0. €3. ALRXANDER AND R. IC. ILER
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DETERMINATION OF PARTICLE SIZES IN COLLOIDAL SILICA BY G. €3. ALEXANDER AND R. K. ILER A Contribution from the Grasselli Chemicals Department, Experimental Station, E. I. du Pont .de Nemours and Company, Inc. Received April 10, 1968
A series of alkali stabilized solutions of colloidal silica consisting of relatively uniform discrete non-porous particles ranging in diameter from about 16 to 60 millimicrons has been prepared and characterized in regard to (a) particle weight by light scattering, (b) particle size as determined from electron micrographs, and (c) specific surface area of the dried silica, measured by nitrogen adsorption. The particles appear to consist of dense amorphous silica, since the “wei ht average” particle diameter, estimated from electron micrographs, agreed reasonably well with the ‘‘weightaverage” particfe diameter calculated from the particle weight determined by light scattering, assuming a density of 2.2 for the silica; the ratio of the two figures ranged from 0.83 to 1.26. The particle diameter calculated from the specific surface area of the dried silica, in five out of six sols also was fairly close to the “surface average” particle size estimated from electron micrographs; the ratio of the two values ranged from 0.75 to 0.89. The uniformity of particle size in the sols is indicated by the fact that the “weight average” particle size, determined from electron micrographs, averaged only 26% greater than the “number average” size; this corresponds to a “weight average” particle weight about twice the “number average.”
Introduction The process developed by Bechtold and Snyder1 has made it possible to prepare silica sols of relatively uniform particle size within the range from about 15 to 130 millimicrons in particle diameter. Colloidal solutions of silica of this type offer an opportunity to compare various techniques for measuring particle diameter or weight, including direct observation of diameter from electron micrographs and calculation from specific surface area and, alternatively, from the particle weight determined by light scattering. For such comparison, a series of six sols, ranging in particle diameter from about 16 to 60 millimicrons, has been prepared, the particle size distribution observed from the electron micrographs, the specific surface area found by the nitrogen adsorption method, and the particle weight determined by light scattering.
The growth of silica particles during the process has been calculated, based on the theory that all of the active, low molecular weight silicic acid fed into the evaporator is deposited on the colloidal particles which are present. Consider 1 unit volume of sol initially in the evaporator, e.g., 1 liter of sol containing G grams of SiOz. Let this contain N particles at time, t , and N oparticles at t = 0. Let the particle weight of the particles be M and M o and diameter of the particles be D and Domillimicrons at time t and t 0, respectively. Let there be X volume of product containing G grams of Si02 per liter withdrawn at time, t; X is proportional to the weight of active Si02 which has been added to the evaporator in time, t . Then N = 6 X loz3(G/M) (1) 5
.The rate of addition of active silica to the system in grams pilr liter per unit time is (dX/dt)G. The rate of increase in weight (W grams) of a single silica particle in grams is dW dX G
dt=dt.Xw Since
Experimental Preparation of Sols.-As a starting material for the preparation of silica particles of desired size, a commercially available silica sol, “Ludox” Colloidal Silica2was employed. A steel, submerged tube evaporator was charged with 12.5 gallons of the starting sol which contained 30% Si02, consisting of particles averaging about 15 millimicrons in diameter, and stabilized with sufficient alkali to give a molecular ratio of SiOz: NazO of 85: 1. The particles were then grown by the process of Bechtold and Snyder,’ using a freshly made silica sol prepared by ion-exchange in accordance with example 3 of their patent. I n this step a solution of sodium silicate having a weight ratio of SiO2:NaZO of.3.25:1,OO was diluted to give a solution containing 2.4% SiOz, which was then passed through a column of the hydrogen form of an ion exchange resin (“Nalcite” HCR). The effluent solution of polysilicic acid was alkalized by the addition of a sufficient amount of the dilute solution of sodium silicate t o give a mole ratio of Si0z:NazO of 8 5 : l . The dilute solution was added to the evaporator a t the rate of 21 gallons per hour. Evaporation was controlled so as t o maintain 12.5 gallons of sol in the evaporator, while at the same time continuously withdrawing product, at the rate of 1.4 gallons per hour from the evaporator. With minor variations, this rate of withdrawal served to maintain the concentration of SiOz in the evaporator very close to 30% by weight, which was followed by noting the specific gravity of the sol. The product sol was collected in fifty-five consecutive onegallon fractions. The turbidity of the product increased gradually and continuously from the first fraction, which was almost clear, to the last, which was extremely white and opaque, though in a thin layer, or when diluted, was transparent but red in transmitted light. Fractions 1 , 9 , 18,27, 35 and 50 were selected for characterization. (1) M. F. Bechtold and 0. E. Snyder, U. 1. Patent 2,674,902. (2) M. Sveda, 8oap and Elanit. Cbemioab (1949).
W = M / 6 X loz3 dW 1 dM _=____ dt 6 X 1023 dt
(3)
Then 1
6x1023
dM
dX
dt = dt
G
M
6 x 1 0 2 3
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whence dM/dt = M(dX/dt) (4) In (M/Mo) = X (5) Since M = 692D3 (assuming the particle has a density of 2.2) In (D3/DOa)= X D 3(2.30) loglo - = X Do
The slope of log particle size versus X according to the above equation is shown in Fig. 1, in comparison with the general slopes of the curves based on particle size data. Characterization of Sols. Particle Weight by Light Scattering.-The method of Stein and Doty3 was emplo ed using the light scattering photometer such as that descriged by Debye4 supplied by the Phoenix Precision Instrument Co., and originally described by Speiser and Brice.6 According to the theory of Stein and Doty, the weight average particle weight, M,, is a function of the turbidity, (3) R. Stein and P. Doty, J . Am. Chem. Soc., 68, 159 (1946) (4) P.Debye, J . Applied Phyr., 17,392 (1946). (6) Speirer and Brioe, J. O p t . 81%dm., 36, 846 (1948).
Dee., 1963 TABLEI PARTICLE SIZEDISTRIBUTION FROM ELECTRON MICROGRAPHS Fraction no. 10“
Total
15
20
25
Percentage of particles within *2.5 millimicrons of given size 30 35 40 45 50 55 60 65 70
75
80
85
90
1 2 7 . 1 28.6 30.5 13.8 9 1 6 . 8 30.3 24.8 17.5 10.2 0 . 4 0.4 18 8 . 7 22,2 2 6 . 9 23.2 17.4 1 . 5 0 27 0 . 7 6 . 0 1 5 . 2 23.6 2 4 . 0 19.4 9 . 5 1 . 4 35 8 . 8 1 . 0 3 . 3 8 . 8 12.7 19.6 20.3 11.4 7 . 9 6 . 2 5 0 . . .. .. .. 0 . 7 2 . 4 2 . 4 2 . 7 2 3 . 2 19.8 12.3 9 . 6 14.3 6 . 8 4 . 8 0 . 3 0 . 7 Includes all visible particles smaller than 12.5 millimicrons in diameter. 7, and the concentration, C, of a polymeric dispersion or solution, according to the equation HC/r = 1/M, 2BCJRT (1) where C = concn. of solute in g. solute per cc. of soh. Ai, is wt. average particle wt. of the solute in g. per mole B is a constant, characteristic of the system, which may be determined by osmotic pressure meafiurement, in ergs x cc./g.2 of solute. = gas constant in ergs/mole of solute per degree of temp. 1 = absolute temperature in degrees T = turbidity (extinction coefficient for scattering a t right angles in reciprocal centimeters)
+
E
particles counted
269 274 275 283 306 293
electron micrographs, and D, being the diameter of a particle having a weight equal to that calculated from the average particle weight estimated for each particle measured in the micrographs. Specific Surface Area.-Samples of the sols were converted to powders by adding concentrated HCl to lower the p H to 8.0, then adding an equal volume of t-butyl alcohol, which brought about incipient gelling. The mass was then dried in air a t 110’ and the specific surface area determined? by nit#rogenadsorption by the method described by Emmett,. 100 90 80
I
I
I
I
(3)
no = refractive index of the solvent dn/dC is equal to the index of refraction gradient between solvent and solution in terms of cc. per g. y = the wave length in air of light used in cent>imeters. No = 6.02 X 1 0 2 3 particles per mole (Avogadro’s number). The calibration of the photometer was based on opal glass standard. Then turbidity measurements were made on polymer solutions a t several concentrations. The particle weight was determined by plott,ing H C / r us. C for the readings taken, and extrapo1at)ing to zero concentration; the values used for no and dn/dC were 1.3338 (at 27’ and a wave length of 547 millimicrons) and 0.076, respectively. The particle weight was determined by the reciprocal of the intercept (see equation 1). Particle Size from Electron Micrographs.-Samples of sols were diluted to 0.1% Si02 or less before application to the supporting screen; negatives were obtained a t 5000 X magnification, and these projected on a screen at an over-all magnification of about 100,000 for measurement. All part,icles in randomly selected areas were classified as being 10, 15, 20, etc., millimicrons in diameter and the distribution of particle sizes (based, in each case, on measurement of about 300 particles) is shown in Table I. As described by Cohan and Watson,6 the number average (Dn), surface average (Ds) and the weight average ( D w )diameters were calculated, D. representing the arithmetic average diameter, and D. representing the diameter of a particle having a specific surface area equal to t.hat calculated from t,hc dist,ribution of diameters observed in the TABLE
11
Nitrogen Light adsorp. scattering Frac- Vol. of Particle Spec. Calcd. Particle Calcd. tion proddiameter surf diani., nt., diam., no. uct, S DO DS DW ,.ai;. D.P millionsb Dm 1 0.08 16.5 19.5 21.2 185 14.7 3.8 17.5 8.3 23 9 0.73 18.8 22.6 25.3 144 18.9 18 1.45 21.1 24.6 27.6 125 21.8 19.5 30 27 2.17 28.4 31.8 34 97 28.1 54.0 43 35 2.80 35.2 41.6 45 84 32.5 100.0 53 4.0 59.2 63.3 67 68 40.0 210.0 66 50
Assuming density of amorphous silica is 2.2, the specific surface area of a silica sphere D. millimicrons in diameter is calculated to be (2700D,-’) square meters per gram. b Assuming density of emorphous silica is 2.2, the particle weight of a particle D, millimicrons in diameter is (690D3). (6)
L. H.Cohan and J. H. L. Wataon, Rubber Ale, 68, 6S7 (1861).
VOLUMES OF PRODUCE. Fig. 1.-1, D, (light scattering); 2, D, (electron mjcrograph); 3, D,(electron micrograph); 4,D, (electron micrograph); 5, D, (nitrogen adsorption); 6, theoretical slope according to equation 6.
Conclusions
D, calculated from the particle weight determined by light scattering was generally somewhat greater than D,, the “weight-average” particle diameter estimated from electron micrographs, except in the case of sols containing an appreciable proportion of particles smaller than 19 millimicrons, which were difficult to measure in the micrographs. 2. The particle diameter Os, in five out of six sols, calculated from the specific surface area of the silica obtained by drying the sol, averaged 17y0 smaller than Os,the “surface-average” particle size estimated from electron micrographs. The sol of largest average particle size, which contained an appreciable fraction of small particles, along with R 1. As shown in Fig. 1, the diameter
(7) P. H. Emmett, “Symposium on New Methods for Particle Size Determination,” p. 95, published by the Amerioan Society for Teatins Materials, Mar, 4, 1941.
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HUGOFRICKE
majority of larger ones, was not included because it was difficult t o measure all the smaller particles. The specific surface area of the silica agreed more closely with D,, the “number average” particle size estimated from electroil micrographs. 3. The ‘‘weight average” particle diameter, D,, measured on micrographs, averaged 26% greater than Dn, the “number average’’ diameter; this corresponds to a “weight average” particle
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weight about twice the “number average” particle weight. The authors wish to acknowledge the work of L. A. Dirnberger in conducting the engineering studies and making mathematical calculations, W. M. Heston, Jr., in preparing samples for characterization, R. E. Lawrence in carrying out the analyses, and Dr. C. E. Willoughby of the Chemical Department in preparing the electron micrographs.
THE MAXWELL-WAGNER DISPERSION IN A SUSPENSION OF ELLIPSOIDS BY HUGOFRICKE Walter B. J a m e s Laboratory of Biophysics, Biological Laboratory, Cold Spring Harbor, N . Y. Received April 14, 1965
The dispersion equations are first derived for the conductivity and permittivity of a suspension of ellipsoids of vanishing volume concentration, in which the components are characterized by both conductivity and permittivity, by extending to complex admittances an earlier treatment2 of this system for the case of ure conductors. The expressions contain 8 “form factor” which is expressed in terms of elliptic integrals. I t s numerical varues are tabulated. For random orientation of the ellipsoidal axis the suspension is shown t o be electricall equivalent t o a simple resistance-capacity network and to be characterized by three dispersion regions inside each of wtich, it behaves quantitatively like a Debye dipole system. This treatment is subsequently extended t o more concentrated suspensions on basis of the semitheoretical general conductivity equation given in rGf. 2.
The problem can be dealt with in a simple manner A heterogeneous system of conducting dielectrics, in which the ratio of permittivity to conductivity is by making use of the theoretical information aldifferent in the different phases, exhibits Maxwell- ready available on the electric behavior of heterogeWagner (M-W)sp4 dispersion in the frequency zone neous systems composed of pure conductors. By where the field shifts over from its low to high fre- introducing complex variables, the conductivity quency course. Below the dispersion region, where equation for such a system is transformed to the the field is determined by the conductivities of the compIex conductivity equation for the same geomeconstituent phases, the conductivity of the system trical system of conducting dielectrics, from which has its minimal value, referred to in the usual kind equation the dispersion equations for conductivity of mixture formula, while the permittivity is greater and permittivity are thereafter obtained by sepathan the (minimal) value obtained when the field is rating real and imaginary terms. By using this determined by the permittivities. The opposite method, Wagner’s treatment of a suspension of condition, minimal permittivity, increased conduct- spheres could have been simplified since the conivity, is found at frequencies above the dispersion ductivity equation for this system was already given region. by Maxwell. The object of the present paper is to extend this Although the M-W effect is of considerable practical interest, its theoretical treatment does, not treatment to a suspension of ellipsoids. The conappear to have been extended beyond the two ductivity equation for this system was dealt with simple systems considered by Wagner,* viz., a strat- by Fricke,2where references to the earlier literature ified body and a dilute suspension of spheres. The will be found. The equation can be derived rigorbehavior of fibrous systems has been discussed ously only when the volume concentration is low, but qualitatively by different a u t h o r ~ , ~by- ~using sup- reference 2 describes also the derivation of a general posedly equivalent electric circuits. A more gen- conductivity equation, which has been found to eral treatment of suspensions which takes into ac- agree well with the experimental evidence. In count both the effect of particle form and higher dealing with the dispersion problem, we shall therevolume concentrations, has recently become of in- fore first consider the case of low volume concenterest in studies of biological materials in the ultra trations and thereafter extend this treatment to high frequency zone.8-10 higher concentrations by using the general conduct(1) Supported by the U. 9. Office of Naval Research. ivity equation of this earlier work. (2) H.Fricke, Phys. Rev., 24, 575 (1924). Theory.-We shall consider a suspension of (3) J. C. Maxwell, “A Treatise on Electricity and Magnetism,” 2nd homogeneous eIlipsoids (axis 2a 2 2b 2 2c) distribEd., Clarendon Press, Oxford, 1881. p. 398. uted a t random in a homogeneous medium. When (4) K. W. Wagner, Arch. Eleklrotech. 2 , 371 (1914). (5) S. Setch and Y. Toriyama, Insl. Phys. Chem. Research, Tokvo the components are pure conductors, the conducBci. Papers, 3 , 283 (1926). tivities of suspension, suspending and suspended (6) D. DuBois, A. I. E. E., 41, 689 (1922). (7) E. J. Murphy, THISJOURNAL, 33, 200 (1929). phases are called IC, ICI and kz,respectively, expressed (8) B. Rajewaky and H. Sohwan, Naturwissenschaflen, 3 5 , 315 -1 ohm-’ crn.-I. In the complex sysin (9 X loL1) (1948). tem, the corresponding quantities are written Q (9) H. F. Cook, Nature. 168,?47 (1951). ?/a,et,,.., where u and E represent conductivity ,(lo) H. F. Cook, Brdt, J . Apzd. Phus.. 2. 295 (1951).
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