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ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
Estimation of Particle Size Distributions from Turbidimetric Measurements K. C. Yang’ and Richard Hogg” Mineral Processing Section, Department of Mineral Engineering, The Pennsylvania State University, University Park, Pennsylvania
A method is described in which simple measurements of the turbldtty of a suspension of fine particles can be used to obtain realistic estimates of the mean and range of particle sizes present. By assuming that the actual distribution can be fitted to a slmple, two-parameter function such as the log normal, the values of the parameters can be estimated from turbidlty data obtained at two wavelengths. Experlrnentally, it has been found that for dispersions of several different minerals, the estimation procedure yields results which agree remarkably well wlth direct measurements using centrifugal sedimentation. Slmulatlon studies, of the effect of significant deviations from the proposed log normal form, lndlcate that the estimation procedure yields results which are reasonably representatlve of the central part of the distribution. There is no evidence of strong bias toward the tails of the distributions.
Light scattering offers an attractive means of studying dispersions of fine particles, particularly when the particle size is too small for other methods such as sedimentation to be used effectively. The principal attractions of light scattering as a method of particle size analysis are: (i) There is no practical lower limit to the particle sizes to which the technique can be applied. (ii) Measurements are rapid and simple. (iii) Sampling problems are minimized since the method can be employed in situ. (iv) Measurement causes no disturbance of the system which is simply illuminated with a beam of light. Unfortunately, however, there are also several disadvantages t o the use of light scattering for particle size measurement. The relationship between the size of a particle and its ability to scatter light is complex and is known only for particles of simple shape. Furthermore, this relationship is not generally unique insofar as similar scattering can result from particles of several different sizes. Nevertheless, light scattering measurements can be used relatively easily t o obtain equivalent spherical diameters for essentially monodisperse suspensions of particles. Scattering techniques can be used to determine average sizes of polydisperse suspensions provided that essentially all of the particles are either much smaller or much larger than the wavelength of the incident light. In addition, an instrument has recently been developed ( 1 ) which uses a complex optical filtering system to evaluate the scattering behavior and can be used to obtain several moments of the distribution. However, it has not yet proved possible to develop general methods for the direct determination of particle size distributions from light scattering measurements alone. Techniques have been described from which size distributions can be evaluated from the combination of light scattering with other methods such as sedimentation ( 2 ) or from scattering by individual particles passing through a beam of light, one Present address, Handy and Harman, 1770 Kings Highway, Fairfield, Conn. 06430. 0003-2700/79/0351-0758$01.00/0
16802
a t a time (3-5). These approaches, however, lose several of the importaiit advantages of light scattering techniques. Specifically, they can no longer take advantage of the speed, simplicity, and range of applicability of a simple measurement of the light scattering behavior of a suspension of particles. Information on the average and approximate range of sizes in a polydisperse suspension can be obtained by comparing light scattering data with theoretical results based on an assumed form for the distribution. In particular, Meehan and Beattie (6) have described a technique which permits a n estimate of the size distribution to be obtained from simple measurements of turbidity at two wavelengths. The purpose of the work described in this paper has been to extend the method of Meehan and Beattie to materials with broad size distribution, to evaluate the limitations of the method, and to present experimental data comparing the estimated size distributions for several materials with the results of direct measurement by centrifugal sedimentation.
THEORY The turbidity T of a suspension of particles is a measure of the reduction in intensity of the transmitted beam due to scattering and is defined by ? = - 1I n - Io
L
I
where 1 is the length of the light path in the suspension and I , and I are the intensities of the incident and transmitted beams, respectively. For a dilute suspension of identical, spherical particles, the turbidity is given by = vdK, (2) where v is the number concentration of particles in the suspension, r is the radius of the particles, K , is the total scattering coefficient, defined as the ratio of scattering cross section to geometric cross section. In general, K , values can be evaluated from the exact Mie (7-9) theory of light scattering by spherical particles. Measurement of turbidity as defined by Equation 2 is not generally feasible. In any practical instrument, the measured intensity is made up of contributions from the transmitted light plus the light scattered in the forward direction in a cone of solid angle o determined by the geometry of the optical system. Equation 2 accounts only for light scattered exactly 0. This problem has in the forward direction, i.e., for w been considered in detail by Gumprecht and Slieprevich (10) who introduced a correction factor, R , defined as R = K,/K, (3) where K , is an effective scattering coefficient defined so as to account for the light scattered in the near-forward direction. Thus, the measured turbidity of a monodisperse suspension of spherical particles of radius r is given by T
-
T
= RK,ar2v
(4)
In the more practical case of a suspension of particles which are not all identical, i.e., when there is a distribution of particle sizes, Equation 4 can be expressed as 0 1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6 , MAY 1979
Kt(m,a) =
r = .rrJmRKtr2du(r) where du(r) is defined as the number of particles, per unit volume of suspension, whose radius lies between r and r + dr. If t h e particle size distribution can be described in terms of a density function n(r)dr, the mass fraction of particles whose radius lies between r a n d r + dr, Equation 5 can be rewritten:
where c is the concentration (mass per unit volume) of particles in the suspension and p is the density of the particles. T h e quantity T / C is known as the specific turbidity of the suspension. It is convenient to express the particle size in terms of the dimensionless parameter a which leads to
In the special cases when all particles are either very much smaller or very much larger than the wavelength of the incident light, the integral in Equation 7 can be related to simple moments of the distribution. Thus, for very small particles, the specific turbidity is proportional to the mean particle volume while a t the other extreme, turbidity is a measure of specific surface mea. For such cases, turbidity measurements yield no further information on the particle size distributbn. In order to apply Equation 7 to the evaluation of real systems, it is convenient to assume a specific form for the size distribution. A suitable form is the two-parameter log-normal distribution defined by 1
n(r)dr = &In
exp[ - 1 / 2 (
In In - In u
759
r50)']^.
r
(8)
u
[I(A,)12 + [WP,)I2 + [R(P,)l2 -2- cn = l [ ~ ( A , ) I 2+(2n (9) + l ) / [ n ( n+ 1112 cy2
The quantities A,, and P, are functions of m and a only and are defined in terms of J-type Bessel functions. The calculations were carried out on a n IBM 370/168 computer a t the Pennsylvania State University. T h e computations, which were performed in double precision arithmetric, were arranged to terminate when the absolute values of each of the four quantities R ( A , ) , [(A,,),R(p,,),and I(P,) became smaller than lo-'. Bessel functions whose orders had absolute values of less than 35 were evaluated using a standard, built-in function. For larger orders, the recurrence formula (11, 12)
where p need not be an integer, was used. Derivatives were determined by means of the relation (11, 12)
The computer program was carefully checked by comparing selected K , values with those published by Gumprecht and Sliepcevich (13) and Pangonis et al. (14). Calculation of Theoretical Turbidities. Theoretical values of the quantity r h / c for dispersions of the minerals used in this study were calculated for different values of the size distribution parameters cyw and u. The values were obtained by numerical integration of Equation 7, with n ( a )as given by the log normal distribution. Typical plots of theoretical i X / c values vs. the dimensionless mean size am are shown in Figures 1 and 2 for several values of the standard deviation U.
where rmis the geometric mean radius and u is the geometric standard deviation. If Equation 8 is expressed in terms of the dimensionless variable a and substituted into Equation 7, the integral can be evaluated numerically yielding a theoretical relationship between the specific turbidity and the parameters of the distribution. Meehan and Beattie (6) described a simple graphical technique which can be used to estimate the parameters from measurements of specific turbidity a t two wavelengths. Briefly, the method involves simultaneous solution of the theoretical expressions obtained by integration of Equation 7. At any given wavelength, the measured turbidity corresponds to pairs of values for the parameters rj0 and u. By plotting theoretical rmvs. u curves using values corresponding to the experimental turbidity, an estimate of the parameters of the actual distribution can be obtained from the intersection of the curves for different wavelengths. This technique will be described in more detail in a subsequent section of this paper. It should be emphasized that these measurements yield only a n estimate of the size distribution since they provide no means of determining the actual form of the distribution. However, for many applications, an estimate of the mean and standard deviation is probably sufficient. Furthermore, in view of the gross approximations (spherical particles, isotropic, homogeneous materials, etc.) implicit in the use of these methods, it is probably unrealistic to expect them to provide more than an estimate of the actual distribution. CALCULATIONS C o m p u t a t i o n of T o t a l S c a t t e r i n g Coefficients. According to the Mie theory, the scattering coefficients, K , can be evaluated from an expression of the form (10)
EXPERIMENTAL Two polystyrene latices (obtained from the Dow Chemical Company) and the minerals: quartz (SOz), anatase (Ti02), gibbsite (AUOH),),and corundum (cu-Alz03)were selected for this study. Quartz suspensions were obtained by the dry grinding of silica sand for 7 h in a mechanized agate mortar, dispersing in a 0.1% sodium pyrophosphate solution and removing the remaining coarse particles ( > l o pm) in a sedimentation column. The anatase was prepared from type T-315 titanium oxide which was supplied by Fisher Scientific Company and was dispersed in double distilled water containing a few drops of 1.5% aerosol OT surfactant solution. The corundum dispersion was prepared from type A-591 alumina also supplied by Fisher Scientific. Three hundred grams of the material were ground in a 1/2-gal. alumina ball mill j, using '/z-in. to 1-in. alumina balls as grinding media. The grinding was carried out wet in a solution of 10 mL glacial acetic acid and 5 g of aluminum hydroxide gel in 400 mL distilled water. After grinding, the coarser particles ( > l o pm) were removed by sedimentation. The finer portion of the suspension was siphoned out and stored in a 1-gal. glass bottle. The gibbsite Al(OH), suspension was prepared in exactly the same manner as the alumina. The starting material was supplied by Atlas-Mason Company. The suspensions were sampled (for turbidity measurement etc.) by carefully pipetting a known volume from the stock suspension. The suspension was thoroughly agitated prior to sampling to ensure uniformity. Suspension concentrations were checked repeatedly by drying and weighing samples. The exact crystalline form of each material was examined by X-ray diffraction techniques. Refractive index values were obtained from tabulations given by Deer, Howie, and Zussman (15) using the averaging procedure described by Jaffe (16). The relevent physical properties of the materials are summarized in Table I.
780
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
Table I. Physical Properties of Materials Investigated average specific refractive gravity, material index g/cmS polystyrene latex, 1.592 1.05
-
LS-i132-E
polystyrene latex, LS-1047-E corundum IAl-0.) I
quartz (sid,) gibbsite ( Al( OH),) anatase (TiO,)
.)I
1.592
1.05
1.767 1.547 1.577 2.537
3.97 2.65 2.42 3.84
I=
2-2
LL
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IO
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DIMENSIONLESS M E A N PARTICLE SIZE, a S 0
0
I
2
Flgure 1. Theoretical light scattering curves for polydisperse suspensions of corundum (a-Al,O,) which conform to the log-normal distribution
I
I
I
3
4
5
CONCENTRATION, C x I O 5 I g m I m l )
Flgure 3. Experimental turbidities of polystyrene
I '
I
1
1
1
1
1
I 6
latex
suspensions
I
1 \
-.-.
-
A
-
g 0.2 CORUNDUM (Q-AIZO,) 0 Xa436rnp 0 Xs546rnp X ~650mp v X '8OOrnr
3
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Flgure 2. Theoretical light scattering curves for polydisperse suspensions of quartz (Si02) which conform to the log-normal distribution
Turbidity measurements were carried out in a Coleman-Hitachi Model 124, double-beam, recording spectrophotometer for various concentrations of the minerals at wavelengths of 436,546,650, and 800 nm. In each case, it was determined that there was no appreciable change in turbidity with time, indicating that effects due to coagulation or sedimentation were not significant. Since the theoretical expressions are valid only for extremely dilute suspensions, when each particle acts as an independent source of scattered light and secondary scattering is negligible, the standard procedure of determining turbidity as a function of concentration and extrapolating to infinite dilution was followed in all cases. Experimental plots of specific turbidity vs. concentration are given in Figures 3 and 4 for the polystyrene latex and the corundum dispersions, respectively.
I
I
2
4
CONCENTRATION,
6
1
I
8
IO
c x 105 (gm/crn3)
Flgure 4. Experimental turbidities of corundum (a-AI2O3) suspensions
For comparison purposes, particle size distributions were obtained for each of the mineral suspensions by centrifugal sedimentation using an MSA-Whitby Particle Sizer Analyzer ( I 7). A t least four replicate tests were performed on each suspension and good reproducibility was obtained in all cases.
RESULTS AND DISCUSSION Evaluation of Experimental Procedure. The experimental methods used in this study were evaluated using resulb obtained for the polystyrene latex suspensions. Average particle sizes for these, essentially monodisperse, systems can be obtained directly from Equation 2 which can be rewritten.
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
761
Table 11. Experimental Results for Polystyrene Latex Beads average particle diameter, pm latex
manufacturers figure
estimated from turbidity
LS-1132-E LS-1047-E
0.091 0.234
0.095 0.231
specific turbidity ( T / c ) ~ = ~ measured theoretical 3.45 x 10-3 21.38 X lo-’
correction factor
3.45 x 10-3 21.50 X lo-’
1.000 0.995
Table 111. Experimental Results for Mineral Suspensions
mineral
corundum ( (Y -AI,03) quartz ( S O , ) gibbsite (Al(OH),) anatase (TiO,)
mean particle diameter, p m estimated from centri fuga1 turbidity sedimentation 0.60 1.25 0.34 0.30
standard deviation, u estimated from centrifugal turbidity sedimentation
0.55 1.18 0.32 0.60
2.84 2.89 2.56 1.54
2.76 2.97 2.30 1.86
Theoretical values of the quantity K , / a as a function of the dimensionless size a are shown in Figure 5 . Experimental average sizes for the two materials, evaluated using Equation 12 and Figure 5 are given in Table 11. These results can be used to obtain experimental estimates of the correction factor R defined by Equation 3. Thus, R can be calculated from where reXpis the experimental turbidity of the suspension and Ttheor is the theoretical value calculated from the known size of the particles. The values of R , so determined, are also given in Table 11. I t should be noted that, in general, R depends on the relative particle size a. Thus, the values show1 in Table I1 are not necessarily valid under all conditions. However, in view of the small corrections involved, it seems reasonable to assume t h a t the correction factor is insignificant for the optical system used in this study. Size Distribution of Polydisperse Mineral Suspensions. For each of the measured turbidities, the theoretical scattering curves (Figures 1 and 2, for example) can be used to obtain a pair of values of the dimensionless median size a 5 0 corresponding to each value of the standard deviation u. Thus, for the corundum system, a relative turbidity ( s X / c ) of 0.5 would be obtained if u = 1.15 and ~ ~ =5 2.0 0 or 9.5. Similarly, if u = 2.50, the same relative turbidity (0.5) would be obtained if a = 2.1 or 9.7. By plotting these theoretical u values vs. the corresponding mean particle radii, a curve is obtained for each measured turbidity. Examples of these curves are shown in Figure 6 for measurements on the corundum system a t four different wavelengths. An estimate of the “true” values of the parameters can be obtained from the intersection of the curves. It should be noted that, in fact, it is necessary to obtain data only at two wavelengths. In the present study, however, we have used four wavelengths as a check on the consistency of the results. Ideally, all four curves should intersect in a single point; the observed discrepancies are surprisingly small in view of the approximate nature of the model. T h e size distributions estimated from the light scattering data are compared to those obtained by centrifugal sedimentation in Figures 7 and 8 for the corundum and quartz suspensions respectively; the log normal distribution parameters are compared for all of the minerals in Table 111. Clearly, the agreement between the turbidimetric and sedimentation results is remarkable. An exception is found, however, in the case of anatase where the light scattering technique appears to underestimate the mean size by a factor
w CORUNDUM b-AI2031
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b
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0.2
0.4
0.6
M E A N PARTICLE R A D I U S , r 5 0 ,
0.8
1.0
(MICRONS)
Flgure 6, Experimental plot for graphical evaluation of the size distribution parameters for corundum (a-A1203)
762
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
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Comparison of the particle size distribution of a corundum (a-Al,O,) suspension as estimated from turbidimetry with direct measurement by centrifugal sedimentation Flgure 7.
U A L SIZE DISTRIBUTION ESTIMATED BY LIGHT SCATTERING
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PARTICLE SIZE (MICRONS)
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of two. The reasons for this discrepancy are not fully clear, though it may very possibly be the result of using improper values for the refractive indices. Titanium oxides have high refractive indices which are known (18) to be strongly dependent on wavelength. E f f e c t of Deviations f r o m t h e Log Normal D i s t r i b u tion. In the previous sections of this paper, it has been shown that, for materials which conform quite closely to the log normal size distribution, reasonable estimates of the parameters of the distribution can be obtained from simple turbidity measurements. Clearly, however, there is no a priori reason to expect any given material to conform to this particular distribution. It is of considerable interest, therefore,
-
Simulation of the result of applying the estimation procedure to particle size distributions which do not conform to the log normal law Figure 10.
to examine the results which would be obtained by applying this technique to particulate systems which do not conform to the log normal distribution. Theoretical turbidities have been calculated, using Equation 7 , for a series of hypothetical distributions which deviate significantly from the log normal form. Examples of two such distributions are shown, as the solid lines, in Figures 9 and 10. These theoretical turbidities have, in turn, been used as “experimental” values in our estimation procedure. The dashed lines in Figures 9 and 10 represent the apparent distributions which would be obtained if we were to apply our estimation procedure to particulate systems whose true size
ANALYTICAL CHEMISTRY, VOL. 51, NO. 6, MAY 1979
distributions were given by the solid curves. In each case, it is found that the estimated distribution conforms quite closely to the central portion of the true distribution and is not strongly biased toward either of the tails. Thus, it appears t h a t the estimation procedure described in this paper can reasonably be applied to real systems which may not necessarily conform to the log normal size distribution. The resulting, estimated distribution will be a kind of "best-fit" log normal approximation to the true particle size distribution. SUMMARY OF ESTIMATION PROCEDURE T h e procedure to be followed in using this approach to estimate particle size distributions can be summarized as follows. (1) Obtain the appropriate values of the light scattering coefficients K , either from tables (see, for example, ref. 13 and 14) or by numerical solution of Equation 9. (2) Determine the theoretical turbidity curves for the material by numerical integration of Equation 7. (3) Measure the specific turbidity of the suspension as a function of solids concentration and extrapolate to infinite dilution. Obtain such measurements for at least two wavelengths. (4) From the measured turbidities and the theoretical curves obtained in (2) above, determine the possible values of the mean size parameter as0 corresponding to each value of the standard deviation u. (5) Plot u vs. mean particle radius for each measured turbidity. (6) Read off the estimated mean and standard deviation for the actual distribution from the intersection of the curves.
763
LITERATURE CITED E. L. Weiss and H. N. Frock, Powder Techno/., 14, 287 (1976). T. Allen, "Particle Size Measurement", 2nd ed., Halstead Press, Wiley, New York, 1975. F. T. Gucker, C. T. O'Konski. H. G. Pickard, and J. H. Pitts, J. Am. Chem. Soc.. 69. 2422 (1949). C. T. O'Konski and G.'J. Doyle, Anal. Chem., 27, 694 (1955). A. E. Martens and J. D. Keller, Am. Ind. W g . Assoc. J., 29, 257 (1968). E. J. Meehan and W. H. Beattie, J. Phys. Chem., 84, 1006 (1960). G. Mie, Ann. Physik, 25, 377 (1908). H. C. van der Hulst, "Light Scattering by Small Particles", Wiley, New York, 1957. M. Kerker, "The Scattering of Light and Other Elecvomagnetic Radiation", Academic Press, New York, 1969. R. 0. Gumprecht and C. M. Sliepcevich. J. phys. Chem., 57, 90 (1953). S. A. Shelkunoff "Applied Math for Engineers arid Scientists", van Nostrand, New York. 1965, p 406. R. Weast, Ed., "Handbook of Tables for Mathematics". 6th ed.,Chemical Rubber Company, Cleveland, Ohio, 1970. R. 0. Gumprecht and C. M. Sliepcevich, "Tables of Light Scattering Functions for Spherical Particles", University of Michigan, Engineerlng Research Institute Special Publication: Tables, Ann Arbor, Mich., 1951. W. J. Pangonis. W. Heller, and A. Jacobson, "Tables of Light Scattering Functions for Spherical Particles", Wayne State University Press, Detroit, Mich., 1957. W. A. Deer, R. A. Howie, and J. Zussman, "An Introduction to the Rock Forming Minerals", Wiley, New York, 1966. H. W. Jaffe, Am. Mineral., 41, 757 (1956). K. T. Whitby, Am. SOC.Heal. Refrig. Air-Cond. Eng., J . Sect., Heat., Piping, Air Cond., June 1965. T. Radhakrishnan, Proc. Indian Acad. Sci., Sec A , 35, 117 (1952).
RECEIVED for review May 18,1978. Accepted January 29,1979. This work described in this paper was supported in part by the National Science Foundation under an Institutional Grant to The Pennsylvania State University. T h e paper was presented at the Eastern Analytical Symposium, New York, November 1973.
Chemically Bonded Aryl Ether Phase for the High Performance Liquid Chromatographic Separation of Aromatic Nitro Compounds Thomas H. Mourey and Sidney Siggia' Chemistry Department, GRC Tower I, University of Massachusetts, Amherst, Massachusetts 0 1003
A chemically bonded aryl ether stationary phase which shows novel selectivity in the reverse-phase liquid chromatographic separation of aromatic nitro compounds has been prepared. Differentiation between nitroaromatic isomers is a result of partltioning and charge transfer interactions. Comparisons wlth octadecyl and phenyl silica show both the differences and the advantages of the aryl ether phase. A modified Hammett equation is used to relate retention to electrophilic aromatic substitution rates for meta- and para-substituted compounds. Absorption spectra of the charge transfer complexes involved are presented, and the significance of the complexation is discussed in terms of its effect on chromatographic behavior.
Selectivity in high performance liquid chromatographic separations is often influenced by the electron donor-acceptor properties of the solute molecules and the bonded phase. While investigating the reverse-phase separation of a number
of aromatic nitro compounds, we hoped to take advantage of the strong electron accepting properties of these compounds by using a stationary phase which contained strong electron donating groups. The resultant charge transfer complexes are well documented in the literature (1-3), and it is basically accepted that there is a partial or complete transfer of electrons from donor to acceptor. Complex stability is a function of the electronic character and number of substituents on the acceptor molecule, their steric effects, and the solvent system employed. The interaction is highly suited for chromatographic application. Complexation kinetics are rapid ( 4 ) ,reversible, and discriminative enough to add functional group selectivity t h a t is not possible in adsorption and partition chromatography. The electron donor of choice for a nitroaromatic selective stationary phase is an aromatic ether. The ether linkage increases the electron density of the aromatic ring. The result is that the phenoxy group is one of the most powerful electron donors for nitroaromatics ( I ) as evidenced by formation of
0003-2700/79/0351-0763$01.00/00 1979 American Chemical Society
