝛾=𝛾𝑛^1∕3 (2.1) with the particle density being𝑛= 6Φ∕𝜋in units of𝜎. From the shape of the potential one can already expect that the coupling parameter 𝛾 needs to grow as fast as𝑒^𝑘 to be large enough for the transition into the solid phase. Only in the limit of weak screening𝑘 → 0, where the1∕𝑟nature of the potential is dominant the slope of the transition line is smaller. Again as expected, for lower densities/volume fractions the transition line goes to higher𝛾 values. Rescaled by the interparticle distance the lines for lowΦ (0.05 and5 × 10⁻⁴) almost collapse, which indicates that they are in the dilute limit, where the hard sphere part of the potential is negligible. On the contrary, coming closer to the MCT glass transition for hard spheres atΦ = 0.516[19] the hard sphere interaction comes into play and lowers the transition line considerably.

A recent work by Yazdi et al. [28] computed the transition line in the dilute limit without considering hard sphere interactions. They used the HNC closure for the computation of structure factors and found

1Note that𝑓(𝑞)is equivalent to the correlation of the density fluctuations on infinite time scales𝑡→∞

2.1 Critical parameters and features at MCT transition lines

Figure 2.1: MCT ideal glass transition lines for different volume fractionsΦas indicated in the legend of the right panel. In the left panel coupling parameter𝛾and screening parameter𝑘are given in units of the hard sphere diameter𝜎(as in Equ.1.1), in the right panel they are scaled to the mean interparticle distance𝑛^−1∕3given by the particle density𝑛= 6Φ∕𝜋. In the rescaled view the curves ofΦ = 0.05and Φ = 5 × 10⁻⁴almost collapse, they are indicating the dilute limitΦ→0of the transition line.

that the glass transition line in units of the interparticle distance (̃𝛾, ̃𝑘) closely resembles the equation

̃𝛾_𝑐(̃𝑘) = ̃𝛾_OCP 𝑒^̃𝑘

1 +̃𝑘+̃𝑘²∕2, (2.2)

where𝛾_OCP is the limit of the so-called one component plasma (𝑘 = 0) where no screening microions are present. Yazdi et al. [28] derived Equ. 2.2 on the basis of the Lindemann criterion [54], which states that a crystal melts if the root mean squared displacement^{√⟨𝑟2⟩}is bigger than a certain fraction of the mean interparticle distance (usually about10%). This is done in the harmonic approximation

̃

𝑢^′′(̃𝑟= 1)⟨̃𝑟²⟩≃ const., with𝑢̃^′′(̃𝑟= 1)being the second derivative of the Yukawa potential at the mean interparticle distancẽ𝑟. With the assumption that⟨̃𝑟²⟩is constant along the transition line, Equation2.2 follows from 𝑢̃^′′(̃𝑟 = 1) = const. It is clear that this approximation is rather imprecise: Not only is the distance to the nearest neighbours assumed to be exactly ̃𝑟 = 1(the mean interparticle distance), interactions with particles further away are also neglected completely.

Taking results from simulations of dusty plasmas (also obeying the HSY potential) Vaulina et al. [55]

found that for ̃𝑘 ≲ 8 Equation2.2 agrees well with thecrystalmelting transition line. They found a value of̃𝛾_OCP = 106(or 172 in units of the Wigner-Seitz radius) for the one-component-plasma limit.

Therefore it is an intriguing result that the MCT glass transition line obtained by Yazdi et al. [28] has the same form, just with a3.45times larger value for the OCP limit: ̃𝛾_OCP = 366. However, the comparison in Figure2.2shows that this simple form does not fit to the results obtained here. Especially for low ̃𝑘, where Yazdi et al. [28] see a good agreement, the shape of Equ.2.2misses the data points. Even more, the OCP limit for this data is rather in the range of̃𝛾_OCP= 150 − 160, half the value obtained by [28]. A reason for this discrepancy could be that the HNC closure relation tends to underestimate the principal peak of 𝑆(𝑞) (see Figure 1.4), which results in an overestimation of the critical values𝛾_𝑐. The here obtained factor of about1.5compared to the crystal melting line seems to be more reasonable, especially due to the fact that MCT usually predicts transitions to be at lower densities or lower coupling strengths compared to simulations and experiments. For instance, the experimental hard sphere glass transition is found at the volume fractionΦ = 0.575[18] which is lower than the MCT predictionΦ = 0.516.

0 1 2 3 4 5 6 7 8

k

10² 10³

10^{4 =5×10}

4

=0.48

equ.2.2,=0.5 OCP=162 equ.2.3, OCP=151, b=0.9,c=0.5

10 ¹ 10⁰ 10¹ 10²

k

10⁰ 10¹ 10²

ex p( k )

Figure 2.2: Trying to fit Equations2.2and2.3to the MCT glass transition lines shown above in Fig-ure2.1. In the right panel all curves are multiplied byexp(−̃𝑘)to get rid of the exponential growth of the curves. Obviously the very simple form given in Equation2.2(black line) does not fit to the results, not even for small ̃𝑘. Curves for higher packing fractionsΦrun almost parallel to the low density line and show the same behaviour as the curve for the low density limit.

Another shape for the melting line in Yukawa systems was proposed in a recent work by Tolias et al. [56]:

̃𝛾_𝑐(̃𝑘) = ̃𝛾_OCP 𝑒^̃𝑘

1 +𝑏̃𝑘+ (𝑐∕2)̃𝑘² (2.3)

They found values of𝑏 = 1.2 and𝑐 = −0.43for the transition from crystal to liquid. For ̃𝑘 ≲ 8the data points obtained here are well represented by Equation 2.3with the choice 𝑏 = 0.9and𝑐 = 0.5. Interestingly, this is quite close to the intrinsic dependency of𝛾on𝑘, conveyed by the definition of the potential (cf. Equ.1.2), which is𝛾 = 𝑙_B∕𝜎𝑍_{ef f}² 𝑒^𝑘∕(1 +𝑘∕2)². This similarity is a hint that the effective charge𝑍_{ef f}(𝑘)at the glass transition line only has weak dependence on the screening parameter𝑘. We will see this also in the later discussions (cf. Section 2.3). Similar to the situation in crystal melting [55,56], also here the fit is only good for low screening̃𝑘 ≲8. For larger̃𝑘the right panel of Figure2.2 reveals a different behaviour which is clearly not explicable by a simple argument like the harmonic approximation. An explanation could be that in the high screening limit ̃𝑘 > 10 the HSY potential resembles more and more the hard sphere potential, so that eventually the growth of̃𝛾_𝑐(̃𝑘)is a bit slower than exponential.

An important question is to what extent the hard sphere part of the potential plays a role. This question is addressed in the following. In Figure2.3we see the dependence of𝛾_𝑐on the volume fractionΦ. For the left panel the screening parameter 𝑘 is held constant in units of the hard sphere diameter, which results in an initially very fast decay of the MCT transition line with increasing density. This is due to the dependency of𝛾on the particle density. For low screening𝑘= 3and in the rangeΦ = 0.3 − 0.45the curve is almost flat, in this range the particle density does not change very quickly. Finally the transition line drops drastically again, when it comes closer to the MCT hard sphere transition atΦ = 0.516. In the right panel of Figure 2.3the fixed parameter is ̃𝑘, which means that the screening length is always the same fraction of the interparticle distance. Therefore the particle density does not play a role. Here, the increase of the volume fraction can be interpreted as an increase of the hard sphere diameter of the particles while the density is kept constant. As expected the transition line stays almost flat until a volume fraction ofΦ = 0.45, where the drop occurs which is caused by the hard sphere interactions.

For a closer look at the shape of the transition lines close to the hard sphere transition limit all curves are presented again in Figure 2.4but now rescaled with a division by𝛾_𝑐(Φ = 0.515), the critical coupling parameter for the highest computed volume fraction Φ = 0.515. In rescaled units (right panel), this

2.1 Critical parameters and features at MCT transition lines

Figure 2.3: MCT ideal glass transition lines for different values of the screening parameter. In the left panel lines are shown for a constant parameter𝑘in units of the hard sphere diameter, in the right panel̃𝛾(Φ)is shown for constant̃𝑘in units of the mean interparticle distance𝑛^−1∕3. In rescaled units the dependency on the volume fraction only sets in when the hard sphere part of the potential is not negligible any more at aboutΦ = 0.45

Figure 2.4: The same transition lines as in Figure2.3above, but now the curves are rescaled with a division by the value for the highest volume fraction𝛾_𝑐(Φ = 0.515). Again, lines for constant𝑘(left panel), where increasingΦ means a density increase, are compared to the lines for constant ̃𝑘(right panel), where increasingΦmeans a diameter increase. Interestingly, there is a region where the critical coupling parameter̃𝛾_𝑐grows with increasing volume fractionΦ.

reveals a growing of the critical coupling parameter ̃𝛾_𝑐 with increasing volume fraction fromΦ = 0.3 to a maximum atΦ = 0.45. This maximum is higher for for stronger screening. More precisely, for

̃𝑘= 8.87it is7 %higher than the base line, while for ̃𝑘= 2.42it is only a3 %effect. It seems that the hard sphere interactions at a volume fraction of about𝜙 = 0.45have a softening effect which is more pronounced if the repulsive interactions are rather short ranged (i.e. for largẽ𝑘). Only aboveΦ = 0.47 the hard cores start to contribute to the stiffness of the system. In principle this means that there should be a reentrant glass-liquid-glass transition from liquid to glass and back to liquid induced by an increase of the volume fractionΦ. However, increasingΦat constant ̃𝛾and constant̃𝑘is equivalent to increasing the radius of the particles without changing the Yukawa part of the potential; experimentally this cannot be achieved. Moreover, it is not clear whether the effect is not a feature that comes in by the rescaled mean spherical approximation (RMSA) used for the computation of the structure factors. Still, it seems interesting to verify if this effect can be seen in simulations. Lai et al. [27,57] observed a similar reentrant transition but they used a differently modified RMSA version, their results are not comparable to the data

presented here. As the left panel of Figure2.4shows, an experimentally more realistic increase of the particle density, while keeping the screening length𝑘and the coupling parameter𝛾 constant, does not yield such a reentrant transition.