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The influence of diffusion on sedimentational particle size analysis
Powder Technology.8
@ Elsevier Sequoia S.A.,
(1973) 13-17 Lausanne -
The influence of diffusion particle size analysis DAVID
W. MOORE*
Georgia Institute of (Received
and CLYDE
Printed in The Netherlands
on sedimentational
ORR,
Jr.
Technology. Atlanta. Ga. (U.S.A.)
January 9.1973;
accepted
February
28,1973)
analysis should be altered to minimize diffusional effects.
Summary Particle size analysis of submicrometer particles by gravity sedimentation is investigated to establish the influence of Brownian motion. Size distribution data from electron microscopic measurements are evaluated in terms of theoretical diffusion equations to determine the effect of diffusion. A parameter, defined as the mean ratio of Brownian to settling motion, is developed to guide the choice of system parameters so that Brownian diffusion effects can be minimized and reliable size results assured.
The basic mathematics of diffusion in the presence of settling was developed more than 50 years ago, although only the differential equation involved has yet been verified experimentallyl. * _This partial differential equation, describing the. time rate of increase in particle concentration as a function of depth in a settling and diffusing system, can be written
ac at
INTRODUCTION
-_=D
Sedimentation analyses based on Stokes’ law provide a convenient method fur determining partf;le size distributions. A lower limit of applicability is established, however, by the equalizing effect of diffusional forces, since Stokes’ law assumes that particles settle only under the influence of field and drag forces. The effect becomes significant when displacement apparticle sedimentational proaches the root mean square Brownian dis placement. It will be shown here how various solutions of the basic equation describing settling and diffusing systems can be employed to predict the influence of diffusion on a sedimentation analysis, and how parameters such as liquid viscosity and density and the time of *Present address: Atlantic Richfield Company, Box 147, Bakersfield, Calif. 93302 (U.S.A.).
THEORETICAL
P-0.
32Lv3C ax*
ax
(1)
where C is the number concentration, D the diffusion constant, t the time, v the sedimentation velocity, and x the depth below the upper surface. Published solutions to eqn. (1) divide into two categories on the basis of chosen boundary conditions: those that assume the top and bottom of the system container to retain, through van der Waals’ forces, all incident particles3 and those that assume top and bottom reflectance of incident particles4 -5. Other assumptions, viz., negligible wall effect, no particle-particle interaction or hydrodynamic influence, settling velocity in accordance with the Stokes-Einstein equation, and initial uniform distribution of particles, are identical in both types of solutions. The solution requiring top and bottom par-
title retention a was used in the eigen-function expansion form* o
C
(%3 2
C,=exp
2 sin (n;iry)[l - (--l)“exp(-1/2a)] n?r[1+1/(4&%%*)]
y =x/L _ I,,,L
18Llj
a=
6KT Lad’gAp
= D/vL
and CO is the initial number concentration of particles of diameter d, g the gravitational acceleration, rl the fluid viscosity, K the Boltzmann cons&t, T the absolute temperature, and Ap the particle density minus the liquid density. For the assumption of top and bottom particle reflectance both the equation’
c exp(y/a) C,=a[exp(l/a)-l] _2 n=1
exp(-anz9*t’)
was in
(%I
eqns. (2) and (3) was lo-*17. This factor was not computed; it was assumed to be zero. The value of y ranged from 0.005 to 0.1, giving a maximum value of lo-aas for the first righthand term of eqn. (3); it also was neglected. To facilitate making comparisons among experimental results, the ratio of Brownkan to settling motion was established. In quantitative terms this ratio is
where
t’ _ &A&”
value of a encountered experimentally 10e3, the maximum value of exp(-1/2a)
where symbols are as given previously_ If the mean particle diameter d of a distribution of particle sizes is used in eqn. (5) instead of d, the diameter of a monodispersion, thereby
+ 16a*7rexp (Q-4$‘)
exp(--an*v’t’)n[l+
[sin(nzy) + 2zna cos(nlry)J (- 1) -“exp(-1/2a)] (1 + 4n2n2a2)*
and a simplified approximation to it5 (4) were employed_ Equations (2), (3) and 4 were solved by digital computer using parameters appropriate to specific experimental systems as described subsequently. For small values of a, as is characteristic of large particles, the two series solutions converge very slowly_ It was found that if it required more than the first 6000 terms of the summation to evaluate either series, then the solutions of eqns. (2), (3) and (4) were virtually the same. Since the largest *An integral solution form for small values of n was also presentedin the referencedwork. It was found to contain a small printing error, the misprint being confirmed by the author. The two v’s in the first group of eqn. (46). p_ 109, should be u’s_ The correct arguments of the error functions in h, should be (2w+u)and (w+u).
(3)
overestimating the true value of u for a polydisperse suspension, then a is a function of t only for a given suspension. The mean value of OL,defined as 7, for a system can be calculated by integrating eqn. (5) over the time required for a sedimentation analysis and then dividing by the elapsed time of the analysis. This gives Y =36 Apg$
2KTq’
( 3nt, J
5
(6)
where t, is the time required for the sedimentation analysis. Finally, when 7 is expressed as a percentage of the total motion, it may be interpreted as the percent average Brownian motion that the mean size particle in the suspension experiences during a sedimentation analysis.
EXPERIMENTAL
Sedimentation data for four submicron par-
15
title systems were studied. These analyses were accomplished using a Model 5000, Particle Size Analyzer (Sedigraph) of the Micromeritics Instrument Corporation, Norcross, Ga. This instrument detects relative mass concentration by monitoring the transmittance of an X-ray beam passing through the sedimenting suspension 6 *’ . Continuous internal computation of Stokes’ law permits it to plot ‘Cumulative Mass Percent Finer” as a function of “Equivalent Spherical Diameter” on an X-Y recorder. During an analysis, the cell containing the suspension is completely sealed and filled; it moves downward at a known mte (ie., the point of mass concentration measurement moves relatively upward) as time progresses in order to reduce the time required for an analysis_ The temperature of the cell and surrounding compartment is controlled to within * 0.5OC, in order to minimize convection currents and, due to the fine collimation of the X-ray beam (0.0051 cm vertical thickness), the vertical position of a 0%pm-diam. particle can be located to within + 0.05 pm. Both sedimentation and electron microscopic sizes for two of the particle materials iron oxide and silver halide - were reported
Fig. 1. Electronmicrographof SiO,, Lot II, particles.
previously’. The other two samples were sibcas prepared specifically for this investigation by reacting tetrapentyloxysilane with water in the presence of ammonias. Figure 1 is a micrograph of some of these particles. As is evident, they are quite spherical and relatively uniform in size. Their size distribution was established with high precision using a Zeiss Particle Size Analyzer (Carl Zeiss, Oberkochen, wuertt.) and measuring the diameters of 1000 (Lot I) and 1494 (Lot II) particles. The iron oxide particles were generally spheroidal and the silver halide particles were cubic with rounded comers. Only 167 and 148 particles, respectively, were measured for the oxide and the halide, so their distribution was established with less reliability; uncertainty limits calculated by standard methods’ are indicated on the figures precenting these results. In order to compare theory with experiment, each distribution of particle size obtained from the micrographs was divided into n groups of essentially monodisperse sizes_ A value of settling time t was chosen which set the distance x that the X-ray beam would have been from the upper surface of the sedimentation had a sedimentation analysis been in progress. This permitted computing by
16
eqns. (2), (3), or 4 the predicted concentration of each monosize group at the chosen time and depth. Each of these R concentrations was then multiplied by the mass fraction represented by the group. The sum of these R products was taken to be the total mass fraction of particles at x and t, and, if sedimentation experiment and theory coincide, this value should be equal to the relative concentration encountered by the X-ray beam corresponding to a particle characterized by the settling velocity x/t. The theoretical size analysis was obtained by choosing several timedepth pairs.
ting
Iron oxide Silver halide Silica, Lot II Silica, Lot I
Figure number of data
Y
Avemge percent Brownian motion k = lOOyl(-y+l)
2 3 4
0.076 0.240 0.336
7.12 19.3 25.2
5
0.579
35.7
RESULTS
Table 1 presents y, the mean value of the ratio of Brownian to settling motion, and the corresponding percent Brownian motion, defined as k, for each experiment, while Figs. 2,3,4 and 5 show the measured and computed results. As noted previously, uncertainty limits are given on the electron microscopy
100
~EquotionsP.lwxi4 Electron
Electron microscope equlltsons 2.3 and 4
100604 Equivalent
03
a2
spherical
100806 Eauivolent
040.3
data in Figs. 2 and 3. Sedimentation results of Figs. 3 and 4 are given as a band in which the measurements fell; this spread arises with silica and not with the other two materials because silica is much less absorptive of X-rays and the limited quantities available made both high instrument sensitivity and relatively concentrated suspensions (1.8 vol. % for Fig. 3 and 1.1 vol. % for Fig. 4) mandatory_ Figures 3 and 4 are presented on a probability grid to show that the theoretical influence of diffusion on a sedimentation analysis is to distribute the particles in such a manner that a normal distribution results. The existence of convection currents was not detected in any of the experimental work. The figures reveal the comparison among rest&s from the electron microscope, actual sedimentation, and theoretical calculations of
and
Equation
spherical
Fig. 2. Size distributions
02
0.1
diarncter.~m
for silver halide.
TiI .
.
%
L
Ow
Sedimentation
OW
microscope
Sedimcntotion
Fig. 3. Size distributions
TABLB 1 Mean and percent Brownian motion Sedimen solid
100
2,3ancl4
Sedimentation
O.IW8
diameter,ym
for iron oxide.
I Equivalent
spherical
Fig_ 4_ Size distributions
diamctqpm
for silica, Lot 11.
17
ElectrOn
micrmcop
Sedimntatiin Eqwtion
Eqrrivolent
spherial
Fig. 5. Size distributions
2 and 3
diometer.Nm
for sil;ca, Lot I.
sedimentation with concomitant diffusion. Since the theoretical curves presume the distribution found by electrc? microscopy to be correct, comparison between curves labeled microscopic and those identified by equation numbers shows the influence to be expected of Brownian diffusion. The close agreement among all results evident in Figs. 2 and 3 arises from the fact that diffusion is of little significance with dense particles of the size indicated. The more pronounced influence for the less dense silica particles (Figs. 3 and 4) is predicted by the theoretical equations and confirmed generally by the measurements. The several solutions to eqn. (1) do not differ significantly except when the percent Brownian motion is rather large, viz. 35.7 % as shown in the last row of Table 1 and on Fig. 4.
CONCLUSIONS On the basis of these measurements, disregarding the effect of Brownian motion on particle size analysis by sedimentation to particle diameters of 0.2 pm appears to be justii-
ied provided the percent Brownian motion, k, is less than about 20%. The approximate equation (eqn. 4) descriL& the influence satiffactorily to k values of :lbou+. 30%. Once this percentage of Brownian motion is exceeded, however, eqn. (2) is recommended, it seeming to agree better with these results than eqn. (3). The selection cf eqn. (2) over eqn. (3) tends to confirm what is visually evident. Le. that particle adhesion to container walls is a factor of some significance, particularly when Brownian motion is high. Reliable particle size analysis by sedimentation should be conducted under conditions such that the value of y, the mean ratio of Brownian to settling motion (eqn. 6), is no larger than 0.2. If in a particular experiment this limit is exceeded, parameters such as liquid density and viscosity, temperature, the field force, or the time of analysis (actually the settling distance) should be adjusted to ensure that -y is reduced to an acceptable level.
REFERENCES J.P. Perrin, Atoms (trans. D.L. Hammick), Constable, London, 1923, pp. 83-108. J.F. Richardson and E.R. Wooding, Chem. Eng_ Sci.. 7 (1957) 51-59. C.N. Davies, Proc. Roy. Sot.. A200 (1949) 100113. N. Masdn and W. Weaver, Phys. Rev., (2) 23 (1924) 412-426. S. Berg, Symp. on Particle Size Measurement, ASTM Spec. Tech. PubI. No. 234 (1958) 143-171. J.P. Olivier, G-K. Hickin and C. Orr, Powder TechnoL. 4 (1970/71) 257-263. W. Hendrix and C. Orr, in M.J. Groves and J-L. Wyatt-Sargent (eds.), Particle Size Analysis 1970, Society for Analytical Chemistry, London, 1972, pp_ 133-146. W. Stcber, A. Fink and E. Bohn, J. Colloid Interface Sci. 26 (1968) 62-69_ D. Montgomery, RubberAge, 94 (1964) 759-761.
@ Elsevier Sequoia S.A.,
(1973) 13-17 Lausanne -
The influence of diffusion particle size analysis DAVID
W. MOORE*
Georgia Institute of (Received
and CLYDE
Printed in The Netherlands
on sedimentational
ORR,
Jr.
Technology. Atlanta. Ga. (U.S.A.)
January 9.1973;
accepted
February
28,1973)
analysis should be altered to minimize diffusional effects.
Summary Particle size analysis of submicrometer particles by gravity sedimentation is investigated to establish the influence of Brownian motion. Size distribution data from electron microscopic measurements are evaluated in terms of theoretical diffusion equations to determine the effect of diffusion. A parameter, defined as the mean ratio of Brownian to settling motion, is developed to guide the choice of system parameters so that Brownian diffusion effects can be minimized and reliable size results assured.
The basic mathematics of diffusion in the presence of settling was developed more than 50 years ago, although only the differential equation involved has yet been verified experimentallyl. * _This partial differential equation, describing the. time rate of increase in particle concentration as a function of depth in a settling and diffusing system, can be written
ac at
INTRODUCTION
-_=D
Sedimentation analyses based on Stokes’ law provide a convenient method fur determining partf;le size distributions. A lower limit of applicability is established, however, by the equalizing effect of diffusional forces, since Stokes’ law assumes that particles settle only under the influence of field and drag forces. The effect becomes significant when displacement apparticle sedimentational proaches the root mean square Brownian dis placement. It will be shown here how various solutions of the basic equation describing settling and diffusing systems can be employed to predict the influence of diffusion on a sedimentation analysis, and how parameters such as liquid viscosity and density and the time of *Present address: Atlantic Richfield Company, Box 147, Bakersfield, Calif. 93302 (U.S.A.).
THEORETICAL
P-0.
32Lv3C ax*
ax
(1)
where C is the number concentration, D the diffusion constant, t the time, v the sedimentation velocity, and x the depth below the upper surface. Published solutions to eqn. (1) divide into two categories on the basis of chosen boundary conditions: those that assume the top and bottom of the system container to retain, through van der Waals’ forces, all incident particles3 and those that assume top and bottom reflectance of incident particles4 -5. Other assumptions, viz., negligible wall effect, no particle-particle interaction or hydrodynamic influence, settling velocity in accordance with the Stokes-Einstein equation, and initial uniform distribution of particles, are identical in both types of solutions. The solution requiring top and bottom par-
title retention a was used in the eigen-function expansion form* o
C
(%3 2
C,=exp
2 sin (n;iry)[l - (--l)“exp(-1/2a)] n?r[1+1/(4&%%*)]
y =x/L _ I,,,L
18Llj
a=
6KT Lad’gAp
= D/vL
and CO is the initial number concentration of particles of diameter d, g the gravitational acceleration, rl the fluid viscosity, K the Boltzmann cons&t, T the absolute temperature, and Ap the particle density minus the liquid density. For the assumption of top and bottom particle reflectance both the equation’
c exp(y/a) C,=a[exp(l/a)-l] _2 n=1
exp(-anz9*t’)
was in
(%I
eqns. (2) and (3) was lo-*17. This factor was not computed; it was assumed to be zero. The value of y ranged from 0.005 to 0.1, giving a maximum value of lo-aas for the first righthand term of eqn. (3); it also was neglected. To facilitate making comparisons among experimental results, the ratio of Brownkan to settling motion was established. In quantitative terms this ratio is
where
t’ _ &A&”
value of a encountered experimentally 10e3, the maximum value of exp(-1/2a)
where symbols are as given previously_ If the mean particle diameter d of a distribution of particle sizes is used in eqn. (5) instead of d, the diameter of a monodispersion, thereby
+ 16a*7rexp (Q-4$‘)
exp(--an*v’t’)n[l+
[sin(nzy) + 2zna cos(nlry)J (- 1) -“exp(-1/2a)] (1 + 4n2n2a2)*
and a simplified approximation to it5 (4) were employed_ Equations (2), (3) and 4 were solved by digital computer using parameters appropriate to specific experimental systems as described subsequently. For small values of a, as is characteristic of large particles, the two series solutions converge very slowly_ It was found that if it required more than the first 6000 terms of the summation to evaluate either series, then the solutions of eqns. (2), (3) and (4) were virtually the same. Since the largest *An integral solution form for small values of n was also presentedin the referencedwork. It was found to contain a small printing error, the misprint being confirmed by the author. The two v’s in the first group of eqn. (46). p_ 109, should be u’s_ The correct arguments of the error functions in h, should be (2w+u)and (w+u).
(3)
overestimating the true value of u for a polydisperse suspension, then a is a function of t only for a given suspension. The mean value of OL,defined as 7, for a system can be calculated by integrating eqn. (5) over the time required for a sedimentation analysis and then dividing by the elapsed time of the analysis. This gives Y =36 Apg$
2KTq’
( 3nt, J
5
(6)
where t, is the time required for the sedimentation analysis. Finally, when 7 is expressed as a percentage of the total motion, it may be interpreted as the percent average Brownian motion that the mean size particle in the suspension experiences during a sedimentation analysis.
EXPERIMENTAL
Sedimentation data for four submicron par-
15
title systems were studied. These analyses were accomplished using a Model 5000, Particle Size Analyzer (Sedigraph) of the Micromeritics Instrument Corporation, Norcross, Ga. This instrument detects relative mass concentration by monitoring the transmittance of an X-ray beam passing through the sedimenting suspension 6 *’ . Continuous internal computation of Stokes’ law permits it to plot ‘Cumulative Mass Percent Finer” as a function of “Equivalent Spherical Diameter” on an X-Y recorder. During an analysis, the cell containing the suspension is completely sealed and filled; it moves downward at a known mte (ie., the point of mass concentration measurement moves relatively upward) as time progresses in order to reduce the time required for an analysis_ The temperature of the cell and surrounding compartment is controlled to within * 0.5OC, in order to minimize convection currents and, due to the fine collimation of the X-ray beam (0.0051 cm vertical thickness), the vertical position of a 0%pm-diam. particle can be located to within + 0.05 pm. Both sedimentation and electron microscopic sizes for two of the particle materials iron oxide and silver halide - were reported
Fig. 1. Electronmicrographof SiO,, Lot II, particles.
previously’. The other two samples were sibcas prepared specifically for this investigation by reacting tetrapentyloxysilane with water in the presence of ammonias. Figure 1 is a micrograph of some of these particles. As is evident, they are quite spherical and relatively uniform in size. Their size distribution was established with high precision using a Zeiss Particle Size Analyzer (Carl Zeiss, Oberkochen, wuertt.) and measuring the diameters of 1000 (Lot I) and 1494 (Lot II) particles. The iron oxide particles were generally spheroidal and the silver halide particles were cubic with rounded comers. Only 167 and 148 particles, respectively, were measured for the oxide and the halide, so their distribution was established with less reliability; uncertainty limits calculated by standard methods’ are indicated on the figures precenting these results. In order to compare theory with experiment, each distribution of particle size obtained from the micrographs was divided into n groups of essentially monodisperse sizes_ A value of settling time t was chosen which set the distance x that the X-ray beam would have been from the upper surface of the sedimentation had a sedimentation analysis been in progress. This permitted computing by
16
eqns. (2), (3), or 4 the predicted concentration of each monosize group at the chosen time and depth. Each of these R concentrations was then multiplied by the mass fraction represented by the group. The sum of these R products was taken to be the total mass fraction of particles at x and t, and, if sedimentation experiment and theory coincide, this value should be equal to the relative concentration encountered by the X-ray beam corresponding to a particle characterized by the settling velocity x/t. The theoretical size analysis was obtained by choosing several timedepth pairs.
ting
Iron oxide Silver halide Silica, Lot II Silica, Lot I
Figure number of data
Y
Avemge percent Brownian motion k = lOOyl(-y+l)
2 3 4
0.076 0.240 0.336
7.12 19.3 25.2
5
0.579
35.7
RESULTS
Table 1 presents y, the mean value of the ratio of Brownian to settling motion, and the corresponding percent Brownian motion, defined as k, for each experiment, while Figs. 2,3,4 and 5 show the measured and computed results. As noted previously, uncertainty limits are given on the electron microscopy
100
~EquotionsP.lwxi4 Electron
Electron microscope equlltsons 2.3 and 4
100604 Equivalent
03
a2
spherical
100806 Eauivolent
040.3
data in Figs. 2 and 3. Sedimentation results of Figs. 3 and 4 are given as a band in which the measurements fell; this spread arises with silica and not with the other two materials because silica is much less absorptive of X-rays and the limited quantities available made both high instrument sensitivity and relatively concentrated suspensions (1.8 vol. % for Fig. 3 and 1.1 vol. % for Fig. 4) mandatory_ Figures 3 and 4 are presented on a probability grid to show that the theoretical influence of diffusion on a sedimentation analysis is to distribute the particles in such a manner that a normal distribution results. The existence of convection currents was not detected in any of the experimental work. The figures reveal the comparison among rest&s from the electron microscope, actual sedimentation, and theoretical calculations of
and
Equation
spherical
Fig. 2. Size distributions
02
0.1
diarncter.~m
for silver halide.
TiI .
.
%
L
Ow
Sedimentation
OW
microscope
Sedimcntotion
Fig. 3. Size distributions
TABLB 1 Mean and percent Brownian motion Sedimen solid
100
2,3ancl4
Sedimentation
O.IW8
diameter,ym
for iron oxide.
I Equivalent
spherical
Fig_ 4_ Size distributions
diamctqpm
for silica, Lot 11.
17
ElectrOn
micrmcop
Sedimntatiin Eqwtion
Eqrrivolent
spherial
Fig. 5. Size distributions
2 and 3
diometer.Nm
for sil;ca, Lot I.
sedimentation with concomitant diffusion. Since the theoretical curves presume the distribution found by electrc? microscopy to be correct, comparison between curves labeled microscopic and those identified by equation numbers shows the influence to be expected of Brownian diffusion. The close agreement among all results evident in Figs. 2 and 3 arises from the fact that diffusion is of little significance with dense particles of the size indicated. The more pronounced influence for the less dense silica particles (Figs. 3 and 4) is predicted by the theoretical equations and confirmed generally by the measurements. The several solutions to eqn. (1) do not differ significantly except when the percent Brownian motion is rather large, viz. 35.7 % as shown in the last row of Table 1 and on Fig. 4.
CONCLUSIONS On the basis of these measurements, disregarding the effect of Brownian motion on particle size analysis by sedimentation to particle diameters of 0.2 pm appears to be justii-
ied provided the percent Brownian motion, k, is less than about 20%. The approximate equation (eqn. 4) descriL& the influence satiffactorily to k values of :lbou+. 30%. Once this percentage of Brownian motion is exceeded, however, eqn. (2) is recommended, it seeming to agree better with these results than eqn. (3). The selection cf eqn. (2) over eqn. (3) tends to confirm what is visually evident. Le. that particle adhesion to container walls is a factor of some significance, particularly when Brownian motion is high. Reliable particle size analysis by sedimentation should be conducted under conditions such that the value of y, the mean ratio of Brownian to settling motion (eqn. 6), is no larger than 0.2. If in a particular experiment this limit is exceeded, parameters such as liquid density and viscosity, temperature, the field force, or the time of analysis (actually the settling distance) should be adjusted to ensure that -y is reduced to an acceptable level.
REFERENCES J.P. Perrin, Atoms (trans. D.L. Hammick), Constable, London, 1923, pp. 83-108. J.F. Richardson and E.R. Wooding, Chem. Eng_ Sci.. 7 (1957) 51-59. C.N. Davies, Proc. Roy. Sot.. A200 (1949) 100113. N. Masdn and W. Weaver, Phys. Rev., (2) 23 (1924) 412-426. S. Berg, Symp. on Particle Size Measurement, ASTM Spec. Tech. PubI. No. 234 (1958) 143-171. J.P. Olivier, G-K. Hickin and C. Orr, Powder TechnoL. 4 (1970/71) 257-263. W. Hendrix and C. Orr, in M.J. Groves and J-L. Wyatt-Sargent (eds.), Particle Size Analysis 1970, Society for Analytical Chemistry, London, 1972, pp_ 133-146. W. Stcber, A. Fink and E. Bohn, J. Colloid Interface Sci. 26 (1968) 62-69_ D. Montgomery, RubberAge, 94 (1964) 759-761.