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 𝛿. One can say that MCT only predicts differences to hard spheres for charge-dominated systems with low enough screening.

Note that for all four different transition points in Table 2.1 there are no qualitative difference in the validity of the TTSP and the factorization law. Further more, both are equally valid for different paths across the transition point. Not only if the transition line is crossed by variation of the coupling parameter 𝛾, but also by varying the volume fractionΦor the screening parameter𝑘.

2.2.4 MCT power laws

With the correlators at hand, the next step is to find out the range of validity for the several MCT predicted power laws. For the𝛼-relaxation MCT predicts a law concerning the relaxation time𝜏_𝛼 ∝𝜖^−γ. Here𝜏_𝛼 is defined as the time where the correlator has decayed to0.01or the MSD has increased to10∕𝜎², both values that are far below/above the critical plateau values. The predicted power law or the𝛽-relaxation time below (“-”) and above (“+”) the transition can be written as𝜏_𝛽^±∝|𝜀|^−𝛿. In the supercooled regime 𝜏_𝛽⁻is defined as the time where the correlator/MSD curve crosses the critical plateau value𝑓_𝑐or(6𝑟^(𝑐)_𝑠 )², respectively. On the glassy side𝜏⁺

𝛽 is the time where the correlator/MSD curve reaches its own plateau value for𝑡→∞within a relative deviation of10⁻⁵.

In the glassy regime, another power law is predicted for the height of the plateaus𝜙_plat =𝜙(𝑡→∞). For the MSD this translates to a law for the squared localization length𝑟²_𝑠, due to its proportionality to the plateau value of the MSD. The combination of Equations1.50,1.52and1.54yields

(𝜙_plat−𝑓_𝑐)

∝√

𝜀 and (

(𝑟^(𝑐)_𝑠 )²−𝑟²_𝑠)

∝√

𝜀. (2.11)

This law is especially interesting for a verification in experiments, because it does not depend on the given system nor on the position of the transition point. MCT always predicts the same proportionality to√

𝜀. Furthermore, it can be useful to determine the actual transition point, which can be done by fitting Equation2.11with𝑓_𝑐/(𝑟^(𝑐)_𝑠 )² andΦ_𝑐/𝛾_𝑐/𝑘_𝑐 as fitting parameters, depending on which control parameter was varied for the transition.

In order to see whether the path across the transition point has an influence on the MCT-predicted validity of the power laws, correlators were computed by varying each of the three parameters𝜙,𝛾,𝑘separately.

All four qualitatively different transition points in Table2.1were investigated.

3They are not used in the following, here they are only presented for the sake of completeness.

2.2 Dynamics near the MCT glass transition

Figure 2.14: Relaxation times and difference of the plateau to the critical plateau for three different paths across the transition point (Φ_𝑐 = 0.25, 𝛾_𝑐 = 1191177, 𝑘_𝑐 = 10.0) varying separately either the coupling parameter𝛾(blue circles), the screening parameter𝑘(green diamonds) or the volume fraction Φ (red squares). Density correlators𝜙(𝑞𝜎 = 5.7, 𝑡, 𝜀)are used to retrieve the values as described in the text. Additionally, for the transition upon varying𝛾, values from the MSD curves are presented for comparison (cyan triangles). Orange lines correspond to power laws given by the exponent𝛿= 1.57for 𝜏_𝛽^±, the exponentγ= 2.39for𝜏_𝛼and the exponent0.5for the plateau difference.

Resulting relaxation times and plateau differences for the average screening case are shown in Figure2.14.

Up to a separation of𝜀 = 0.01, all power laws are well fulfilled independent of the varied control pa-rameter. For the variation of the screening parameter𝑘(green diamonds), a remarkable deviation from the𝜏^±

𝛽 and𝜏_𝛼 power laws is seen for a separations|𝜀_𝑘| > 0.01. Varying parametersΦand𝛾 the scope is somewhat larger, up to|𝜀_Φ,𝛾| ≃ 0.1. The power law for the plateau values (lower right panel) has a rather small range of validity. Far all three different control parameters it is only useful up to𝜀= 0.01. A further observation, illustrated in Figure2.14, is that the power laws are equally fulfilled if relaxation times and plateau differences are derived from MSD curves.

Another interesting question is whether the ranges of validity for the power laws are different at the four different transition points listed in Table2.1. Relaxation times𝜏_𝛼and plateau distances𝜙_plat−𝜙_𝑐 for all four transition points are compared in Figure2.15. Coupling parameter𝛾is varied for the three charged systems, volume fractionΦis varied for the hard sphere system. While the power law for the𝛼relaxation is valid up to|𝜖|≃ 0.1in the charged systems, one can see a significant deviation for hard spheres already for|𝜖| ≃ 0.05. For the power law of the plateau differences, deviations are seen earlier for both, hard spheres and the charged system with very low screening. Averagely screened and the moderately low screened systems show validity up to|𝜖|≃ 0.05, the other two only up to|𝜖|≃ 0.005.

−10⁻⁴

Figure 2.15: Structural𝛼-relaxation times and distances of glassy plateaus to the critical plateau for the four transition points in the four different systems mentioned in Table2.1. Varied parameter for the charged system is the coupling parameter𝛾, for the hard sphere system it is the volume fractionΦ. Lines are not fits, they represent the corresponding power laws with exponents𝛿given in Table2.1for the left panel and a square-root-law with exponent 0.5 for the right panel.

As a conclusion one can say that except from the somewhat different values of the exponents, MCT predicts no qualitative difference in the dynamics close to the ideal glass transition for charged hard sphere Yukawa (HSY) systems compared to the pure hard sphere system. Power laws, time temperature superposition law and factorization laws have similar ranges of validity as it is the case for the hard sphere system.

Im Dokument Glass transition in charged colloidal suspensions (Seite 47-52)