Langmuir 1995,11, 3401-3407
3401
Precise Analysis of the Turbidity Spectra of a Concentrated Latex U. Apfel, K. D. Homer, and M. B a l l a u P Polymer-Institut, Universitat Karlsruhe, Kaiserstrasse 12, 76128 Karlsruhe, FRG Received January 23, 1995. In Final Form: June 16, 1995@ A precise analysis of turbidity spectra of a concentrated polystyrene latex (number-average diameter of particles: 68 nm) for a range of wavelength between 400 and 1100 nm is given. By suitable definition, the measured turbidity of the nonuniform system as function of 1 (1is the wavelength in the medium) is factored into a “measured” integrated form factor &(A2) and a “measured” integrated structure factor Z M ( ~ ~(c, C is)the weight concentration). The normalized scattering cross sections &(a2) evaluated from the experimental specific turbidities extrapolated to vanishing concentration are in full agreement with the theoretical results provided by the Mie theory. The turbidities measured at finite concentrations(up , C ) furnishes the structure factor S(q) [q = (4d1) sin(8/2);8: to 8.2 wt %) yield the function Z M ( ~ ~which is the scattering angle] in the region of low scattering angles. In particular, it can be demonstrated that S(0) as well as a,the first coefficient of the expansion of S(q)in terms of even powers of q, can be obtained with good accuracy without the need of assuming a specific model of particle interaction.
Introduction In a previous publication’ it has been shown that turbidimetry is well-suited to the study of particle interaction in concentrated colloidal suspensions. The analysis given in ref 1extends earlier work of Vrij and co-workers2p3and rests on the factorization of the specific turbidity TIC(c is the weight concentration) into an “integratedform factor”&(Az), and an “integrated structure factor”, Z(Az,c). Thus the conventional analysis5 of the scattered light in terms ofthe magnitude of the scattering vector q [q = (4n/A) sin(8/2);8 is the scattering angle, and A is the wavelength in the medium] is replaced by a rendition in terms of functions of the reciprocal quadratic wavelength. In particular, it can be ~ h o w n l -that ~ the function Z(A2,c)may be expanded in powers of The leading term of this expansion is S(O), the structure factor S(q) at vanishing scattering angle. Turbidimetry therefore provides a means to obtain the long-wave part of the structure factor of colloidal suspensions. As has been shown this method is practically insensitive toward multiple scattering. Therefore it is well-suited to study concentrated, strongly turbid systems. Also, the effect of the size distribution of the particles under consideration can easily be incorporated into the analysis by definition of a ”measured” function &(i12,c) analogous to the definition of a “measured structure factor” sM(q) used for the interpretation of angularly-resolved light scattering.6 The method worked out in ref 1has been applied to a narrowly distributed polystyrene latex. Data of the integrated structure factor Z(L2,c)for a range of wavelengths between 400 and 800 nm have been obtained. Assuming hard-sphere interaction of the particles, the data obtained could successfully be interpreted in terms of an effective diameter of interaction. Since the range of wavelength used previously was rather small, the determination of S(0)by extrapolation
* Author to whom all correspondence should be addressed. Abstract published in Advance A C S Abstracts, September 1, 1995. (1)Apfel, U.;Grunder, R.; Ballauff,M. Coll. Pol. Sci. 1984,272,820. (2)Jansen, J. W.;de Kruif, C. G.; Vrij, A. J. Colloid Znt. Sci. 1988, 114, 492. (3)Rouw, W.; Vrij A.; de Kruif, C. G. Colloids Surf 1988,31,299. (4)Penders, M.G.H. M.; Vrij, A. J. Chem. Phys. 1990,93,3704. ( 5 ) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. @
to infinite wavelength could only be done with limited accuracy. Measurements extending to the near infrared would clearly be helpful in this regard. Also, the previous analysis’ is restricted to the interpretation in terms of the Rayleigh-Debye (RD) approximation of light s ~ a t t e r i n g . ~ In this paper the theory given in ref 1 is extended to include the treatment of spherical particles in terms of the Mie t h e ~ r y .Furthermore, ~ the range of wavelength is extended to 1100 nm to facilitate the extrapolation of S(0). The treatment is applied to a narrowly distributed polystyrene latex of 68 nm number-average diameter. It will become apparent that the turbidimetric analysis furnishes the low-angle part ofthe structure factor without application of any particular assumption regarding the interaction of the particles.
Theory Monodisperse Systems. For a system of monodisperse interacting colloidal particles of number density N the Rayleigh ratio, R(8),is given by7p8
where k is the magnitude of the wave vector given by 2nnd1,. The functions Sl(f3)and S2(f3)are defined as
and
where the functions n,,(cos 8 ) and zn(cos0) are related to the associated Legendre polynomials of first kind, P,(cos e), by COS 8 ) = P,(COS @/sin e and Z,(COS e) = ~P,(cos e)/d8 and n is the number of poles. Expressions for the coefficients a , and b, are found in ref 7. The turbidity z given by (6) D’Aguanno,B.; Klein, R. J. Chem. Soc. Faraday Tram. 1991,87, 379. (7)Kerker, M. The Scattering ofLight and Other Electromagnetic Radiation Academic Press: New York, 1969.
0743-7463/95/2411-3401$09.00/0 0 1995 American Chemical Society
3402 Langmuir, Vol. 11, No. 9, 1995 z = % l R ( q ) sin 8 de
(4)
o9
,
Apfel et al. I000
800
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70
may by suitable definition be factored into a form contribution &(A2) and the integrated structure factor Z(A2,c):
where u is the diameter of the particles, m = nJno their relative refractive index, no the refractive index of the medium, c denotes the weight concentration, and K* is an optical constant defined by
with epbeing the density of the particles. The normalized cross section &(A2) accounts for the wavelength-dependence of the total scattering cross section at finite wavelengths. By virtue of its definition, it is normalized to unity for vanishing k2.Within the RD approximation it can be obtained by integration of the form factor of a homogeneous sphere over the solid angle; in the case of Mie scatterers, &(A2) is also a function of the relative refractive index m. Its relation to the specific turbidity (z/c), for an infinitely dilute suspension is given by
Here C,,denotes the total scattering cross section of the particles and Mptheir mass. The function Z(A2,c)can be obtained from experimental data by
For size parameters ngu/Aothat are not too large (see the discussion of this point in ref 11, the integrated structure factor may be analyzed by series expansion in powers of (nJAo)2(cf. eq (9) of ref 1). Expanding up to the order A-6 we have
Z(A2,c)= S(0)+ 8a 488
-[P5
nn4 + &..2..0](+J
+
non ”[ 5 1 1 +~ 25aR44,0+ $R2g,o - i a ( R 2 g , o ) 2 ] ( x+) 4
4
6
Here the optical radius of gyration, Rg,0,and the fourth moment, R44,0, have to be replaced by the respective to the expressions given by the Mie t h e ~ r y . In ~ ,contrast ~ Rayleigh-Debye limit, Rg,oand the correspondinghigher moments depend on the relative refractive index (see the discussion of this point in refs 7 and 8). The quantities a,/3, and y are the coefficients of the expansion of the structure factor S(q) in even powers of q as discussed recently (see ref 1). From inspection of eq 9 it becomes obvious that terms of the order A-4 and higher contain expressions referring (8)van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1957.
0 ’
0
01
02
03
I
04
(no/@ io5 I nm-2
Figure 1. Comparison of the integrated structure factor (cf. eq (8))of homogeneous, hard spheres (solid lines) and hollow, hard spheres (dashed lines) calculated accordingto eq (9). The numbers in the graph indicate the volume faction of the spheres in the system; the numbers on the top of the graph display the respective wavelengths in vacuo in nm.
to the intraparticular interference; i.e., the factorization of dc into a pure form part and a part solely referring to interaction is only given in the long-wavelimit. To assess this problem in more detail, it is expedient to compare the function Z(A2,c) for monodisperse, homogeneous, hard spheres to the result obtained for hollow hard spheres of same size. The calculation can be done using eq 9 by insertion of the optical radii of gyration and the corresponding fourth moments of homogeneous and hollow spheres obtained within the RD approximation. A calculation using the respective Mie terms is not necessary since the optical moments in eq (9)present only corrections of higher order. The comparison for spheres of 120 nm diameter is shown in Figure 1. It demonstrates the influence of internal structure on Z(A2,c)to be limited to wavelengths below approximately 650 nm, even for objects so markedly different with regard to their inner structure. Therefore the factorization of the measured turbidity into an integrated form part &(A2) and the integrated structure factor Z(A2,c)is a viable approach for the particles sizes under consideration here. In consequence, plots ofZ(A2,c) vs (nJAOl2 should be suitable to extract the long-wave part of the structure factor from experimental data if the range of measurements can be extended to the near infrared. Polydisperse Systems. For a polydisperse system with N, being the number density of particles of size i , a “measured integrated structure factor” &(A2,c) can be defined by1S2
where theZi(8)denote the scattering intensities of species i as defined by eq (1). The quantity S(q) is a suitable
Langmuir, Vol. 11, No. 9, 1995 3403
Analysis of the Turbidity Spectra of Latices average of the structure factor defined by eq (27) of ref 4 with the scattering amplitudes fiBi(q) of particles being replaced by the respective expression given by the Mie
800
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In the limit of sufficiently small concentrations, Z(A2,c) as well as &(A2,c) may be rendered approximately by
Z,(A2,c)-l
= 1+ 2B,,,c
(11)
where the apparent second virial coefficient Bapp depends on ~ a v e l e n g t h . l -Plots ~ of the reciprocal turbidity versus concentration in the dilute regime can be used to obtain (d c 10.
2 0.9
2
0%
With the analogous definition of a “measured”normalized cross section &&I2), the specific turbidity at infinite dilution (cf. eq (7)) for a polydisperse sample follows as
10%
\
u p 70nm
20%
\ \
\
\ \ \
0.8 0
0.5
R.0
1.0
(noiho)zio5 i nm’?
Hence, extrapolation of the specific turbidity to inf-inite wavelength leads to the “turbidity-average diameter”, a,:
The term turbidity-average diameter has been chosen because a, governs the leading term of the specific turbidity at infinite dilution (cf. eq (12)). It must be distinguished from the respective weight-average diameter defined by
Figure 2. Comparison of the normalized scattc
L U ~ UUJJ
section Qh1(A2) (cf. eq (12)) calculated for spheres by the Rayleigh-Debye approximation (uniform system, dashed lines) and the Mie theory (Gaussian size distribution, solid lines). The turbidity average size (see eq (13)) as well as the relative refractive index m have been kept constant for all calculations. The numbers in the graph refer to the standard deviation ofthe size distribution of the spheres; the numbers on the top of the graph display the respective wavelengths in vacuo.
function Z M ( A ~ ,according C) to eq (10)can be calculated in terms of the Rayleigh-Debye approximation. Model calculations reveal that only if the diameter of the spheres exceeds 200 nm must the amplitude functions be derived from the Mie theory.
Experimental Section i
and a similarly defined number-average diameter, a,. In order to discuss the function &&I2) defined in eq (12) in more detail, Figure 2 displays this quantity as a function of (nJAoI2.The calculations refer to spheres (m= 1.20) of turbidity-average diameter 70 nm. To model the influence of polydispersity a Gaussian size distribution has been assumed with standard deviations indicated in the graph. The dashed curve shows the result obtained for uniform spheres within the RD approximation (eq (8) of ref 1). This comparison shows that already spheres of this size must be treated within the frame of the Mie theory when discussing the dependence of the specific turbidity on wavelength. Also, a finitepolydispersity may partly mask the difference between the Rayleigh-Debye and the Mie result. A precise analysis of the integrated form factor Q&A2) may thus give information about the size distribution of the spheres provided the dispersion of m is known with great precision (see below.) The discussion of Figure 1has shown that the influence of the optical moments R2,,o and R 4 4 ,on ~ the integrated structure factor (eq (9)) is practically negligible for the particle size under consideration here. Hence, despite the fact that the calculation O f &(A2) must be done within the frame of the Mie theory, the amplitude functions necessary for the calculation of S(q) for the measured
The PS-latex used herein was prepared using a standard emulsion polymerizationof styrene (Fluka,299.5%;destabilized, dried, and distilled prior to use) a t 80 “C under an atmosphere of nitrogen. The recipe contained 50 g of styrene in 525 g of water (distilled twice); 0.312 g of K&Os (Fluka, recrystallized twice from H2O) was used as initiator and 1.4965 g of sodium dodecyl sulfate (Merck; ’99%) as surfactant. The latex was M KC1 solution. dialyzed extensively against 2.5 x The number- and the weight-average diameter as determined by transmission electron microscopy are on= 68.1 nm and ow= 71.3 nm. The particle size distribution could be approximated in the limits of error by a Gaussian distribution with a standard deviation of 8.7 nm. The extinctions of the latex with reference to water were measured using a Perkin-Elmer Lambda 2 s U V M S spectrometer operating a t a scan speed of 480 n d m i n . By varying the scan speed it was assured that the measured turbidities are independent of this parameter. To account for the strong variation of the optical density, five different optical path length (5,2, 1, 0.2, and 0.1 cm) were used. Onlyextinctions for which thevalidity of the Lambert-Beer law could be validated were taken into account (see the discussion of this point in refs 1 and 2). The independenceof the given turbidity of optical path length is strong proof that multiple or fonvard scattering may be dismissed indeed: both effects should lead to a decrease of the turbidity with increasing path length of the light which is not observed. Since water exhibits a non-negligibleabsorption around 1000 nm, a correction for the effect of the background is applied for the region between 700 and 1100nm. At higher volume fractions (a) ofthe latex the extinction ofwater in the sample is diminished by a factor of (1- @), Since the reference is pure water, the extinction of the sample must be corrected by addition of W E w
Apfel et al.
3404 Langmuir, Vol. 11,No.9, 1995
2 7~10.'~
------7
1.5
1.4
c
g
. s
-
24x10'"
5
1.3
3 4
. m
'f
* Q 2
r
r
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P
2 Ixio-"
0
c / g/l
Figure3. Extrapolation ofthe experimental specificturbidities to vanishing concentration according to eq (11): Circles, 1070 nm... crosses, 700 nm; triangles, 500 nm.
where I is the length of the optical path and E , the extinction mnffi&mt nf niirn w n t m n t n
&ven waveleneth
Results and Discussion Scattering Cross Sections. Figure 3 shows that the specific turbidities (zlc), at vanishing concentration can be obtained with very good accuracy. For the sake of comparison, the turbidities are scaled with the fourth power of the wavelength to account for the leading term in eq (7). Although the concentrations are quite small, there is clear evidence for the onset of nonlinearity at concentrations above 1%. This shows that in the regime of concentrations displayed in Figure 3 the third vinal coefficient already gives a small but non-negligible contribution. Therefore all data were fitted by adding a term quadratic in concentration to eq (11). To.check the internal consistency of the data in this regime of low concentrations, the reciprocal integrated structure factor calculated (cf. eq (8)) from the data displayed in Figure 3 is plotted in Figure 4. Since eq (11) suggests that all alterations of ZM(A~,C) are embodied in the apparent second virial coefficient, the concentration axis has been scaled by Bapp.The latter quantity is given by the slopes of the curves in Figure 3 at vanishing concentration divided by the respective intercepts. The dashed line has exactly a slope of 2, as suggested by eq ( 1l),the solid line is a second-order polynomial fit in concentration through all data points. It is obvious that in this regime of concentration Z(A2,c) is in good approximation a function of B,,,c only since all points are lying on a single curve. The deviations from the linear relation eq (11) show, however, that a linear extrapolation to vanishing concentration is not admissible and a polynomial fit as applied in Figure 3 must be used. The influence of the third virial coefficientseen in this regime of concentration renders the determination of the apparent second virial coefficient a rather difficult task and it is obvious that the latter quantity should be measured at much lower concentrations by conventional static light scattering. On the other hand, concentrations employed in the present experiment are still low enough, allowing
0.05
0.10
0 15
0.20
c Ba,
Figure 4. Test of eq (11) for the dilute regime: reciprocal integrated structure factor versus the concentration scaled by the apparent second virial coefficient.Key: circles, 1070 nm; crosses, 700 nm; triangles, 500 nm; the solid line presents a second-orderpolynomial fit in c ofthe data, and the dashed line is a straight line with a slope of 2 (see the text for further explanation). the extrapolation of the specific turbidities to vanishing concentration with excellent accuracy. For the calculation ofthe normalized cross section QM(A~) according t o eq (12) the dispersion of m must be known with utmost accuracy (see eq (6)). For water we used the data given for 25 "C in l i t e r a t ~ r e For . ~ the extended range of wavelengths under consideration here (1100 nm 2 A. 1 400 nm) the refractive index can be described by the empirical relation no2= A BIA2 - CIA4 DIA6 - EIA8 (A = 1.72969, B = 3.41948 x lo4 nm2, C = 1.02982 x 1O'O nm4, D = 1.73695 x 1015nm6, E = 1.05980 x 1020 nm8). Using the Cauchy relation of polystyrene taken from the literature (see the discussion in ref 1)and the standard deviation of the particle size distribution as determined by electron microscopy (see above), the turbidity average diameter of the latex spheres can be fitted to eqs 12 and 13. From this fit the turbidity-average diameter (cf. eq (13)) results to 67.9 nm, which compares favorably with the data furnished by electron microscopy. The dashed line shown in Figure 5 displays the best fit obtained by this set of data, whereas the circles refer to the experimental specific turbidities. For better comparison of theory and experiment throughout the entire range of wavelengths, the specific turbidities have been scaled by (AJn0)4to account for the strong variation with 1of the leading term in eq 12. It is evident that semiquantitative agreement of theory and experiment can be reached (see the discussion of this point in ref 1). Nearly full agreement of theory and experiment (see the solid line in Figure 5) can be obtained by a modification of the Cauchy relation for the refractive index n, of polystyrene (np2= A B/A2 CIA4;A = 2.4556; B = 2.1935 x lo4nm-2; C = 1.3315 x lo3nm-4). Hence, specific turbidities measured for latices ofknown diameter and particle size distribution may be used to evaluate the refractive index of polymers, as suggested by Devon and
+
+
+
+
(9) D'Ana-Lax, E.,Ed.Taschenbuch fur Chemiker und Physiker, 3rd ed.; Springer: Berlin, 1967.
Langmuir, Vol. 11, No. 9, 1995 3405
Analysis of the Turbidity Spectra of Latices 600
500
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I
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3.0~10~ 0
.E
m-
,+
2
29x105
rlr
2 8x10~
16
2 7x10'
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04
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12
(no/h,,)2 io5 I nmQ
Figure 5. Specific turbidities scaled by A4 versus A-2 obtained for the polystyrene latex under consideration and comparison with the theoretical cross section calculated by the Mie theory (eq (12)). For all calculations the experimental standard deviation has been taken. The dashed line denotes the fit of the data using the dispersion of the relative refractive index given by ref 9; the solid line shows the fit by the improved Cauchy relation given in the text. The numbers on the top of the graph display the respective wavelengths in vacuo in nm.
Rudin'O (see also the discussion of this point in ref 1). In the course of the present study it became apparent, however, that for a precise determination of the Cauchy relation of the latex spheres, the calculation of the specific turbidity must be done within the frame of the Mie theory, even for size parameters 2nalA below unity. Thus, using the Cauchy relation of polystyrene obtained1 previously through the Rayleigh-Debye approximation willonly lead to qualitative agreement with experimental data. From the comparison of the Cauchy relations of water and polystyrene it becomes obvious that the marked minimum of the scattering cross sections (see Figure 5) is due to a strong variation of the relative refractive index m in this region of wavelengths. Given the above Cauchy relations, the measured function &,(A') can be calculated from the specific turbidities by resort to eq (12). The specific turbidities are mainly determined by the cube of the turbidity-average diameter a, (see eqs 12 and 131, whereas the normalized scattering cross section &M(A') presents only a correction for the effect of finite wavelength. Therefore &&') is multiplied by the cube of a, to ensure a meaningful comparison for different fits. The respective result is displayed in Figure 6. The dashed curve gives the best fit obtained assuming monodisperse particles; the solid line refers to the result for a standard deviation of 8.7 nm determined experimentally. It can be seen by this comparison that the effect of a small but finite breadth of the size distribution is not negligible. Given the accurate dispersion relations for the relative refractive index m such a comparison can be used in turn to obtain the standard deviation of the size distribution.ll The present analysis has shown, however, (10) Devon, M. J.; Rudin, A. J. Appl. Polym. Sci.1987,34,469.
(11)Melik, D. H.; Fogler, H. S. J. ColloidZnterfaceSci.1983,92,161, and further literature cited therein.
io5 /
Figure 6. Normalized scattering cross section Q&Iz) (see eq (12))obtained from the polystyrene latex (circles) and by the Mie theory. The dashed line displays the fit obtained for a monodisperse system; the solid line results for a nonuniform system with the standard deviation taken from an experiment. For better comparison, Q d A z ) has been scaled by the cube of the turbidity average diameter a, (eq (13))of the spheres. For all calculations, the improved Cauchy relation has been used (cf. Figure 5).
that both the dispersion relations as well as the polydispersity will influence &,(A2). Hence, a precise determination of either quantity may be severely hampered by the interrelation of both effects. Interaction of Particles. For particle diameters below 100 nm, the discussion of Figure 2 has shown that regardless of the inner structure of the particles, the structure function Z(A2,c) of monodisperse systems is determined mainly by S(0) and the first coefficient a. Therefore it should be possible to extract the low-angle part of the structure factor by extrapolating Z(A2,c) to infinite wavelength. Figure 7a,b displays the integrated structure factor obtained for the PS-latex under consideration. The dashed lines are second-order polynomial fits in powers of (nJAJ' as suggested by eq (9). In the regime of small concentrations the determination of Z(A2,c)is less secure, but above 1wt % the integrated structure factor can be obtained with very good accuracy. Due to the extension of the range of wavelengths to 1100 nm, the extrapolation of S ( 0 ) and the determination of the coefficient a (6. eq (9))can be done in a much more accurate fashion than previously1. The S(O),,, and &xp obtained from this fit for different latex concentrations are gathered in columns 3 and 4 of Table 1, respectively. The size distribution of the latex under consideration here is narrow and the effect of polydispersity can be dismissed within limits of error (see the discussion of Figure 2 of ref 1). Therefore the intercepts of Z(A2,c) determined by the above fit (see the dashed lines in Figure 7a,b) do not contain appreciable contributions caused by polydispersity. As a consequence, the analysis of Z(A2,c) can be done in terms of the Weela, Chandler, and Anderson perturbation theory.12J3 As outlined in ref 1the effective volume fraction and the effective diameters of the hard (12) Andersen,H. C.; Weeks,J. D.;Chandler, D. Phys. Rev. 1971,A4, 1697.
Apfel et al.
3406 Langmuir, Vol. 11, No. 9, 1995 600
800
1.0
400
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800
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40C
f
1.0 gll
33.9 g/l
9.5 gll
09
48.6 gll 4 5 g1l
.-!
08
9.4 g/l
N '
14.2 gll
5
N
0.5
N?
38.4 gll
4
-5
N
07
0.3
31.6 gll
06 /
05
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12
0.9
(noo/ho)2io5 / nm.'
n" ,i I
0
03
0.6
09
12
(n,/ho)2 IO5 Inm-2
Figure 7. Integrated structure factor Z&z,c) as function of for different concentrations indicated in the graph. The symbols refer to the experimental data evaluated according to eq (8);the numbers in the graph refer to the latex concentration in gram per liter. The solid lines display the theoretical results calculated (eq (10))for a system of polydisperse hard spheres. The dashed lines refer to a second-order polynomial fit suggested by eq (9). The resulting S(O)., and o+ obtained from both fit procedures are gathered in Table 1. The numbers on top of the graph denote the wavelengths in vacuo m nm. Table 1. Analysis of the Structure Factor of the PS-Latex in the Long-Wave Region no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
cond
S(O),,," GX.4 aWcab1 S(0)p.f
(a)
0.487 0.969 1.46 1.95 2.92 3.85 4.85 6.79 9.55 14.5 19.4 24.2 33.9 48.6 68.4 82.0
0.990 0.975 0.962 0.955 0.927 0.920 0.881 0.840 0.782 0.697 0.617 0.550 0.438 0.317 0.207 0.155
nm2
nm2
9 19 30 31 63 52 110 151 205 253 294 308 298 241 159 108
9 26 39 42 70 68 109 138 175 212 235 244 237 200 144 112
0.989 0.974 0.961 0.953 0.929 0.917 0.884 0.844 0.789 0.704 0.627 0.560 0.447 0.322 0.209 0.154
dp.fl nm
@p.f
98.4 103.6 103.8 100.3 101.1 97.4 101.8 101.0 100.9 100.2 100.1 100.1 99.7 99.3 98.8 98.6
0.0014 0.0033 0.0050 0.0060 0.0092 0.0108 0.0156 0.0219 0.0298 0.0443 0.0592 0.0737 0.1025 0.1447 0.2002 0.2388
ad1 nul2 -
11 26 40 45 67 72 106 133 167 205 228 239 234 200 145 111
a S(O),,, and G,,were obtained from the polynomial fit (see the dashed lines in Figure 7a,b) suggested by eq (9). awcA was calculated from the effective diameter obtained by the condition s(0) = s h s ( 0 ) ; see the text for further explanation. s(o)py, dpy, Qpy, and apy were obtained by the fit of the experimental data by the Percus-Yevick structure factor of hard spheres; Qpy is the effectivevolumefraction ofhard spheres. dpy is the effectiveradius
of hard spheres see the text for further explanation.
sphere reference fluid can be calculated directly from the experimental S(0) values. Thus, the first expansion coefficienta w c A can be calculated in terms of the PercusYevick theory (see column 5 of Table 1). For small (entries 1-6, Table 1)and very high concentrations (entries 15 and 16, Table 1)the first expansion coefficientsG~obtained experimentally practically agree with respective figures for a w C A . For the intermediate range of latex concentrations, a w c A is slightly lower than aexp. This is expected from the deductions of Niewenhuis (13) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986.
et al.14and underscores the validity of the perturbative approach. It also demonstrates that the accuracy of the present set of data is good enough to obtain not only S(0) but also the expansion coefficient a (cf. eq (9)). As already outlined in ref 1 the integrated structure factor can also be interpreted in terms of a given model for particle interaction. For the present latex, the structure factor of choice is given by the Percus-YevickVrij15J6theory developed for a system of nonuniform hard spheres. This is due to the fact that the ionic strength of the latex is quite high, leading to an electrostatic repulsion of the spheres raising steeply at rather short distances." Also,the turbidimetric method discussed herein explores the long-wavelength range of the structure factor. Therefore the present experiment is more sensitive to the balance of repulsive and attractive forces than to the details of repulsive interaction.l8 The solid lines in Figure 7a,b refer to the fit of the Percus-Yevick-Vrij structure factor already applied previous1y.l Since the PS-latex under consideration here is narrowly distributed, the fit of the integrated structure factor obtained (see Figure 7a,b) by assuming monodisperse hard spheres leads to practically the same results as the fit by the Percus-Yevick-Vrij model. Also, as shown by Figure 7a,b the fit of the hard-sphere model (solid lines) nearly coincides with the result of the polynomial fit in even powers of (nJ&)(dashed lines). Table 1gathers S(O), and the effective diameter ( d w )as well as the effective volume fraction and the expansion which results from the fit of polydisperse coefficient (am) hard spheres to the experimental data. By comparison of S(O),, and S(O), (Table 1)it becomes (14) Nieuwenhuis, E. A.; Pathmamanoharan, C.; Vrij, A. J. Colloid Interface Sci. 1981,81,196. (15)Vrij, A. J. Chem. Phys. 1979,71, 3267. (16) van Beurten, P.; Vrij, A. J. Chem. Phys. 1981, 74, 2744. (17) Goodwin, J. W., Ottewill, R. H., Parentich, A. Colloid Polym. Sci. 1990,268, 1131. (18) Salgi, P.; GuBrin, J.-F.; Rajagopalan, R. ColloidPolym. Sci. 1992, 270, 785.
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Analysis of the Turbidity Spectra of Latices obvious that these two values agree very well. Also, the difference of awcA and am is marginal. Therefore modeling the interaction of the latex particles in the long-wave region of the structure factor in terms of a system of hard spheres appears to be a valid approach. It is evident, of course, that the slight decrease of the effective diameter (column 7of Table 1)with increasing concentration directly points to the “softness” of the electrostatic repulsion. In this context it must be remarked that the effective diameters of interaction as gathered in Table 1 are approximately small (ca. 100 nm) and the present experiment therefore explores only the long-wave part of the structure factor. It is obvious that much larger spheres cannot be treated with the approximation eq (9). In this case, S(0) and in particular the coefficient a are not available by a simple extrapolation procedure as shown in Figure 7. But even in this case, a comparison of the measured zM(a2,c)and a given structure factor S(q) can be done as outlined above. Thus, if strong forward scattering can be excluded, turbidity should be still a viable method to explore particle interaction in concentrated latexes. A detailed analysis of latexes with larger diameters is under way.
Conclusion The extension of the turbidimetric analysis to include wavelengths up to 1100 nm leads to an accurate deter-
mination of the “measured”normalized cross section &M(A2) (cf. eq (12)) of the latex system. By comparison with the cross section provided by the Mie theory, the particle size distribution can be analyzed. For finite concentrations the measured integrated structure factor zM(a2,C) of latex suspensions in the region of low q is obtained. For the latex under consideration here (number-average diameter: 68 nm), the wide range of wavelengths employed herein allows the extrapolation of S(0) and the determination of the first expansion coefficient a with very good accuracy. The method works best at higher concentrations (> 1 wt %). Hence, turbidimetry furnishes precise information about particle interactions in concentrated systems where multiple scattering prevails. Since the low-angle region of the structure factor is sensitive to longrange attractive forces, turbidimetry may be the method of choice when studying systems with more complicated potentials of particle interaction.
Acknowledgment. Financial support by the Bundesministerium ftir Forschung und Technologie, Projekt “Mesoskopische Systeme”,by the AIF,project 9749, and by the Otto-Rohm Gedachtnisstiftung is gratefully acknowledged. LA950049W
