Realistic colloidal systems

2.3.1 Unreachable transition line

Up to now the description of the screened Coulomb potential of the HSY system was considered as being defined by the coupling parameter𝛾that determines the strength of the potential and the screening parameter𝑘that determines its range. However, in real experimental colloidal systems one rather works with the effective charge𝑍_{ef f}and the concentration𝑐₁₁of added salt ions. As one can see in the definition of𝛾 in Equ. 1.2,𝛾 not only depends on𝑍_{ef f} but also on𝑘. A tuning of𝛾 by adjusting 𝑍_{ef f} without changing 𝑘is not possible. The screening parameter𝑘 depends not only on the concentration of salt ions𝑐₁₁, but also on the concentration of counterions, which is proportional to𝑍_{ef f}. This results in the situation that not any desired value for𝛾and𝑘is accessible by variation of𝑍_{ef f} and𝑐₁₁.

In order to learn what𝛾 values are accessible by variation of the effective charge𝑍_{ef f}, one can use the definition of 𝛾 in Equ.1.2to obtain𝑍_{ef f}(𝛾, 𝑘). After inserting this into the definition of the screening parameter in Equ.1.3, one obtains:

2.3 Realistic colloidal systems

Figure 2.16:MCT transition lines for two different volume fractionsΦtogether with curves for the cou-pling parameter𝛾(𝑘)in realistic systems including counterions and added salt ions of molar concentration 𝑐₁₁. The particle diameter is always set to𝜎 = 150nm. Curves are shown for two different dielectric constants𝜖. At the low volume fraction ofΦ = 5 %the transition line is unreachable for a system with a low dielectric constant (e.g. particles in a CHB/Decaline mixture), while it is still reachable for the system with a high dielectric constant (e.g. particles in water). Note that increasing the diameter has the same effect as increasing the dielectric constant. Adding salt ions increases the minimum value for the screening parameter and therefore reduces𝛾(𝑘)for low𝑘values.

The second equation (2.13) shows that for a real, physically meaningful value of the coupling parameter 𝛾(𝑘), the screening parameter 𝑘must have a minimum value, which is determined by the added salt concentration𝑐₁₁.

Going along the curve𝛾(𝑘) to larger values can be done by increasing𝑘 via an increase of the effec-tive charge𝑍_{ef f}. As one can infer from Equations2.12and2.13,𝑘(𝑍_{ef f})is monotonically increasing and therefore also𝛾(𝑘(𝑍_{ef f}))is monotonically increasing. In Figure2.16we see 𝛾(𝑘)for two typical particle-solvent combinations together with the corresponding MCT glass transition lines, that were al-ready presented above. The solid phase is not always reachable, especially at low volume fractionsΦ(left panel). The𝛾values required for the transition can be so high that𝛾(𝑘)never crosses the transition line.

For example for a suspension at volume fractionΦ = 0.05 with dielectric constant𝜀 = 7and without any salt ions𝑐₁₁= 0, the MCT glassy phase is never reached, not even for arbitrary high effective charge numbers.

From Figure2.16one can deduce that a higher dielectric constant 𝜀enhances the possibility to reach high𝛾 values and cross the transition line for a large enough number of effective charges𝑍_{ef f}. This seems counterintuitive because a higher dielectric constant should imply a better screening of charges.

For pure Coulomb potentials without screening by free ions this is certainly true, the potential energy is simply reduced by a factor𝜀⁻¹. But in this case, including the effect of free salt and counter ions in in an effective Yukawa-like potential, this is not the whole story.

In order to understand this, one needs to recall that𝜀is inversely proportional to the Bjerrum length𝑙_B. To get two like-charges initially separated by in infinite distance as close together as the Bjerrum length, an energy of𝑘_B𝑇 is required. An equal definition is that𝑘_B𝑇 is required to infinitely separate two opposite charges that were initially at a distance of𝑙_B. So a smaller Bjerrum length (larger𝜀) means that less energy is required to bring like-charges close to each other, and as a consequence they can move more freely. Furthermore, the attraction of opposite charges is weaker, so that the thickness of the double layer around the colloids becomes larger. Both these factors taken together, the screening length𝑘⁻¹is larger for a smaller Bjerrum length (larger𝜀). In other words, the screening is weaker. Therefore, in systems with a larger dielectric constant𝜀, both the counterion and the salt concentration can be a lot higher,

while the screening is still weak enough. This explains that systems with a higher dielectric constant are more likely to be glassy.

Since dimensionless length units are used in the above considerations, “small Bjerrum length” actually means small Bjerrum length compared to the particle diameter𝜎. This implies that big particles are an advantage: The glass transition is reached more easily in systems with big particles. Altogether, the key parameter is the ratio 𝜎∕𝑙_b, which is proportional to the product𝜎 ⋅𝜀. The higher, the more likely a colloidal system can be glassy.

Additional salt ions (𝑐₁₁>0) lower the possibility to get into the solid phase by enforcing a lower limit to the screening parameter𝑘. They result in a sharp decrease of𝛾(𝑘)at low𝑘(see Figure2.16. This is not surprising since more screening reduces the range of the particle repulsions, making the system softer.

2.3.2 Reentrant transition

A closer look at Figure2.16shows that there is indeed the possibility to get two crossings of a curve𝛾(𝑘) with the MCT glass transition line. For instance, the two intersections of the blue line𝛾(𝑘,Φ = 0.05, 𝜎= 150nm, 𝜖= 80, 𝑐₁₁ = 0)with the green transition line in Figure2.16indicate a reentrant transition from liquid to glass and back to liquid: Going along𝛾(𝑘)from left to right (which means increasing𝑍_{ef f}) there is one intersection at low𝑘(low𝑍_{ef f}), going from liquid to glass, and another intersection at higher𝑘 (higher𝑍_{ef f}), going from glass to liquid. The physical reason why the glassy phase is left again is that an increase of the effective charge leads to a higher counterion density and thus to an increased screening, such that the potential eventually becomes too soft for the system to stay solid.

In Figure 2.17we see MCT transition lines for the two systems with different dielectric constants that were presented already in Figure 2.16. In the case of a high dielectric constant𝜖 = 80, like there is in water, one obtains a regime of effective charges 600 < 𝑍_{ef f} < 6000, where very dilute systems (Φ < 0.05) are glassy, even if a considerable amount of salt is added. The self dissociation of water yields a value of 𝑐₁₁ ∼ 0.1µmol∕l, which means that at 𝑐₁₁ = 10µmol∕l the salt concentration is 100 times higher than in fully deionized water. For the case of a lower dielectric constant𝜖 = 7(e.g. in the solvent CHB/Decaline), but with the same particle size𝜎 = 150nm, the lowest possible volume fraction

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

Figure 2.17:Reentrant transition by varying the effective charge𝑍_{ef f}. MCT transition linesΦ(𝑍_{ef f})two realistic experimental systems are shown for two different added salt concentrations𝑐₁₁. Left and right panel represent systems with different dielectric constants𝜖but the same particle diameter𝜎. Transition lines were computed with the methods described in section2.1.

2.3 Realistic colloidal systems

Figure 2.18:MCT transition lines for variation of the added salt concentration𝑐₁₁. Again, two different systems are compared, but now both, the dielectric constant𝜖and the hard core diameter𝜎are differ-ent. They represent typical colloidal suspensions used in light scattering measurements and confocal microscopy, respectively. Left panel applies to PS particles in water, right panel to PMMA particles in a CHB/Decaline mixture. Note that the typical minimal ion concentration in water is𝑐₁₁ = 0.1µmol∕l (self-dissociation of water), while it can be two orders of magnitude lower in the other solvent. Transition lines were computed with the methods described in section2.1.

for the transition isΦ ≃ 20 %. This much larger than for the other system. However, the required number of effective charges to make the system solid is smaller, a value of𝑍_{ef f} ≃ 300is large enough.

For low𝑍_{ef f} the Yukawa part of the potential becomes negligible and the transition is observed atΦ = 0.516, which is the value for pure hard spheres. On the other hand, going to very high𝑍_{ef f}the screening due to the counterions becomes so strong that again very high volume fractions are required for the transition. Interestingly, both diagrams in Figure 2.17 show a small range, where the transition line Φ(𝑍_{ef f})drops dramatically upon increasing𝑍_{ef f}.

Unfortunately, a variation of the effective charge cannot be realized in a controlled way in an experiment with colloids. It is possible to synthesize particles with a different amount of ionizable surface groups but one cannot change this number in situ, at least not without also changing the concentration of salt ions. But exactly that would be necessary to observe the reentrant transition during a measurement.

2.3.3 Phase diagrams for varying the salt concentration𝑐₁₁

A parameter that is more easy to modify in an experiment is the concentration of additional salt ions𝑐₁₁. Staying at the same volume fractionΦone can induce a transition from solid to liquid by adding salt ions to the solvent. Figure2.18shows volume fractions and corresponding salt concentrations for typical systems used in experiments: PS particles in water and PMMA particles in a CHB/Decaline mixture.

The minimum volume fraction for the transition strongly depends on𝑍_{ef f} and a variation of𝑐₁₁allows one to increase the necessary volume fraction up to the hard sphere limit atΦ = 0.516.

2.3.4 Power laws for varying the salt concentration𝑐₁₁

Another interesting question is whether the MCT power laws already presented above in Section2.2.4 are predicted to be valid as well upon changing the added salt concentration and how big their ranges of validity are. Figure2.19shows𝛼and𝛽relaxation times and plateau differences𝜙_plat−𝜙_𝑐for two typical

−10⁻⁴

𝛽/𝜏_𝛼 and distance of the plateau height to the critical plateau height 𝜙_plat−𝜙_𝑐for varying the added salt concentration𝑐₁₁(expressed by𝜀= (𝑐_11,𝑐−𝑐₁₁)∕𝑐_11,𝑐). Other sys-tem parameters correspond to the same two typical experimental syssys-tems as in Figure2.18, MCT glass transition points are given in the legends. Straight lines are power laws with exponents calculated from the critical exponent parameters𝜆 = 0.747(blue lines) and𝜆 = 0.718(green lines), the exponent for the plateau differences is always0.5. All values are derived from the coherent density correlators as de-scribed in section2.2.4(cf. Figures2.14and2.15). The range of validity of the power laws is comparable or even better than for a variation of theoretical parameters𝛾or𝑘(cf. Fig.2.14).

experimental systems at different separations𝜀 = (𝑐_11,𝑐−𝑐₁₁)∕𝑐_11,𝑐. The power laws for the relaxation times are again predicted to be valid up to a separation of𝜀= 0.1and the plateau difference power law at least up to𝜀= 0.01.

As a conclusion one can say that measuring relaxation times and or plateau heights of correlators during a controlled modification of the salt concentration should be a very good way to test the predictions of MCT.