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Autocorrelation Example: Viscosity Titration

The QuantumXpert provides an ideal method of measuring solvent viscosities in volumes as low as 25µL.

In this experiment, solvent viscosities from 1.15 – 20 cP were used to simulate diffusion measurements of protein-sized particles from 2,000 – 10,000,000 Daltons. The relationship between correlation time (τD) and viscosity was found to be linear over the entire experimental range.

See the following sections for detailed information:

Theoretical Backgroundback to top

Physical Properties that Determine τD

The QuantumXpert provides a measure of the size of particles in a given sample by calculating the correlation time, τD, which is determined by the diffusion coefficient, D, of the fluorescent particles in a sample according to the following equation:

equation

where ω is the instrument’s beam radius.

The diffusion coefficient of a particle is in turn determined by two properties: the viscosity of the solvent, η, and the hydrodynamic radius of the particle, Rh. This relationship is described by the Einstein equation below:

equation

where kB is the Boltzmann constant (1.38x10-23 J/K), T is the temperature, and the particle is assumed to be spherical.

This relationship means that increasing the solvent viscosity two-fold has the same effect on D, and therefore on τD, as increasing the particle radius two-fold.

Rhodamine 6G Properties

From the well-determined diffusion coefficient of Rhodamine 6G in water, 2.8x10-6 cm2/sec (η = 1.0 cP at 20°C), we can calculate that Rh(R6G) = 7.7x10-10 m.

Experimental Methodback to top

Sample Preparation

Solutions of glycerol in water were made at 10 different weight percents shown in Table 1. Rhodamine 6G was then added to a concentration of 1.5nM. The viscosities of these solutions (shown in Table 1) were then measured with a Brookfield DV II+ viscometer.

Measurement Conditions

The fluorescence correlation functions were measured with a QuantumXpert spectrometer at a laser attenuation of 1.5 OD and measurement time of 60 seconds.

Table 1. Sample viscosities.

Glycerol % (w/w)Viscosity (cP)
01.15
202.00
302.65
403.70
454.80
505.65
557.65
609.85
6513.7
7020.0

Resultsback to top

Table 2, below, lists the calculated values of D and effective particle radius (if viscosity were constant at η = 1). The fitted values of τD are also listed with the standard error and sample size.

Table 2. Effective radii and correlation times at varying viscosities.

Viscosity (cP) D (cm2/sec) Rh (m) τD Std Error (N)
1.15 2.4x10-6 8.9x10-10 0.00155 5.26x10-5 (10)
2.00 1.4x10-6 1.5x10-9 0.00276 1.73x10-4 (10)
2.65 1.1x10-6 2.0x10-9 0.00400 1.66x10-4 (10)
3.70 7.6x10-7 2.8x10-9 0.00642 4.97x10-4 (9)
4.80 5.8x10-7 3.7x10-9 0.00672 3.49x10-4 (8)
5.65 5.0x10-7 4.4x10-9 0.00863 3.29x10-4 (10)
7.65 3.6x10-7 5.9x10-9 0.0114 8.71x10-4 (10)
9.85 2.8x10-7 7.6x10-9 0.0127 8.74x10-4 (6)
13.7 2.0x10-7 1.1x10-8 0.0223 0.00192 (5)
20.0 1.4x10-7 1.5x10-8 0.0281 0.0015 (10)

The plot of τD vs. viscosity (Figure 1) shows that the relationship between τD and viscosity is linear over the entire experimental range.

Figure 1. Correlation time vs. viscosity and effective radius.

The effective particle radii range from Rh = 8.8x10-10 – 1.5x10-8 m. For proteins, this corresponds to a molecular weight range of approximately 2,000 – 10,000,000 Daltons (assuming spherical particles with density of 1.2 g/cm3).