PARTICLE SIZEDISTRIBUTION
Sept., 1955
84 1
PARTICLE SIZE DISTRIBUTIONS FROM ANGULAR VARIATION OF INTENSITY OF FORWARD-SCATTERED LIGHT AT VERY SMALL ANGLES1i2 BY
J. H. CHIN13c. M. sLIEPCEVICH4 AND M. TRIBUS~
Department of Chemical and Metallurgical Engineering, Universitu of Michigan, Ann Arbor, Michigan Received February 66,1066
A mathematical basis of a method of determination of the size distribution in olydispersions based on the angular variation of the intensity of forward-scattered light is presented. An integral form& was first derived from the modified Bouguer-Beer6 light transmission equation to enable the computation of the size distributions of particles, large in comparison with the wavelength of the incident light, from transmission measurements at various values of the forward-scattering half angle. The accuracy of the derived integral formula was checked graphically against typical distribution curves with good results. Three possible experimental techniques of applying the integral formula are discussed, a lens-pinhole, a moving pinhole, and a micro-densitometric method. In the latter two, the integral formula was modified so that the relative size distribution could be computed from measurements of the angular variation of the intensity of the forward-scattered light within a half angle of about three to four degrees.
Particle sizes in polydispersed systems can be determined by means of light transmission measurements combined with differential settling.’ However, for steady-state flow systems and for systems undergoing changes other than gravity settling, ie., evaporation or agglomeration, this method presents difficulties which are amenable only to empirical approximation.* Since a technique based on light-scattering alone would circumvent these difficulties, a method based on the variation of the forward-scattering half angle was derived from the consideration of the modified Bouguer-Beer light transmission equation
where Fa is the incidenh parallel flux, ergs/sec. F is the flux after traversing a distance, t,through the dispersion and is picked up by the measuring system Kt is the Mie theory total Scattering coefficient and is a function of the particle diameter, 0;the wave length of the incident light in the surrounding medium, h; and the index of refraction of the particle relative to that of the surrounding medium, m R is a factor allowing for the geometry of the optical system employed to measure the transmitted flux N is the no. distribution function of the dispersion and is the no. of particles of average diameter D per unit volume of dispersion per unit range of particle diameter, or fNdD = c n i
For particles large in comparison with the wave length of the incident light and for a measuring system which has a small value of the forwardscattering half-angle e, according to the diff raction theory the factor R may be obtained by the expression’ R = R ( 4 = (1/2)[i Jo2(ae) JP(,~)I (2) where a! is nD/X and JO (ae)and J1 (a!@ are BesseI functions of the first kind and of orders zero and one, respectively. For a given index of refraction of the particle and for a given wave length of the incident light, the total scattering coefficient K t is a function of the particle diameter alone. Rather than using the distribution function N ( D ) , it is more convenient to define a distribution function in terms of a. Thus
+
+
N(D)dD = N(oc)doc
(3)
Equation 1 may thus be rewritten, for a given index of refraction m, and noting that D = Xa/n
Equation 4 is an integral equation. The Mellin transformation theorem9 states that the quantity K t ( o ) N (a)a2in equation 4 may be calculated from an integral of the form
i
where ni is the no. of particles of size i per unit volume of dispersion. (1) Presented before the twenty-ninth National Colloid Symposium which was held under the auspices of the Division of Colloid Chemiatry of the American Chemical Society in Houston, Texas, June 20-22, 1955. (2) This work was performed in partialfulfillmentof therequirements for the Degree of Doctor of Philosophy in Chemical Engineering at the University of Michigan, Ann Arbor, Michigan. (3) Graduate Fellow, Department of Chemical and Metallurgical Engineering, University of Michigan, Ann Arbor, Michigan. (4) Department of Chemical Engineering, University of Oklahoma, Norman, Oklahoma. (5) Department of Engineering, University of California, Loa Angela, California. (6) The Bouguer-Beer transmieaion equation is sometimes called the Lambert-Beer law. (7) R. 0. Gumprecht and C. M. Sliepcevich, THISJOURNAL, 67, 90 (1953). (8) J. A. Consiglio, Ph.D. dissertation, University of Michigan, Ann Arbor, Michigan, 1953.
where h(a0) is a kernel to be determined. Development of Theoretical Equations.-The direct solution of the integral equation 4 is difficult. However, equation (4) may be changed to a new form by the process of differentiation with respect to e and multiplication with a function of e, thereby a solution may be obtained as follows. Substituting for R(ae) as given by equation 2 and differentiating equation 4 with respect to e, gives J12(d)] Kt(
a ) N ( cu)ru2da
By using the differentiation and recursion formulas for the Bessel functions, the above equation may be (9) I. N. Snedden, “Fourier Transforms.” MoCraw-Hill Book Co., Inc., New York, N. Y., 1951, p. 7.
J. H. CHIN,C. M. SLIEPCEVICH AND M. TRIBUS
842 simplified to
Vol. 59
X M
U
n
x
Hence Differentiating equation 7 with respect to 0 TitchmarshlO gives the formula
0
20
80 120 160 200 240 280 320 360 ff.
-Typical
where Yv(xu) is Bessel function of the second kind and of order v. The above formula is true for v 2 0, if tf(t) is integrable over (0, a). Therefore, by letting v = 1, x = a, u = e, t = a’,and f(z) = Kt(a)N(a)a Kt(a)N(+ = -zrr
K
. )
particle distribution: 1, Kt( a)N(a)a2 X 10-4; 2, K ~ ( ~ ) N ( ~ ) .
Jl(ore)Yl(cre)Ode
$ [ e ~ ~ * ( ~ ’ e ) i ~ ’ ~ ~ ( ~ ’ ) ~ ( ~(’I O))~ ’ d ~ ’ Let I
Then according to equations 8 and 10 Kt(ol)N(a)a =
Hence
- 27r10an[(s)J,(
K
Kt(a)N(a)az = 27r
0
Yl(ae)ede (12)
0.01
0.02
0.03
0.04
0.05
e. Fig. 2.--61(0) for typical particle distribution.
iM(e)[-J,(aYe)Yl(ole)(ae)ld0
i
(13)
M(0) may be calculated from experimental data according to equation 11, i.e., readings of the measured radiant flux F contained in cones of varying half-angle 0. Once M(0) is obtained as a function of 0, Kt(a)N(a)a2may be calculated from the integral formula 13 by inserting, one at a time, a series of values of a into the kernel [-J1(ae)Yl(aO)(a0)],graphically integrating the right-hand side of equation 13, and obtaining the value of Kt(a)N(a)a2corresponding to the particular value of a used for the integration. From K t ( a ) N ( a ) a 2the number distribution N ( D ) may be easily computed. The feasibility of computing Kt(a)N(a)a2from equation 13 was successfully tested with a typical distribution function, K t ( a ) N . (a)d = .4a3e-(Pa)’. With p = 0.00884, A = 0.20, and X = 5460 A., the distribution function Kt(D). N ( D ) has a peak at D = 14 p and the corresponding weight concentration (specific gravity = 1) is approximately 3 grams per cubic meter. Figures 1, 2 and 3 show the distribution functions, M(B), and the kernel x J l ( z )Y l ( x ) , respectively. The graphical integrations of equation 13 for two values of a(a = 100 and a = 200) are given i n Figs. 4 and 5. The comparisons between the assumed and calculated values of K t ( a ) N ( a ) a 2are P
Kt ( 4N ( P )u,’ From typical distribution function, A a % - ( p a ) 2
K t (or)N(a)a* from eq. 13
100 200
91548 70229
92200 71800
(10) E. C . Titchmarsh, Proc. London Math. Soc., 83, XXIII (1925).
I
I
I
j 1
2
3
4
5
6
7
8
9
,
I
X. Fig. ~.-ZJ~(Z)YI(Z).
Experimental Methods of Applying the Integral Formula.-To apply the integral formula, three possible experimental methods are considered. (1) Lens-Pinhole Method.-The lens-pinhole optical system’ is shown in Fig. 6. From the lens law
where f is the focal length of the lens, X is the object distance of the particle, and X’ is the image distance. By similar triangles
where r is the radius of the receiving pinhole and h is the radius of the receiving lens. From equations
I
843
PARTICLE SIZEDISTRIBUTION
dept., 1955
D,
FOCAL PLANE
.-
Fig. 6.-Lens-pinhole
system.
1
9 (DL - b ) 5;
or 0
0.01
0.02
0.03
0.04
e. Fig. 4.-Graphical
integration for typical particle distribution (CY = 100).
- 2tr D L F T + ~
(17)
the photocell will receive all the light with halfangle $ S r/f. Here D L is the diameter of the receiving lens, b is the diameter of the light beam, and t is the distance between the most remote particle to the receiving lens. The radiant flux which passes through the receiving pinhole is F. By varying the radius of the receiving pinhole, a series of F’s with different half-angle e, may be measured. M(8) may then be computed from equation 11. (2) Moving Pinhole Method.-If a moving tiny pinhole instead of a pinhole with varying aperture is used, the first differentiation step in computing M(0) from equation 11 may be eliminated. From Fig. 7, the radiant flux received by the tiny pinhole with diameter a is given by a
aZAF
5.=-
A( ara) a2AF =8rAr
From equation 16 r = f6anda = f4
(19)
Therefore +z AF --
$J=
88 Ae
If the moving pinhole is very small AF AB
- N -
dF de
Hence dF 8e d0
(b2 $--0
0I
.02
Using 5, the first differentiation step in obtaining M(e) according to equation 11 is eliminated.
03
8. Fig. 5.-Graphical
integration for typicaI particle distribution (CY = 200).
14 and 15
Therefore h
=S= tanB-6
(16)
for small 8. Consequently, if the light is scattered only once to the receiver, the half-angle 8 is independent of the position of the particle in the light beam and is characterized only by the ratio of the radius of the receiving pinhole to the focal length of the receiving lens, according t o equation 16. It is evident from Fig. 6 that as long as
Fig. 7.-Moving
pinhole system.
(3) Micro-densitometric Method.-If a piece of photographic film is placed on the focal plane of the receiving lens (Fig. 8) and a beam of parallel
844
J. H. CHIN,C. M. SLIEPCEVICH AND M. TRIBUS
monochromatic light is passed through the dispersion in front of the lens, the distribution of the blackness or density is somewhat like that given in Fig. 8b. The central peak density is the result of the parallel beam of light and the light scattered at 6 = 0". The density outside the central peak is the result of the light scattered with forward halfangle 6. The densities on the photographic film may be measured by means of a micro-densitometer. If the film is calibrated, the relative illumination E on each position of the film may be determined. The relationship between the illumination E and the radiant flux F may be obtained as
The radiant flux falling over a ring of radius r and width dr on the film is d F = 2nrEdr
(22)
Therefore
L.
F = 2 r l Erdr
+ F8
(24)
where Fais the flux falling on a very small circle of radius 6. M(6) may be computed using equations 11, 23, and 24. It may be noted that the radiant flux 5 in equation 21 may be converted into the illumination E by dividing 5 by the area of the moving pinhole
Thus the micro-densitometric method and the moving pinhole method are equivalent. Greater accuracy is possible with these two methods than with the lens-pinhole method because as pointed out above one differentia.tionstep is eliminated. By combining equations 11 and 23, the following expression for M(0) may be obtained.
FOCAL PLANE
IO1
Fig. 8.-Micro-densitometric
syatem.
.
For dilute dispersions, F increases very slowly while E changes rather rapidly with 6; therefore, as a very good approximation M(6) may be written as
If only the relative distribution of sizes is needed, the value of the constant 87r2f2/tX2Fsis immaterial. Since the integral method is derived using the principle of diffraction theory for which Kt = 2, equation 13 may then be written N(a)a2 = tX2Fs YPJ* de II_ (e3E)[-Jl(o~e)Y~(o~e)(ae)Ide
(23)
The total radiant flux over a circle with radius r is then
VOl. 59
(27)
If the diameter of the moving pinhole (assumed very small) is a, then 32r2f2
N(cr)a2 = tX2a2F8 -Jo
m
d
(835) [ - ~ ~ ( ~ e ) y ~ ( ~ e ) ( ~ e ) i d e
(28)
The relative distribution may then be obtained as N ( D ) D ~=
clam 2
(835) [-J,(ole)Y,(oLe)(oLe)]de (29)
where C is a proportionality constant. The flux 5 measured by the moving pinhole as a function of 6 is a measure of the angular variation of the intensity of the forward-scattered light. The relative number distribution and volume distribution may be readily computed from the relative N ( D ) D 2 distribution curve. Thus the integral method provides a convenient means of determination of particle size distributions. Acknowledgment.-The Eastman Kodak Company provided a Fellowship Grant for the year 1951-1952 and the Horace H. Rackham School of Graduate Studies of the University of Michigan provided the Horace H. Rackham Predoctoral Fellowship Grants for the years 1952-1954.
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