253
Time Evolution of Average Size and Number Density
A Method for Rapid Measurement of Particle Size and Relative Number Density of Particles in Suspension. I' N. Ben-Yosef, 0. Glnlo, D. Mahlab, A. W e b , Division of Applied Physics, The School of Applied Science and Technoiogy, The Hebrew University of Jerusalem, Jerusalem, israei
and S. Sarlg" Casaii institute of Applied Chemistry, The School of Applied Science and Technology, The Hebrew University of Jerusalem, Jerusalem, israei (Received March 20, 1975; Revised Manuscript Received October 8, 1975) Publication costs assisted by the Casaii institute of Applied Chemistry
I t is shown that by using a multifilter spectrum analyzer one can obtain, by means of light beating spectroscopy techniques, the time evolution of average size and number density of the scatterers. The time evolution is obtained at a rate of 1 min-l. By numerical calculation it is shown that even when the scatterers are not monodisperse the spectrum can be used to calculate the average size with satisfactory accuracy.
Introduction In order to test the nucleation process of strontium sulfate in a saturated solution in the presence of a small amount of a polymer which retards the precipitation, there was a need to measure the time evolution of the size distribution of the microcrysta1s.l Size measurement of microscopic particles in solution can be measured by light beating spectroscopy. This powerful tool was employed in the present case after certain modifications, due to the experimental requirements, were introduced. The aim of the present communication is to show the experimental system and the data interpretation process by which the time evolution of the average size and number density of the microcrystals was measured at a rate of 1 min-l. Measurement of the size of microscopic particles in solution by the measurement of the scattered light spectrum is an accepted and well tried method.2 Briefly, the optical spectrum of coherent light scattered by spherical particles of size r is given by
Where N is the number density of the scatterers, A ( r ) is the scattering amplitude, D the diffusion coefficient, and q the scattering vector
where no is the refractive index of the solvent, the wavelength, 0 the scattering angle, and 00 the light frequency. The measured quantity in light beating spectroscopy is the power spectrum obtained from the optical spectrum by convoluting I ( w ) with itself. Particle size is derived from measurement of the diffusion coefficient which is size dependent. In the present case, the aim is to measure the time evolution of the size distribution of microcrystals growing in the solution. This requirement imposes two difficulties: (a) fast measurement, since the sample evolves with time the spectrum has to be measured faster than any time constant related to the time evolution; (b) data interpretation, as the microcrystals are not necessarily monodisperse, the spectrum obtained may not be a simple Lorent-
zian so that the size is not simply related to the spectral shape.
Experimental Section These problems were partially solved in the system to be described. The optical part of the spectrometer used is similar to those already reported in the l i t e r a t ~ r e The . ~ laser used was a Spectra-physics Model 124A He-Ne laser, while the photomultiplier used was the RCA 7265 model. For a stationary sample the easiest way of measuring the photomultiplier current power spectra is to use a spectrum analyzer, as reported in ref 4. For a time-evolving sample this approach is unacceptable because the time of measurement needed is too long. The time of measurement has to be taken into account due to the signal-to-noise ratio of the final signal. As has been shown,2 this ratio depends on the square root of the time of measurement at a fixed frequency. In the case of swept frequency, the sweep rate has to be slow enough; in practical cases, the total time of measurement is about 20 min. In the present case, the rate of the time evolution of the sample was such that the spectrum had to be obtained in a time of 1 min. For this purpose a multifilter spectrum analyzer (MFSA) was designed and built. The MFSA is composed of eleven parallel channels, one of them monitors the d.c. component of the photomultiplier output, whereas the other ten are composed of active band pass filters having different central frequency but equal transmission curves. The output of each filter is rectified and integrated, so that the output is composed of eleven capacitors whose potential is proportional to the rms value at the band passed by the respective filters. Each a.c. channel has a width of 10 Hz, and the time constant of each integrating circuit is such that for 1 min integration the capacitor potential is more than 95% of its maximum value. At the end of the measuring time (1 min) the voltage of the capacitors is relayed through a multiplexer to a recorder; after the recording, each memory is shortened and a new measuring sequence starts. In the present system, the frequency range covered starts at 15 Hz with a 30-Hz step to 285 Hz. Using this system, the power spectra of a time-evolving sample can be obtained, provided the rate of change of the sample is not faster than 1 min. The Journal of Physical Chemistry, Vol. 80, No. 3, 1976
Sarig et al.
254
The output of the MFSA includes the power spectra as well as the shot noise level using the d.c. channel data. For single size scatterers, the data interpretation is straight forward: the signal is processed as a Lorentzian, its half-width found, and the particle size calculated. T h e system was checked and calibrated using commercially available polystryrene latex spheres of known size; the size and their monodispersivity were checked and measured using a scanning electron microscope. The results show 5% accuracy (in particle size).
Data Interpretation When the scatterers are size distributed the relation between the power spectrum and size distribution is more complex. If the size distribution is given by d N = G(r) dr, Le., a t size r d r there are d N particles, the optical spectrum is:
TABLE I:'
~ . ~ ~ . . ~ ~ ~
..
~
..~.
~
width,
Calcd diameter,
WlOI,
U
U
90
0.060
0.228 0.446 0.631 0.825 1.067
14 11 5 3 0.7
A"
Half-
diameter, U
0.200 0.400 0.600 0.800 1.000
0.120 0.180 0.240 0.300
Rel.
a The calculation was performed on ideal spherical parti cles, size distributed according to eq 4,with parameters as stated in the table. The refractive index used in the calculation of the scattering amplitude was n = 1.5983,the re. fractive index of hulk strontium sulfate. The particles were assumed to he suspended in water, n = 1.333. i
+
the power spectrum is the convulution of I ( w ) with itself, and the zero frequency centered result is P(w)=
In the integral 9 is given as before, D(r) is the diffusion coefficient given by kTIGmqr, where T is the solution temperature, q the solvent viscosity, and r the particle radius. A ( r ) is the scattering amplitude which depends on size, refractive index of particle and solvent, wavelength. and scattering angle." The integral equation relating between the size distribution function G(r) and the power spectra P ( u ) is not easily inverted and it seems that no unique solution exists. Nevertheless, some approximation of the average size of the scatterers can be obtained from numerical calculations of eq 2 starting with an assumed size distribution. T o check the accuracy of a monodisperse approximation for size distributed scatterers, numerical calculations of eq 2 and 3 were performed. Assuming that the scatterers are size distributed according to the log-normal distribution, i.e. G ( r )= 1 exp [-;In 2r
(3'1
i Fwre 1. Scanning elechon microscope photograph of the final strontium sulfate microcrystals, Note that the size distribution is nar-
row.
(4)
Using the calculated values of A ( r ) one calculates the optical spectra using eq 2, and by numerical convolution the power spectra are obtained. The best single Lorentzian is fitted to that function, Le., data of size distributed particles are analyzed as if due to monodisperse particles. From the best fit obtained one calculates the particle size, which can be compared to the average size with which the calculation was started. In Table I the results of such a comparison are shown. The first column represents the average diameter used in eq 4, the second column is the size distribution width (3096 of the average size in all cases), whereas the third column gives the diameter calculated from the power spectra which were analyzed as if it is due to monosize particles, as previously described. One observes that the original average size and the calculated size do not differ by more than 15% in the size range of (0.2-1.0)r m for a distribution of appreciable width. As shown in Figure 1. the scatterers in the present case do not possess a wide distribution so that one can calculate that the measured spectra can be analyzed as
1
1
I
l
,
1
l
l
1
1
1
1
1
I
,
Zemlevd~DCChChl ChZ C h 3 C h 4 C h 5 Ch6 Ch7 Ch8 Ch9 ChlO Zero *vel
Flgure 2. Recwding of the MFSA output at intervals of recording is shined upward by a fixed amount.
1 min.
Each
255
Time Evolution of Average Size and Number Density
Relative Number Density Measurement By integrating eq 1 over all the frequencies one obtains the total light intensity scattered. It is observed that this intensity is proportional to Y A ( r ) 1 2 .T o obtain the absolute light intensity one has to consider the geometry of the experiment as well. For an experiment, such as discussed previously, where the number density and average size change with time, the relative number density of the scatterers can be calculated from this relation. The total light scattered into the photomultiplier is proportional to the d.c. current. Using the previous proportionality one can write
:o'i x L
o
c
031
a
I ( t ) = KNJA(r)J2
Flgure 3. Graphical representation of the reciprocal of the power spectrum vs. the square of frequency. Note the near linearity.
if due to monodisperse scatterers and the calculated size represents the average size within the limits of 20%.
Data Analysis As described, the scattered light spectrum was obtained from the MFSA on a recorder. A set of consecutive measurements recorded in this way is shown in Figure 2. each recording is further analyzed as a Lorentzian. The recorded data are the square root of the power spectra and the shot noise. The level of the shot noise is known from the d.c. level. Using these data, one writes the power spectra P(w) as follows:
P(,) = S(w)2 - N(w)2
(5)
where S ( w ) is the recorded signal and N(w) the shot noise (slightly dependent on frequency due to the system transfer characteristics). Since P(w) is a Lorentzian, one designs A as its half-width and obtains
where B is a constant. By graphical analysis of l/P(w) (measured quantity, eq 5 ) as a function of w2 one obtains A. Since A = 2Dq2(eq 3) and q is known, one calculates D and from it the size r . Furthermore, the nonlinearity of the data is an indicator of the distribution width of the scatterers.
where I ( t ) is the measured current, K an unknown constant, N the number density of the scatterers, and A ( r ) the scattering amplitude. As described previously, r is obtained experimentally, and A ( r ) can be calculated. I ( t ) is measured directly, so that KN can be calculated as a function of time, to obtain the relative number density of the scatterers.
Conclusion I t has been shown that by using the MFSA one can obtain the time evolution of the scattered light. For the case under discussion, where the size distribution width does not exceed 30% of the average size, one can analyze the data as if it is due to monodisperse particles. Using this approximation one obtains the average size and the relative number density of the scatterers as a function of time. The time evolution of the average size and number density of the evolving microcrystals in a retarded precipitation of saturated strontium sulfate solution are summerized and discussed in the second part of this artic1e.l References and Notes (1) The second part of this article is entitled A Mechanism for Retarded Precipitation Based on the Time Evolution of Particle Size and Relative Number Density, S. Sarig and 0.Ginio, J. Pbys. Cbem., following article in this issue. (2) H. 2. Cummins and H. L. Swinney, "Progress in Optics", Voi. VIII. E. Wolf. Ed., North Holland Publishing Co., Amsterdam, 1970. (3) S. B. Dubin, J. M. Lunaceck, and G. B. Benedeck, Proc. Natl. Acad. Sci., USA, 57, 1164 (1967). (4) N. Ben-Yosef, S.Zweigenbaum, and A. Weitz, Appl. Pbys. Lett., 21, 436 (1972). (5) M. Kerker, "The Scattering of Light", Academic Press, New York, N.Y., 1969.
The Journal of Pbysical Cbemistry, Vol. 80, No. 3, 1976
