86
Anal. Chem. 1986, 58,86-90
(14) Sepaniak, M. J.; Vargo, J. D.; Ketter, C. N.; Maskarinec, M. P. Anal. Chem. 1984, 58, 1252-1257. (15) Pang, T. J.; Morris, M. D. Appl. Spectrosc. 1985, 39. 90-93. (16) Leach, R. A.; Harris, J. M. J . Chromatogr. 1981, 278, 15-19. (17) Jones, L. M.; Leroi, G. E.; Myerhoitz, C. A.; Enke, C. G. Rev. Scl. Instrum. 1984, 55, 204-209. (18) Maimstadt, M. V.; Enke, C. G.; Crouch, S. R. “Electronicsand Instrumentation for Scientists”; Benjamin Cummings: Reading, MA, 1981; Chapter 14. (19) Brigham, E. 0. “The Fast Fourier Transform”; Prentice-Hail: Engiewood Cliffs, NJ, 1974. (20) Yariv, A. “Optical Electronics”, 3rd ed.; Hoit, Rinehart and Winston: New York, 1985.
(21) Lee, S. H. “Optical Information Processing”; Lee, S. H., Ed.; SpringerVeriag: New York, 1984. (22) Phillips, G. R.; Eyring, E. M. Anal. Chem. 1983, 55, 1134-1138. (23) Campbell, N. A. Appl. Stat. 1980, 29, 231-237. (24) Bevington, P. R. “Data Reduction and Error Analysis for the Physical Sciences”; McGraw-Hili: New York, 1969.
RECEIVED for review May 22,1985. Accepted August 19, 1985. This research was supported by BRSG SO7 RR07093-17 awarded by the Biomedical Research Grant Propam, Division of Research Resources, National Institutes of Health.
Sizing Synthetic Mixtures of Latex and Various Colloidal Suspensions by Photon Correlation Spectrometry Muriel Cintr6, Sylvain Cambon, Dominique Leclerc, and John Dodds*
Laboratoire des Sciences du G h i e Chimique-CNRS-ENSIC, 1, rue Grandville, 54042 Nancy Cedex, France
A Maivern 4600 photon correlation spectrometer has been used to measure the size of submicrometer particles uslng data treatment by the method of cumuiants and by the exponentlai sampling method. Blnary mixtures of standard latex particles, 1091399 nm and 220/945 nm, together with two simulated wide distributions, both 91 mm to 945 nm, have been investigated together with various real coiloldai suspensions, such as paint, mlik, lymph, cement, and formazlne. The cumuiants method is found to be unsuitable for binary mixtures, whereas the exponential sampling method applied to wide distributions and real systems can glve more Informatlon in the case of double population systems.
Colloidal suspensions, defined by IUPAC as those containing molecules or polymolecular particles having at least one dimension between 1 nm and 1 bm, are of growing importance in fine chemical processing and in biochemical and biomedical applications. This comprises a wide diversity of systems including mineral microparticles, emulsions, biomolecules, and microorganisms, but a common requirement for correct use in industrial and research applications is a means of characterizing their properties and, of particular importance, is a means of measuring their particle size. The determination of the average size of submicrometer particles is now possible by several new techniques such as hydrodynamic chromatography and field flow fractionation; photon correlation spectrometry (PCS) offers many advantages over these methods. The technique is finding wide application in the study of colloids, for example, viruses, Pusey ( l ) and , the conformation of DNA molecules, Jolly et al. (2). Photon correlation spectrometry is only one of the many names used for the technique. Others are, dynamic light scattering, intensity fluctuation spectroscopy, or quasi-elastic light scattering. The method is based on the measurement of fluctuations in the intensity of light scattered from a suspension of particles undergoing Brownian movement. Analysis by autocorrelation leads to a value of the coefficient of Brownian diffusion that can be related to particle size by the Stokes-Einstein equation. Detection of such fluctuations in light requires a photomultiplier with a sufficiently rapid response detecting emission from a sufficiently small volume for the effects not to be 0003-2700/86/0358-0086$01.50/0
smoothed out. The theory of the method is given in standard works, such as Berne and Pecora (3),or is discussed extensively in recent symposia, Dahneke ( 4 ) ,which should be consulted for full details. The application of the method is now well established for measuring monosize or narrow distributions of particles. At the present time the central problem is the application to multimodal or wide distributions and in the practical application of what involves very sophisticated numerical analysis. In the case of a suspension of monosize particles the time base autocorrelation function of the fluctuations in intensity of light scattered from a suspension is a single exponential. The determination of the time constant of the autocorrelation function then leads to a value of the diffusion coefficient and hence the particle size. In the case of a suspension containing a range of particle sizes, the autocorrelation function is the superposition of several exponentials and the normalized correlation function becomes
Here r is a function of the Brownian diffusion coefficient and the magnitude of the scattering vector F(T) is a normalized distribution function containing details of the size distribution. The problem is therefore to invert eq 1to obtain F ( r ) from measurements of G(T). Unfortunately the problem is illconditioned, no solution is guaranteed, if a solution exists it may not be unique, and the convergence to a unique solution may not be uniform. Several methods have been proposed for obtaining a solution and have been reviewed in Chu et al. ( 5 ) and Dahneke (4). Here we use two methods: the Pusey cummulants method and the exponential sampling technique. The Pusey cummulants method is now well established and fully detailed elsewhere (I). Briefly it yields the first two moments of the required distribution function, the first moment giving the average diffusion coefficient and the second moment a measure of the polydispersity of the sample (called here the Pusey polydispersity factor). The method is well adapted to narrow distributions but fails for wide and multimodal distributions. This is the basic resident numerical treatment in the Malvern 4600 apparatus used in this work. The exponential sampling method is an alternative treatment adapted for wider distributions. 0 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 1, JANUARY 1986 CONTROLLED TEMPERATURE RATU --
MEASURING, CELL
Table I. Results from Malvern Photon Correlation Spectrometer (PCS) for Monosize Latex Particles (Dow Latex and Rhone Poulenc Estapor)
FilMP
P \
__
1
-
.. .....
FILTER LASER
FOCUS
~
_-_____._ ~
, DIAPHRAGM SLIT
---
WRRELATOR
makers ref size, mm
PCS measurement size, Pusey index of polydispersity nm
109 f 2.7
114.1
0.065
220 f 6.5
226.8 226.2 225.1
0.053 0.013 0.040
550
553.2 546.4 551.2
0.065 0.043 0.093
794 f 4.4
829.0 807.4 806.8
0.065 0.015 0.007
945
925.6
0.042
I
r I - 1i
PMI
87
I
c
I
U
Flgure 1. Schematic diagram of the photon correlation spectrometer apparatus.
The method is based on the numerical inversion of Laplace transforms and the determination of the function F ( r J in a finite assembly of i points spaced in a geometric progression with a dilatation factor of 6
urnax is the threshold frequency below which the eigenvalue xu, becomes less than the experimental noise. The complete function F ( r )is then constructed by interpolation. A complete description of the exponential sampling treatment is given in McWhirter and Pike (6) and Ostrowsky et al. (7). This recently developed method is available for the Malvern 4600 as the program POLYBAS. It does not require an a priori knowledge of the unknown distribution, but the parameters have to be found by trial and error from successive numerical tests. The results require evaluation and the parameters require modification until they have an internal numerical and physical coherence. Here we present results for different colloidal systems using data analysis both by this method and by the method of cumulants.
EXPERIMENTAL SECTION The Malvern 4600 photon correlation spectrometer used in this work is a commercial instrument (Malvern Instruments, Ltd., Malvern, Worcestershire, England) and is shown schematically in Figure 1. It comprises the following: the spectrometer having He-Ne laser 5 mW wavelength 6328-A Hughes 3225 H-PC polarized horizontally; a temperature-controlled measuring cell with a refractive index matching bath and filtration system; An EM1 9863 photomultiplier on a goniometer mounting with rotation 0-180"; a digital correlator Loglin 47027 with 64 delay channels based on a Commodore 8031 microcomputer. The laser beam is focused by a convergent lens on the sample suspension contained in a rectangular cell placed in the temperature controlled index matching bath at the center of the goniometer. The light diffused is transmitted through an adjustable slit and diaphragm at the front of the photocathode of the photomultiplier used as a photon counter. The time varying signal is amplified and sent to the correlator. In this work we used rectangular fluorescence cells as sample holders which precluded measurements at angles other than 90". Methodology. Care must be taken to filter the liquid used in the index matching cell before experiments and to ensure that the laser and photomultiplier are correctly aligned. A severe problem in using PCS is in preparing samples free from other contaminating particles such as those in ambiant air and in the solutions used for dilution. This need arises from the strong diffusion from large particles which greatly affects the measurements. The samples used in this work were prepared in
a laminar flow cabinet using water from a Millipore MiUiQ system, that is prefiitered then demineralized by a double passage through mixed bed ion exchange columns, followed by a column of active carbon and finally filtered with an 0.22-pm membrane filter. The correlograms are obtained under control of the microcomputer by fixing the experimental parameters: experiment duration, sample time, clip level, number of experiments to be used to obtain an average, rejection factor used in the calculation of this average. Two modes are available: linear mode giving 64 linearly arranged delays with an extra delay of 1064 before the last four channels to give a far point count; log mode with 32 channels in logarithmic progression, which is determined to be an optimum arrangement for polydispersed systems and for use with the exponential sampling method. The program calculates the diffusion coefficient, the mean diameter, and the index of polydispersity by the Pusey cumulants method and gives various data values that allow an evaluation of the results of the measurements (signal/noise ratio, number of channels used in the fit, ratio of the calculated mean intensity to that measured in the last channel, --).
RESULTS AND DISCUSSION Three series of experiments have been performed: measurements on series of binary mixtures of standard latex particles, measurements on two different mixtures of seven different latexes to simulate wide distributions, and measurements on various real colloidal systems. The results are given in the accompanying tables and figures. Binary Mixtures. Table I gives results of size analysis by PCS of the latexes used in the experiments. These are Dow (Dow Chemical, Midland, MI) and Estapor (RhBne Poulenc, Aubervilliers, France) latexes available commercially. It can be seen that the measured values of average diameter are reproducible and correspond to the values given by the makers as determined by electron microscopy. In addition the Pusey index of polydispersity is low in all cases, as is expected for these narrow distribution standard particles. However when these particles are used in binary mixtures, the results are not as good. Figure 2 shows how the average diameter and the polydispersity, as calculated by the cumulants method, vary with the volume ratio of large to small particles in two different sets of specially prepared binary mixtures. We used mixtures of 109 nm and 399 nm particles and 220 nm and 945 nm particles. That is diameter ratios of 3.7 and 4.3, respectively. The results show that the mean diameter is not a linear function of the composition, neither in terms of volume, projected area, nor number fraction. In both sets of mixtures the PCS analysis gives a value less than the calculated linear mean. The Pusey index of polydispersity varies with the relative proportions of the two sizes in the mixtures in a more
ANALYTICAL CHEMISTRY, VOL. 58, NO. 1, JANUARY 1986
88
1OOOl
i
0.4
Table 11. Compositions of Synmthetic Mixtures of Seven Standard Latex Particles
4,
O
l " " . ' . . ~ l 0 0 10 20 30 40 50 60 70 80 90 100 % large 100 90 80 70 60 50 40 30 20 10 0 %small
nm
A, %
B, %
91 220 399 481 550 794 945
3.9 7.7 19.2 38.4 14.2 7.7
2.3 14.8 21.6 22.6 21.6 14.8
2.9
3.1
Table 111. Malvern PCS Results for Synthetic Mixtures of Seven Standard Latex Particlesu
prepared
results, nm exponential sampling
W t percent o
109.399nm
o
220.945 nm
Figure 2. Pusey cumuiants analysis. Mean diameter and Pusey index of polydispersity as a functlon of composition for two binary mixtures of standard latex particles 109-399 nm and 220-945 nm.
cumulants
ends
A B
91-945 91-945
62-1178 98-835
mean
A B
485 488
417 375
466 417
stand dev
A B
117 188
188 137
(0.21)b (0.22)b
Analysis by cumulants and the exponential sampling method. Pusev index of DolvdisDersitv.
%t
xt '
o
o
l Mixture A
9-
Mixture B
DP
Flgure 4. Schematic diagram of the shape of two simulated wide distributions prepared by mixing seven different polystyrene latex standards.
1
1001
399
109
J99
Flgure 3. Analysis by exponential sampling method of five binary mlxtures of standard polystyrene latex particles 109 nm and 399 nm.
complicated fashion, being a curve with a double maximum. This same behavior is found with both sets of mixtures used
but the positions of the maxima and the intermediate minimum are not in the same place. Similar results have been reported in the literature by Daniels and Etter (8) using a Coulter Nano-Sizer, but no satisfactory explanation is yet available other than the obvious one that the cumulants method is not applicable in this sort of situation. Clearly the cumulants method is of little value in characterizing such systems. The same data on binary mixtures have also been analyzed by the exponential sampling method. Figure 3 regroups the results for the 109 nm/399 nm mixture and shows that a good correspondence is obtained. The method does find the size of the two components in the population and their respective proportions. Simulated Wide Distributions. In a further examination of the analysis of mixtures of standard particles we have simulated two wide distributions by mixing seven different latexes in two different ways. These mixtures, called A and B, have the composition shown in Table 11. The schematic diagram of Figure 4 shows that each mixture has the same volume mean diameter and spread but in one case the distribution is concave (A) and in the other case it is convex (B). The results of analysis of these mixtures, treated as if they were continuous distributions, are given in Table 111. It can be seen that the cumulants method gives an average diameter slightly closer to the required value than does the exponential sampling method. The spread of the distribution is found by the exponential sampling method while the polydispersity index does not seem to be very signifieant. A value of about
ANALYTICAL CHEMISTRY, VOL. 58, NO. 1, JANUARY 1986
EXPONENTIAL
WEIGHT
OISTRIBUTION DATA FILE DATAFILE 7
.-
D E L T A FN.
t
IIEIGtiT
273
7 66 0
638
25
50 116
qn --
116
2 73
EOUlVALEh? PARTICLE -0iAMETER
89
638 NM.
Figure 6. Analysis by exponentlal sampling method of micelles in Pasteurized mllk. D E L T A FN.
220 LIZ
Dp (nm)
EXWNENTIAL WEIGHT DISTRIBUTION DPTA FILE :DATAFILE 4 1
MIXTURE A
@
WEIGH1
2 87
1448 OMEGA M A X .
A-
DIL. FRCT.
220
772
1448
EOUNALENT WWICLE DIAMETER NM
Flgure 7. Arlalysis by exponential sampling method of formazine precipitate.
+-\
T
8
/
-, /
-
50-
\ \
\
/
/ -
-
\
/
/
\
\
Dp (nmi
MIXTURE B
@ Figure 5. Analysis by exponential sampling method. Resuks for two simulated wide distributions prepared by mixing seven different polystyrene latex standards.
Table IV. Malvern PCS Results for Various Colloidal Suspensions mean diameter, nm
Pusey index
520.0
0.26
394.4 400.7 476.8 375.8
0.24 0.19 0.20 0.15
full cream 1 / 2 cream*
250.0 204
0.24 0.25
UHT inilk full cream 1 / 2 cream lymph EBC formazine*
368.0 353 202 572.8
0.20 0.21 0.18 0.22
colloidal sample white emulsion paint cements alumina 1 alumina 2
silica 1 silica 2
pasteurized milk
0.2 is of the same order as found for the binary mixtures, which has a lesser difference between the largest and the smallest size. Figure 5 shows the simulated distribution and the results from the exponential sampliqg method in graphical form. It can be seen that the concave distribution of the mixture A is relatively well represented but the convex distribution B is not. The exponential arrangement of numerical parameters does not give sufficient points in the decreasing part of the curve to give a good idea of the shape in this case. Real Colloidal Suspensions. Various colloidal suspensions and emulsions have been analyzed by PCS and the
resulta from the cumulants method are given in Table IV. As can be seen they include paint, cement dust, milk, lymph from a Wistar rat, and formazine precipitate. These results &rein agreement with values obtained by hydrodynamic chromatography (9) and also, in the case of the cement, by electron microscopy (IO). The particles found in the lymph sample are probably chylomicrons formed with a triglycerol core surrounded by lipoproteins, which are generally taken to have a size of 0.2 pm, as is found here. The milk samples are of commercial full fat and half cream milk either pasteurized or ultrahigh temperature (UHT). The particles found are probably casein protein micelles and the differences found indicate the effects of the processing method used. The formazine precipitate’was prepared by the standard procedure laid down by the European Brewery Convention (EBC) for calibrating turbidimeters. Analysis of the data by the exponential sampling method merely confirms thesd results bringing no new information except in the two cases marked with an asterisk; the pasteurized half cream milk and the formazine are found to be bimodal distributions. As shown in Figure 6, the pasteurized milk has two populations of protein micelles at about 110 nm and 640 nm. Figure 7 shows that the formazine has populations at about 400 nm and at 1400 nm. Analysis of these two samples by hydrodynamic chromatography detected the smaller of the two components in each case and the larger component was probably filtered out (9).
CONCLUSIONS The measurements reported here with known samples and real colloidal suspensions demonstrate that PCS can give useful results in characterizing submicrometer particles in suspension. The cumulants method is a rapid technique for analysis in terms of mean diameter and polydispersity index which is valid for monomodal systems. However it only allows the “detection” of wide distribution without giving any quantitative information about the spread of the size distribution. This emphasizes that the cumulants analysis applied to PCS data, as with most other submicrometer sizing methods, is not capable of detecting foreign contaminant particles such as agglomerates or flocs in suspensions of single size particles. The problem is however crucial in practice as real suspensions always have dispersion in size and are usually contaminated at least to a small extent and it is very necessary to have quantitative information.
90
Anal. Chem. 1988, 58, 90-93
In the case of the prepared binary mixtures we have found the exponential sampling technique to give good results that lend support to the detection of two populations in the milk sample and the formazine precipitate. The exponential sampling method is found to be a useful complement to the cumulants method but it requires a trial parametrage and a response after an evaluation of the results and can be sensitive to the choices made. In the analysis of unknown wide distributions it seems reasonable to rely on the cumulants method, even with the attendant imprecision on the spread, except where the exponential sampling technique gives a clear indication of more than one population of particles. In this latter case it has demonstrated that the cumulants method cannot be relied on. In any event the two methods are complementary and furthermore use should be made of any information contributed by other methods of size measurement.
LITERATURE CITED (1) Pusey, P. N.; Koppel, D. E.; Schaeffer, D. W.; Camerini-Otero, R. D.; Koenig, S. H. Biochemistry 1974, 73, 952-954. (2) Jolly, D.; Elsenberg, H. Biopo/ymers 1970, 15, 61-95. (3) Berne, B. J.; Pecora, R. "Dynamic Light Scattering and Its Applications"; Wiley: New York, 1976. (4) Dahneke, B. E. "Measurement of Suspended Particles by Quasi Elastic Light Scattering"; Wiley: New York, 1983. (5) Chu, B.; Gulari, E.; Guiarl, Er. fhys. Scr. 1979, 19, 476-485. (6) McWhirter, J. G.; Pike, E. R. J . fhys. A : Gen. fhys. 1978, 7 1 , 1729, (7) Ostrowsky, N.; Sornette, D.; Parker, P.; Pike, E. R. Opt. Acta 1981, 28,1059-1070. (8) Daniels, C. A.; Etter, A. A. Powder Techno/. 1984, 3 4 , 113-1 14. (9) Leltzeiement, M.; Larson, K.; Dodds, J. Analysis 1984, 12, 260-265.
(IO) Relave, C., personal communication.
RECEIVED for review March 15, 1985. Accepted August 13, 1985. We wish to acknowledge the financial support of the Ministere de 1'Industrie e t de la Recherche through the research program on colloid filtration.
Depth Profile of Silver in a Matrix of Silicon Dioxide by Rutherford Backscattering Spectrometry John A. Leavitt Department of Physics, University of Arizona, Tucson, Arizona 85721
David K. Rollins and Quintus Fernando* Department of Chemistry, University of Arizona, Tucson, Arizona 85721
Rutherford backscattering spectrometry has been used to obtaln the average depth profile of elemental silver embedded In a matrix of the compound silicon dioxlde. General equatlons have been developed to obtain an accurate depth proflle of a high concentratlon of an element In thin-flim samples. These equatlons have been applied to an Ag In SIO, sample, 5800 A in thickness, wlth a measured maxlmum average concentratlon ratio, N*9/Nsn2 = 0.60 (which corresponds to 1.1 X 10'' Ag atoms/cm3). Results of detailed calculatlons uslng the general equatlons are compared with results obtalned by uslng the usual low-concentration approximation. The latter are In error by up to 31 % for the hlgh-concentratlon ratio of Ag:SIO, present In this sample.
The introduction of atoms and ions at concentrations greater than a few atomic percent into a solid substrate is an important technique that has found increasing use (1-4). A knowledge of elemental depth profiles in the resulting solid is frequently required in order to correlate changes in the chemical and physical properties of the solid with the concentration ratios present at various depths (5,6). Rutherford backscattering (RBS) spectrometry (7), utilizing 4He+beams with energies near 2 MeV, is a well-established technique for obtaining elemental depth profiles for small concentrations of heavy atoms in light substrates. High-impurity concentration distributions, however, are difficult to determine experimentally if the concentration changes rapidly with depth (3, 7-10). In such cases, the stopping power of the composite target changes rapidly with depth and the backscattered energy-to-depth conversion normally assumed for the target material is altered. Further, a nonuniform change in the concentration of the impurity produces a nonlinear depth scale 0003-2700/88/0358-0090$0 I .50/0
that distorts the observed backscattered impurity profile so that this observed profile does not accurately represent even the relative impurity concentration as a function of depth in the target. In this work, we have obtained the average concentration of the heavy atom, Ag, in the light host matrix, SiOz, as a function of depth for a sample containing a high, rapidly changing, Ag concentration. We have developed the equations necessary for the depth profiling of the Ag. We have applied these equations to the experimental RBS spectrum of the sample, and we have compared the resulting profile with the profile predicted by making the usual low-concentration approximation.
EXPERIMENTAL SECTION Schematic diagrams of the experimental setup and apparatus used for obtaining the RBS spectrum of the Ag in SiOzsamples are shown in Figures 1 and 2. The 4He+beam, produced by the physics department 6-MV Van de Graaff accelerator (High Voltage Engineering Corp.), was normally incident on the sample. The energies of the backscattered (by 170.0 k 0.5') 4He+ions were measured by the silicon surface barrier detector (Ortec Basr 014-025-100, with 15-keV resolution, subtending 0.78 X at the target) and associated amplifying and pulse-height analysis equipment (Figure 2). The spectral data were stored in an IBM PC and plotted with an X-Y digital plotter (IBM 749). The entire sample chamber was electrically isolated and served as a very deep Faraday cup for measurement (Figure 2) of the number of 4He+ ions incident on the target. A Bi RBS standard (11) was used to check the efficiency of charge collection. A thin film of Ag in SiOzwas bonded to an aluminum substrate and a 100-A layer of gold was deposited on the front surface to increase the conductivity and reduce beam damage to the sample. When an electron photomicrograph of the thin film was taken, the center of the Ag "layer" appeared to be about 3500 A below the SiOzsurface. It is difficult, however, to obtain quantitative information about the average concentration of Ag vs. depth from the photomicrograph. Such information can be ob-
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0 1985 American Chemical Society
