Anal. Chern. 1983, 55,2165-2170
2165
Simultaneous Determination of Particle Size and Density by Sedimentation Field Flow Fractionation J. J. Kirkland" and W. W. Yau E. I . d u Pont de Nemours & Company, Central Research & Development Department, Experimental Station, Wilmington, Delaware 19898
I n sedimentation fieid flow fractionation (SFFF), particle density normally must be available for calculating particle size from retention data. However, if the denlsity of the material to be analyzed is not known, it is feasible to measure the density accurately by direct SFFF. Retention data from separations carried out with several different mobile phase densities actually permit the simuitane!ous calculation of particle density and particle mass or sizo. This approach Is feasible since SFFF retention is predictably related to the density difference between the particles aind the mobile phase used for the separation. A new approach has been devised to make generally useful the slmultaneoius measurement of particle density and size for monodisperse or polydisperse samples, utilizing time-delayed, exponentHal force-field decay field-programmed SFFF (TDE-SFFF). Densities of unknown particles can be measured at high accuracy with precisions approaching 0.1 O/o in favorable cases.
Sedimeintation field flow fractionation (SFFF) is an analytical technique that can be used to separate particles in a dispersion according to particle size, or macromolecules in a sample solution according to their molecular weight (1, 2). Separation occui-Bin an open flow-through channel that rotates in a centrifuge, as illustrated in Figure 1. An imposed centrifugal force causes solute particles heavier than the liquid mobile phase medium to sediment radially outward against the wall of the channel. However, this buildup of particles near the channel wall is resisted by normal diffusion in the opposite direction. Because of higher diffusion rates and lower Sedimentation rates, smaller and lighter particles are not forced as close Lo the wall as larger and heavier particles. Under this balance of !sedimentation and diffusion rates, particles of different masses reach sedimentation equilibrium with different characteristic layer thicknesses, resulting in different mean distances from the analytical wall (e.g., and CB in Figure 1). This difference in particle sedimentation equilibrium is magnified by the superposition of a transverse, laminar-flow velocity gradient in the open SFFF channel. Since flow velocity is slower near the wall, heavier and bigger particles are intercepted by slower-moving flow streams and will elute from the channel later than simaller particles. Therefore, in an SFFF fractogram of detected particle elution peak vs. particle retention times, earlier-eluting peaks correspond to lighter and smaller particlles followed by particles of increasing masses or sizes. The quantitative aspects of SFFF retention have previously been described (1-4). Applications have now been expanded to include a variety of analytical problems involving organic and inorganic colloids, polymer lattices, and biopolymers (3-6). I t has been demonstrated that SFFF can be used as a high-resolution particle size analyzer for samples of uniform particle density ( 4 , 6, 7). In cases where particle density is known, the SFFF fractogram can be transformed into quantitative particle size distribution curves ( I , 4, 6 ) , and particle
size averages of various statistical significances can be calculated from the differential or the cumulative particle size distribution curves (6). On the other hand, when the particle density is not known, direct determination of particle size from the SFFF retention data is no longer possible. It previously has been suggested, however, that it is feasible to use SFFF to determine particle density at several mobile phase densities (3,4). This proposed method later was fully documented and successfully tested in constant-field SFFF (CF-SFFF) experiments using polystyrene latex standards of very narrolw particle-size distributions (7). Unfortunately, this CF-SFF'F density-measuring method is of limited utility, since it is only applicable to samples that have relatively narrow particle-size distributions. Many practical particle size problems involve samples with broad particle size distributions that require SFFF experiments with field programming features ( 2 , 4 , 6 , 8, 9).
The intent of this work is to extend the method of simultaneous particle density and particle size determination so it can be generally useful for SFFF experiments with any mode of sedimentation force field, either constant field or field programmed, and to samples of any type with narrow or broad particle size distributions. The usefulness of the new general method is illustrated by results obtained on practical samples with time-delayed exponential-decay field programmed SFFF (TDE-SFFF) experiments (8, 9).
THEORY Earlier attempts to simultaneously determine particle density and size by SFFF were strictly based on peak retention. For a retained particle in a CF-SFFF experiment, it is well established (1,2 ) that nGW t , = to-Apd;
36kT
where t, is the peak retention time (min) of the solute particle, to is the elution time (min) of the unretained solvent peak, G is the gravitational force field (cm/s2), W is the channel thickness (cm), lz is the Boltzmann constant (1.38 X 1W6 (gcm2)/(s2-deg)),T i s the absolute temperature (K), Ap is the density difference between sample component and mobile phase (g/cm3), and d, is the particle diameter (cm). In eq 1
G = w2r and
AP = I P S - Pol where w is the centrifuge speed (rad/s), r is the radial distance (cm) from the centrifuge rotating axis to the SFFF channel, p s is the density of the sample component (g/cm3), and p o is the density of the mobile phase (g/cm3). When two different mobile phase densities are used, with the other CF-SFFF experimental conditions kept constant, it is expected that solute retention times will change directly in proportion to Ap changes -t r = 2 - APZ t,l APl
0003-2700/83/0355-2165$01.50/0 0 1983 American Chemical Society
(2)
2166
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
W
Figure 1. Sedimentation FFF in a centrifuge.
Equation 2 was the basis for the original method proposed for determining particle density pe by SFFF (4). If the densities of the two mobile phases used to obtain the retention data are known, then the unknown value for pa can be calculated from eq 2 by using the observed t , value in two experiments. One of the deficiencies of the original proposal for determining density involving the use of only two mobile phase densities is the relative inability to compensate for common experimental uncertanties in retention data. This deficiency was eliminated in an improved method (7) where several mobile phase densities are used to improve the precision of determining the pavalue. The basic expression for this improved method is obtained by a simple rearrangement of eq 1
(3) where the “+” and ‘‘-* signs in eq 3 correspond to the cases of pa < po and ps > po, respectively. It is implied in eq 3 that a plot of the mobile phase densities po vs. the corresponding particle retention t, values should yield a straight line with an intercept on the density axis equal to the solute density pa. While each individual pair of po and t, values may not fall exactly on the expected straight line, experimental uncertanties are much reduced when several pairs of data points are used to establish a best-fit straight line in determining the intercept, or the pavalue. An analogous relationship can be made by rearranging the basic TDE-SFFF retention expression (8, 9)
(4) to obtain
where there is only one d, value for a monodisperse sample, and solute peak retention time is expected to be a good representation of sample retention time t,. However, with a polydisperse sample of broad particle-size distribution, a single d, value is not sufficient to characterize the particle size of the sample. Depending on the type of particle-size distribution in “real” samples, the SFFF elution profile can have broad, skewed, bimodal, or other shapes. Obviously, sample elution characteristics in such cases defy description by a single retention parameter (e.g., elution peak retention time, tr). There exist many different statistical d, averages for a single sample with finite particle size distribution. Examples of commonly used statistical averages include median, number average, and weight average d, values (8, 9). Each of these average d, values reflects features of part of the sample’s particle-size distribution. In theory, there is a corresponding t, value for every average d, value to satisfy the following relationships:
for CF-SFFF
for TDE-SFFF
where d, is a particular type of average d, value, and f, is an average retention time corresponding to 2., It should be noted, however, that a practical problem is the difficulty of identifying t, values on the SFFF fractogram for a broad particle-size distribution, especially for experiments where different mobile phase densities are used. It is apparent, therefore, that it is not possible to use existing SFFF methods to determine the density of samples with broad particle-size distributions. We present now a new approach to overcome the problems of previous methods by eliminating the need for identifying t, values in eq 6 and 7. Method Using Assumed paValues. The validity of this new approach is based on the fact that the fundamental SFFF separation parameter is the effective particle mass, which is directly proportional to the product (Apd;). This value always appears as an inseparable entity in all SFFF retention equations (e.g., eq 1 and 4). It is expected, therefore, that each retention time on an SFFF elution fractogram is in direct correspondence with one value for (ApdP3). The value of this fundamental quantity remains the same regardless of different Ap values in the SFFF particle size calculations. Thus
with where T is the matching time-delay and time-decay constant (min) in the TDE-SFFF separation, Go is the gravitational force field at the start of field programming, and the other symbols are the same as defined before. Therefore, eq 5 predicts that a plot of po vs. the exp(t,/T) values for the TDE-SFFF experiments should result in a straight line with an intercept corresponding to pa. Sample Polydispersity Problem. Methods to determine solute density paby using either eq 3 for CF-SFFF or eq 5 for TDE-SFFF are strictly limited to samples that are monodisperse or have very narrow particle-size distributions (e.g., polystyrene latex standards). The basis for successful application of these approaches is that the t, value used in the equations is in direct correspondence with the true sample particle diameter d, value. This does not present a problem
(AP)o = IPs,o - Pol
where pa,ois the assumed particle density and d,,o is the calculated particle diameter value based on the assumed ps,o value for particle density. Rearrangement of terms in eq 8 gives
Equation 9 gives a clear indication that the effect of using an assumed instead of the accurate particle density in the particle size calculation is to cause the calculated d,,o values to differ from the true d, values by a simple factor containing the density term. For samples containing particles of ho-
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
mogeneous density ps, the ratio between the d,,, and d, values is expected to be constant for all particle sizes a t every retention time on the SFFF fractogram. With samples of homogeneous particlle density, therefore, eq 9 should accurately relate the average particle size of the sample as well. It is expected that all the average d, values should also differ by a constant factor
7
I
I
I
I
2167
I
31 PARTICLE DIAMETER AVERAGE 0 NUMBER
(10) where dp,ois the calculated particle diameter average based on the assumed pS,, value and d, is the true average particle diameter of the same statistical averaging type. Equation 10 is generally applicable to all types of particle diameter averages. Furthermore, it is also generally applicable to any type of force-field programming used in SFFF separations, as long as the programmed force field is properly accounted for in the SFFF particle size calculations. Under the normal condition of ( P , , ~- po) having the same sign as (p, -- po) (particle more dense than the mobile phase), Equation 10 can be expressed as
The form of eq 111suggests that a plot of po values against the quantity in the brackets, [ (p,,, - po)d,,3], should produce a straight line, with the intercept a t the density axis giving an accurate measure of ps. In practicing the method, a reasonable value of is arbitrarily assumed. This pa,, value is used with known po values for several mobile phases to calculate the dP,o of the sample for any specified type of statistical particle size average. Generally, the more data points used in the plot to determine ps, the better the precision of the determined ps value. In practice, four to five uniformly spaced mobile phase density values appear to produce reasonably precise density results. As predicted in eq 11 (and verified by experiments), the arbitrarily assigned particle density P , , ~value should have no effect on the calculation of the actual particle density value. However, in instances where the initially assumed ps,ovalue is very different from the actual pS,, value, the difference between the calculated d , , and the actual d, can be large enough to cause some inaccuracy in the final result. This inaccuracy is the result of errors associated with the assumed particle diamete1 value in transforming the raw fractogram obtained by light mattering detection to the true differential particle-size-distribution plot (e.g., using the Mie scattering theory (6)). When this happens, a repeat cdculation and plot can improve the accuracy of the ps value by using the initially determined p, value as ps,o for the second calculation. Steric Effects. When the size of a solute particle d, becomes an appreciable fraction of the particle layer thickness 1 in an SFFF experiment, particles will elute earlier than expected because of a steric wall effect (IO). This effect on SFFF retention was accounted for in this study to ensure the accuracy of the padetermination. To account for steric effects in TDE-SFFF, numerical computer calculations were used (11). Corrections for steric effects can be effectively carried out on particles until the transition point between normaland steric-effects is reaclhed (IO).
EXPERIMENTAL SECTION Data reported in this study were obtained on an SFFF instrument based on a Sorvall Model RC-5 Superspeed centrifuge (Du Pont Clinical and Instrument Systems Division, Wilmington, DE) (6, 12). A “floating” channel, with dimensions of 0.025 >< 2.54 X 43.7 cm, was used (12). The SFFF instrument was equipped with a water-cooled rotating face-seal (12),a Du Pont
Figure 2. Measurement of polystyrene latex density by constant-field SFFF: “floating” channel, 0.023 cm thickness; mobile phase, 0.1 56 AerosoCOT; sample, 25 pL, 0.025% 0.176-pm polystyrene latex standard (Dow) in mobile phase: injection, 0.1 mL/min for 10.0 min; flow rate, 4.0 mL/min; rotor speed, 8000 rpm; detector, turbidimetry at 300 nm; assumed density for calculations, 1.10 g/cm3; mobile phase density modifier, glycerine.
850 microprocessor-controlledsolvent delivery system, a “Varichrom” UV spectrophotometric detector (Varian Instruments, Walnut Creek, CA), and a Model AH-6UhPa-N6O remote airactuated sampling valve (Valco Instruments, Houston, TX). A MINC-023 computer (Digital Equipment Co., Maynard, MA) wws used to control the rotor speed, collect elution data, and calculate sample particle size distribution (6). The general procedures of sample injection, relaxation, and subsequentseparation have been previously reported ( 4 , 6). The aqueous mobile phase used most in this study contained 0.1% Aerosol-OT surfactant (Fisher Scientific, Pittsburgh, PA). A 0.1% solution of “Micro” detergent (International Products Corp., Trenton, NJ) was used for the diesel exhaust soot samples. Glycerine was used as a modifier to make up mobile phases with the desired densities. Polystyrene latex standards were obtained from Dow Diagnostics (Dow Chemical Co., Midland, MI). Other experimental particulate samples were secured within Du Pont. Diesel exhaust soot samples were provided by K. K. Unger of Johannes Gutenberg University, Mainz, G.F.R.
RESULTS AND DISCUSSION Validity of Method with Polystyrene Latex. Validity of the general method to determine ps by SFFF was tested with a 0.176-pm polystyrene latex standard. Data in Figures 2 and 3 show that the proposed method described by eq 11 works equally well for constant-field and time-delay exponential-programmed force field SFFF. Data for the CF-SFFF experiment were obtained at a constant rotor speed of 8000 rpm. TDE-SFF’F data were obtained with an initial rotor speed of 8000 rpm and a time delay/decay constant T = 4.0 min. As anticipated by eq 11, the CF-SFFF and TDE-SFFF experiments produced data with the expected linear relationship. Intercepts of data in Figures 2 and 3 correspond to densities of 1.0504 f 0.0008 and 1.0496 f 0.0008 g/cm3 for the CF-SFFF and TDE-SFFF experiments, respectively. These values compare favorably with the Dow-reported value of 1.05 g/cm3 and the value of 1.051 g/cm3 obtained by Giddings et al., with constant-force-field experiments (7). An arbitrarily assumed particle density value of 1.10 g/mL was used to calculate the values plotted in Figures 2 and 3. All values represent the average of duplicate runs, and corrections for steric effects were applied in all cases where applicable. As suggested by eq 11,the general method for measuring particle densities herein proposed is expected to be valid for
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
2168
4.0-
PARTICLE DIAMETER AVERAGE 0 NUMBER v SURFACE 0 SP SURFACE WEIGHT A VOLUME 0 TURBIDITY A MEDIAN
PARTICLE AVERAGES 0
3.0-
0
NUMBER WEIGHT TURBIDITY
2.0 -
1.0-
p, =1.050f0.001
01 1 I I I I I 0.99 1.00 1.01 1.02 1.03 1.04 1.05 106 MOBILE PHASE DENSITY,po,g/crn3
1-
Figure 3. Measurement of polystyrene latex density by TDE-SFFF. Conditions are given in Figure 2, except: initial rotor speed, 8000 rpm;
g/cm:
time delayldecay constant, 7 = 4.0 min.
Table I. Polystyrene Latex Density Data (Least-Squares Fit) particle diam av'
a
MOBILE PHASE DENSITY,
density, g/cm3 CF-SFFF TDE-SFFF
number surface sp. surface weight turbidity median
1.0501 * 1.0501 i 1.0504 ?r 1.0505 i 1.0502 + 1.0509 *
av
1.0504 i 0.0008
0.0009 0.0009 0.0008 0.0007 0.0008 0.0007
1.0496 i 1.0496 i 1.0496 i 1.0496 i 1.0496 * 1.0494 i
0.0011 0.0010 0.0008 0.0008 0.0007 0.0006
1.0496
0.0008
i
SFFF. Conditions are given in Figure 2, except: rotor speeds, 3500-6250 rpm with increaslng mobile phase density; flow rate, 2.75 mL/min at highest mobile phase density only; sample, 25 pL of 0.3% latex in mobile phase; detector, turbidimetry at 254 nm; assumed density for calculations, 1.30 g/cm3.
r = 10.0rnin
Calculated according to ref 8.
PARTICLE DIAMETER AVERAGE 0 NUMBER v SURFACE PllPFArF
t
any specific particle average d, used in the plot for the density measurement. This feature of the method is clearly demonstrated in Figures 2 and 3 where plots of number, weight, and turbidity average d, values all produce straight lines converging to a single density value of high precision. (Values for other particle size averages also converging at the same density value were not included in Figures 2 and 3 for clarity.) This is not a surprising observation since the polystyrene latex standard has a very narrow particle size distribution and would be expected to have a comparable homogeneity in particle density. Density values obtained by both CF- and TDE-SFFF for different particle diameter averages d, are almost identical and impressively precise, as illustrated by the data summarized in Table I. The excellent agreement in density values obtained by using different particle size averages confirms the density homogeneity of this polystyrene latex sample. Density of Polychloroprene Latex. The proposed method is also applicable to samples with moderate or wide particle size distributions. T o illustrate this feature a polychloroprene sample with a polydispersity value (ratio of particle weight average diameter to number average diameter, clpw/clp,J of 1.08 was selected for study. This polydispersity reflects a moderate distribution in particle diameter. Determination of density values for the polychloroprene latex sample by CF- and TDE-SFFF produced values of 1.218 and 1.225 g/cm3, respectively, as illustrated by the plots in Figures 4 and 5. Data in both plots were based on an arbitrarily assumed value of 1.30 g/cm3. The CF-SFFF measurements were carried out with constant rotor speeds of 3500-6250 rpm and TDE-SFFF data were obtained at initial
Po,g/cm3
Figure 4. Measurement of polychloroprene latex by constant-field
2
9"
12 A
X
0
10
VOLUME TURBIDJTY
'0.
9"
m o I,"
1
6
4
1
1225 t 0.005g/crn~
2
O' 1 bo
1J
.I I
1.10 120 MOBILE PHASE DENSITY, PO,g/cm3
1
130
Flgure 5. Measurement of polychloroprene latex by TDE-SFFF. Conditions are given in Figure 2, except: initial rotor speeds, 3000-9000 rpm with increasing mobile phase density; flow rate, 2.0 mL/min at two highest mobile phase densities only; sample, 10 pL of 0.3% latex dispersion.
rotor speeds of 3000-9000 rpm with a time delay/decay constant of 10.0 min. The measured density values for this sample compare favorably with the reported nominal density value of 1.23 g/cm3 for polychloroprene (13). Again, close convergence of the plots in Figures 4 and 5 to essentially the same particle density ps value is observed. Polychloroprene particle density results by CF-SFFF and TDE-SFFF using different particle diameter averages are summarized in Table 11. The precision of measured density values for polychloroprene is somewhat poorer than that found for the polystyrene latex, partly because of the much broader particle-size distribution of the polychloroprene latex com-
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13,NOVEMBER 1983
7r-
Table 11. Polychloropropane Latex Density Data (Least-Squares Fit) density, g/cm3
particle diam
CF-RFFF
av a
a
number surface sp. surface weight volume turbidity median
1.23:L 1.226 1.215 1.21'1 1.222 1.20!3 1.20!3
av
'TDE-SFFF
i
0.008
f
0.002
i
0.005
1.228 1.227 1.224 1.223 1.226 1.224 1.220
1.218 i 0.009
1.225
5
2169
0.011
i 0.009 i 0.006
* 0.003
* 0.015
X
,
0.015 i 0.017 i 0.017 i 0.016 i: 0.018 i 0.017 ?
5
I
\
5 I'
