In this section we present an analysis of the experimental data for a dispersion of core-shell particles with a concentration of 13.01wt%. By considering the high frequency data points we obtain:
η∞ωRHD0
kBT ≈ 0.34. (4.6)
HereRH denotes the hydrodynamic radius andD0 the self-diusion coecient of the spheres in the dilute regime,kBdenotes the Boltzmann constant andT the temperature of the sample. We use this fact to optimize our tting procedure:
• We choose ²and x˜as described above.
• We choose vσ in such a way that we hit the high frequency data point forG0(ω).
• We choose Γ in such a way that equation (4.6) is satised. With this choice we hit the high frequency data point forG00(ω).
• We choose x= 100.
• We consider this choice as initial values and by independent varying of all parameters we obtain the best t by eye.
Following [18], we choose v2c = 2.00 for all ts. The gures 4.3 to 4.8 show the reduced shear stress σRkB3HT as a function of the Peclet numberP e0≡ γR˙D2H
0 and the corresponding reduced moduli
G0,G00R3H
kBT as a function of ωRD02H for dierent eective packing fractions φef f. The corresponding tted parameters are shown in table 4.1. The data points for the ow curves were obtained with decreasing shear rates. The data points where the relative dierence between the measurement with increasing shear rates and the measurement with decreasing shear rates dier more then 50% are shown with dierent symbols. This hysteresis becomes weaker for higher eective packing fractions and is an indication of crystallization in the sample at low shear rates. In our tting procedure we do not take account of these data points. Then we obtain qualitatively and quantitatively good ts for all ow curves. For low eective packing fractions (gure 4.3, gure 4.4 and gure 4.5) we obtain also a qualitatively and quantitatively good t for the moduli at suciently large frequencies. At low frequencies, because of the increasing elasticity and dissipation, the experimental data suggest crystallization in the sample. As shown in gure 4.6, gure 4.7 and gure 4.8, for high eective packing fractions we obtain qualitatively and quantitatively good ts for the storage moduli at suciently high frequencies. At high frequencies we also obtain a qualitatively but not quantitatively good t for the loss moduli. This could probably be explained by the fact that the colloidal particles do not behave like ideal hard spheres at high eective packing fractions, see gure 4.9. The fact that, compared to the theory, at low frequencies the experimental data show a higher dissipation and a lower elasticity remains unclear.
-6 -4 -2 0 log10(Pe0)
-3 -2 -1 0 1
log 10(σR3 H/k BT)
strong hysteresis
φeff=0.527
-4 -2 0 2
log10(ωR2H/D0) -4
-2 0 2
log 10(G’, G’’ R3 H/k BT)
G’’
G’
Figure 4.3: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.527. The red lines show the t with the extended F12( ˙γ) model. The corresponding tted parameters are shown in table 4.1. The data points for the ow curves were obtained with decreasing shear rates. The data points where the relative dierence between the measurement with increasing shear rates and the measurement with decreasing shear rates dier more then 50% are shown with dierent symbols. This hys-teresis and the increasing elasticity and dissipation at low frequencies indicate a crystallization in the sample. For suciently large shear rates respectively frequencies we obtain a good t.
-6 -4 -2 0 log10(Pe0)
-2 -1 0 1
log 10(σR3 H/k BT)
strong hysteresis
φeff=0.540
-4 -2 0 2
log10(ωR2H/D0) -1
0 1 2
log10(G’, G’’ R3 H/kBT)
G’’
G’
Figure 4.4: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.540. We obtain qualitatively the same behavior as explained for gure 4.3. For suciently large shear rates respectively frequencies we obtain a qualitatively and quantitatively good t.
-6 -4 -2 0 log10(Pe0)
-2 -1 0 1
log 10(σR3 H/k BT)
strong hysteresis
φeff=0.567
-4 -2 0 2
log10(ωR2H/D0) 0
1 2 3
log 10(G’, G’’ R3 H/k BT)
G’’
G’
Figure 4.5: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.567. We obtain qualitatively the same behavior as explained for gure 4.3. Compared to the two lowest eective packing fractions the hysteresis is weaker but the t-range for the loss modulus is smaller.
-6 -4 -2 0 log10(Pe0)
-1 -0.5 0 0.5 1
log 10(σR3 H/k BT)
φeff=0.580
-4 -2 0 2
log10(ωR2H/D0) 0
1 2
log 10(G’, G’’ R3 H/k BT)
G’’
G’
Figure 4.6: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.580. We obtain qualitatively and quantita-tively good ts for the storage modulus at suciently high frequencies. At high frequencies we also obtain a qualitatively but not quantitatively good t for the loss modulus. This could prob-ably be explained by the fact that the colloidal particles do not behave like ideal hard spheres at high eective packing fractions. The fact that, compared to the theory, at low frequencies the experimental data show a higher dissipation and a lower elasticity remains unclear.
-6 -4 -2 0 log10(Pe0)
-0.5 0 0.5 1
log 10(σR3 H/k BT)
φeff=0.608
-4 -2 0 2
log10(ωR2H/D0) 0
1 2 3
log 10(G’, G’’ R3 H/k BT)
G’’
G’
Figure 4.7: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.608. We obtain qualitatively the same behavior as explained for gure 4.6. Compared to the packing fraction of 0.580the t-range for the moduli is larger. The ow curve behaves glassy.
-6 -4 -2 0 log10(Pe0)
-0.5 0 0.5 1 1.5
log 10(σR3 H/k BT)
φeff=0.622
-4 -2 0 2
log10(ωR2H/D0) 0
1 2 3
log 10(G’, G’’ R3 H/k BT)
G’’
G’
Figure 4.8: The reduced shear stress as a function of the Peclet number and the corresponding reduced moduli for an eective packing fraction of 0.622. We obtain qualitatively the same behavior as explained for gure 4.7. The ow curve behaves glassy.
0.52 0.54 0.56 0.58 0.6 0.62 φeff
-0.04 -0.03 -0.02 -0.01 0
ε
Figure 4.9: The separation parameter shows a non-linear dependence on the eective packing fraction. This indicates that, especially at high packing fractions, the colloidal particles do not behave like ideal hard spheres.
φef f ² x˜ vkσR3H
BT
ΓR2H
D0 x η∞ωkRHD0
BT η∞γ˙ RHD0
kBT
0.527 −0.04 0.30 18 23 110 0.23 0.63 0.540 −0.022 0.35 25 33 100 0.27 0.64 0.567 −0.0025 0.40 35 43 110 0.33 0.73 0.580 −0.0002 0.40 50 60 100 0.33 0.75 0.608 0.0003 0.40 77 84 90 0.37 0.83 0.622 0.0006 0.40 90 88 90 0.41 0.92
Table 4.1: The tted parameters for the extended F12( ˙γ) model. For all ts we used v2c = 2.00. While x is varying only 10% around its mean value of 100and x˜ reaches a saturation value of 0.40, all other parameter vary monotonically with φef f.
Chapter 5
Conclusion
5.1 The β -relaxation process
The shape of the solutions of the β-scaling equation under shear can be described by dierent generalized power series and an exponential function. Four of these power series are necessary to describe the dynamics in the liquid region. The dynamics in the transition region can be described by two power series. In the yielding glass region, three power series and an exponential function are necessary to describe the dynamics on all time scales. The short time asymptotes and the shear-dominated long time asymptotes are not dependent on the separation parameter.