# Viscosity Measurements of Transparent Mixtures

Prepared by Mathias Reufer at LS Instruments AG

June 2012

Introduction

Diffusing Wave Spectroscopy (DWS) is a powerful optical technique primarily used to study the rheological properties of turbid sam­ples. The method is based on the analysis of the fluctuations of laser light that is scattered mul­tiple times within a sample [1,2]. The DWS RheoLab from LS Instruments is a versatile tool to conduct such measurements.

Because DWS requires sufficient  turbidity to ensure strong mul­ti­ple scattering of la­­ser light, samples such as creams, milk, yo­ghurt, lotions, shampoos, paints and ceramics are well suited. How­ever, if the sample is transparent, some sample pre­­paration is required prior to measurement.

The purpose of this application note is to show how one can pre­pare a trans­parent sample by adding tracer particles to obtain suf­ficient turbidity for DWS measurements. As a model system, we use mixtures of glycerol with water at different mixing ratios and compare the measured viscosities with tabulated (published) va­lues. Moreover, we explain how one can measure the transport mean free path l* over an ex­tended range of turbidity. This is the distance after which a scattered photon has totally lost its original propagation direction. It thus indicates how turbid a sample is. The smaller l* is, the stronger the scattering. Accurate values of l* are necessary not only for extracting viscoelastic in­formation from DWS measurements, but also for many other applications. It is the most important optical parameter for turbid samples.

Sample Preparation

Mixtures with different ratios of glycerol and water were prepared. Tracer particles, consisting of polystyrene beads with a diameter of 980 nm dispersed in water, were added. We used a tracer particle concentration of approximately 2 wt% (so­lid content) for all samples. Table 1 sum­ma­rizes the different samples S1 to S5.

The choice of the tracer particle concentration is connected to the cuvette size. The cornerstone of accurate DWS measurements is ensuring applicability of the diffusion approximation which requires that L ≥ 5 l*, where L is the thickness of the cuvette [1]. In the case of dilute systems l* is inversely proportional to the tracer particle concentration. Furthermore, l* is dependent on the re­fractive index of the solvent nsolv (see Table 1). In our system, an increase in glycerol con­cen­tration leads to an increase in nsolv which decreases the optical contrast of the tra­cer particles. Thus, an increase in the glycerol concentration yields an in­creased l*.

 sample [glycerol] nsolv CR l* η - wt.% - kHz µm mPa·s S1 30.9 1.372 346 224 3.12±0.03 S2 48.6 1.396 409 283 6.20±0.08 S3 64.7 1.419 470 357 16.8±0.6 S4 77.9 1.439 506 407 55.9±1 S5 86.5 1.453 539 459 183±5

Table 1. Properties of the samples S1 to S5: weight fraction of gly­cerol in the solvent, refractive index nsolv  of the solvent from tables [3], count-rate CR measured with DWS RheoLab, obtained l* based on CR, viscosity η measured with DWS RheoLab at a temperature of 20°C.

Determination of l*

The transport mean free path l* can be calculated based on Mie-theory (e.g. implemented in the web based scattering calculator [4]). For accurate results however, the precise particle concentration must be known. One must also pay attention to use volume fraction (not weight fraction) for the calculation!

To circumvent these problems, the DWS RheoLab offers the possibility to measure l*based on a reference sample thereby eliminating the need to know the con­centration of the scattering particles. The reference sample must consist of dispersed monodisperse particles of well-known size (e.g. determined by static or dynamic light scattering or purchased from a reliable particle supplier). Typically one will use the same or similar particles at approximately the same concentration for both the reference and the sample. To determine l* the following equation can then be applied:

$$l^{\ast }_{sample}\approx \left ( \frac{CR_{sample}}{CR_{ref}} \right )\cdot l^{\ast }_{ref},$$

(1)

where the count rates CR are simply the number of photons per second measured in transmission by the optical detector of the DWS RheoLab. The count rates of the sample CRsample and the reference CRref must be determined for cuvettes of equal thicknesses L and a constant setting of the optical attenuator.

This approach is accurate under the condition that L ≥ 5 l*. Further­more CRsample and CRref may notdiffer by more than 20 %. However, in our example the resulting CRs range from 346 to 539 kHz for S1 to S5 (see Table 1) because of the different refractive index of each sample. We thus choose a set of 5 reference samples, with different concentrations of polystyrene beads (diameter of 980 nm) dispersed in water to never have a difference of more than 20 % in CR between sample and reference. Table 2 summarizes the properties of the different reference samples R1 to R5.

 reference sample [tracer particle] CR l* - wt% kHz µm R1 2.07 271 174 R2 1.70 326 210 R3 1.32 399 276 R4 0.95 497 392 R5 0.69 574 522

Table 2. Properties of the reference samples R1 to R5: tracer particle concentration,

count rate CR and transport mean free path l* measured with DWS RheoLab.

The use of the appropriate reference sample (R1 to R5) for each sam­ple (S1 to S5) is sufficient to reach an accuracy of about 10 % in the determination of l*. However, we can further increase the accuracy by using an interpolation scheme. The measured count rates CR and transport mean free paths l* allow us to establish the calibration curve shown in Figure 1, which is a fit of the five data points from the  reference samples with a second order polynomial.

Based on this calibration curve we determine l* of S1 to S5 with in­ter­polation (see Table 1). Note that this curve holds only for a given laser intensity. The intensity of the laser may not be changed during the measurements on R1 to R5 and S1 to S5. In most cases a single reference sample is sufficient when measuring several samples.

Only when the count rate (i.e. l*) varies over a wide range among the dif­ferent samples, as in this particular case, several re­ference samples are re­com­mended.

Figure 1. Calibration curve obtained from the reference samples R1 to R5 (squares)

and a fit (line) using a second order polynomial.