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E. A. HAUSER AND H. K . SCHACHMAN
PARTICLE SIZE DETERMINATION OF COLLOIDAL SYSTEMS BY T H E SUPERCENTRTFUGE’ E. A. HAUSER
AND
H. K. SCHACHMAN
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts Received September 20, 1959
The possibility of determining particle size or particle size distribution in colloidal systems must today be considered a problem of utmost importance to the colloid chemist. Desirable as this information is, its attainment is in many cases only approximated, unless highly involved or costly methods are employed. This paper outlines a method whereby it is believed that particle size determinations of colloidal system9 can be simply and accurately obtained. First let us briefly examine the limitations of the most outstanding methods now in use. A simple convenient optical method is the direct count of reflection disks in the slit ultramicroscope. From this number and the specific gravity and concentration of the substance in suspension, it is a simple matter to evaluate the size of the particles. The method is strictly accurate only as long as monodisperse systems are being examined, and even then only a well-trained observer will be able to obtain reliable results. These limitations rule out exact determinations in most colloidal systems, which rarely are monodisperse. Methods involving kinetics and mechanics such as measurements of sedimentation rates when applied to colloidal systems are vitiated by Brownian movement and adhesion effects. Also, unless gravity is replaced by forces of higher magnitude they are extremely time-consuming. The Svedberg (6) and his collaborators were the first to apply OdBn’s work on sedimentation in an extremely strong centrifugal field. McBain (4), B e a m ( l ) ,and others have also contributed much in this direction. These ultracentrifugal methods possess a high degree of accuracy and are valuable for many purposes. However, machines of the Svedberg type are expensive both in initial cost and in operation and are, therefore, available only to a few laboratories in the entire world. McBain’s ultracentrifuges of the spinning-top type (opaque as well as transparent) are reasonably priced and simple to operate. However, they too are of extremely small capacity and do not permit a more or less continuous production of reasonable quantities of individual fractions. Lenoir (3) extended Svedberg’s principles to apply to the supercentri1 Presented before the Division of Colloid Chemistry a t the Ninety-eighth Meeting of the American Chemical Society, held in Boston, Massachusetts, September, 1939.
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585
fuge manufactured by The Sharples Corporation of Philadelphia, Pennsylvania. Hauser and Reed (2), continuing this work, systematically developed a method of obtaining particle size distribution by the use of the supercentrifuge. This method overcame the limitations of availability and cost of equipment and permitted particle size determinations to be performed simultaneously with a reasonable degree of fractionation. The mathematics of this method, however, still are tedious and extremely involved. Because the factors of availability, price, and continuous fractionation even in highly colloidal systems are of such importance, it was felt that a method of particle size determination, by means of the supercentrifuge, which presented no real mathematical difficulties nor time-consuming operations would be of real value. THEORY UNDERLYING USE O F T H E SUPERCENTRIFUGE
Although it is well known that few colloidal systems consist of spherical particles, mathematical evaluation taking account of shape is as yet beyond our abilities. To permit evaluation of all systems the standard of an “equivalent spherical diameter’, (e.s.d.) was set up. The e.s.d. of :t particle is an arbitrary dimension equal to the diameter of a sphere which sediments a t the same rate as the particle under consideration. This convention permits solution of the mathematical problems and gives relative values as long as due cognizance of morphological considerations is taken. Prior to a mathematical analysis of the problem, a logical analysis should be of value in outlining the variables to be encountered. Since the particles in suspension generally are denser than the dispersion medium they will settle towards the walls of the centrifuge bowl under the influence of the centrifugal force. The three major controlling factors for any sedimentation are the density difference between the disperse and continuous parts of the system, the magnitude of the gravitational or centrifugal force applied, and the kind and magnitude of any other forces applied. (Hauser and Rtled, applying Stokes’ law for falling bodies and Newton’s law for viscous flow have already solved the problem involving these factors in the sedimentation equation.) In addition, large particles will be thrown out on the liner sooner than would small particles, and, consequently, the large particles will settle a t the bottom of the bowl and the small ones a t the top. Turning from general considerations to a closer examination of a particle as it passes through the bowl, it is apparent that the path must be completely described by two velocity vectors of the particle, one parallel to and one perpendicular to the axis of rotation of the bowl. Also, if external conditions such as type arid rate of feed, temperature, angular
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E. A. HAUSER AND H. E. SCHACHMAN
velocity, etc., are maintained constant, the point a t which a particle hits the wall will be a function of the point of departure and the size of the particle. Coming back to a mathematical viewpoint, Y = f ( D , X O ) ,where Y is the distance up from the bottom of the bowl a t which the particle settles, D is the equivalent spherical diameter of the particle, and X Ois the distance from the axis of rotation a t which sedimentation started (figure 1). MOTOR COURlffi
FIG.1. Path of particles in the bowl of
D
a
Sharples supercentrifuge
x AND
Y As a result of previous work done on a similar centrifuge, the assumption can be made that streamline flow is in effect with the streamlines parallel to the axis of rotation. Obviously there are some slight errors in that assumption, owing to end effects a t the bottom and top of the bowl. The possibility of turbulent flow a t the very bottom of the bowl, due to the speed of input through the small entrance nozzle, is eliminated by taking the entrance plane a small distance above the nozzle. This seems permissible because the turbulent region is probably confined to the space in which the straightening vanes act. If the suspensions used throughout all the work on the centrifuge are sufficiently dilute so that they can be considered truly viscous fluids obeying Newton’s law of viscous flow, we obtain, by following the derivation of the velocity equation as given by Lamb, an equation for the velocity of any streamline parallel to the axis of rotation. By the derivaTHE EVALUATION O F
AS A FUNCTION O F
PARTICLE SIZE IN COLLOIDAL fiYsTEM18
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tion of OdBn, using Stokes' law for falling bodies, an equation for the velocity component perpendicular to the axis of rotation is obtained. Now, the two velocity components can be combined to find D as a function of X o and Y .
At constant values of &",in., 7, Ap, and w and integrating between the limits X = X o , Y = O'and X = R2, Y = Y the desired equation is obtained.
where Y = the distance from the bottom of the bowl a t which the particle strikes, the distance from the axis of rotation a t which the particle starts, D = the equivalent spherical diameter of the particle, K = a constant, depending on the dimensions of the bowl (1.log), t l = the viscosity of the dispersion fluid in poises, Qmin. = the rate of flow of the suspension in cubic centimeters per minute, A p = the difference in density between the disperse and the continuous phases, the distance of the overflow weir from the axis of rotation in centimeters, Rz = the radius of the bowl lass that of the liner in centimeters, t = time, xo = the distance from the axis of rotation a t which the particle starts, and w = the angular velocity of the bowl in radians per second. While the above equation fulfills the task of obtaining Y as a function of Xo and D, simple determinations of particle size are still impossible, since the equation is implicit in X Oand therefore has no exact solution.
x=
DEVELOPMENT OF THE FINAL EQUATION
If the conditions be imposed that t,he system under consideration be well defined (particle sizes covering but a small range) and that passage
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E. A. HAUSER AND H. K. SCHACHMAN
through the centrifuge be a t an appropriate rate, the alignment of particles will approach a continuous distribution from large particles a t low values of Y to small particles a t high value$ of Y . Thus, contrary to the first assumptions, at a given point on the bowl there is essentially but one particle size or, conversely, a particle a t a given value of Y could have started only from a unique value of X o . Hence the further limitations have permitted the elimination of D as an independent variable, so that Y = ~ ( X Oholds ) with but negligible error. In the transformation of
Y
2.0
1.2
1.6
.8
4
0
YO
FIQ.3 FIQ.2. Plot of C against Y FIQ.3. Showing the paths of the various particles in the bowl and the relationship between X oand Y
equation 2 to the form Y = f(XJ it is important to note that Qmin.must be proportional to D2,a fact which is of considerable value in the fractionation procedure. Justification of this new equation is found in the work of Reed ( 5 ) . H e found that the slope of the cumulative weight per cent curve wm directly proportional to the square of the diameter of the particle and inversely proportional to the rate of flow of the suspension. Furthermore, if the fractionation is carried out so as to conform to the limitations imposed,-a well-defined range of particles and uniformity of the material
PARTICLE SIZE IN COLLOIDAL SYSTEM8
sedimented on the liner,we have
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the slope of this curve is constant. Therefore, &min.
= aD2
(3)
a being a constant.
FIG.4. Circular slide rule for solving the equations C = MD*Y or D = 4 C T Y
Since X o depends only on Y , the quantity in brackets of equation 2 can be assigned numerical values for all values of Y . This quantity has been called C and a plot of C versus Y made (figure 2). Figure 3 has been constructed to show the paths of the various particles in the bowl and the relationship between X o and Y . Calling the product of all the other constants in equation 2 1 / M (modulus of sedimentation), a final simple equation is reached:
Y = C/MD2
(4)
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E. A. HAUBER AND H. K. BCHACHMAN
or, since D is what is desired,
D = dC/MY
(5) TO facilitate the solution of this equation, an alignment chart in the form of a circular slide rule has been constructed (figure 4), with the aid of which particle sizes can be calculated in a few minutes by anyone. SAMPLE CALCULATIONS
T o illustrate the method the particle size of a fraction from a bentonite suspension will be calculated. Data on the fractionation are as follows: temperature of suspension = 72°F.; 7 = 0.00953 poise; w = 51,500 R.P.M. = 5390 radians per second; density of bentonite = 2.78; Ap = 1.78 g. per cubic centimeter; Qmin. = 41.75 cc. per minute. ( A ) The first step is to calculate M , the modulus of sedimentation for the run. 46Ap~'lO-'~- 46 X 5390' X 1.78 X M = 0.00953 X 41.75 7Qmin.
M = 0.0000597 ( B ) The next step is to pick a value of Y a t which we desire to obtain the size of the particle. Y = 2 is chosen. (C) Picking Y fixes X o and C. The value of C is read off the graph of C versus Y (figure 2). It is 0.072. (D)The slide rule is now used to find D. The value of C (innermost scale) is brought opposite the value of M (outermost scale). The pointer is now moved to the value of Y on the second outer scale and the value of D read off opposite it on the second inner scale. It is found to be 24.6 millimicrons. To avoid mistakes in reading the slide rule it is advisable to have the two sets of scales on the slide rule drawn in two distinctly different colors. ( E ) For particle sizes of this fraction a t other points the process is repeated, utilizing the desired values of Y . FRACTIONATION PROCEDURE
Inasmuch as the derivation of the final equation was made possible only by a number of assumptions, it is necessary to carry out the actual experimental work carefully so that all these are valid. The layer of material sedimented out must not be too thick and must be of uniform depth for the entire length of the liner. The suspension fed into the centrifuge should contain only a limited range of particle sizes. To secure these conditions the fractionations are carried out in two stages. First, a run is made under appropriate conditions of concentration and rate of flow. Extreme care as to constancy of flow and speed of
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rotation and to thickness of sedimented layer is not required in this first run. The bottom portion of this layer is rejected, since the first run is made with a polydisperse sybtem and all the ‘particles larger in size than called for by the calculation will deposit in the layer closest to the bottom of the bowl. The rest is redispersed and rerun. During the rerun all possible precautions are taken in regard to the points mentioned above. The very bottom portion of the sedimented layer (for example, a strip of 1 cm. height) is again rejected,.as a precautionary measure. The rest is divided into the requisite number of fractions. If the amount of suspended material is very great in the range being sedimented, it is advisable to rerun the effluent liquor, adding the additional sediment of that secured in the first run before going through with the redispersion and so forth. To get still finer fractions the effluent liquor is now run through the centrifuge a t a slower rate and the above procedure repeated at this new rate of flow. By suitably varying rotational speed and rate of input of suspension any desired degree of separation can be secured and calculations made as to the size of the particles so obtained. The authors wish to express their sincere thanks to Professor R. D. Douglas of the Department of Mathematics of the Massachusetts Institute of Technology for his valuable advice and assistance. Their thanks are also due to Mr. L. L. B a r d of the Department of Chemical Engineering for his assistance in the preparation of this paper. They also wish to express their appreciation to The Sharples Corporation of Philadelphia, Pennsylvania, for putting a steam-driven supercentrifuge a t their disposal to make this work possible. REFERENCES (1) BEAMS,J. W.:Rev. Sci. Instruments 6, 299 (1935). (2) HAWSER, E. A., AND REED,C. A.: J. Phys. Chem. 40, 1169 (1936). (3) LENOIR,W. F.: Master’s Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1936. (4) MCBAIN,J. W., AND O’SULLIVAN, C. M.: J. Am. Chem. Soc. 67,2631 (1935). (6) REED,C. E . : Doctor of Science Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1937. (6) SYEDBERQ, THE: Ind. Eng. Chem., Anal. Ed. 10, 113 (1938). This paper is an up-to-date summary of Svedberg’s work and contains also an excsllent summary of the literature.
