Oct., 1959
ABNORMALITIES IN LOG-NORMAL DISTRIBUTION OF PARTICLE SIZE
1603
tions are collected,28-82 then it is the concentration tubes in a swinging bucket rotor offers the possidistribution that is determined. But with only a bility of converting a preparative ultracentrifuge few fractions, it is instead the quantity distribution. into an absorption optics analytical ultracentrifuge For this case the apparent boundary position rep- for those laboratories not so equipped, or for those resentation v*, in effect, treats the tube as a series of who choose to use several different wave lengths on two-compartment cells with the level of the separs- the same sample. tion or partition varying. Of course, if the initial DISCUSSION concentration is unknown, then one of the other formulas would be used. A s one gets further into VERNEN. SCHUMAEER (University of Pennsylvania, Philathe study of the specific agent, certain short-cut delphia) (communicated).-In reference t o your comment the development of the two generalized differential equaprocedures are possible (such as using the J c dv di- that tions 9 and 10 depended on induction, I wonder if you have rectly or using just a two-compartment cell). considered substituting dv/dr = h y / 2 and s = (dr/dt)/g Eventually, the agent may be isolated in pure form into equation 3 and rearranging to g(dv/dr)dc,/dt c dand possibly studied optically in the analytical ul- (gdv/dr)/dt = 0 which is the perfect differentiald(c,gdv/drJdt = 0. Then consider two general classes of cells and fields tracentrifuge. such that either gdvldr av or a constant. The comparison made in the experimental secR. TRAUTMAN.-NO,we have not made that substitution. tion between the optically scanned curve and the you substituted for the s in the plateau region primes assay of pipetted fractions could be made here on Since should appear on your terms to give d(c,g’dv‘/dr‘)/dt = 0. the model systems because of the absence of any (Primes really should be used in equation 3, but note that contaminants. However, the scanning technique only the cases were considered for which the term in the could be used to locate known s-rate markers or to square brackets is independent of r’). It is not immediately how to go from your equation to equations 9 and determine whether there were contaminants of sim- apparent 10 which involve V. But perhaps this can be done and thus ilar s-rate t o the unknown. Furthermore, the provide a deductive derivation. Thank you for the suggescombination of the scanning device and quartz tion.
+
THE INTERPRETATION OF ABNORMALITIES IN THE LOG-NORMAL DISTRIBUTION OF PARTICLE SIZE BY RIYADR. IRANI Research Department, Inorganic Chemicals Division, Monsanto Chemical Company, St. Louis 66, Missouri Recebed April 8 , 1969
I t is well accepted that the distribution of particle size, in many rocesses of growth or breakage, obeys the log-normal law. I n the present paper, basal cases where %on-ideal” log-normal Iistribvtions prevail are treated, and simple methods are resented for their resolution into the “parent” distributions. The cases of limited formation, multiple formation rates and Eoundary conditions, and artificial separation are discussed. De ending on whether the size distribution is on a number, surface or weight basis is it shown that abnormal log-normal distri&utions may or may not be detected.
Introduction The log-normal distribution law has been applied by several authorsi-* to the distribution of sizes of particles obtained by crystallization and/or crushing. Hatchg and later Ames, et aL18showed that if the particle size distribution gives a straight line on a number basis when plotted on log-probability graph paper, then the size distribution by weight or surface area is a parallel straight line on the same coordinates. Kottler’o has published an excellent discussion on “the goodness of fit and the distribution of particle sizes.” Kottler” also showed how to handle (1) P. Drinker, J. I n d . H y g . Toz., 7 , 305 (1925). (2) J. B. Austin, Ind. Eng. Chem., Anal. Ed.. 11, 334 (1939). (3) T. Hatch and S. Choate, J. Franklin Inst., 207, 369 (1933). (4) B.Epstein, ibid., 244,471 (1947); J . Appl. Phys., 19,140 (1948); Ind. Eng. Chem., 40, 2289 (1948). (5) F. Kottler, J. Franklin Inet., 250, 339 (1950); 260, 419 (1950). (6) G. W. Phelps and 8. G. Maguire, J. Am. Cerarn. SOC.,40, 403 (1957). (7) 0.Menir. H. P. House and C. M. Boyd, ORNL 2345, Chemistry-General, Oak Ridge, Tenn., Atomic Energy Commission Unclassified Report. ( 8 ) D. P. Atnes, R. R. Irani and C. F. Callis, THIS JOURNAL, 63, 531 (1959). (9) T. Hatch, J. Franklin Inst., 2 l S , 27 (1933). (10) F. Kottler. ibid., 251, 499 (1951): 251, 617 (1951). (11) F. Kottler, THIS JOURNAL, 66, 442 (1952).
algebraically the size distribution encountered by Loveland and TrivelliI2 during the study of photographic emulsions, where it was found that the simple log-normal distribution does not fit the data. This article presents an interpretation and discussion of possible basal cases whereby a modified log-normal distribution is obeyed. The interpretations are particularly useful for powders obtained through fractionation, spray drying and/or controlled formation. They are also useful in the measurement of particle size through the use of the Tyndall spectra (the variation of scattered light with wave length) where Heller and othersi3-lS have shown that the tool is powerful for monodisperse systems and would be more generally applicable if the distribution function of the particles had been known a priori. Experimental The particle size distribution measurements were made according to previously described techniques.8
Limited and Unlimited Formation.-KottleP was the first to treat size distribution from a kinetic (12) R. P. Loveland and A. P. H. Trivelli, J . Franklin Inst., 204, 193 (1927): 204, 377 (1927); THIS JOURNAL, 61, 1004 11947). (13) W. Heller and E. Vasay, Phys. Rev., 63, 65 (1943). (14) W. Heller and E. Vassy, J. Chem. Phys., 14,565 (1946). (15) A. B. Loebel, Ind. Bng. Chem., 61, 118 (1959).
RIYARD R. IRANI
1604
Vol. 63
where f(x) is the probability of occurrence of size x, and M and u are the geometric mean diameter -
43t
L
0.1
1
1
1
I
I
I
1 2 5 10
I
,
I
/
I
,
30 50 70
,
I
9095
\ (
1
and geometric standard deviation,8 respectively. From equation 5 we note that f(x) + 0 as x approaches either xo or xm. Equation 5 can be integrated to yield P the per cent. greater than diameter x
I
99 99.9
% greater than. Fig. 1.-Limited
and unlimited growth.
L
The special cases of equation 6 are as follows: Case 1.-If xo = 0 and xm 4 a , then equation 6 reduces to the simple log-normal distribution that gives a straight line on log probability axis as shown in Fig. 1 for M = lop and u = 2. This case corresponds to unlimited formation. Case 2.-Figure 1 also illustrates the case when M = lop, u = 2, xm+ m but with xo = 7 p . The curve is shown to be asymptotic to 7 p ; the points are calculated from equation 6. Here if (x - 20) rather than x had been plotted on log probability paper, a straight line would have been obtained. The case with xo > 0 has been used by several authors of whom we can mention Sheppard, et a1.,16 and Gaddum.” 0.1 1 2 5 10 30 50 ’70 90 95 99 99.9 Case 3.-When 2, # ~3 a plot similar to the % greater than. The Fig. 2.-Artificial separation: case 7, if 10 p ? x > 100 p one shown as case 3 in Fig. 1 is obtained. removed; case 5, if x < 10 p removed; case 6, If x > 100 p curve is always asymptotic toward the upper limit diameter, as illustrated in Fig. 1 for xm = 20 p, removed. M = lop and u = 2. If (xzm/(xm- z)) rather than point of view. I n general, it can be assumed that x had been plotted on log-probability paper, a the size x of a particle grows or diminishes accord- straight line would have been obtained. Case 4.-If xo = 7 p , xm = 20p, M = lop and ing to the equation u = 2 the plot shown in Fig. 1 is obtained. The interesting point here is that similar plots do not deviate significantly from a straight line (shown where t , the time of growth’or disappearance is dotted) and when the data are obtained experimennormally distributed tally it is not justifiable to use the 4-parameter fit indicated above, but rather two new parameters, j ( t ) = 1/.\/27; exp( --t2/2) (2) namely, M’ = 1 2 . 6 ~and U‘ = 1.18. The imporwhere f ( f ) is the probability of a particle growing tance of this conclusion is that although most powor diminishing for a time interval t. The function ders are formed according to the kinetic equation d(x) can be expressed as describing case 4, the evaluation of experimental data indicates simple unlimited-growth type log normal distribution. Thus, although a particular (3) substance under specified conditions has a specific where xo and x m are the minimum and maximum rate constant K the values of M’ and c’ can be sizes formed, respectively, while K is a velocity varied by changing xo,e.g., nucleation or finer grindconstant of formation. It is significant to note ing, and xm, e.g., changing the time function by that when x is equal to either xo or x m then .2: be- shortening or prolonging the particle formation comes time independent. If equation 3 is com- period. bined with equation 1 and integrated then For cases 2 and 3 xo and xm are located easily from the asymptotical character of the curve. Artificial Distributions.-Cases 2 through 4 cover the modifications in the log-normal distribution where a and b are constants. 6 ) S. E. Sheppard, E. D. Wightman and A. P. H. Trivelli. P h o t . If equation 4 is substituted into equation 2, then J.,( 149, 134 (1925). it can be shown that (17) J. H. Gaddum, Nature, 166,463 (1945).
Oct., 1959
ABNORMALITIES IN LOG-NORMAL DISTRIBUTION OF PARTICLE SIZE
1605
due to variables controlling the formation of the 60 particles. However, after formation, powders are 40 generally tampered with, either through dust separation by cyclones or removal of coarse particles i (or combinations of these and similar operations) as - 20 described in cases 5-7 below and illustrated in z Fig. 2. 2 g 10 Case 5-1f particles below a certain diameter 8 s x1 are removed, e.g., cyclone dust removal, a curve 0 asymptoting toward x1 is obtained. In Fig. 2, 2 1 = lop, M = 3 2 p , u = 4. 4 Case 6.-If particles above a certain diameter N, are removed, e . g . , precise sieving or collecting 2 the fines in a cyclone, a curve asymptoting toward 1 2 5 10 30 50 70 YO95 99 xm is obtained. In Fig. 2, x,,, = loop, ilf = 32p, yb by weight greater than. u = 4. experiCase 7.-This is a combination of cases 5 and 6. Fig. 3.-0, for Po = 55%; 0 , for P o = 50%; 0, mental points. In Fig. 2, xm = loop, x1 = lop, M = 3 2 and ~ u = 4. Case 7 is similar to case 4 in that a straight line, shown dashed in Fig. 2 , can adequately represent the data. The original distribution call be obtained for case 6 by first estimating the “per cent. finer than” (100- P o )that would have been observed had there been no removal of particles bigger than the diameter exhibiting an asymptote, and then dividing the “per cent. finer than” by 100/PO a t various diameters to get the original distribution exhibiting log normal behavior. The value of P a can be estimated within 10% by extrapolating the straight line observed at low diameters to the diameter exhibiting an asymptote. If the chosen value of P o does not render a straight line the fit can be 1 2 3 4 5 F 7 S 9 10 improved either by increasing or decreasing P o as Diameter, f i . illustrated in Fig. 3. The data shown in Fig. 3 are Fig. 4.-Hi~tograins for cases 2 and B with 11.1 = 6.5 ant1 for a sample of clay that had been treated to reu = 2.31. move large particles; it is shown that by making 100 only two approximations one can obtain the parent 80 size distribution curve and also the amount of clay GO that had been removed above 45p. -10 Case 5 can be treated in the same manner us Case 6 except that P o is used in place of (100 - Po). d A question that arises a t this point is whether 30 Cases 5 , 6 and 7 are significantly different from those 9 of 2, 3 and 4. It turns out to be that they are 2 10 different and that they can be distinguished easily 6 8 from a histogram plot. For cases 2, 3 and 4 the 0 size-frequency plot has a definite slope at 20 and 4 5 - as compared with cases 5, 6 and 7 where a discontinuity in the size-frequency curve is obtained a t x1 and xm. This phenomenon is illustrated in 2 Fig. 4 for cases 2 and 5, where it can be easily seeii 1 2 5 10 30 50 70 90 95 99 !)O.‘J that although x1and zowere identical for a particuyo by weight greater than. lar distribution, the two cases gave histograms Fig. 5.-Heterogeneous distribution case Sa. exceedingly different near x1 a d ZO. Heterogeneous Distributions.-Kottlerll has discussed the problem of homogeneous vs. bi-modal diff erent rates and boundary conditions existing during the formation of the particles, e.g., different heterogeneous particle size distributions from an shape particles and/or crystal habits. Thus, if algebraic-statistical point of view. Although the one had an nth-modal distribution, meaning n approach is rigorous, it is time consuming; and, due to the unavoidable experimental error in size populations, then P can be expressed as determination, it becomes less advantageous as compared to the semi-graphical approach giveu i= 1 L In ui J below. (7) Heterogeneous particle size distributions generally arise either due to mixing of powders or due to where the subscript i refers to a particular popula13
&-
RIYARDR. IRANI
1606
Vol. 63
characteristic of the plot is that it asymptotes a t its upper and lower levels to two non-horizontal lines, these lines being the parent distributions (equation 7). As a n example the experimental points shown in Fig. 5 are treated as follows to obtain the parent distributions. The data are plotted on a sizefrequency histogram as shown in Fig. 6 to obtain M 1= 5p and M z = 19p. Equation 7 becomes (8)
where f is the fraction fl. The next step is to get a first approximation of ul and uzfrom the asymptotes 0 4 8 12 16 20 24 28 32 36 of the experimental points and compute the value Diameter, p . Fig. 6.-Histogram of heterogeneous distribution case 8a. off at various values of P. If the value off turns out to be independent of P and x, then the estimated values of u1 and uz are accurate. Otherwise successive approximations are necessary. Table I demonstrates the relative constancy of f for the datfa in Figs. 5 and 6 after only two successive approximations to obtain ul and uz. The calculated f values a t the lowest values of x tend t.0 be off due to the high dependence upon (100 - P ) shown in equation 8; a t low values of x, the value of P usually cannot be determined with enough accuracy, so that the calculated f can be ignored in that region. Case 8b.-If the parent distribution curves intersect on a log probability plot, a curve similar to that shown in Fig. 7 is obtained. However, in 0.1 1 2 5 10 30 50 70 90 99 this case the experimental points asymptote at yo by weight greater than. both ends of the distribution toward the original Fig. 7.-Heterogeneous distribution case 8b : 0 , measured; parent distribution with the higher u. The asymp0, calculated. totes do not approach specific diameter values as However, the point of 7 was observed in case Sa. I inflection turns out to be a n important point since both parent curves must go through it. Therefore, to resolve the example given in Fig. 7, the data are first plotted on a histogram (as was done in Fig. 6) to obtain M1 and Mz which in this case turns out to be 3.0 and 10.5p, respectively. The point of inflection can then be located very accurately by drawing the tangents to the curve and locating the point of maximum change of slope, 14p in this example. M I and Mz are located on 50% probability and connected to the point of inflection by 20.1
I
1 2 5 10
30 50 70
90
98
99.9
% greater than. Fig. &-a, weight convert of number data; 0, number data (cintel); 0, weight data (sedimentation).
tion. However, in most practical cases n = 2 and in the following special cases it will be also assumed that xoi = 0 and xmi+ a . I n both cases a good estimate of M1and M z can be made from the size-frequency histogram as illustrated by Whitby la for a sample of flour. Case 8a.-If the parent distribution curves do not intersect on a log probability plot, a curve similar to that shown in Fig. 5 is obtained. The (18) K. T. Whitby, Hentino, Piping,Air Conditioning, 27, No.6,139 (1955).
TABLE I
HETEROGENEOUS DISTRIBUTION EXAMPLE FOR CASE 8a 5
Phe4
/odod.a
3 98.3 0.85 4 91 .51 5 77 .46 6 65 .45 8 52 .48 9 49 .48 10 46 .51 13 30 .51 15 33 .52 20 21 .54 30 7.5 53 40 2 5 46 Av.~ 0 50 rfr 0.03 a Assuming MI = 5 p , M z = 19 p , UI = 1.28, 02 = 1.56. Ignoring the value a t x = 3.
ABNORMALITIES IN LOG-NORMAL DISTRIBUTION OF PARTICLE SIZE
Oct., 1959
1607
straight lines to obtain UI = 6.0 and uz = 1.47. The values o f f are then computed from equation 9 a t various measured values of P and x
butions are combined, the treatments presented for cases 8a and 8b do not apply directly and the complexity of the problem of resolution increases “exponentially.” P = 50 - 1OOf erf Number-Surface-Weight Distributions.-As was shown previously8 the value of P depends to a (9) large extent on whether it is referred to a number, The calculated values o f f are shown in Table 11, surface or weight basis. Thus, if one had the hypoand are erratic for the low values of P due to the thetical case of equal numbers of spherical particles high dependence of f on the measured values of having 1 and lop diameters the contribution P; unavoidable experimental errors in P can cause of the small particles to the then mixture is (l/2) on a variations in f as shown for x = 14p and x = 17p. number basis, (l/lol) on an external surface basis However, if the average value of f is taken and and (l/lo~l) on a weight basis. When abnormal Pcelcd compared with Pmeasd the average difference is distributions are encountered they may be im2 which is definitely within the 5 for portant only on a number basis and not on a surface the experimental error. Thus, from only one esti- or weight and vice versa. Figure 8 illustrates mate an excellent resolution was obtained. For all this point basis, clearly. The number-size distribution the resolutions we have done it was not necessary shows definite bi-modality while the weight conin any situation to do more than three approximavert of the data and the measured weight size distions. Table 111 shows the agreement between tribution agree with one another and show log calculated f and known values of f for synthetic normal behavior in all practical regions. The powdered mixtures. dotted line shown in Fig. 8 is where the weight-size distribution starts deviating from a straight line. TABLE I1 Many other similar examples have been observed HETEROQENEOUS DISTRIBUTION EXAMPLE FOR CASE8b for cases 2-8. dP) P(masd.) f(oalad.)’ P(aalod.)b AC Acknowledgment.-The author wishes to thank 3.0 71 0.58 69.0 2.0 Dr. D. P. Ames and Dr. C. F. Callis for helpful dis4.0 66 .60 65.3 0.7 cussions and Mr. W. W. Morgenthaler for making 6.0 56 .63 57.6 1.6 some of the measurements. 9 .o 42 .62 42.3 0.3 12.0 28 .63 28.3, 0.3 DISCUSSION
*
*
30.4 0.4 14.1 4.1 .... 10.8 4.3 .... 8.5 4.5 35 .... 5.1 2.1 A v . ~0.61 =k 0.02 Assuming M I = 3.0 p, Me = 10.5 p , U, = 6.0, ua = 1.47. Assuming a and f =: 0.61. PEalod Pmeasd. For 1 4 p > 2 > 3 p . 14.0 17.0 20 25
20 10 6.5 4.0 3.0
( .77) ( .06)
-
TABLE I11 HETEROQENEOUS DISTRIBUTION foaled.
fknom’
faalcd.
fknmn”
0.50 0.50 0.18 0.20 .61 .67 .72 .75 .30 .33 .66 .67 .85 .80 a From synthetically mixing known weights of two powders.
In some rare cases where limited growth distri-
M. J. VOLD (University of Southern California).-To what extent does the log normal distribution depend critically on the assumed form of 9(2)7 And have you found any instances in which departure from the log normal distribution could be due to +(z) having a different form? R. R. IRANI.-The dependence of the log normal distribution law on the assumed form of ~ ( z lies ) only in the proper choice of XO and X,. During our investigations which covered hundreds of powdered materials we have found that the particlesizedistribution can be very well represented by one of the cases described in this article. W. HELLER(Wayne State University).-How sensitive are thef values to choice of the parameters needed for their calculation? R. R. IRANI.-ID the worst case we have had to make three approximations to arrive a t f values that do not deviate by more than 2 per cent. and are random in character. W. HELLER.-DO you think there is any chance of applying your method of analysis to systems with three components? R. R. IRANI.-Yes, but adding three new parameters complicates matters.
