**Quantification of the Gel Point**

The software package of the *DWS ResearchLab* automatically extracts the mean square displacement, *<Δr ^{2}(t)>*, from the measured correlation function

*g*. This is a measure for the space within the sample that the scattering particles explore in a given time.

_{2}(t)If the scatterers are sufficiently monodisperse and if they are not part of the overall network, one can relate the mean square displacement to the rheological quantities *G’* and *G”*, the elastic and the viscous modulus respectively. Obviously, yoghurt or milk do not obey these requirements. However, even if we are limited to the mean square displacement in this specific case, we nevertheless gain valuable information about the sample.

For samples that show a fully decaying correlation function, the accessible time window is limited by the dynamics of the scatterers and the effective turbidity, which can be expressed as *L/l**, where L is the thickness of the cell. By using two different samples cells with *L* = 5 mm (symbols) and *L *= 2 mm (lines) we explore up to seven orders of magnitude in time, as shown in Figure 5. Upon lowering the pH we see a slow down of the dynamics in the system: the slope and the absolute value of the mean square displacement decrease. A pseudo-plateau develops at longer times (Figure 6).

**Figure 5. **Mean square displacement of milk at pH 4.95 measured with DWS

*in a 5 mm (symbols) and a 2 mm cell (line).*

**Figure 6. **Mean square displacements for acidifying milk at various pH values

*(legend), including fits of Eq. 2 (lines) to the data.*

The experimental data in Figure 6 are fitted with a stretched exponential (Eqn. 2):

\( <\Delta r^{2}(t)>=\delta ^{2}(1-e^{-\left (\frac {time}{\tau } \right )^{p}}), \)

(2)

where *δ* is related to the free space per scatterer in the aggregated network, *t *is the characteristic relaxation time, and *p* an indication of how diffusive the motion of the scatterers is [5,6].

Approaching the isoelectric point of casein (pH ≈ 4.6) both the proteins and the protein-stabilized fat droplets start to aggregate. When the aggregates become space-spanning, the sample gels to form yoghurt. To characterize this point, we follow the exponent *p* determined from the fits in Figure 6 with decreasing pH, as summarized in Figure 7. For liquids the mean square displacement linearly increases with time, which corresponds to *p* = 1. This is clearly the case for fresh milk with a pH around 7.

* Figure 7. p as a function of decreasing pH for acidifying milk.*

The onset of aggregation concurs with the decrease of* p* at pH 5.2, which agrees well with the qualitative picture obtained from the half time of the correlation function in Figure 4. For particle gels the gel point is defined as the moment where *p* drops below 0.7 [5,6]. Thus, we determine the gel point of acidifying milk at pH ≈ 4.9.

**References**

^{[1] D.A. Weitz, and D.J. Pine, Diffusing-Wave Spectroscopy. In Dynamic Light Scattering; Brown, W., Ed.; Oxford }

^{ University Press: New York, 652-720 (1993).}

^{[2] M. Alexander, and D.G. Dalgleish, Diffusing Wave Spectroscopy of aggregating and gelling systems,Current }

^{ Opinion in Colloid & Interface Science 12, 179-186 (2007).}

^{[3] H. Ruis, K. van Gruijthuijsen, P. Venema, and E. van der Linden, Structure-rheology relations in sodium caseinate }

^{ containing systems, Langmuir 23, 1007-1013 (2007).}

^{[4] P. Zakharov, F. Cardinaux, and F. Scheffold, Multispeckle Diffusing-Wave Spectroscopy with a Single-Mode }

^{ Detection Scheme, Physical Review E 73, 011413 (2006).}

^{[5] A.H. Krall, and D.A. Weitz, Internal dynamics and elasticity of fractal colloidal gels, Physical Review Letters 80, }

^{ 778-781 (1998).}

^{[6] S. Romer, F. Scheffold, and P. Schurtenberger, Sol-Gel Transition of Concentrated Colloidal Suspensions, Physical }

^{ Review Letters 85, 4980-4983 (2000).}