High-Concentration Submicron Particle Size Distribution by Dynamic Light Scattering: Power spectrum development with heterodyne technology advances biotechnology and nanotechnology measurements M. N. Trainer and P. J. Freud Application Note SL-AN-05 Revision D Provided By: Microtrac, Inc. Particle Size Measuring Instrumentation
DYNAMIC LIGHT SCATTERING (DLS) is a well-established technique for measuring particle size over the size range of a few nanometers to a few microns; however, at high sample concentrations severe limitations are placed on the DLS measurement. This paper discusses the causes of the high concentration limitations, the means of overcoming the limitations, and the results of measurements at high concentrations. DLS determines particle size from the analysis of the Brownian motion of suspended particles. Light scattered from a moving particle has a Doppler light frequency shift imparted to it. Scattering from a group of particles will have a distribution of shifts from the randomly moving particles. Figure 1 illustrates two measurement configurations that can be employed to extract the Brownian motion information from the frequency shifted, scattered light. Homodyne detection shown on the left extracts the shifts by the interference between the light scattered from each particle with the light scattered by the rest of the particles. The interference or selfreferencing removes the high optical frequency, leaving the lower shift frequencies. This is the conventional photon correlation spectroscopy (PCS) geometry, usually measured at a 90 scattering angle. HOMODYNE INCIDENT LIGHT PARTICLES HETERODYNE INCIDENT LIGHT PARTICLES i s i s i o i s DETECTOR DETECTOR 2 2 P( ) = < i > P( ) = i < i > s 2 ) 2 o s 2 ) 2 kt = D q 2 D = 6 a q = 4 sin( / 2) Figure 1: Comparison of homodyne and heterodyne configurations used for dynamic light scattering. Io represents the reflected, reference laser. IS represents the scattered light from particles. P( ) represents the distribution of frequencies related to the distribution of particle size 2
In the heterodyne measurement, shown on the right, a portion of the incident light is diverted from the scattering particles and then mixed with the scattered light. The unshifted diverted light acts as a reference or local oscillator (hence heterodyne) for the shifted scattered light from each particle. In both cases, the interference signal frequencies are indicative of the Doppler frequency shifts (due to particles' Brownian motion) and are the basis of the particle sizing measurement of DLS. The expressions shown in Figure 1, P( ), represent the power spectrum of the distributions of frequencies. These are Lorentzian functions of angular frequencies,, for both homodyne and heterodyne detection. The particle size is determined by the analysis of these power spectra. The constant,, appears as a characteristic frequency of the response and is a function of the wavelength,, scatter angle,, and the diffusion coefficient, D. The diffusion coefficient,d, depends on the temperature, T, suspending medium viscosity,, and particle radius, a. Note that there are two differences between the power spectra for the two cases. First, the homodyne detection power spectrum depends upon twice the characteristic frequency,, while the heterodyne detection power spectrum depends upon just. Second, the homodyne power is proportional to the scattered light intensity squared, (Is) 2, while the heterodyne power is proportional to the product of the scattered intensity and reference intensity, (Is) (Io). When measuring in the homodyne mode, only the homodyne expression applies. Conversely, when measuring in the heterodyne mode, a mix of the two modes will apply. The effect of mixing the two modes together will be to mix power spectrum of characteristic frequency,, and power spectrum of characteristic frequency, 2, with a resulting ambiguity of particle size. The mixing will depend upon the relative magnitude of each mode. By providing a large reference intensity, the heterodyne mode can be made large enough to dominate the mixture of the two. The Microtrac NANOTRAC operates primarily with heterodyne detection. Figure 2 is a diagram of the NANOTRAC measurement system. A magnified view shows the region of optical interaction with the suspended particles. The analyzer has an optical waveguide immersed in the suspension. The 50- m diameter waveguide delivers the input beam to the sample and collects the backscattered light from a region within 100 urn of the waveguide-medium interface. The Fresnel reflection at the interface between the waveguide and the medium is mixed with the backscattered light. The reflection is un-shifted in frequency and provides the local oscillator for heterodyne detection. 3
FIBER OPTIC LASER OPTICAL SPLITTER CONNECTION STAINLESS STEEL PARTICLE PHOTO Laser Scattered Reflected WAVEGUIDE 100 SUSPENDED Figure 2: Diagram of the implementation of the Controlled Reference Method used in the Nanotrac - The term relates to the use of a reflected beam as a reference oscillator to isolate the scattered, frequency-shifted light. The waveguide includes the optical splitter, fiber optic connector and the enclosed stainless steel probe. Two important features should be noted. First, the high-intensity reflected reference allows the heterodyne component to dominate the power spectrum. Second, the very short pathlength, the 100- m that the scatter signal has to travel, minimizes multiple optical scattering effects even at high particle concentration. Also shown in Figure 2 is the balance of the measurement system for the analyzer. The tip of a three-port surface waveguide is immersed in a sample cell containing the suspended particles. The directional Y optical splitter delivers the input from the laser diode to the tip and returns the scatter and the reflected beams to the photodetector. The power spectrum of the interference signal is calculated with dedicated high-speed FFT (fast Fourier transform) digital signal processor hardware. The power spectrum is then inverted to give the particle size distribution. In considering the effects of high concentration on the dynamic light scattering particle size measurement, a number of mechanisms that alter the apparent particle size can be considered. The possibilities listed in Figure 3, are divided into two groups: those which increase the apparent size and those which decrease the apparent size. Van Der Waals Forces Agglomeration Increase Apparent Size Other Interparticle Attractive Forces Decrease Apparent Size Interparticle Repulsive Forces Multiple Scattering Homodyne (self-beating) Mixing Figure 3: Mechanisms for concentration dependence of apparent size in Dynamic Light Scattering measurements 4
Interparticle attractive forces such as Van Der Waals result in a net apparent size increase with concentration. Agglomeration as concentration increases would also result in an apparent size increase. The occurrence of agglomeration can usually be established since it tends to be irreversible. Repulsive forces result in an apparent size decrease as concentration increases. Two instrumentation effects would cause apparent size decreases as concentration increases. Multiple scattering results from light interacting with particles multiple times. Change of Measured Size as Sample Concentration Changes Relative Size Relative Size 1.2 1.0 0.8 0.6 0.4 0.2 0 1.2 1.0 0.8 0.6 0.4 0.2 0 1 1 10 100 1000 10,000 100,000 Concentration, ppm Nanotrac 150 PCS Competitor 0.705 Polystyrene 10 100 1000 Concentration, ppm Figure 4: Concentration dependence of apparent size for polystyrene samples The scattering and re-scattering causes multiple frequency shifts at different angles. The net effect is an apparent size reduction. A second effect occurs if heterodyne detection is used. Interference from the homodyne component with its doubled characteristic frequency will be present and will cause a decrease in apparent size. The effect will depend on how large the homodyne component is relative to the heterodyne component. The concentration dependence of the apparent size of a number of polystyrene sphere samples has been determined using the Microtrac NANOTRAC and a PCS instrument. The comparison is shown in Figure 4. The PCS instrument utilizes 90 5
scattering, homodyne detection, and a cuvette sample cell with a scatter pathlength of approximately 0.5 cm. The effects of multiple scattering are evident for the PCS long path measurement. A large decrease in apparent size is seen at relatively dilute samples. The short path NANOTRAC measurement shows no change in apparent size for the 200-nm polystyrene sample in Figure 4. For larger 460-nm and 705-nm polystyrene, the PCS measurement again shows the effects of multiple scattering at its upper concentration limit, while the NANOTRAC continues to show a concentration-independent size. Another advantage of the heterodyne technique is illustrated in Figure 4 for the 460- nm and 705-nm measurements at the low concentration end. The improved signalto-noise ratio of the heterodyne measurement allows the NANOTRAC to measure at lower concentrations, while the PCS instrument fails to measure at the same concentrations due to the inherently lower signal level of the homodyne measurement. The comparisons shown in Figure 4 illustrate that a short-path method is required for high-concentration dynamic light scattering size measurements in order to avoid multiple scattering and the resulting size errors. At the highest concentration range, only the short-path method of the NANOTRAC measurement is considered. With the concentration extended to 30% for 200-nm polystyrene, Figure 5 illustrates that the analyzer continues to measure concentration-independent size up to a few percent concentration, while above the few percent concentration range, the apparent size increases, reaching a value 30% higher than that of the nominal size. Relative Size 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.01 0.1 1.0 10 100 Volume Concentration, % Figure 5: Concentration dependence of apparent size for 200nm To determine the sources of the high-concentration size dependence, one additional aspect of the measurement system needs to be considered. Although the immersed waveguide measurement emphasizes the heterodyne contribution in the measured power spectrum, a homodyne component is mixed in that will shift the apparent size to a smaller value. The question is, how large is the shift? The probe medium reflectance determines the magnitude of the reflected reference and the magnitude 6
of the heterodyne component. The reflectance depends upon the difference between the index of refraction of the waveguide and of the suspending medium. The dependence is shown in Figure 6. For a silica waveguide, i.e., fiber optic, with an index of 1.46, the reflectance decreases as the medium index approaches the silica index. When the two indices are matched, the reflectance is zero and the NANOTRAC measurement would be pure homodyne with no heterodyne component. The apparent size would be half the nominal size (Carbon tetrachloride, toluene, and trichloroethylene are examples of suspending media that almost match a silica optical waveguide). 0.025 0.020 PROBE 0.015 Sapphire REFLECTIVITY 0.010 0.005 Silica 50% 200nm 0.000 1.30 1.35 1.40 1.45 SUSPENDING MEDIUM INDEX 1.50 Figure 6: Reflectance dependence on suspending medium index for silica and sapphire waveguide probes. Equivalent points for 50% A probe has been developed for use by the analyzer that avoids this problem. Sapphire, with an index of 1.76, is employed as a tip on the silica waveguide. The sapphire tip interfaces with the suspension and eliminates the concern of index matching. Figure 6 illustrates the reflectance curve versus medium index for sapphire. Even if the suspending medium is water, the effective index of the suspension can take on a high value at high particle concentration. The suspension takes on an index that is approximated by a volume average between the particle index and the suspending medium index. The lower plot of Figure 6 illustrates the case of polystyrene, with an index of 1.59; mixed with water, with an index of 1.33. The net index results in index matching with a silica interface at concentrations of 50%, while the sapphire-tipped probe maintains a high value of reflectance even at the highest concentrations of polystyrene. With the sapphire-tipped probe, the particle size was measured for 200-nm polystyrene up to 30% concentration. As shown in Figure 7, the measured size approaches a linear dependence on concentration in this concentration range. The homodyne mix correction, calculated for the decrease in probe reflectance with concentration, would produce a net size decrease as shown in this figure. 7
1. 200 NM POLYSTYRENE MEASURED RELATIVE DIAMETER 1. 0. CALCULATED HOMODYNE MIX 0. The cause of the difference between the measured size and the corrected size is the concentration dependence of interparticle interaction, an inherent property of the suspension. Concentration-dependence measurements were made with the analyzer on three sizes of polystyrene. The relative size, that is, the measured size relative to the low concentration size, is plotted in Figure 8. The three sizes of polystyrene fit a common linear dependence on concentration indicating a common interparticle interaction for these similar particle suspensions. 1.25 0 1 VOLUME CONCENTRATION, Figure 7: Concentration dependence of apparent size for 200nm polystyrene. 2 3 1.20 RELATIVE 1.15 APPARENT 1.10 1.05 140 204 394 LINEAR FIT 1.00 0 5 10 15 VOLUME CONCENTRATION, % 20 Figure 8: Concentration dependence of apparent size for three sizes of polystyrene (140nm, 200nm, and 394 nm). 8
In summary, the authors have shown that high concentration particle size measurements can be made using dynamic light scattering. The Microtrac NANOTRAC incorporates a direct-immersion waveguide with its inherent short optical pathlength to minimize multiple scattering errors. The sapphire probe with high tip reflectance provides a high heterodyne reference intensity, minimizing the homodyne mixing error. Interparticle interaction is the principal mechanism for size concentration dependence in the very high-concentration region, with three different polystyrene samples showing a common concentration dependence. 9
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