Measurement of Particle Size Distributions with a Disc Centrifuge

Chapter 9

Measurement of Particle Size Distributions with a Disc Centrifuge

Downloaded by NORTH CAROLINA STATE UNIV on August 2, 2012 | http://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch009

Data Analysis Considerations Michael J . Devon , Theodore Provder , and Alfred 1

2

Rudin

3,4

Dow Chemical Canada Inc., P.O. Box 3030, Sarnia, Ontario N7T 7M1, Canada The Glidden Company, 16651 Sprague Road, Strongsville, O H 44136 Guelph-Waterloo Centre for Graduate Work in Chemistry, Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 1

2

3

The disc centrifuge with an optical detector measures particle size distributions of various types of particles over a wide dynamic size range, .01 to 50μmbased on Stokes' Law for centrifugation. Generally the extinction efficiency of particles is a function of particle size, refractive index and an optical correction based on Mie scattering theory is required to ensure that the photodetector response is proportional to particle concentrations. It is shown that such corrections can be simplified if experimental conditions are adjusted to prevent the largest particles from sedimenting in short spin times. When this is done, data acquired on an equal time interval basis can be used to obtain valid particle size distribution averages. The disc centrifuge (1,2) with an o p t i c a l detector i s an excellent instrument for the measurement of p a r t i c l e size d i s t r i b u t i o n s of species with sizes from several micrometers down to less than 0.1 μια. A small sample i s injected into the center of a spinning disc containing a known volume of f l u i d . The p a r t i c l e s sediment toward the outer edge of the rotor where they pass through the l i g h t beam of the o p t i c a l detector. The hydrodynamic sizes of the p a r t i c l e s that are being detected can be calculated from Stokes* Law, as described below, and the time between i n j e c t i o n and the a r r i v a l of the p a r t i c u l a r species at the detector l i g h t beam. The instantaneous output of the detector i s used to estimate the number of p a r t i c l e s i n the beam, and thus to measure the p a r t i c l e size distribution. It i s recognized, however (3-5), that the uncorrected output of the detector i s proportional to the number of p a r t i c l e s only for large p a r t i c l e s or for a very narrow p a r t i c l e size d i s t r i b u t i o n . 4

Corresponding author

0097-6156/91/0472-0134$06.00/0 © 1991 American Chemical Society

In Particle Size Distribution II; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.


9. D E V O N E T A L .

135

Particle Size Distributions with Disc Centrifuge

Generally the extinction efficiency of p a r t i c l e s i s a function of p a r t i c l e size and refractive index as well as the wavelength of the l i g h t beam. A v a l i d o p t i c a l correction method has been proposed (6) but i t s application i s somewhat tedious without the approximation of a single wavelength computation of extinction efficiency for the whole d i s t r i b u t i o n . In this a r t i c l e we delineate the experimental conditions under which the o p t i c a l correction may be simplified allowing for data a c q u i s i t i o n on a straight time basis and simplifying data c o l l e c t i o n . The disc centrifuge method of p a r t i c l e size analysis without the use of e x p l i c i t o p t i c a l corrections i s shown to provide r e l i a b l e information for the p a r t i c l e size d i s t r i b u t i o n s of various polymer emulsions. Theory The disc centrifuge operates by forcing p a r t i c l e s r a d i a l l y outward through a spin f l u i d under high centrifugal force. The p a r t i c l e s s e t t l e at rates determined by their sizes and d e n s i t i e s . At a s p e c i f i c r a d i a l distance the p a r t i c l e s interrupt a l i g h t beam and the p a r t i c l e size and r e l a t i v e concentrations of the p a r t i c l e s are calculated from known parameters. Particle settling is described by Stokes* Law for centrifugation. D

where t D U Lp

• = =

7j = R = Ro =

2

=

6.299 x 109iylog(R/Ro)

(

1

)

centrifuge time i n minutes p a r t i c l e diameter i n micrometers centrifugal speed i n rotations per minute density difference between p a r t i c l e s and spin f l u i d in g/mL spin f l u i d v i s c o s i t y i n poises fixed distance from the center of the disc cavity to where the photodetector i s located starting distance of p a r t i c l e s from the center of the disc cavity, determined by the volume of spin f l u i d used i n the rotor.

The most generally applicable method for t u r b i d i t y axis c a l i b r a t i o n i n the turbidity-time raw data plot of the disc centrifuge i s the technique described by Oppenheimer (6). The t u r b i d i t y , T, of a uniform dispersion illuminated by monochromatic l i g h t with incident intensity I i s given by 0

I_ = e-(rZ) lo

,

(2)

where I i s the transmitted intensity and Z i s the path length. From e q . ( 2 ) : T(t) = _ In = _ In _ t I(t) H T(t)


(3)

136

P A R T I C L E S I Z E D I S T R I B U T I O N II

where T(t) i s the transmittance at time t . The t u r b i d i t y i s related to the p a r t i c l e diameter D, and to the number of p a r t i c l e s in the l i g h t path at time t by the expression (7) r

(

t

)

(2.303)D3F (D) Qext ( D . m . X )

-


where FH(D)

^

N

16

( 4 )

£J12£JI(R/R )2 0

=

(dn/dD) is the d i f f e r e n t i a l number d i s t r i b u t i o n of p a r t i c l e s between D and D+dD at the center of the detection zone Qext(D,m,X)= the extinction c o e f f i c i e n t which includes absorption and scattering effects m = r e l a t i v e refractive index; r a t i o of p a r t i c l e refractive index to that of the medium. X = wavelength i n the medium.

Time i s related to diameter, D, i n eq. ( 4 ) through Stoke's Law, eq. ( 1 ) . For a fixed detector position at radius R , the t u r b i d i t y i s proportional to the d i f f e r e n t i a l volume d i s t r i b u t i o n D 3 F N ( D ) . The

term R 2 £ n ( R / R )

i n the

0

denominator of

eq.(4)

is

a radial

d i l u t i o n factor. The t u r b i d i t y w i l l decrease as the p a r t i c l e s spread r a d i a l l y outward i n the disc c a v i t y . In the disc centrifuge experiment raw data are obtained as a function of time. For a fixed detector p o s i t i o n , R , and fixed spin f l u i d volume corresponding to R , the denominator i n e q . ( 4 ) i s a constant. The procedure for estimating Qext involves calculations using Mie theory ( 8 ) for given values of diameter, wavelength and r e f r a c t i v e index r a t i o of the polymer and spin f l u i d . A computer program eliminates the more tedious aspects of curve f i t t i n g i n the c a l c u l a t i o n of Qext. Each polymer has a different r e f r a c t i v e index. Qext i t s e l f i s also a function of the p a r t i c l e size and the wavelength of the l i g h t source. For polychromatic l i g h t , which normally i s used, i t i s usually necessary to integrate the product of Qext and the wavelength response of the instrument over the range of wavelengths. Oppenheimer ( 6 ) showed, however, that for low r e l a t i v e refractive indices a single-wavelength computation of Qext i s a good approximation. This approximation to the integrated extinction c o e f f i c i e n t w i l l be designated as *Qext(D,m). The c a l c u l a t i o n of the extinction e f f i c i e n c i e s for a wide range of diameters is a slow process for a microcomputer. A l s o , the wavelength dependence of refractive index i s not readily available for a l l materials that may be analyzed on the instrument. An assumption made i n disc centrifuge analysis is that the t u r b i d i t y is proportional to the negative logarithm of the transmittance: 0

T a - InTS ( l - T ) -

I

o

'

I

(

t

)

.

for

(

T

'

1

)

2

«1

(5)

The percentage error i n this assumption i s given by: Z Error = {[InT-(T-l)]/InT} x 100


(6)

9. D E V O N E T A L .

137


If the transmittance l e v e l i s 80Z, the error w i l l be less than 10Z in t u r b i d i t y . The actual expression used for the normalized d i f f e r e n t i a l volume d i s t r i b u t i o n u t i l i z i n g eqs.(5) and (4) i s as follows: D3F (D)dD = {[(I -I(t))/I )/Qext(D,ni)}dD N

0

0

,

(7)

I {[(I -I(t))/I ]/Qext(D.m)}dD


H

0

0

where DL and DH are the smallest and largest p a r t i c l e diameters i n the sample p a r t i c l e size d i s t r i b u t i o n . For p a r t i c l e s of constant density the normalized d i f f e r e n t i a l volume d i s t r i b u t i o n can be considered the normalized d i f f e r e n t i a l weight d i s t r i b u t i o n of p a r t i c l e sizes, F (D), W

F (D)dD - D3Fn(D)dD.

(8)

w

The formulae for the c a l c u l a t i o n of d i s t r i b u t i o n averages are shown in Table I both i n discrete and integral form where FN(D) i s defined by eq.(7) for integration i n diameter space. Equation(4) can be rearranged and set up for integration of the t u r b i d i t y function i n time space. From Stoke's Law, e q . ( l ) D2t

= K

(9)

dt = -(2K/D3)dD where K i s a constant.

,

(10)

U t i l i z i n g eq.(4) and the d e f i n i t i o n of

Qext(D,m), F (D)D3dt N

16£R2(R/R ) T(t) dt 0

=

— —

—

(11)

(2.303)Q t d>.m> ex

Substitution of eqs.(10) and (5) into e q . ( l l ) and normalization yields F (D)dD = { [ ( I - I ( t ) ) / I ] / Q e x t ( t , m ) } d t N

0

tf

j:

to

0

?

(

1

2

)

{(I -I(t))/Qext(t,m)}dt 0

where to and tf correspond to the time of appearance of the largest and smallest p a r t i c l e diameters i n the sample, respectively, i n the center of the detection zone. Thus the product of the diameter and the t u r b i d i t y extinction coefficient r a t i o yields the number average diameter i n time space. D

N

=

PH ftf DF (D)dD = {[(I -I(t))/I ]D(t)/Qext(t,m)}dt J L Jto N

0

0

D

ftf

{[(I -I(t))/I ]/Qext(t,m)}dt 0

Jt

0

0


(

13)

138


It follows that the weight average diameter i n time space i s given by T>H | D[D3F (D)]dD =^ o

{[(I -I(t))/I ]D (t)/Qext(t,m)}dt

D

{[(I -Kt))/I ]/Qext(t,m)}dt

t f


N

L

0

0

4

(14)

t

f Jt tf

0

0

0

In this work the proportionality between the integrated extinction coefficients ( Q e x t ) and the slope of a Stokes Law plot ( i . e . p a r t i c l e diameter versus spinning time) i s investigated for a range of diameters. The manner i n which the extinction coefficient varies with diameter ( i . e . increasing Qext with increasing diameter up to 1.1 fim) i s similar to the manner i n which time and diameter are related ( i . e . increasing inverse time with increasing diameter) through Stokes* Law ( e q . l ) ) . The range over which this relationship holds true i s determined by the conditions used i n the disc centrifuge measurement. It turns out that this range i s dependent on the time for the i n i t i a l appearance of the sample at the photodetector. The relationship that i s observed between Qext and time of appearance of the p a r t i c l e s at the photodetector under favorable operating condition permits elimination of the e x p l i c i t expression for the extinction efficiency from eq.(12) and the s i m p l i f i c a t i o n of the data handling i n this analysis. We have calculated the extinction coefficients for a number of different types of latex p a r t i c l e s (polystyrene, p o l y ( v i n y l acetate) and a c r y l i c copolymers) and then used these coefficients to calculate d i s t r i b u t i o n averages for different polymers that have been analyzed with the disc centrifuge. These averages are then compared to those that are computed without the e x p l i c i t use of the extinction c o e f f i c i e n t , but a simplified form of the extinction coefficient. Ultimately the r e l i a b i l i t y of the disc centrifuge for latex p a r t i c l e size analysis i s determined by the precision and accuracy of the results obtained. We have determined the minimum number of points required for precise measurements and are able to confirm accuracy by measurements on standards of known diameter. Experimental P a r t i c l e size measurements were performed with an ICI-Joyce Loebl Disc Centrifuge Mk III with the photodetector attachment. A very important step i n the operation of the centrifuge i s the formation of a density gradient within the spin f l u i d to allow better and more e f f i c i e n t separation of the suspended p a r t i c l e s . The buffered l i n e start method has been widely used (9). For this work, however, the external gradient method of Holsworth and Provder (10), was preferred because of i t s s i m p l i c i t y . A hypodermic syringe i s used to form the density gradient external to the disk. For spin conditions i n which 15 mL of aqueous spin f l u i d were used:



9.

139


DEVON E T A L .

exactly 15 mL of water were drawn into a 25 mL syringe. A i r bubbles were then expelled from the syringe. With the needle pointing down, an additional 1 mL of methanol was drawn into the syringe. This entire volume was then injected into the spinning disc cavity of the disc centrifuge. Centrifuge speeds were chosen so that p a r t i c l e s passed the detector at times between 1 and 25 minutes after i n j e c t i o n . Speeds were used 3200 to 8534 rpm, depending on the p a r t i c u l a r sample. Commercial a c r y l i c , v i n y l acetate copolymer and polystyrene latex samples were diluted to between 0.25 and 0.52 weight concentrations with a 802 water - 202 methanol mixture. The spin f l u i d i n a l l cases was water and the density gradient within the spin f l u i d was formed with methanol. The output of the o p t i c a l detector was acquired with a minicomputer data a c q u i s i t i o n system and converted to p a r t i c l e size d i s t r i b u t i o n s . D i s t r i b u t i o n averages are calculated according to the formulae l i s t e d i n Table I . Calculations of the integrated extinction e f f i c i e n c i e s , Q e x t , for the p a r t i c l e size range of 100 to 1000 nm were done at 50 nm i n t e r v a l s with a microcomputer by the method described by Oppenheimer (6). The output of this program was f i t t e d to a t h i r d degree polynomial to allow c a l c u l a t i o n of Qext for each measured diameter. The c a l c u l a t i o n of Qext was done for several aqueous spin f l u i d s and polymer types. The wavelength dependence of the r e f r a c t i v e indices of the spin f l u i d s was taken from the l i t e r a t u r e (11) . Refractive indices and densities of solvent mixtures are best determined experimentally. However, the refractive index of the mixture, rjm may be calculated i f the density of the mixture, pm, the densities of the components of the mixture, p\ pi, and the r e f r a c t i v e indices of the components, rji, Tjz, are known accurately (12) . The simplest formula to use i n this case i s the empirical Gladstone-Dale equation: f

t

1 (JKl) = (Wl//?l)(J7l-l) + (W2//>2)(72-l)

,

(15)

p* where wi and W2 are the weight fractions of the components of the spin f l u i d mixture. The wavelength dependence of the refractive index of the polymers also was taken from the l i t e r a t u r e (13), where possible, or measured v i a the method of Devon and Rudin (14). Results and Discussion We have investigated the proportionally between the extinction c o e f f i c i e n t s and the slope of the Stokes' Law p l o t . Stokes' Law plots for polystyrene and polymethyl methacrylate at a given set of spin parameters are shown i n Figure 1. Figure 2 i l l u s t r a t e s the relationship between p a r t i c l e diameter and the integrated extinction c o e f f i c i e n t . The polystyrene integrated extinction c o e f f i c i e n t includes data for three different spin f l u i d s (water, 42 aqueous sucrose and 72 aqueous methanol).


140


TABLE

I

FORMULAE FOR THE CALCULATION OF DISTRIBUTION AVERAGES


(A general d e f i n i t i o n of a s t a t i s t i c a l moment of p a r t i c l e taken about

size

zero i s EmDiJ

where m i s the number f r a c t i o n of p a r t i c l e s per unit volume of the sample with diameter Di. NUMBER

MEAN

Ui DN

EniDi

Uo SURFACE

Em MEAN

5 _/ \ \uTy U2

VOLUME

D

/

1/2

/ ^Em

\ ')

/ yEni

V /

EniDi2

\

U3

SPECIFIC

U Dss

EniDi3

1/3

SURFACE

[/DF (D)dD]i/2 N

/3

[/D3F (D)dD]i/3 N

MEAN

EmDi3

3

*

/D3F (D) d D / / D 2 F (D) dD

= U

WEIGHT

N

N

EniD

Measurement of Particle Size Distributions with a Disc Centrifuge

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