Glass transition in binary mixtures

Theoretical calculations for binary mixtures were done with the intent to do comparisons to the diffusing wave spectroscopy (DWS) light scattering experiments performed with polystyrene (PS) particles in wa-ter (cf. Chapwa-ter8). Since one cannot derive detailed information about the structure of the samples from DWS measurements, the only way to do a comparison is to use structure factors from MC simulations for exemplary systems similar to the experimental ones. In experiments the path into the glassy phase usually goes via deionization: Salt concentration is reduced to a very low value𝑐₁₁<0.5µmol∕l so that the screening of charges becomes low enough to arrive deep in the glassy phase.

Similar to the samples in the DWS experiments, simulations were performed for particles with the diam-eters𝜎_𝐴 = 124nm and𝜎_𝐵 = 196nm suspended in water (𝜀 = 80). Moderate but realistic values were chosen for the effective charges:𝑍_{ef f,𝐴} = 380and𝑍_{ef f,𝐵} = 600. The ratio𝛿_𝑠 =𝑍_{ef f,𝐴}∕𝑍_{ef f,𝐵} = 0.633 of the effective charges is equal to the ratio of the diameters, a choice that follows Schöpe [62], who found the effective charge number of small PS particles (⪝500nm) to be proportional to the particle diameter.

Experimentally it is more convenient to keep the volume fractionΦof a suspension constant at a certain value, a later tuning is very hard to realize and requires a very good accuracy in the determination of the volume of removed or added solvent. By using two suspensions at the same volume fraction it is possible to mix them arbitrarily without changingΦ. Therefore, in th computations the volume fraction is held constant at a low value ofΦ = 5 %and only the added salt concentration𝑐₁₁is varied to find the MCT glass transition. As described above MC simulations were used for the computation of both, the partial structure factors𝑆_𝛼𝛽(𝑞)and the pair distribution functions𝑔_𝛼𝛽(𝑟)(see Equations1.64and1.65). No signs of crystallization were found with the common bond orientational order criteria (cf. Chapter5.4.1), the crystallinity was always below1 %, which is the same as for monodisperse systems deep in the liquid phase.

Within a relative separation of𝜀(𝑐₁₁) ∼ 1 %, MCT glass transition points were determined using the partial structure factors as input to the MCT equations. The same methods as described above for the monodisperse systems were used for the computations, except that now, in the binary case, there are three different components AA, AB, BB for the correlators, two different tagged correlators𝜙_𝐴and𝜙_𝐵and two MSD curves for A and B particles. In Table2.2critical parameters of those binary systems are shown

𝑥_𝑠 𝑐₁₁[µmol∕l] 𝛾_𝐴[µm] 𝛾_𝐵 [µm] 𝜅[1∕µm] 𝑥̄_𝑠 𝑟_𝑚[µm]

0.0^∗ 2.77 n.a. 21.35 10.04 0.0 0.43

0.2 4.93 11.83 22.55 11.14 0.06 0.41

0.3 4.63 11.84 22.59 11.17 0.10 0.39

0.4 4.63 11.90 22.83 11.37 0.14 0.38

0.5 4.35 11.93 22.96 11.49 0.20 0.37

0.6 4.05 11.99 22.18 11.67 0.28 0.35

0.7 3.92 12.10 22.62 12.02 0.37 0.34

0.8 3.22 12.18 22.92 12.27 0.50 0.32

Table 2.2:Parameters at the MCT glass transition for the simulated binary system of charged PS particles in water at different number fractions𝑥_𝑠of the small particles. The total volume fraction is alwaysΦ = 0.05, the particles have diameters𝜎_𝐴= 0.124µm and𝜎_𝐵= 0.196µm and effective charges𝑍_{ef f,𝐴}= 380 and𝑍_{ef f,𝐵}= 600. Listed are the additional salt concentration𝑐₁₁the partial coupling parameters𝛾_𝐴and 𝛾_𝐵, the screening paramter𝑘, the packing contribution𝑥̄_𝑠of the small particles and the mean interparticle distance𝑟_𝑚(including both particle species). The monodisperse system marked by∗is computed using MPB-RMSA structure factors, all other systems were simulated to obtain𝑆(𝑞).

together with the parameters for a monodisperse system of only the big B particles (computed using MPB-RMSA closure relation). Definitions of coupling parameters𝛾_𝐴,𝛾_𝐵 and screening parameter𝜅 in binary systems are given in Equations1.60and1.61. Another important parameter for binary systems is fraction of small particles. It can either be given as a number fraction𝑥_𝑠or a packing contribution𝑥̄_𝑠:

𝑥_𝑠= 𝑁_𝐴

𝑁_𝐴+𝑁_𝐵, 𝑥̄_𝑠= Φ_𝐴

Φ_𝐴+ Φ_𝐵 (2.15)

Here 𝑁_𝐴 and 𝑁_𝐵 are the number of small and big particles, Φ_𝐴 and Φ_𝐵 the corresponding volume fractions of the total volume.

MC simulations also allow the computation of a mean squared displacement using the number of MC cycles as the lag time. In Figure2.21we see MC MSDs of the small particles in the mixture with50 %

10² 10³ 10⁴

t [MC cycles]

10^-2 10^-1 10⁰ 10¹

­

r

2 A

( t )

®

/r

2 m

10.07.0 4.32.5

c

11

[µmol/l]

1.2

Figure 2.21: Monte-Carlo-MSDs of the small particles in the binary mixture with𝑥_𝑠 = 0.5(see Ta-ble2.2). Lengths are normalized by the mean interparticle distance𝑟_𝑚. Note that the maximum dis-placement𝛿_𝑟,maxin the simulations is adapted to get an acceptance rate of the MC moves of∼ 20 %. It therefore varies from0.55µm for𝑐₁₁= 10.0µmol∕l down to0.43µm for𝑐₁₁= 1.2µmol∕l. However, the variation of𝛿_𝑟,maxcannot explain the slowing down by more than one order of magnitude. The slowing down is clearly a sign of glassy dynamics.

2.4 Binary systems

Figure 2.22: Critical coupling parameters for the binary systems presented in Table2.2once in units of µm plotted against the fraction𝑥_𝑠 of the small particles (left panel) and once in units of the mean interparticle distance𝑟_𝑚plotted against the packing contribution𝑥̄_𝑠of the small particles.

small particles. Although there is no bifurcation yielding a partition into liquid and glassy systems a dramatic slowing down of the dynamics is clearly visible. While MCT predicts the transition at𝑐11 = 4.3µmol∕l, the MC simulations still see a supercooled system. This is not surprising as it is expected that the theory usually predicts the transition to occur earlier, i.e. for parameters that still correspond to liquid systems in experiments and simulations.

Due to the difference in the particle diameters of the two species, a variation of the number fraction of small A particles from𝑥_𝑠= 0.0to𝑥_𝑠 = 0.8corresponds to a variation of their packing contribution from

̄

𝑥_𝑠 = 0.0only up to𝑥̄_𝑠 = 0.5. The size difference is also responsible for an increase of the total particle density𝑛= (𝑁_𝐴+𝑁_𝐵)∕𝑉, as it is the volume fraction that is kept constant. The density increase reduces the mean interparticle distance𝑟_𝑚 from0.43µm in the monodisperse case (only B particles,𝑥_𝑠 = 0.0) down to0.32µm for the case with many small A particles𝑥_𝑠 = 0.8. With particles coming closer for increasing𝑥_𝑠 the potential at the glass transition is allowed to become softer. That’s why the critical screening parameter𝜅 increases slowly, as can be seen in the left panel of Figure 2.22(red circles).

Coupling parameters𝛾_𝐴and𝛾_𝐵 also show a small increase which is mainly due their dependence on𝜅 (𝛾 ∝𝑒^−𝜅). Since the density of counterions also increases together with the increasing total particle den-sity, the critical concentration𝑐₁₁of added salt has to be reduced from4.93µmol∕l down to3.22µmol∕l going from𝑥_𝑠= 0.2to𝑥_𝑠 = 0.8, in order to maintain the slow increase of the critical𝜅.

Right panel of Figure2.22shows the critical parameters in units of the mean interparticle distance 𝑟_𝑚 plotted against the packing contribution of the small particles𝑥_𝑠. Here it becomes clear that with an increasing fraction of small particles, the screening actually needs to be reduced to reach the glass tran-sition. This can explained not only by the smaller effective charge of the small particles leading to a smaller partial coupling parameter𝛾_𝐴, but also by the smaller value of the product𝜎_𝐴⋅𝜖of diameter and dielectric constant that effectively increases the screening if the particle volume fraction is kept constant (see section2.3.1).

Since structure factors for the monodisperse system were computed using MPB-RMSA the value𝑐₁₁ = 2.77µmol∕l is about50% lower than one would expect from continuing the series of values for the binary mixtures to the case𝑥_𝑠 = 0.0. This is not unexpected regarding the differences between MC and MPB-RMSA structure factors (see Fig.2.20). Looking at the parameters𝛾_𝐵 and𝜅the discrepancy is actually quite small: For𝜅there is only a difference about5 %between𝑥_𝑠 = 0.0(MPB-RMSA) and𝑥_𝑠 = 0.2, for 𝛾_𝐵 it is about10 %. Interestingly, at the volume fractionΦ = 5 %a monodisperse system of only small A particles (𝑥_𝑠= 1.0) has too many counterions for a transition, the system is not predicted to be glassy even for𝑐₁₁ = 0.

10 20 30 40 50 60

Figure 2.23:Critical partial structure factors and pair distribution functions corresponding to the systems introduced in Table2.2. Colors correspond to different fractions𝑥_𝑠of small particles, as indicated.

For the systems closest to the transition, partial structure factors and pair distribution functions as defined in Equations1.64and1.65are presented in Figure2.23. Especially in𝑔(𝑟)one can see that there is no actual change in the degree of the correlations when going to a higher fraction of small particles 𝑥_𝑠. Even the different partial pair distribution functions𝑔_𝐴𝐴,𝑔_𝐴𝐵and𝑔_𝐵𝐵are very similar, the only notable difference is that the principal peak is higher for the big particles which is likely due to the stronger repulsion. Going to higher𝑥_𝑠one can see the effect of the density increase. It shifts the principal peak of 𝑔_𝛼𝛽(𝑟)to lower𝑟with a very small increase of its height. For𝑆_𝛼𝛽(𝑞)this is seen as a shift of the principal peak to larger 𝑞. Since structure factor amplitudes of𝑆_𝐴𝐴 𝑆_𝐵𝐵 go linearly with density (cf. Equ.1.64) the density changes of small and big particles are reflected in the height of the peaks. The peak height of the mixed structure factor𝑆_𝐴𝐵 is proportional to𝑥_𝑠(𝑥_𝑠− 1)𝑛, therefore it slightly goes up and down during the increase of𝑥_𝑠from𝑥_𝑠 = 0.2to𝑥_𝑠= 0.8.

Similarities in the structure for different mixing ratios𝑥_𝑠 become even more evident when everything is rescaled to the mean interparticle distance 𝑟_𝑚. Figure2.24exhibits total structure factors and total pair distribution functions of the system in rescaled units. For these quantities all particles are included ignoring their type (cf. Equ.1.66). Now the structural curves lie almost perfectly on top of each other, even the monodisperse curve coming from MPB-RMSA is very similar. Only a closer look can reveal that there is a minimum in the height of the principal peak near𝑥_𝑠 = 0.6and𝑥_𝑠 = 0.7. This could be a hint that there is an optimum mixing ratio around𝑥_𝑠 ∼ 0.65, where it is somehow easier to reach the glassy phase (smaller screening lengths, i.e. larger𝜅s are allowed). For mixtures of hard spheres small effects of the mixing ratio on the critical volume fraction at the transition have been observed [63]: Depending on the size ratio of the two species Φ_𝑐 increases or decreases by a few percent. However, one cannot observe a minimum at𝑥_𝑠 ∼ 0.65in the critical parameters𝛾_𝐴,𝛾_𝐵,𝜅for this system (cf. Figure2.22), it is likely to be masked by the increase of the particle density and the counterion density.

Figure2.25presents the non-ergodicity parameters (NEPs) of the glassy systems closest to the transition.

Increasing the fraction of small particles partial NEPs 𝑓_𝐴𝐴 of small particles develop more and more distinct peaks, while the peaks of𝑓_𝐵𝐵 become more and more washed out. For the low fraction𝑥_𝑠 = 0.2 of small particles 𝑓_𝐴𝐴 is very close to the corresponding tagged particle NEP, which means that correlations of the small particles with themselves are very weak in this case. The same is valid for the particles in the limit𝑥_𝑠 = 0.8. Like the total structure factor, the total NEP (again computed ignoring the different particle species), shows no clear evolution upon change of 𝑥_𝑠. Even a close look at the

2.4 Binary systems

Figure 2.24: Total structure factor𝑆_tot =∑

𝛼,𝛽𝑆_𝛼𝛽 =𝑆_𝐴𝐴+ 2𝑆_𝐴𝐵+𝑆_𝐵𝐵 and total pair distribution function 𝑔_tot =∑

𝛼,𝛽𝑥_𝛼𝑥_𝛽𝑔_𝛼𝛽 =𝑥²_𝑠𝑔_𝐴𝐴+ 2𝑥_𝑠(1 −𝑥_𝑠)𝑔_𝐴𝐵+ (1 −𝑥_𝑠)²𝑔_𝐵𝐵 corresponding to the partial functions shown in Figure2.23, here in units of the mean interparticle distance𝑟_𝑚. Insets are a zoom in to the principal peaks.

Figure 2.25: Normalized non-ergodicity parameters (NEPs) in units of𝑟_𝑚for the systems close to the transition described in Table2.2. Partial NEPs are normalized as𝑓_𝛼𝛽 =𝐹_𝛼𝛽∕√

𝑆_𝛼𝛼𝑆_𝛽𝛽, the total NEP is given by𝑓_tot =∑

𝛼,𝛽𝐹_𝛼𝛽∕∑

𝛼,𝛽𝑆_𝛼𝛽= (𝐹_𝐴𝐴+ 2𝐹_𝐴𝐵+𝐹_𝐵𝐵)∕(𝑆_𝐴𝐴+ 2𝑆_𝐴𝐵+𝑆_𝐵𝐵).

principal peak shows now trend, deviations might rather come through the fact that the transition point only determined with a rather low accuracy of about1 %. The total NEP for the monodisperse system with only big particles fits very well to the curves for the binary system, except for low𝑞̃. However, deviations for𝑞 <̃ 5could well be a result of the finite size of the systems in MC simulations.

Localization lengths𝑟_𝑠close to the transition point can be derived from the plateau values of the MSDs.

Results for different 𝑥_𝑠 are shown as blue diamonds and green squares in Figure 2.26. The method also suffers from low accuracy for the critical parameters, therefore the error bars are quite large. Error estimation is based on the average distance of the plateau levels of the two glassy systems closest to the transition (Δ𝑐₁₁ = 0.01µmol∕l). As expected, the increase of the density with increasing𝑥_𝑠leads to a decrease of the critical localization length. When normalized to the mean interparticle distance𝑟_𝑚 this decrease almost disappears for the small particles while it remains notable for the big particles. A physical reason could be that due to their larger size the big particles “suffer” more from the density increase, so that their cages effectively become smaller compared to those of the small particles. Similar as in the

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 2.26: Localization lengths of small (A) and big particles (B) upon variation of the fraction𝑥_𝑠 of the small particles. Values are shown directly at the glass transition and also somewhat deeper in the glass for a constant added salt concentration of𝑐₁₁ = 3.0µmol∕l. In the right panel the localization lengths are normalized to the mean interparticle distance𝑟_𝑚and plotted against the packing contribution

̄

𝑥_𝑠of the small particles. Error bars are due to the error in the determination of the transition point (see text).

monodisperse systems, the localization length at the transition is about 10 % of the mean interparticle distance, which corresponds to the Lindemann criterion for the melting transition. Since a delocalization of the small particles would be enough for the system to melt the melting criterion is mainly a criterion for the small particles, the localization length for the big particles is considerably lower.

Figure2.26also shows what happens when the concentration of added salt𝑐₁₁ is kept constant for the different mixtures at a value where the system is in the glassy phase for all 7 different mixing fractions 𝑥_𝑠.⁵ Although there is a density increase with increasing𝑥_𝑠, cage sizes are only slightly affected. One can even see a minimum at 𝑥_𝑠 = 0.7. For higher𝑥_𝑠 it seems that the softening effect of the stronger screening (more counterions due to the higher density) overrules the effect of the higher density.

Like it was done before for monodisperse systems the finally presented investigation is on the validity of the power laws in this binary system. Results are plotted in Figure2.27. Relaxation times and plateau height differences are derived from the MSD curves for the big particles using the definitions described in section2.2.4. The uncertainties introduced by computing the structure factors from MC simulations manifest themselves in a relatively large scattering of the relaxation time values especially close to the transition. This is not surprising since in this regime the MSD is very sensitive to small errors in𝑆_𝛼𝛽(𝑞). Nevertheless the power laws give a good description for the evolution of the relaxation times close to 𝜖 = 0. For the plateau differences presented here (lower right panel in Fig.2.27) the agreement is worse.

But this is not unexpected. In the discussion of monodisperse systems, we have seen that deviations from the square-root law for the plateau differences become evident for𝜀 > 0.01. Most of the values presented for the here presented binary system are indeed for separations from the transition point larger than𝜀= 0.01.

Being still too far away from the transition point it is not possible to give a good estimate for the MCT exponent parameter 𝜆. However, since there is a certain agreement of the relaxation times with the exponents coming from the monodisperse case𝑥_𝑠 = 0.0(magenta lines in Figure2.27) one can suspect that the exponent parameter for the binary systems should also be around𝜆∼ 0.74. As seen in Figure2.10 this value corresponds to a system with a low screening screening parameter̃𝑘 ≲5.

5For𝑥_𝑠= 0.8the mixture is already quite close to the MCT glass transition point.

Im Dokument Glass transition in charged colloidal suspensions (Seite 59-65)