Particle Size Analysis with a Disc Centrifuge - ACS Symposium Series

Chapter 12

Particle Size Analysis with a Disc

Centrifuge

Importance of the Extinction Efficiency

Downloaded by UNIV OF ARIZONA on August 2, 2012 | http://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch012

Bruce B. Weiner, D. Fairhurst, and W. W. Tscharnuter Brookhaven Instruments Corporation, 750 Blue Point Road, Holtsville, NY 11742

Comparison of standard samples is made between mass distributions obtained gravimetrically and those obtained from a disc centrifuge photosedimentometer with full extinction efficiency corrections. The excellent agreement between theory and experiment suggests that accurate results are readily obtainable for many types of materials. Some form of light scattering is used in most particle sizers. For particles less than a few microns in size light scattering corrections are necessary to obtain accurate size distribution information. Calculations of the full extinction efficiencies for correcting turbidity data obtained with a broad spectrum source are reviewed. Examples for carbon black, quartz powder, and polystyrene latex are given. Most particle sizing instrument specifications concern size range, reproducibility, resolution, ease-of-use, and, occasionally, accuracy; accuracy in determining the size, not the amount. Yet the amount is just as important as the size. The aim of an accurate particle size distribution measurement is to produce either a differential or cumulative size distribution with the amount at each size in terms of either the volume, mass or number of particles. This information is needed to calculate the common statistics that characterize the distribution. These statistics~the mean is the simplest example-are just as much a function of the amount as they are of the particle size. Yet most attention is focused on the sizing capabilities of a technique. Most commercially available submicron particle sizing instruments either use light scattering to determine both the size and the amount or just the amount. The former category includes photon correlation spectroscopy (PCS) and multiangle light scattering (MLS). The latter category includes disc centrifuge photosedimentometry (DCP), cuvette centrifuge photosedimentometry (CCP), sedimentation field-flow fractionation (SdFFF), and two forms of chromatography: hydrodynamic ( H D C ) and capillary hydrodynamic fractionation ( C H D F ) . In all these techniques, except PCS, a light intensity is measured. It must then be converted into size distribution information, and this process involves 0097-6156/91/0472-0184S06.00/0 © 1 9 9 1 American Chemical Society

In Particle Size Distribution II; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.


12. WEINER ET AL.

185

Importance of Extinction Efficiency

light scattering corrections. With PCS the intensity autocorrelation function is measured, and, for broad distributions, light scattering corrections are then applied to transform from intensity-weighted to volume- and number-weighted distributions. Light scattering corrections are of two types: the extinction efficiency and the particle scattering factor. Extinction efficiency is associated with turbidity measurements. It is an important correction for D C P , C C P , SdFFF, H D C , and C H D F measurements. The scattering factor is associated with the angular pattern of intensity measurements. It is an important correction for PCS and M L S measurements. Both types of corrections are calculated from Mie scattering theory or, in certain limiting cases, from simpler analytic functions. This paper will focus on the extinction efficiency and its importance in the determination of particle size distributions. A spherical model for particle shape is assumed not only for simplicity but also because light scattering corrections are then readily calculable from Mie theory and its limiting forms. Fortunately, many submicron applications involve either spheres (latexes, liposomes, monoclonal antibodies) or relatively compact shapes approximating spheres (silver halides, ceramics, some pigments).

Theory A brief review of the well-known theory for extinction efficiencies 1 is presented here. For sufficiently dilute systems the intensity I of the transmitted light is related to the incident intensity lo by,

I = l exp[-TL] 0

(1)

where L is the path length and r is called the extinction coefficient in the light scattering literature and the turbidity in much of the chemical and particle sizing literature. Turbidity arises from two sources: absorption and scattering. Separately or in combination these two sources are the cause for the extinction of the transmitted light. Turbidity is related to the number of particles N per unit detected volume, the particle's geometric cross-sectional area A , and the extinction efficiency Qext by,

r = NAQext

(2)

A l l of these variables are a function of particle size D . (Here D is the diameter of the assumed sphere. In general D is some characteristic length, often an equivalent spherical diameter determined by the particular technique used to measure it.) In particular the area A is proportional to D ~ 2. The extinction efficiency is a function oft), the wavelength of light in the medium J^, the particle refractive index np, and the refractive index of the suspending medium no. The two refractive indices are also, in principle, a function of . In practice, however, they are often taken as constant, especially np, since the wavelength dependence may not be known. Absorption is accounted for by specifying the imaginary part of both refractive indices, although choosing a medium that absorbs is usually counterproductive. Given \ , np, and no the numerical value of Qext is calculated for each D using spherical M i e theory.


186

by,

P A R T I C L E S I Z E D I S T R I B U T I O N II

According to Mie theory for spheres, the extinction efficiency is given Qext = (2/x*)*I(2n+1)*Re(an+b )


n

(3)

Here x, the size parameter, is equal to T T D / J ^ . Both an and bn, the partial wave scattering amplitudes, and, therefore, Qext, are functions of x and m, where m = np/no. The scattering amplitudes can be written in terms of RiccatiBessel functions. Numerical algorithms are used to calculate the final results. The algorithms in the appendix of the Bohren and Huffman book in reference 1 are particularly useful for this purpose. Commercially available D C P instruments use a tungsten-halogen lamp, a broadband source. This type of source is very compact and stable, reaches operating conditions in a few seconds, is inexpensive, rugged, and produces an intense beam of light. Since it is not monochromatic the calculation of an appropriate extinction efficiency is more difficult. However, once calculated, its use in Equation 2 is straightforward. If a light source is not monochromatic, then Qext is replaced by a weighted average. The weighting function is proportional to the product of the wavelength dependence of the source and detector. Following Oppenheimer2 we write the average, integrated extinction efficiency as Q*ext = B - J > U )-QextU)

(4)

where the asterisk denotes the integrated efficiency, B is a normalizing factor, and P(J^) is the product of the source and detector wavelength sensitivities. In principle these sensitivities should be measured for each sourcedetector combination. Oppenheimer shows a measured curve for a particular tungsten-halogen lamp and photodiode detector used for his D C P measurements. In practice the source dependence is reasonably well described by the Planck blackbody radiation law modified by the wavelength dependence of the uartz envelope used to house the lamp. A n d the detector dependence is ominated by the near-infrared wavelength dependence of the silicon photodiode and the glass material covering the active part. The P(X ) is sufficiently similar in either case that the final, corrected distribution results agree to within experimental error. Limiting cases are useful as guidelines. For large particles Qext approaches the Fraunhofer value of 2. For small particles in the Rayleigh regime Qext varies as D for strong absorbers like carbon black and D ^ 4 for nonabsorbers like polystyrene latex. Thus, Qext as a function of D must increase, then peak, then approach a constant value. For strong absorbers the approach is nearly linear. For weak absorbers and for nonabsorbers two kinds of periodicity are apparent provided m is roughly less than or equal to 2.5. The low frequency periodicity is termed the interference structure. The high frequency periodicity is termed the ripple structure, and it is appears for m greater than roughly 1.3 or 1.4. In this study m

o c

0.004

0.10

Slope = 3.05 140nm < d < 320nm

0.20

0.30

0.40

D i a m e t e r (/xm)

Figure 4. Log-log plot of the integrated extinction efficiency of PS latex in as a function of diameter from 140 to 320nm.


190


selected particles. Emulsifier-free particles were selected because it was felt that they would be easier to dry. To determine the percent solids a stock suspension was made from the concentrate by accurately diluting 2g with 20g of milliQ water. Twenty drops were then weighed on an aluminum dish. The sample was then evaporated slowly at room temperature to constant dry weight over a period of one week. The mean and standard error of 7 samples for each lot are shown in Table I.

Table I. Solids Content of Two Standard Latexes


Lot 10-65-53, Std. 1 Labeled

9.30 ± 0.10% w/w

Measured

8.90 ± 0.02%

Lot 10-95-38, Std. 2 10.10 ± 0.10% 9.98 ± 0.02%

A l l subsequent mixtures were prepared gravimetrically using the measured percent solids for calculations. PCS results on the unmixed polystyrene samples were obtained as pooled averages from 8 runs on one instrument and 6 runs on another. The instruments were two Brookhaven BI-90 submicron particle sizers. Ppoled data showed no instrument-to-instrument variations. Each run took just over 4 minutes. Samples were made by diluting one drop of the concentrates into 5ml of aqueous solutions of lOmM NaCl and 0.1% v/v Triton X-100. A l l solutions were prepared using milliQ water, p H 5.3, which had been filtered through a 0.2/JLfilter.Sample cells were cleaned repeatedly using the same diluent. The individual and mixed latexes were each run twice on two different disc centrifuge photosedimentometers, Brookhaven Instruments model B I D C P . Pooled data showed no instrument-to-instrument variations. Samples were prepared as follows: 1 drop of the concentrate was diluted into 3ml of a 0.2fi filtered aqueous solution of 0.1% v/v Triton X-100 followed by 30 seconds of sonication, after which 3ml of M e O H was added. A n external gradient was formed using 15ml of milliQ water and 1ml of M e O H as described previously. Disc rotation speed was 8000RPM, and run times varied from about 12 minutes for the larger particles to about 60 minutes for the smaller particles. Temperature was monitored and recorded throughout the runs. Deviations of no more than two degrees centigrade were noted. The mean temperature was used for calculating fluid viscosity and density.

Results and Discussion Data analysis for a disc centrifuge using the line start method is well known7. In particular, all the data presented here were analyzed using the Treasure8 correction. The Treasure correction arises from the finite size range in the detector due to the finite detector aperture. Treasure was the first to note that, apart from any variation of Qext with diameter, this leads to a volume rather than ^surface area distribution as might be expected from a cursory examination of Equation 1.


12. W E I N E R E T A L .

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It is difficult to prove that a submicron particle sizing method is accurate using carbon black. Such particles are neither spherical nor solid, and carbon black standards are not available. Thus, proving that extinction efficiencies have been properly accounted for is impossible. However, the BCR66 quartz powder standard and polystyrene standards do provide a means for testing. B C R Results. The BCR66 standard consists of 11 data points from about 0.35/ti to 3.5/x,. The cumulative percent undersize values are known to within several percent. These values are an average from several laboratories using the Andreasen pipette method. Although tedious and difficult for small particles that also diffuse, this method relies on direct gravimetric analysis to determine the weight of particles that has sedimented a known distance. Figure 5 shows the mass percent undersize versus the Stokes Diameter for BCR66. The filled circles represent the standard data. The smooth curve was obtained using the D C P method described in the experimental section and the integrated extinction efficiency shown in Figure 2. The agreement is excellent. The dashed curve shows the results calculated without any extinction correction. A significantly more coarse distribution is obtained. Since the refractive index or PS is close to that of natural quartz, it might be expected that the full PS extinction correction would yield acceptable results. As shown by the dotted curve in Figure 5, it does not. Allen7 also measured the particle size distribution of BCR66 using a disc centrifuge fitted with a tungsten-halogen lamp and silicon diode detector. Instead of correcting theoretically for the extinction efficiency as we have done in this paper, he chose to calculate the extinction efficiency as a function of diameter given the BCR66 size distribution and the uncorrected D C P data. In other words, he constructed a calibration curve. Figure 6 in the Allen paper shows the extinction curve constructed for calibration. It does not resemble our Figure 2. Figure 7 in the Allen paper shows the cumulative percent undersize versus diameter for BCR66 with and without the extinction correction. Even with the extinction correction the cumulative distribution does not agree nearly as well with the standard B C R 6 6 data as our data shown in Figure D. These differences are puzzling. Latex Results. The results of particle size measurements on the individual latex standards are shown in Table II.

Table II. Unmixed Latex Standards: Diameter Std. 1(nm)

Std. 2(nm)

LABEL

300 ± 1

121 ± 1

PCS

313 ± 1

151 ± 1

DCP

316 ± 2

151 ± 2

Technique



192


Stokes Diameter

(/xm)

Figure 5. Cumulative undersize distribution by mass for BCR66, a natural quartz powder standard, in as a function of diameter and different optical corrections. The data was measured with a disc centrifuge. 280

P o w e r Law E x p o n e n t , n

Figure 6. Weight average diameter of a bimodal suspension as a function of the power law exponent used to calculate the correction due to extinction efficiency. The raw data was measured with a disc centrifuge using a 23.9%/76.1% by weight mixture of 151/314nm PS in H 0 . 2



12.

WEINER E T A L .

0.00

193


0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

S t o k e s D i a m e t e r (/xm) Figure 7. Cumulative undersize distribution by mass o f a typical carbon black in as a function of diameter and different optical corrections. The data was measured with a disc centrifuge.

The uncertainties are the standard deviation of the mean values. PCS measurements in this size range are least subject to error. The D C P results depend on the density used for the particle. Here a value of 1.045g/cm3 was used for both samples. Furthermore, this same density has been used in our laboratory on several Duke Scientific latex standards. A n d agreement between the labeled values and those found with the D C P are usually within 2%. O n this basis we find the supplier's values are suspect. Further measurements made by the supplier on Std. 2 revealed an error in calibration. A new value of 149nm was subsequently reported. Std. 1 was never remeasured by the supplier. Mixtures of these two latexes were made with nominal 2:1,1:1, and 1:2 ratios by weight. More exact ratios were calculated by weighing the samples used to make the mixtures. These exact ratios, along with the weight average diameters calculated gravimetrically and those obtained from the D C P , are presented in Table III.

Table III. Mixed Latex Standards: Weight Average Diameters Mixture

DCP Results(nm)

Grav. Results(nm)

69.1/30.9

265 ± 5

264 ± 3

46.5/53.5

225 ± 2

227 ± 3

23.9/76.1

195 ± 2

190 ± 3

The D C P results were calculated using the sizes determined from each mixture and the extinction efficiency shown in Figure 3. The gravimetric results were calculated using the values 314.5nm and 151nm for the diameters and the


194



percents byweight determined gravimetrically and shown in the first column of Table III. The agreement is excellent with the worst case differing by only 2.5%, yet still within one standard deviation of the mean. The other two cases agree to better than 1%. Oppenheimer2 also tested his extinction efficiency calculations using a mixture of latexes. His sizes also varied by a ratio of about 2:1 as ours do; however, he chose sizes in a range over which the extinction efficiency is nearly linear in the diameter. The extinction efficiency in the size range we have chosen varies as the cube of diameter. A given relative error in size would lead to three times the same relative error in the calculated extinction efficiency. A n d the final weight-average diameter would show a much greater sensitivity to the extinction correction. The excellent agreement obtained in this work is a more sensitive test of the extinction efficiency calculations than that of Oppenheimer. Effects of Ignoring the Treasure Correction and Refractive Index Variations. The weight average diameter for the 23.9/76.1 mixture was recalculated from the D C P data using different power laws for the extinction efficiency. The results are shown in Figure 6. Using the full extinction correction the answer is 195nm in excellent agreement with 190nm obtained gravimetrically as shown in Table III. The same result is obtained using an exponent of 3 in the power law as shown in Figure 6. The vertical bars represent the 1% random error in the mean value obtained in this work. The horizontal bars represent the error in the power law exponent obtained by varying the refractive index by ± 0.01. A variation of this magnitude would occur by ignoring, as we have done, the wavelength dependence of the refractive index in carbon black or quartz powder; or that due to temperature differences of many degrees; or that due to slight chemical heterogeneities from sample-to-sample. Oearty these errors yield results within the experimental error of the measurement. Thus, they can be ignored. The Treasure correction varies linearly with the diameter. Ignoring this correction is equivalent to changing the power law exponent by one. Here the systematic errors are clearly much greater than the random errors, and ignoring this correction is never justified. Carbon Black. Figure 7 shows the mass percent undersize versus the Stokes Diameter for a carbon black sample. The smooth curve was obtained using the integrated extinction efficiency shown in Figure 1. The filled circles represent values calculated assuming Q varies as D ^ 1. The agreement is excellent as it should be since the full curve in Figure 1 is very nearly linear up to several hundred nanometers. The dashed curve represents the uncorrected results. As expected, a coarser distribution results.

Conclusions Extinction efficiency corrections are crucial for accurate interpretation of D C P measurements and for any other technique which utilizes a turbidimetric method of detection. Indeed, without appropriate extinction efficiency corrections particle size distribution data are of questionable value. These corrections can be calculated sufficiently accurately for a broadband source using the bulk refractive index for polystyrene latex and natural quartz. The excellent agreement obtained in this work between theory and experiment suggests that other materials are also amenable to this treatment.


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Literature Cited 1.

7. 8.

Van de Hulst, H.C. Light Scattering by SmallParticles;Dover Publications: New York, 1981. Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by SmallParticles;Wiley-Interscience Publications: New York, 1983. Oppenheimer, L.J. J.Colloid and Interface Sci. 1983, 92, 350. Medalia, A.I.; Richards, L.W. J. Colloid and Interface Sci. 1972, 40, 233. Kerker, M. The Scattering of Light and other Electromagnetic Radiation; Academic Press: New York, 1969; p. 324. Boundy, R.H.; Boyer, R.F. Styrene: Its Polymers, Copolymers and Derivatives; American Chemical Society Monograph Series; Reinhold Publishing Corp.: New York, 1952; p. 524. Holsworth, R.M.; Provder, T.; Stansbrey, J.J. In Particle Size Distribution: Assessment and Characterization; Provder, T., Ed.; ACS Symposium Series 332; American Chemical Society: Washington D.C., 1987; Chapter 13. Allen, T. Powder Technology 1987, 50, 193. Coll, H.; Haseler, S.C. J. Colloid and Interface Sci. 1984, 99 591.

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Particle Size Analysis with a Disc Centrifuge - ACS Symposium Series

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