Critical non-ergodicity parameters and critical structure

In the computation of the transition lines the critical non-ergodicity parameter𝑓_𝑐(𝑞)(NEP) is a by-product because it is required to test whether a system is ergodic or non-ergodic. One can think of it as the minimum remaining correlation of the density fluctuations for𝑡→∞of any glassy system. According to MCT, any system in the glassy phase must show a larger correlation (𝑓(𝑞) > 𝑓_𝑐(𝑡)for all𝑞) and for any supercooled or liquid system the correlation vanishes for𝑡→∞.

In the left panel of Figure 2.5 one can see how𝑓_𝑐(𝑡) evolves for a dilute system upon increasing the screening parameter ̃𝑘together with ̃𝛾 while staying on the MCT transition line. Interestingly, towards higher screening the curves converge to the same curves that one obtains for the glass transition of hard spheres at a volume fractionΦ = 0.51585. Looking at the shape of the potentials at the same parameter values in Figure2.6, indeed one can see that even in the dilute case forΦ = 5 × 10⁻⁴ one obtains an effective hard sphere potential with radius𝑟_𝑚in the limit̃𝑘→∞. Curves for higher volume fractions are not shown here, they do not reveal any qualitative difference.

The here used MPB-RMSA closure and the PY (Percus-Yevick) closure yield the same results for the critical NEP of hard spheres. The latter closure is the de-facto standard for hard spheres. Both curves for Φ = 0.51585lie almost perfectly on top of each other (cf. black line and black dashed lines in Figure2.5).

This is good news, since otherwise a study of the transition by varying the hard core volume fractionΦ cannot be very accurate.

The most obvious feature of the evolution of the critical NEP upon increasing the screening parameter (which means decreasing the screening length) is a gradual shift in the low-𝑞 part. The correlation on length scales longer than the interparticle distance stays higher if the screening is stronger. Note that this is not explained by a decrease of the compressibility. Even contrary, the compressibility along the

0 5 10 15 20 25 30

Figure 2.5: Critical non-ergodicity parameters 𝑓_𝑐(𝑞)̃ along the transition line for a constant volume fractionΦin the dilute limit (left panel) and for a constant screening parameter ̃𝑘. The black line and the black dashed line, which lie almost perfectly on top of each other, give𝑓_𝑐in the limit of having no charges, once computed with the MPB-RMSA closure and once with the PY closure, commonly used for hard spheres. Length unit for𝑞̃is again the mean interparticle distance, which makes it easier to compare the curves at different volume fractions.

2.1 Critical parameters and features at MCT transition lines

Figure 2.6: Curves of the dimensionless potential𝑢(̃𝑟)̃ in units of the mean interparticle distance for the same parameters along the transition line as in Figure2.5. Left panel: For high screening lengths the potential converges to a hard sphere potential with the core diameter being the interparticle distance

̃𝑟= 1. Right panel: The hard core of the potential only play a role at very high volume fractions. Even forΦ = 0.515a quite strong repulsive part is needed for the transition to a solid.

transition line actually grows with increasing𝑘as seen from the limit𝑆(𝑞 → ∞)in Figure2.8. The explanation is that the coupling of particles at larger distances is stronger for weaker screening because the range of the pair potential𝑢(𝑟)is larger. This leads to a better propagation of short distance fluctuations to longer length scales. The result is a lower correlation of the density fluctuations for low𝑞, and this is what one observes in the low-𝑞limit of the curves in Figure2.5.

A gradual approach to the critical NEP for pure hard spheres with increasing̃𝑘can also be seen at shorter length scales (𝑞 ≳̃ 8). However, up to a value of ̃𝑘 = 10, there is not much change at all compared to the OCP limit (̃𝑘 = 0). Between ̃𝑘 = 0.1 and ̃𝑘 = 10 one can even see a decrease of 𝑓_𝑐(𝑞) on the right shoulder of the principal peak. The principal peak itself does not change much: A very small movement towards higher𝑞 between ̃𝑘 = 0.1and ̃𝑘 = 10 points to a decreasing distance to the next nearest neighbours with increased screening. Additionally there is a slight broadening of the principal peak together with small decrease of the peak height. This is a hint to a small reduction of the long range order and an enhancement of the short range order (cf. Fig.2.8). The peak height of the critical NEP is always very close to the one for the pure hard sphere transition. This again underlines the importance of the principal peak of the structure factor𝑆(𝑞)for the glass transition, that is located at the same𝑞value.

0 5 10 15 20

Figure 2.7: Structure factor𝑆(𝑞) and pair distribution function 𝑔(𝑟)for three qualitatively different transition points: Very low screening (OCP limit, red line), average screening (green line) and for full screening (only hard sphere interactions, blue line). The high principal peak of𝑆(𝑞)in the OCP limit corresponds to a low principal peak but slightly higher next order peaks in𝑔(𝑟).

3

Figure 2.8: Features of the structure factor𝑆(𝑞)and the pair distribution function𝑔(𝑟)along the tran-sition lines for constantΦ (left panel) and constant̃𝑘(right panel). Black circles and squares denote the values for the pure hard sphere transition. Coming closer to the hard sphere potential one can see a decrease of the long range order (decreasing peak of𝑆(𝑞)together with increasing peak with) and an increase of the local order (increasing peak of𝑔(𝑟)together with decreasing peak width). The isothermal compressibility, which is proportional to𝑆(𝑞 → 0)increases as the range of the repulsive interaction becomes smaller.

The right panel of Figure2.5shows the evolution of the critical NEPs upon increasing the volume fraction Φwhile keeping the screening parameter constant at a low value of̃𝑘= 2.42. The picture is very similar to the one for constantΦ. However only very close to the pure hard sphere transition line forΦ≳ 0.45 there is a notable change of the curve shapes. This is not surprising compared to the transition lines in Figures2.3and2.4. One can conclude that the hard sphere interaction only has an effect forΦ≳0.45. In Figure2.8we see the evolution of some of the features of𝑆(𝑞)and𝑔(𝑟)along the transition line as the potential approaches the limit of pure hard core interactions. A decrease of the principal peak of𝑆(𝑞) together with an increase of the peak width indicate that the system’s long range order is declining, while an increase of the principal peak of𝑔(𝑟)together with a decrease of the peak width show that the short range order is rising. To further illustrate this, Figure2.7also shows the full curves of𝑆(𝑞)and𝑔(𝑟)at three different characteristic transition points.

It is also interesting to note that the principal peak height𝑆(𝑞_max)at the transition is usually well above 2.85, which is the well-known Hansen-Verlet criterion [58] for crystallization in colloidal systems. This is also a hint that charged monodisperse systems have a tendency to crystallize already at considerably lower couplings𝛾than those of the MCT transition line computed here. Therefore one has to impede the system from crystallizing to obtain a sample in the glassy phase.

In comparison to the work of Yazdi et al. [28], the results obtained here are very similar. But since they use the HNC closure in the computation of the structure factors, their critical NEPs do not converge to the critical NEPs of the PY closure hard sphere transition. Instead, in the case of strong screening, they strongly overestimate𝑓_𝑐(𝑞)for low𝑞values. Since structure factors for hard spheres computed with the PY closure relation are used very often in the literature²this convergence with the use of the MPB-RMSA closure is indeed a great improvement.

2mainly because they are quite close to results from simulations of hard sphere systems [59]

2.1 Critical parameters and features at MCT transition lines 2.1.4 Critical localization lengths and critical exponent parameters

The ideal MCT glass transition line also implies a transition from delocalized particles in the liquid to localized particles in the ideal glass. One can define a localization length as the long time limit of the mean squared displacement (see also Figure1.5):

𝑡→∞lim⟨𝑟²(𝑡)⟩= 2𝑑𝑟²_𝑠, with space dimension𝑑 (2.4) 𝑟_𝑠 can be interpreted as the size of the cage, in which the particles are trapped in the glassy state. From Equation1.42one can derive a way to compute𝑟_𝑠via the incoherent NEP𝑓^(𝑠)(𝑞), which is the long-time limit of the tagged particle correlator𝜙_𝑠(𝑞, 𝑡)(also named self-intermediate scattering function𝐹_𝑠(𝑞, 𝑡)):

𝑟²_𝑠 = 2𝑑lim

Apart from this way, one can also do a Fourier transform to compute the self part of the van Hove function 𝐺_𝑠(𝑟, 𝑡→∞)from𝐹_𝑠(𝑞, 𝑡→ ∞)(cf. equs.1.12and1.14). Since𝐺_𝑠(𝑟, 𝑡)is the probability function for

In the present analysis the method via the van Hove function was found to be more numerically stable, because the very low 𝑞 limit of𝑓^(𝑠)(𝑞) is prone to numerical errors. The incoherent NEP 𝑓^(𝑠)(𝑞) is computed as the fixed point of an equation similar to the one for the coherent NEP𝑓(𝑞)(cf.1.45) with a memory function𝑀_𝑠{𝑓(𝑞), 𝑓^(𝑠)(𝑞)}. Again, Th. Voigtmann’s MCTSolver was used for the computations.

0 20 40 60 80 100

Figure 2.9: Critical localization length ̃𝑟^(𝑐)_𝑠 along the transition lines presented above in Figures2.1 and2.3. Left panel is for constant volume fractionΦ, right panel for constant screening parameter ̃𝑘. Black dashed line and black circle indicate ̃𝑟^(𝑐)_𝑠 for the pure hard sphere transition. All data is given in units of the mean interparticle distance𝑛^−1∕3. The results for𝑟^(𝑐)_𝑠 are very sensitive to the separation from the glass transition line. 5 outliers had to be removed because their system parameters were obviously not close enough to the transition. A relative error of about1 %has to be assumed for the values given here. The remaining noise on the curves illustrates the size of that error.

0 20 40 60 80 100

˜ k

0.71 0.72 0.73 0.74 0.75 0.76

cr iti ca l e xp on en t λ

^{Φ =5×10−4}

Φ =0.5

HS-PY

0.0 0.1 0.2 0.3 0.4 0.5

Φ

0.72 0.73 0.74 0.75

0.76 ˜k=2.42

˜k=4.03

˜k=6.45

˜k=8.87

HS-PY

Figure 2.10: Critical exponent parameter 𝜆 along the transition lines for which critical localization lengths are presented in Figure 2.9above. Black dashed line and black circle are again values for𝜆 at the MCT glass transition for pure hard spheres. The computation of𝜆is even more sensitive to the separation from the actual transition that it is the case for𝑟^(𝑐)_𝑠 . An error of 1-2%has to be assumed for the given values, which is obvious from the noise on the curves.

Resulting localization lengths are shown in Figure2.9 for systems along the glass transition lines, that were presented in the sections before. Note that with about1 %the assumed relative error is relatively large. The results for𝑟^(𝑐)_𝑠 are very sensitive to the actual separation of the system from the transition line, which could only be approached within a relative error of10⁻⁴. Nevertheless, the evolution of the critical localization lengths coming closer to the transition for pure hard spheres via ̃𝑘→ ∞or viaΦ →0.516 approaches nicely the values obtained for hard spheres within the PY closure (𝑟^(𝑐)_𝑠 = 0.75[49]).

Compared to pure hard spheres, the softness of the repulsive potential in general allows for the particles to be somewhat less localized in the glassy state. Effectively, the cages where particles are trapped are slightly bigger in charged systems. Similar as in the results before, hard sphere interactions only play a big role on forΦ≳0.45, they significantly reduce the critical localization length. For very low screening 𝑘 ≲ 5 one can see another decrease of ̃𝑟^(𝑐)_𝑠 , for Φ = 0.5even below the hard sphere value. A reason could be that now the long range Coulomb interactions come into play in a way that not only the nearest neighbours but also particles further away contribute to the stability of the cages [28]. For comparison;

Yazdi et al. [28] obtained qualitatively similar results, but they did not consider hard sphere interactions and therefore did not compute the dependence of𝑟^(𝑐)_𝑠 onΦ.

With a somewhat larger relative error of about 1-2%it was also possible to determine the MCT exponent parameter𝜆. From the knowledge of𝜆, the exponents for all power law predictions and asymptotic laws within MCT can be calculated (cf. section1.3.3). The computation of𝜆itself is via derivatives of the memory function evaluated for the critical non-ergodicity parameter𝑓_𝑐(𝑞)(more precisely, the so-called stability matrix, see [12,19,49]). Therefore it is even more sensitive to the separation from the transition point than the critical localization length. In Figure2.10one can see similar trends for𝜆as for𝑟^(𝑐)_𝑠 : As the potential approaches the hard sphere limit, also𝜆approaches its hard sphere glass transition value of𝜆 = 0.735(cf. [19]). Interestingly, for̃𝑘 ∼ 5the curves𝜆(̃𝑘)cross the hard sphere value. Therefore, at this point, a charged system should behave very similar to a hard sphere system. The corresponding screening length ̃𝑘⁻¹ of about 1∕5of the mean interparticle distance is very common in experimental charged colloidal systems. This implies that experimental investigations of the glass transition in charged colloidal suspensions might not see big differences to the behaviour of hard spheres.

2.2 Dynamics near the MCT glass transition

2.2 Dynamics near the MCT glass transition

2.2.1 Computation of density correlators and mean squared displacments

Knowing the critical parameters(Φ_𝑐, 𝛾_𝑐, 𝑘_𝑐) for a transition point one can go on with an investigation of dynamic density correlators𝜙(𝑞, 𝑡). Of particular interest is the supercooled regime and the glassy regime, shortly below and above the transition. To this end one has to solve the full dynamic MCT Equation1.39for an overdamped system. Required input is again the structure factor𝑆(𝑞)and the particle density𝑛. But now, in order to give the correlators the appropriate time scaling, an additional input is necessary: The effective diffusion constant𝐷₀ that describes the short time Brownian motion. In order to stay general, dimensionless time units are used by setting𝐷₀∕𝜎² = 1. With this scaling, the MSD at time𝑡= 1for a freely diffusing particle becomes⟨𝑟²(𝑡 = 1)⟩= 2𝑑𝐷₀𝑡= 6𝜎², where𝜎is the hard core diameter.

After solving the MCT equation for the coherent correlators𝜙(𝑞, 𝑡)one can go further with the compu-tation of the incoherent (or tagged particle) correlators and then use Equation1.42to obtain the mean squared displacement (MSD). Computations were done with Th. Voigtmann’s MCTSolver using the same 𝑞grid as for the computation of the transition lines. A special scheme is used for the time domain to allow a coverage of the dynamics for many orders of magnitude in time. Due to the integro-differential nature of the MCT equation ( a consequence of the long-lasting memory in glasses), solving has to be done iteratively. The knowledge of previous times is required to compute future times. Computations are done block-wise for 256 equally-spaced𝑡values in each block. All blocks start at𝑡= 0, but the time-spacing is increased for each new block. Starting with very small time stepsΔ𝑡= 10⁻⁶in the first block,Δ𝑡is doubled in each of the following blocks. Like this, any block always contains 128 times where𝜙(𝑞, 𝑡)is known (from the computation of the previous block) and 128 new times for which new values are numeri-cally computed. Of course there is no previous block for the first block, therefore the first 128 times of the first block are computed by usage of the short time expansion for𝜙(𝑞, 𝑡), which is given in Equation1.25.

To get somewhat more reliable results, the accuracy of M. Heinen’s MPB-RMSA structure factor solver was pushed to a relative error of10⁻⁵by choosing transition points where the solver is numerically more stable (difficulties occur mainly in systems with very low screening).

2.2.2 Density correlators and mean squared displacements near the glass transition In the following relative separations𝜀_𝑥from the critical parameters at the transition point are given by (see also Equ.1.46):

𝜀_Φ = Φ − Φ_𝑐

Φ_𝑐 or 𝜀_𝛾 = 𝛾−𝛾_𝑐

𝛾_𝑐 or 𝜀_𝑘 = 𝑘_𝑐−𝑘

𝑘_𝑐 (2.7)

For the results presented here, two of the three parameters are kept constant and the separation from the transition point is obtained by the changing the other one. In Figure2.11we see a representative set of correlators and MSD curves near a transition point upon a change of the coupling parameter𝛾. The qualitative picture is the same as for the transition in a pure hard sphere system where the volume fractionΦ is the varying parameter. Coming closer to the critical point from the liquid side, both, the correlator and the MSD develop a plateau that is left at later and later times the smaller the separation from the transition point. On the glassy side the correlation does not decay to zero any more. The plateau is reached at earlier and earlier times the deeper the system is in the glassy phase. Consequently, the plateau height is increasing for the correlator and decreasing for the MSD curves. In the physical

4 2 0 2 4 6 8 10

log

₁₀

(

^D_σ2⁰

t ⁾

0.0 0.2 0.4 0.6 0.8 1.0

φ ( qσ = 5 . 7 ,t )

-1 -3 -6 -9 -11

+11 +8 +6

4 2 0 2 4 6 8 10

log

₁₀

(

^D_σ2⁰

t ⁾

10

^-3

10

^-2

10

^-1

­

r

2

( t )

®

/σ

2

-1 -3 -6 -9 -11

+11 +8 +6

Figure 2.11: Density correlators (left panel) and MSDs (right panel) upon crossing the transition point (Φ_𝑐 = 0.25, 𝛾_𝑐 = 1191177, 𝑘_𝑐 = 10.0) of an averagely screened, charged system by changing the cou-pling parameter𝛾. Scattering vector𝑞is chosen close to, but somewhat to the right of, the principal peak of𝑆(𝑞). Numbers𝑛indicate the separation𝜀_𝛾 from the transition point𝜀_𝛾 = sign(𝑛)10^{−|𝑛|∕3}. Negative 𝑛refers to liquid side and positive𝑛to glassy side. Bold dashed line is the critical correlator for𝜀_𝛾 = 0.

picture of particles trapped in cages, one can say that a particle needs more and more time to escape the cage as the system is supercooled. After crossing the transition, the deeper the system is in the glassy phase the smaller becomes the size of the cages.

2.2.3 Time temperature superposition principle and factorization law

One of the important MCT predictions is the time temperature superposition principle (TTSP, cf.1.3.3).

It states that correlator curves near the𝛼-relaxation (second step of the curves in Fig.2.11) lie on top of each other when rescaled by their relaxation time 𝜏_𝛼 (cf. Equ. 1.48). MCT goes even further and predicts a power law for the relaxation times𝜏_𝛼(𝜀) ∝𝜀^−γ, such that taken together one should be able to superimpose the curves by rescaling them as

𝜙(𝑡∕𝜏_𝛼, 𝜀) ∝ 𝜙(𝑡∕𝜀^−γ, 𝜀) = 𝜙(̃𝑡).̃ (2.8) Figure2.12shows nicely that both, TTSP and𝛼scaling law are valid up to a separation of about𝜀_𝛾 = 0.1 for the curves already presented above. As expected, the TTSP is constrained to times𝑡that correspond to the long time tail of correlators and MSD curves.

For times 𝑡near the edge of the step in the curves, the so called 𝛽-relaxation regime, MCT predicts a factorization law (cf. Equ.1.50), which can again be expressed as curves lying on top of each other. For density correlators one has to compute

(𝜙(𝑡∕𝜏_𝛽, 𝜀) −𝑓_𝑐)

∕√ 𝜀 ∝ (

𝜙(𝑡∕𝑒^−𝛿, 𝜀) −𝑓_𝑐)

∕√

𝜀 = 𝑔̂_±(̂𝑡), (2.9) where𝑓_𝑐 = 𝑓_𝑐(𝑞) is the critical NEP for the corresponding𝑞 value, which is simply the height of the plateau of the critical correlator (𝜀 = 0). The master function𝑔̂_±(̂𝑡)has no further dependence on the separation 𝜀, except that for ̂𝑡 ≳ 10⁻² (times after the first step) it discriminates between supercooled (“-”) and glassy (“+”) since there is no further decay in the ideal glass.

2.2 Dynamics near the MCT glass transition

Figure 2.12: Validity of the time temperature superposition principle shown by rescaling the curves in Figure2.11using𝑡→𝑡∕𝜀^−γwithγ= 2.39.

Figure 2.13:Validity of the factorization law, again for the correlators and MSD curves already presented in Figure2.11using Equation2.9and2.10. 𝛿= 1.57was used for the rescaling. Curves on the glassy side (positive numbers 𝑛) are shifted by -5 for rescaled correlators in the left panel and by +1.5 for rescaled MSD curves in the right panel. Note that forlog₁₀̃𝑡 <−2supercooled and glassy curves would superimpose, if they were not shifted.

For the MSD one has to use the critical localization length𝑟^(𝑐)_𝑠 to superimpose the curves, more precisely one uses the plateau value of the critical MSD curve(6𝑟^(𝑐)_𝑠 )²: Plots of the𝛽-relaxation master curves in Figure2.13nicely reveal the symmetry of the correlators below and above the glass transition and they show that the time range for the validity of the factorization law grows the closer the system is to the transition point. For𝜀 < 0.001the curves agree in a time range covering about five orders of magnitude.

type Φ_𝑐 𝛾_𝑐 𝑘_𝑐 𝜆 γ 𝛿 𝑎 𝑏

very low screening 0.05 356.485 0.2 0.772 2.66 1.70 0.294 0.522

low screening 0.25 950.091 3.0 0.753 2.55 1.65 0.303 0.553

average screening 0.25 1191177 10.0 0.719 2.39 1.57 0.319 0.608

hard sphere 0.51585 0 0 0.735 2.46 1.60 0.312 0.583

Table 2.1:Critical Parameters and MCT exponents for four qualitatively different transition points.

Exponentsγand𝛿have a direct relation to the exponent parameter𝜆via Equations1.58,1.57and1.52.

Table2.1shows𝜆,γand𝛿for a selection of three qualitatively different transition points. Additionally values for the exponents𝑎and𝑏describing the asymptotic curves in the𝛽-relaxation regime are exhib-ited.³ Systems with average screening have exponents that are very similar to those for the glass transition of hard spheres. Only for low and very low screening one obtains considerably larger values for𝜆,γand

Table2.1shows𝜆,γand𝛿for a selection of three qualitatively different transition points. Additionally values for the exponents𝑎and𝑏describing the asymptotic curves in the𝛽-relaxation regime are exhib-ited.³ Systems with average screening have exponents that are very similar to those for the glass transition of hard spheres. Only for low and very low screening one obtains considerably larger values for𝜆,γand

Im Dokument Glass transition in charged colloidal suspensions (Seite 42-0)