1768
M. WALES
Vol. 66
PARTICLE SIZE DISTRIBUTION I N RUBBER LATEX BY M. WALES Shell Development Company, Emeryville, California Received March 11, 2966
Turbidity spectra were measured for various polyisoprene latex samples in the region 0.6-1.1 p on a Cary Model 14 spectrophotometer. The validity of the measurements was checked by comparison with turbidity spectra of polystyrene latices of known diameter. Results were interpreted in terms of a weight median particle diameter and a breadth parameter for an equivalent log normal distribution. These results were compared with particle count os. diarileter from Coulter counter measurements. The actual particle size distributions are quite closely log normal, at least on the high diameter end.
Introduction This paper is concerned with the general problem of rapid, routine particle size analyses in aqueous dispersions, particularly in latex produced artificially from synthetic cis-polyisoprenes. Data on comparison of turbidity spectra with Coulter counter1-* results on Hevea and synthetic cispolyisoprene Iatices will be presented. Experimental Turbidity Spectra.-Recently, Wallach, Heller, and Stevenson4 have prepared tables for use in interpreting turbidity spectra. These tables were not available a t the time this work was done. The interpretation of our turbidity data was done in much the same way as suggested by these authors, except that the data were not normalized to a reference wave length and that a log normal distribution of diameter was employed as a standard distribution function instead of the function used by Wallach, et al. In our work, tables of T / C (turbidity/concentration) were calculated on a Bendix G-15D computer at each of four standard wave lengths for a range of the two distribution parameters in the log normal function using the Mie theory.5 A brief outline of these calculations is given in the Appendix. A simplified representative plot of these values is shown for X = 0.65 p in Fig. 1. Here p iEthe breadth parameter of the log normal and is equal to a42, where u is the standard deviation and D,, is the weight median diameter. (Diameter at the 50% by weight point.) Then, from the value, of T / C a t each of the four wave lengths, a curve of p us. D,, is determined. The intersection of these four curves fixes the parameters D,“,and p for an equivalent log normal distribution, as in Fig. 2. It must be borne in mind in using these curves that for dispersions of very small particle size, the size distribution has no effect on the turbidity, which is determined solely by the diameter equivalent to the weight average particle weight, rather than the weight median diameter. (Rayleigh scattering, D 5 0.1 A). A criterion for this condition is, of course, the constancy of the quantity (X“/c)/(n I n the calculation of these families of T / C curves, the effect of dispersion was taken into account, in the refractive indices of both phases. The refractive index of Hevea, as measured by Wood, et was used. The refractive index of water was calculated from a relation in the International Critical Tables.7 The resulting values of m for the rubberwater system can be fitted by least squares t o a function of the form n
m = - = 1.134406 no
+ 0 .OO 152547 A2
0,000149881 + 14
H. Berg, ASTM Special Publication No. 234 (1958). E. Kubitschek, Nature, 182,234 (1958). F. T. Mattern, F. 8. Brackett, and B. J. Olson, J . A p p l . No. 10, 1 (1957). (4) M. L. Wallach, W. Heller, and A. F. Stevenson, J, Chem. Phys., 34, 1796 (1961). ( 5 ) G. Mie, Ann. Phys., 25, 377 (1908): H. C. Van de Hulst, “Light Scattering by Small Particles,” John R7iley and Sons, Inc., New York, N. Y., 1957. (6) L. A. Wood and L. W. Tilton, J . Research A’atl. Bur. Standards. 43, 57 (1949). (7) “International Critical Tables,” McGraw-Hill Book Co., New York, N. Y., Vol. V I I , p. 13. (1) R. (2) H. (3) C. Physiol.,
where n is refractive index of particles, no that of the medium, and X is the wave length in microns, in vacuo. A small correction was made in calculating T / C , for the fact that latices were diluted for measurement in 0.1% Triton X-100, rather than in pure water, but this was insignificant. The density of rubber was taken as 0.906 g./cc. Measurement of Turbidity.-The turbidity of latex dilutions of 2-15 X 10-0 g./ml. solids was measured in 10-cm. cells in a Cary Model 14 spectrophotometer, in the range X = 0.6 to 1.1 p . No change of T / C with concentration was observed in this range. (At very small en displacements T / C tended to be larger than a t large displkements, but the same effect was found with the ratio of absorbance to concentration for aqueous CuSO,.) This instrument waa selected because in this wave length range its optical system has a very small angle of acceptance at the detector, minimizing reception of forward scattered radiation,*^^ and the beam has to pass through a number of slits. Furthermore, the instrument has a wave length dispersion, Ax, of 37 8.l mm. slit. The instrument is of the double beam type, with blackened cell chambers. A solution of O.lY0 Triton X-100 in water served as the reference. In order t o test the reliability of these measurements, the turbidity of a number of Dow standard polystyrene latices was measured over the wave length range 0.6 to 1.1 p. Results are shown in Table I. Agreement with theory for the 0.814 p latex is almost perfect; in the case of the 0.088 p latex the material seems t o behave more like a monodisperse material of 0.083 U, than 0.088 p.lo The deviations between the theoretical and measured values of T / C may also be connected in some cases with the value of AX/X; i.e., one cannot distinguishll in fact between a dispersion in diameter ( D )and between a dispersion in a, where
In the case of the 1.17 p latex, some forward scattering radiation may have been received, because of its large particle size.8 The deviations from theory in this case cannot be explained from the deviations from monodispersity found by Coulter counter measurements.’* (Last row of Table I.) We are indebted to Dr. J. W. van der Hoff of Dow Chemical Company for these latices. This is Sample 1 of reference 12. In order to compare theory with experiment it was necessary to estimate values of m for the polystyrene-mater system. The refractive index of polystyrene us. wave length was obtained from industrial research.I3 For polystyrene
m
=
1.18392
+ 0.00281227 + 0.000309505 A2
where 1 is wave length in vucuo, microns. polystyrene was talien as 1.057.
A4
(3)
The density of
(8) R. 0. Gumprecht and C. M. Sliepcevich, J . Phys. Chem., 6 9 , 849 (1955). (9) W. Heller and R. M. Tabibian, J . Colloid Sci., 12, 25 (1957). (IO) I n this connection see G. Dazelic and J. P. Xratohvil, J . Colloid Sci., 16, 561 (1961). (11) M. L. Wallach, private communication. (12) M. Wales and J. N. Wilson, Rev. Sci. Instr., 32, 1132 (1961). (13) Polaroid Corporation, “Optical Plastic Material: Synthesis, Fabrication, and Instrument Design,” OSRD Report No. 4417 (1946).
PARTICLE SIZEDISTRIBUTION IN RUBBER LATEX
Oct., 1962
1769
TABLE I DOW MONODIBPERXE POLYBTYRENE LATICEB A. D = 0.088p 0.60 0.65 0.70 0.75 0.80 0.85 0 . 9 0 0.95 1.00 1.05 1 . 1 0 685 489 361 275 215 171 138 115 97 83 71 90 74 61 716 520 384 290 223 175 138 108 850 616 456 348 265 208 165 132 108 88 73
'hRBII>ITY ~ ~ X S W R E M E N T ON B
X1, P
Av. T / C , 4.5-13.5 X lo-* g./ml. Four concn. Theoretical r / c . D = 0.083 p D == 0.088 p Av. r/c, 2-4 X lov6g./ml. Theoretical 7 / c
B. D = 0.264 p ; T / C multiplied by 10-3 Four concn. 9.65 7.66 6.25 5.11 4.26 3.55 2.95 2.47 2.13 1.80 1.53 9.74 7.90 6.48 5.36 4.43 3.68 3.09 2.59 2.18 1.84 1.57 C. D = 0.365 p ;
Av. T/C, 8-20 X 10-4 g./ml. concn. Theoretical r / c Av. T / C , 7-11 X 10-6 g./ml. Theoretical r / c
multiplied by l o A 3
15.6 12.3 9.88 8.18 7.04 6.03 5 . 2 3 4.54 4.03 3.74 3.18 16.7 13.2 10.6 8.75 7.30 6.17 5.31 4.64 4.08 3 . 6 0 3.11
D. D = 0.814 p ; T / C multiplied by IO-' Two concn. 3.50 3.02 2 . 6 2 2.30 2.05 1.82 1.63 1.46 1.31 1.19 1.07 3.62 3.11 2.69 2.36 2.07 1.82 1.62 1.45 1.33 1.21 1.11
E. Sample 1; D = a t 10-6 g./ml. 3.12 Theoretical r / c 3.89 T / C from Coulter counter distribution12 T/C
T/C
Four
1.171 p ; T / C multiplied by IO+ 2.93 2.72 2.53 2.34 2.17 1.99 1.82 1.68 1.56 1.45 3.60 3.28 2 . 9 4 2.68 2.42 2.20 2.00 1.82 1.66 1.52 3.32 2.59 1.98 1.54
Coulter Counter Measurements.-The Coulter counter is an instrument in which colloidal particles flow throu h a small orifice in very dilute suspension. Simultaneouiy an electric current is flowing through the orifice and aa each particle passes, a momentary change in voltage drop occurs, which is amplified and counted. B means of a discriminator circuit it is possible to count o n g those pulses greater than some adjustable arbitrar threshold, the sire of the pulse being nearly proportionaf to particle volume. A known small volume of suspension is automatically metered through the orifice. The instrument is described more fully e l s e ~ h e r e . ~ - ~ . ~ ~ With a 30 p orifice, the instrument can detect particles not smaller than 0.5 p in diameter. This restricts its applicability to latices of small particle size. Preparation of Latex for Measurements.-Suspensions first were diluted in 0.1% Triton X-100 (a non-ionic surfac$ant) and then a very small known quantity of this dilution was added with stirring via a gravimetrically calibrated Calab micro-syringe to 250 ml. of the electrolyte, 2% NaCl containing 0.1% Triton X-100 as a dispersant. It is believed that the ver low final particle concentration (of the order of 1 p.p.m. Ctex solids or less) plus the presence of relatively large amounts of surfactant are effective in suppressing coagulation, ?t least for times an order of magnitude greater than the time of memurement. No evidence of instability, as shown by a systematic drift of count with time, has ever been noticed. The electrolyte always was filtered through Millipore filters of 0.3 or 0.45 p pore size. The 250 ml. graduated sample beaker of the apparatus was filled with filtered electrolyte, after washing with the same. Then a background count was made over the whole range of thresholds. The measured amount of latex dilution then wm added, and the final count made over the range of thresholds, the background count being subtracted, If necessary, more latex dilution was added to the beaker to obtain counts a t high thresholds and the resulting count scaled appropriately to tie in with the previous lower concentration count. Usually this scaling was done by repeating several counts at thresholds previously counted and employing the ratios of coincidence corrected counts.
Results and Discussion A representative cumulative number distribution for ti Hevea latex, obtained with the Coulter counter, is shown in Fig. 3. The general form of these curves for the polyisoprene latices is the same (14) Operating Manual, Coulter Industrial Sales Gorp., P. 0. Box 22, Elmhurat, 111.
3t
8=Q
I
i' 2 2 X
-Y 1
0
0
0.5 1.0 1.5 2.0 Weight median diameter, p. Fig. 1.-Turbidity vs. mean diameter and breadth parameter a t X = 0.65 p.
as in Fig. 3. The curves through the data were obtained by fitting the points to a cumulative lognormal distribution in threshold, where the threshold, t, is proportional to particle volume.
where N is the true count a t threshold t , No is the total number of particles in the metered volume which passes through the orifice, to is the number median threshold, and Pt = 3p. The P notation refers to the probability integral
The fitting was done by a machine procedure for a number of assumed values of No, subject to the coiistraint that
M. WALES
1770
Vol. 66
1.2 I 1 ,! i since we are dealing with a lower counting limit of I -
0.5 p and values of D,, which usually are considerably higher than this. I n Table I1 is shown a comparison between values of D,, and p obtained from the Coulter counter and from turbidity spectra, for a number of polyisoprene and Hevea latices. TABLE I1 D~STRIBUTION PARAMETERS FOR VARIOUS LATICES D,,,. wt.
diameter median
0.7
0.9
0.8
Dm,P. of guide curves to characterize a latex, Hevei NC400.
Fig. 2.-Use
Diameter, 0.94
0.44
f 1041
EL.
2.02
\o Computer Fit Dwm = 0.82
= 0.609 = 398,000
NO
4
\
\
101
I
I
I
10-2 Fig. 3.-Cumulative
,,,,,I
1
\
!
Q\,,, , !
10-1 1 10 Threshold, t. number distribution for Hevea latex NL 76.
where V is a known quantity, the volume of particles in cubic microns per metered volume. The residual sum of squares also was obtained, but this was insensitive to the choice of No. Since the fit of eq. 4 to the data was uniformly good, the values of D,, and p obtained from this type of curve fitting were taken as characteristic parameters for each latex sample. Note that for the Hevea latices, about 30% by number of the sample is accessible to the counter. The weight percentage is, of course, much higher,
Latex
Polymer
1 2 3 4 5 6 7 NL 78 NL 356
Polyisoprene Polyisoprene Polyisoprene Polyisoprene Polyisoprene Polyisoprene Polyisoprene Hevea Hevea
8. distribution
breadth j3 from TurTur- regresCounter bidity Counter biditv s1on
--in
p-
0.92 1.02 1.31 1.23 1.82
0.84 0.94 1.34 0.97 1.05 .98 1.48 1.06 1.05 .81 1.25 0.90 1.18 .70 1.00 .72 2.0 .98 1.10 .79 . , 0.51 . . 1.1 .79 . , 1 . 2 . . 1.25 .90 0.82 0.78 . 6 l 0.92 .66 0.83 0.72 .65 1.05 .76
Latices 6 and 7 were studied by electron microscopy, giving for latex 6, a number median diameter of 0.127 p, D,, of 0.53 p, from which p = 0.98. Latex 7 had D,, of 1.4 p from electron microscope measurements. Note that there is a slight tendency for turbidity spectra to give low weight median diameters compared to the counter, but that the parameter p obtained from turbidity measurements is uniformly high. It is believed that the reasons for this finding are (a) the wave length dispersion AA for the Cary spectrophotometer is not zero; (b) forward scattered radiation is being received by the detector to some extent. Use of a wave length band of finite width would tend to have little effect on apparent diameter a t a mean diameter where turbidity is not increasing very rapidly with diameter as is the case with these materials, but would produce an apparent widening of the distribution by a dispersion in a. The second effect would lower the apparent turbidity more a t lower wave lengths than a t higher ones and would tend to lower D,, if the system lies below the maximum in r / c us. A and to raise p. The second effect should lower the apparent turbidity increasingly severely as particle size increases.* If anything, p values from the Coulter counter have been suspected as being too large.Iz Finally, since the Coulter counter actually traces an appreciable part of the particle size distribution function, the p parameter derived from it is preferred. We find that if p from the counter is plotted us. the turbidity @, the results can be represented quite well by a linear regression, using the method of least squares p,, -0.008 0.728Pturb O.7288turb (7) afore data than are presented here were used in obtaining this relation, twelve comparisons in all. Values of p calculated from the turbidity values, by this relation, are shown in the last column of Table 11.
+
Oct., 1962
PARTICLE SIZE DISTRIBUTION IN
Nearly all the polyisoprene samples in Table I1 were prepared as part of a polyisoprene latex reRonay, W. M. Sawyer, search program by G. K. E. Manchester, N. C. May, and W. C. Simpson of these Laboratories. Electron microscope data on two of these materials were made available through the courtesy of A. M. Cravath of these Laboratories. Appendix
s.
Turbidity Relationships in Polydisperse Suspensions.For a monodisperse suspension
rKD2N ,. = E-1I n -IIo 4
(8)
I
where 1 = length of scattering path, 10and I are incident and emergent light intensities, D is particle diameter, N = number of particles/cc., and K is an involved function’s.’*
K
=
fh4
(9)
where m = n/no as previous1 defined and CY = moD/X, X measured in vacuo. For poly&perse suspensions ?r
7
= -
C KiDi2Ni
(10)
4 i
which becomes, in terms of concentration, c, in:g./ml.
where p is particle density and yi is the weight fraction of species i in the disperse phase. For the case of a continuous distribution
where f(D ) is a differential weight distribution function in D normalized to unity. A number distribution function
was chosen, where
and
(15)
D,, = With this choice, eq. 12 becomes
Here (./c)i refers to T / C for a monodisperse suspension of diameter D. The integration of eq. 16 for various values of D,, and B a t different wave lengths was carried out numerically by Weddle’s Rule on the Bendix computer using the values of ( ~ J c ifrom ) Table 111, which here have been rounded off t’o save space. A triangle approximation was used for m the portion of the integral from D = 5.4 to D The values in Table I11 had in turn been machine cal=i
.
(15) “Tables of Scattering Functions for Spherical Particles,” National Bureau of Standards, Applied Math. Series 4, U. S. Govt. Printing Offioe, 1948. (16) W. S. Pangonis, W. Heller, and A. Jacobson, “Tables of Light Scattering Functions for Spherical Particles,” Wayne Univ. Press, 1957.
RUBBERLATEX
1771
TABLEI11 CALCULATED TURBIDITIES OF MONODISPERSE POLYISOPRENE LATICES AT FOUR STANDARD WAVELENGTHS X T / C , cm.*/g. Diameter#
0.65
Wave lengtha 0.80 0.95
0.00745 0.00317 ,0548 ,0240 .I584 .0735 .2980 .I505 .4364 .2415 ,5688 .3307 .7236 ,4147 .8981 .5055 1.056 .6118 1.195 ,7235 1.340 .8245 1.484 .60 .9148 1.618 .65 1.006 1.745 1.100 .70 .75 1.865 1.191 1.981 1.276 .80 .85 2.094 I. 359 .90 2.197 1.438 .95 2.290 1.513 1.00 2.383 1.587 1.05 2.469 1.660 2.543 1.10 1.728 2.614 1.789 1.15 1.20 2.679 1.849 1.30 2.786 1.964 1.40 2.865 2.059 2.921 1.50 2.144 1.60 2.949 2.214 1.70 2.952 2.269 1.80 2.932 2.314 1.90 2.893 2.341 2.00 2.830 2.359 2.10 2.752 2.362 2.20 2.659 2.354 2.30 2.551 2 334 2.40 2.431 2,303 2.50 2.307 2.263 2.60 2.173 2.213 2.70 2.036 2.157 2.80 1.900 2.090 2.90 1.763 2.021 3.00 1.626 1.943 3.40 1.150 1.605 3.80 0.8232 1.259 4.20 .6600 0.9513 4.60 .6279 ,7186 5.00 .6707 ,5756 5.40 .7250 ,5125 Both expressed in p, A in vacuo. 0.05 .10 .I5 .20 .25 .30 .35 .40 .45 -50 .55
0.00157 .01209 ,0382 .0819 .1397 ,2037 .2660 ,3248 ,3851 ,4353 .5303 .6077 ,6781 .7418 ,8043 .8691 .9346 .9974 1,057 1.115 1.171 1,225 I . 277 1.329 1.427 1.513 1.597 1.669 1.734 1.792 1.841 1.882 1.916 1.942 1.958 1.969 1.972 1.968 1.956 I . 940 1.915 1.888 1.723 I . 503 1.259 1.020 0.8080 0.6391
1.10
0.000864 ,00672 ,0216 .0476 .0844
,1288 .1762 .2223 .2659 ,3093 .3567 ,4101 .4676 .5243 .5763 .6237 .6696 ,7167 .7651 .E4130 .8589 ,9029 * 9457 ,9874 1.067 1,144 1.217 1.281 1.343 1.401 1.451 I . 499 1,540 I . 577 1.608 1.634 1.657 1.672 1.684 1.691 1.693 1.691 1.640 1.534 1.386 1.213 1.033 0.8572
culated from K(a,m)from the Mie relations. The reduction of these to an algebraic form suitable for computation is too long t o reproduce here; Intercom 201 was used and the Bessel functions involved and their derivatives were obtained from J*a,z(or) and J * t { ~ ( aby, ) repeatedly using the recursion relationships between successive cylinder functions. Values of K(a,m) for m = 1.20 and CY = 0.2, 3.0, and 10.0 were compared with the tables of Pangonis, Heller, and Jacobson1* in checking out the program. Agreement t o at least four significant figures was obtained.
Acknowledgment.-The author is iiidebted to S. J. Rehfeld for experimental assistance.
As. WALEB
1772 DISCU8SIOK
V. K. LA MER (Columbia University).-In a paper to appear shortly in J . Colloid Sci., Wachtel and I find agreement with Dr. Wales that the Coulter counter is a very rapid and accurate method of getting size distribution curves of fine and, in some cases, electrically charged emulsion droplets in the range 0.4 to 1.0 p. After working the last fifteen years to determine particle size distributions from light scattering data alone in the absence of knowledge of the distribution function, I have abandoned these attempts as too tedious and unrewarding for the effort spent in favor of direct counting methods which extend light microscopic methods t o much lower values.
R. D. VOLD (University of Southern California).-Do you have definitive evidence that your system is free of aggregates? If there should be any aggregation it is necessary to know the refractive index of the aggregate, the density of the aggregate, and the internal structure of the aggregate in order to derive ita scattering coefficient and, since this information is generally not derivable from the observed data, interpretation in terms of a physically real system becomes nearly impossible.
Vol. 66
M. W A L E S .would ~ like to emphasiee that there is no attempt here to determine a detailed particle size distribution by light scattering, but merely t o derive a parameter by which one can rank various preparations in their order of polydispersity. Furthermore, for the. distribution of sizes to have an effect on the turbidity, it IS necessary that the ratio of article diameter to wave length be such that one be not too from the maximum in T / C (see Fig. 1). I n regard to aggregation, if this were such that “interpretation in terms of a physically real system were impossible,” the data of Table I could not have been obtained. These results are in agreement with those of Heller, et al., J . Colloid Sei., 11, 195 (1956); Bateman, ed al., ibid., 14,308 (1959); and DeieliC., et aE., ibid, 16, 661 (1961). JVe also have found that mean diameters of two Dow latices by the Coulter counter, using suspensions of known solids content, agree quite well aFith optical and electron microscope values. Also from dilution data in the Coulter counter, we find that concentration-dependent aggregation is most certainly absent. Changing the mode of dilution and the electrolyte does not change Coulter counter results for polystyrene or rubber latices. Finally, all of these measurements were made in the p.p.m. range of solids content or less, and in the presence of a large amount of surfactant.
kr
