Langmuir 1988, 4, 668-676
668
Hard-Sphere Colloidal Silica Dispersions. The Structure Factor Determined with SANS C. G . de Kruif,*t W. J. Briels,t R. P. May,* and A. Vrijt Van’t Hoff Laboratory, Padualaan 8, University of Utrecht, 3584 CH Utrecht, The Netherlands, and Institut Laue Langevin, Rue des Martyrs, 156X-38042 Grenoble, France Received May 4, 1987. I n Final Form: October 8,1987 Small-angle neutron-scattering experimentswere performed on the D11 instrument at Grenoble. The colloidal dispersions in cyclohexane studied contained silica particles sterically stabilized by odadecyl chains terminally g-rafted to the surface. From dilute dispersions the mean particle radius and the radius distribution were obtained. With these single-particle parameters the scattering behavior can be modeled with a hard-sphere interaction potential at all volume fractions up to 4 = 0.4-0.5. Polydispersity effects on the representation of the structure factor were explicitly taken into account. These results confrm and amplify previous results on the modeling of a silica dispersion as a hard-sphere supramolecular fluid.
I. Introduction The thermodynamic and structural equilibrium properties of a colloidal dispersion can be modeled as a supramolecular fluid14 by viewing the colloidal particles as (supra) atoms dispersed in a continuous background. This approach appears to be highly successful, and therefore statistical mechanical theories developed for simple fluids are also applied extensively in modern colloid science. In a particular simple model-the hard sphere-the pair interactions between the particles are zero except for center to center distances smaller than d, the hard-sphere “diameter”, where the pair potential rises to infiiity. The applicability of the model is based on the fact that at moderate and high densities the steep repulsive part of the pair potential dominates the influence of the pair interactions on the fluid “structure”. When this repulsion rises to several k T over a distance small compared to the particle diameter, the repulsion can hardly be distinguished from an effective hard-sphere r e p u l s i ~ n . ~ In our laboratory we work on a model colloidal system which consists of silica spheres sterically stabilized by a dense layer of short aliphatic chains that are terminally grafted to the surface of the spheres.* These particles have a refractive index of -1.44 and are dispersed in a lowdielectric organic solvent, e.g., cyclohexane. From previous studies using techniques like light scattering, sedimentation, dynamic light scattering, and rheometry,48*’we found that these colloidal particles behave as hard spheres. These studies are concerned with both (fluid) dynamic properties and thermodynamic properties such as osmotic compressibility (kT ( d p / d ~ ) ) . For studying the supramolecular fluid structure, e.g., the average position of particles with respect to a given particle, one can use light-scattering methods, at least in principle. In practice problems arise with concern to accessible (limited) Ka space, where K = 4anX-lsin (t9/2) is the length of the scattering vector and a is the particle radius. Furthermore, at high volume fractions multiple scattering blurs the picture. With the availability of the small-angle neutron scattering technique these problems can be attacked more successfully. Limited beam time on the D-11 spectrometer at the Institute Laue Langevin (ILL) at Grenoble, France, was available for the measurements of the static structure factor of silica dispersions in cyclohexane as a function of volume fraction. We studied two silica samples named SJ4 and SP23. The particles have an average radius of 20 and University of Utrecht.
* Institut Laue Langevin.
26 nm, respectively. The SJ4 system was used before, also in SAXS measurements. The SP23 system is used for rheometry and adhesive hard-sphere phase separation experiments also in the Van’t Hoff laboratory. In this paper we present a short review of the theory pertinent to SANS, sample preparation, data reduction, and Results and Discussion. Particular emphasis will be placed (taking fully into account polydispersity effects) not only on the particle-scattering functions but on the structure functions as well.
11. Scattering Equations Excellent treatments on the scattering of cold thermal neutrons by colloidal matter are given by Jacrotg and Hayter.lo The transparency of colloidal matter to cold thermal neutrons is usually very high. When the neutrons are viewed as electromagnetic waves (X(0) 1 nm) it is understandable that the resulting scattering equations strongly parallel those in the Rayleigh-Gans-Debye theory of light scattering (see, e.g., Van de Hulstll and Kerker12). (1)Vrij, A.; Nieuwenhuis, E. A.; Fijnaut, H. M.; Agterof, W. G. M. Faraday Discuss. Chem. SOC.1978,65,7. (2) Dickinson, E. In Colloid Science; Everett, D. H., Ed.; Specialist Periodical Report, The Royal Society of Chemistry: London, 1985;Vol. x, P 3. (3)Ottewill, R. H. Concentrated Dispersions i n Science and Technology of Polymer Colloids; Poehlein, G. W., Ottewill, R. H., Goodwin, J. W., Eds.; 1983;Vol. 11. (4)De Kruif, C. G., Jansen, J. W., Vrij, A. Physics of Complex and Supramolecular Fluids; Safran, S. A., Clark, N., Eds.; Wiley: New York, 1987. (5)McQuarrie, D. A. Statistical Mechanics; Harper and Row: London, 1973. (6)Vrij, A.; Jansen, J. W.; Dhont, J. K. G.; Pathmamanoharan, C.; Kops-Werkhoven, M. M.; Fijnaut, H.M. Faraday Discuss. Chem. SOC. 1983,76, 19. (7)De Kruif, C. G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1985,83,4111. (8)Van Helden, A. K. Organic Silica Dispersions. Preparation, Characterization, and Particle Interactions. Ph.D. Thesis, Utrecht, 1980. Van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354. (9)Jacrot, B. Rep. Prog. Phys. 1976,39,911. (10)Hay-&, J. B. Physics of Amphiphiles-Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985. (11)Van de Hulst, H.C. Light Scattering by Small Particles; Dover: New York, 1981. (12)Kerker, M. The Scattering of Light and other Electromagnetic Radiation; Academic: New York, 1969. (13)Porod, G. Kolloid 2. 1976,124,83. (14)Harris, N. Small Angle Neutron Scattering from Colloidal Systems; Ph.D. Thesis, Oxford, 1980. (15)Moonen, J. Ph.D. Thesis, Utrecht, 1987. (16)Vrij, A. J. Colloid Interface Sci. 1982,90, 110.
0743-7463/88/2404-0668$01.50/00 1988 American Chemical Society
Langmuir, Vol. 4, No. 3, 1988 669
Hard-Sphere Colloidal Silica Dispersions For the reader's convenience, we reproduce here some of the basic equations and definitions. The scattering length of atoms of type i is given by f k The bar indicates an averaging over a scattering probability distribution. The volume-averaged scattering length is then
p(r)d3r=
Zfi; i in d3r
(1)
For monodisperseparticles, the normalized scattering ratio can be written as
R ( K ) = p(pP - pJ2V,2P(K) S(K) + B'
160
8C
0 diameter ( n m )
Figure 1. Histogram of size distribution from TEM; r = 18.4 f 3.3 nm for SJ4.
(2)
Here p is the number density of colloidal particles, and pp and pa are the (average) scattering length density (for neutron scattering) of particle and solvent. V is the volume of the particles (i.e., the volume of liqu&f that is replaced by the particle). The scattering vector K is given by =io(3) K = lZl= ~ P / sin X (812) (4)
z
Here the yector io points in the direction of the incident wave and ksin that of the scattered wave, with the enclosed angle 8. X is the wavelength of the radiation. The background term B'contains contributions from the (coherent and incoherent) solvent scattering and from the incoherent scattering of the silica. The function P ( K ) is called the particle form factor and depends on the scattering length distribution within a particle and on its shape. For a homogeneous sphere
P ( K ) = ([3(sin K a - K u cos K u ) ] / ( K u ) ~ ) ~( 5 )
diameter (nm)
Figure 2. Histogram of size distribution from TEM; r = 23.5 2.8 nm for SP23. tracted from the scattered intensities. For a dilute dispersion where S ( K ) = 1 the small K a limit or Guinier limit is given by
R(K) = PVp2(PP- P,) 2e- ( P r g P / 3
(10)
For a homogeneously scattering sphere the radius of gyration rg is related to its geometrical radius rp as
rg = r p ( 3 / 5 ) 1 / 2
(11)
Making a Guinier plot, Le., In ( R ( K ) )versus P, one thus finds rg from the slope and from the intercept value
R(K=O) = p(pP - d2Vp2
The function S(K) is called the structure factor or structure function and describes interference effects of correlations between particle positions
S ( K ) = ( 1 / w ( E E e i k * ( F-i
(6)
For polydisperse systems with particles having a distribution in particle sizes and particle interactions, polydisperse forms P ( K ) and S(K)can be formulated. P ( K ) is well-known, and S ( K ) we found for hard spheres by using the Percus-Yevick equationl8 and MC simulati~ns.~' From eq 2 it follows that the scattering intensity of the particles is proportional to the number density. At high K values S ( K ) goes to unity whereas the form factor P(K) decays with 1 / K 4 independent of the precise form of the particles under the condition that the scattering density change a t the particle surface is abrupt. This is called Porod's law:13 lim R ( K ) = p(A'/K4) + B'
K-m
R ( K ) K 4 = pA'
+ B'K4
(7)
(8)
In practice we subtract the solvent scattering from R ( K )
[ R ( K )- R c H x ( K ) ] K = ~ pA'+ BK4
(9)
where B (= B '- RcHx(K))is the incoherent background correction due to the particles (minus that of the replaced solvent). So a plot of [R(K) - RCHX]K4 against K 4 has a slope B and an intercept proportional to p , the number density. This is called a Porod plot. The background B is sub(17) Frenkel, D.; Vos, R. J.; de Kruif, C. G.; Vrij, A. J . Chem. Phys. 1986,84, 4625. (18) Van Beurten,P.; Vrij, A. J. Chem. Phys. 1981, 74, 2744.
where Vp = 4/3srp3.So, given absolute scattering length densities and the volume fraction, one calculates a particle radius, rp. The slope of the Guinier plot gives a radius (5/3)1/zrr 111. Experimental Section By dispersingweighed amounts of dry silica particlea in weighed amounts of cyclohexane, we prepared samples of known mass fractions. At 25 "C the density of cyclohexane is 0.775 g whereas that of the silica spheres is m1.6 g cm-3 and is to be determined from the experiment. Size distributions determined from TEM photographs of the particles are shown in Figures 1 and 2. In previous experiments it appeared that scattering from samples in quartz cells with a 1-mmpathway resulted in multiple scattering. Experimentswith sample pathways of 0.5 and 0.2 mm showed that at 0.5 mm hardly any and at 0.2 mm no multiple scattering is present. The presence of multiple scattering was judged from measurements of the particle form factor of a large (SJ18 76-mm radius) particle. The blurring of the minima in the P ( K ) is clearly visible when multiple scattering is present. Furthermore, we checked with the criteria developedby Harris." For the measurements reported here we used 0.1-mm quartz Hellma cells type 121. Measurementswere made at 25 "C on the D11 instrument at the ILL in Grenoble. We used two detector-collimator combinations. For small K values we used the detector at 10 m and a collimation distance of 10.0 m. For larger K values, partly overlapping with the first ones, we used a setup with the detector at 3.50 m and a collimation of 5.0 m. The neutron wave length used was 1.00 nm (AA/A is 9%, fwhm). Scattering intensity data were collected on the two-dimensional detector of the spectrometer.
670 Langmuir, Vol. 4, No. 3, 1988
de
Table I. ExDerimental Results on the Silica Systems 554 and SP23 Porod Guinier C d 106A B intercept slope
m
Kruif et al.
In l/S(K=O)
SJ4 0.005 0.01 0.03 0.1 0.2 0.3 0.4 0.5
B3 B4 17 X1 X2 X3 6 19
0.00987 0.01974 0.05590 0.1763 0.3416 0.4760 0.5902 0.6691
0.0077 0.0155 0.0448 0.152 0.330 0.503 0.683 0.829
0.0048 0.0096 0.0277 0.093 0.200 0.304 0.409 0.493
0.03 f 0.01 0.057 f 0.01 0.189 f 0.01 0.637 f 0.02 1.45 f 0.06 2.14 f 0.04 2.86 f 0.03 3.50 f 0.02
0.16 f 0.01 0.082 f 0.008 0.059 f 0.008 0.114 f 0.008 0.061 f 0.02 0.096 f 0.014 0.076 f 0.012 0.014 f 0.008
0.004 0.01 0.03 0.1 0.2 0.3 0.4 0.5
NO
0.0092 0.0207 0.060 0.191 0.338 0.479 0.590 0.692
0.071 0.0162 0.0479 0.162 0.315 0.486 0.644 0.811
0.004 0.010 0.030 0.102 0.197 0.306 0.409 0.519
0.026 f 0.006 0.055 f 0.006 0.145 f 0.03 0.588 f 0.02 1.07 f 0.03 1.65 f 0.04 2.19 f 0.04 2.99 f 0.05
0.101 f 0.003 0.085 f 0.003 0.060 f 0.014 0.045 f 0.011 0.065 f 0.021 0.024 f 0.016 -0.048 f 0.018 -0.016 f 0.021
9.705 9.628 9.600 (9.504
f f f f
0.022 0.025 0.019 0.010)
mean
-8808 f 139 -8841 f 153 -8667 A 120 (-7805 f 68)
mean
9.644 f 0.05
-8772 f 100
0.097 0.466-0.494 1.42-1.39 2.47-2.34-2.42 3.20-3.25-3.30 3.95-3.89
SP23 8 75 76 77 B1 3 11
IV. Data Reduction The raw data from the detector are radially averaged and stored in an i n t e n s i t y 4 file. Similar files are made for measurements on solvent, empty cell, background noise (Cd file), and water. Finally the transmission (2'') of all samples is measured also. Since the scattering intensity of water is exactly known and practically not K dependent, the resulh on water are used first to linearize the detector and then to put the measurements on a absolute scale. R(K) =
(Isample
- Tsampl$emptycell
- (1- Tsample)1Cd)/pH20
(13)
This reduction is usually done at the Grenoble facility. All the data including the raw data were stored on tape and treated further on our lab computer. Results of these treatments are collected in Table I. The first two columns designate each sample, and the third column designates its mass fraction m and concentration c (mass/volume): mP
m= mp
+ mCHX
;
mP
c= mp/dp
+~CHX/~CHX
(14)
Here mp and m C H X are the masses of silica particles and cyclohexane in the dispersion. By use of d, 1.6 g cm-3 and dCHX= 0.775 g approximate volume fractions are calculated as indicated in the sample designation. Concentrations calculated from actual densities obtained with Porod's law are given in column 4. This density of the silica particles was determined by using eq 8 and by making a Porod plot, Figures 3 and 4. In these figures the relatively large but constant solvent scattering is already subtracted from the measured Rayleigh ratios. Fitting the high K values to a linear equation gave ordinates and slopes as reported in columns 6 and 7. The slope B in a Porod plot (see eq 8) is taken as a background correction of mainly incoherent scattering, while the ordinate of the plot is proportional to the volume fraction of the particles. So pA' = A = constant X c. The volume fraction is defined as
4, = q p c = (l/d,)c
10.243 f 0.012 10.251 f 0.013 10.183 f 0.017 10.236 f 0.059
-13394 f 96 -13841 f 99 -13806 f 130 (-13948 f 375)
mean
mean
10.228 f 0.020
-13680 f 200
+
+
+ +
+&-+-+-+-+-+-
I 1
Figure 3. Porod plot, R(K)K4 against K4 for 554. Linear
least-squares fits of high K values are the drawn lines. The coefficients of these fits are given in columns 6 and 7 of Table I. The RcHx(K) was first subtracted from R(K).
(15)
where qp is the specific volume or the reciprocal mass density of the silica particles. With the definition of the mass fraction, eq 14 and 15 can be combined to give
+
+ X - . - L - p l A - I
f
0.15 0.60 1.30 2.16 2.85 3.60
U
.l
.2
.3
.4 K7nm4
1
Figure 4. Porod plot, R(K)K4 against K4 for SP23. Linear least-squares fits of high K values are the drawn lines. The coefficients of these fits are given in columns 6 and 7 of Table I. The R ~ H ~ (was K )first subtracted from R ( K ) .
Langmuir, Vol. 4, No. 3, 1988 671
Hard-Sphere Colloidal Silica Dispersions
I
A
' 9
-
0
.2
.4
m/A
.6
4
.a
Figure 5. Mass density of SJ4 from Porod plots. Mass fraction divided by A from Porod plot against mass fraction according to eq 16;qp(SJ4) = 0.57 0.02 cms g-l.
*
-
0.02
0.04
0.06 K2/(nm2)
0
0.08
Figure 7. Guinier plot of 554. In [(R(K,c)- B -R(K,solvent))/c] against @. Since the plots virtually coincide they are shifted vertically by a few units. Results of the linear least-squares fit as indicated by the drawn line are given in columns 8 and 9 of Table I.
of them S(K)=1 at all K. We used the second one for dividing out the particle form factor because of the better counting statistics in the scattering pattern. S(K-0) values are determined from a plot of In S ( K ) against Icz, which is up to relatively high K values and volume fractions as a linear function of P.The resulting values for In ( l / S (K=O))are given in column 10 of Table I.
_-
0
.2
-
.6
.4
.8
m/A
Figure 6. Maes density of SP23 from Porod plot. Mass fraction divided by A from Porod plot ainst mass fraction according to eq 16; qp = 0.66 0.02 cm8gy
*
So plotting the mass fraction divided by A against mass fraction gives (from the slope and ordinate) the ratio qcHx/qp.It should be noted that the value of qp thus found represents a mean value over the particle. These plots are given in Figures 5 and 6, and the least-squares fit leads to qp(SJ4)= 0.57 f 0.02 cm3 g-' and qp(SP23) = 0.66 f 0.02 cm3 8'. These values are used to calculate mass concentration, c, as given in column 4. The experimental structure factor of the colloidal dispersion of concentration, c, is calculated from R(K,c)- B ( c ) - R(K,CHX) claw (17) s(K'c) = R(K,clow)- B(clow)- R(K,CHX) c where cbwis a sample with a low volume fraction such that S(K,c)=l for all K. In this equation, R(K,c) is the measured intensity of the sample a t concentration c (see eq 12). B(c) is the background correction in the Porod plot after subtraction of the solvent, as obtained from the slopes in Figures 3 and 4. There is no difference in the scattering pattern of the three lowest volume fractions, showing that for al'l three
V. Results and Discussion Porod Plots. 1. Background Scattering, In Figures 3 and 4 we give a few representative Porod plots. Before calculating the ordinate values we have subtracted the constant solvent scattering from R ( K ) ,which implies that the remaining background scattering is mainly due to the difference in the incoherent scattering of the silica particles and the solvent. The results of the least-squares fitting of these auxiliary data are given in Table I, columns 6 and 7.
The background scattering is small, at least when compared to the scattering intensities at small K. Nevertheless, if this background is not subtracted, one finds structure factors which do not tend toward unity with increasing scattering angle. Furthermore, structure factor values at high K values will slope upward or downward. There is some systematic trend in the background scattering with concentration. The results in column 7 are consistent with the fact that hydrogen contributes more to incoherent scattering then does Si02. From all experimental scattering intensities we subtracted the background values as given in column 7. 2. Mass Densities. Plotting the A values from the Porod plots according to eq 16 leads to Figures 5 and 6. From this we calculate the specific volumes as qp(SJ4) = 0.57 f 0.02 cm3 g-' and qJSP23) = 0.66 f 0.02 cm3 g-l. The procedure is used to obtain a consistent set of intensity levels and concentrations for the dilution range. The two auxiliary values €or the specific volumes of the two silicas differ more than expected, for which we have no ready explanation. Determinations in a pycnometer give qp = 0.62 f 0.02 cm3 g-' for both silicas. The Porod values apparently contain some systematic error. Nevertheless, we used these values for qp to calculate c, as given in column 4 of Table I.
672 Langmuir, Vol. 4, No. 3, 1988
t I
0.02
0.04
de Kruif et al.
0 A
-
I I
0.06
008
.2
0
.4
K2/(nd)
Guinier Plots. The results in Figures 8 and 9 show that for the three lowest concentrations the particles behave as independent scatterers. So-called Guinier fits result in the data in columns 7 and 8 of Table I. The data in parentheses are not taken into consideration because they are obviously influenced by structure effects, which come into play for volume fractions well below 10%. From the slopes we calculate values for the radius of gyration: rg = 16.2 f 0.2 nm for SJ4 and rg = 20.3 f 0.3 nm for SP23. From these values we calculate the radius rp(rg) = 20.0 f 0.2 nm for SJ4 and rp(rg) = 26.1 f 0.3 nm for SP23. Contrast variation measurements of Moonenlbgave for the scattering length density p(SJ4) = (1.93 f 0.08) X cm-2. With the help of eq 11we calculated rp = 19.6 nm whereas Moonen found the value 19.1 nm again, showing the good reproducibility of experimental results. In the above calculations we left out the contribution to the scattered intensity of the stabilizing layer on the surface. Although there is a slight difference in scattering length density of octadecyl chains and cyclohexane, the overall contribution of the layer to the scattered intensity is negligible. The difference between the radii rp(r ) and rP(VJ may be ascribed to polydispersity of the particfe size, because they are derived from different moments of the particle size distribution. For a narrow width of the distribution one may write I J ~= f ; n ( l + u2)(1/2)n(n-1) (184 ((r- i;)/P)2 (18b) where u is the standard deviation. Using these definitions one can derive rp(Por) = ~ (+1u2)1/2 rp(Vp)= ~ ( +1 u 2 ) 5 / 2 rp(rg)= ~ (+1u2)13/2 a2 =
where rp(Por) is the so-called Porod radius and i; is the number average radius as determined, e.g., from electron microscopy. P(K).Form Factor. Modeling the particles as spheres one can easily calculate the scattering intensity of a diluted dispersion of spheres by using the equation
-
CPir,;pi(K) i
d
4
.6
K/(nml)
Figure 8. Guinier plot of SP23. In [(R(K,c)- B - R(K,solvent))/c] against p. Since the plots virtually coincide they are shifted vertically by a few units. Results of linear least-squares fit as indicated by the drawn line are given in columns 8 and 9 of Table I.
NK)
A
(19)
Figure 9. P(K), form factor of SJ4. Plot of experimental scattering intensities against K on a semilog scale. The dashed line is the optimal fitting result, see text.
Z
’6-
T
4:
2-
0I
0
I
.2
I
I
I
I
.6
.4
1
.8
K/(nm+)
Figure 10. P(K), form factor of SP23. Plot of experimental scattering intensities against K on a semilog scale. The dashed line is the optimal fitting result, see text.
where the form factor Pi(K) is obtained from eq 5. We used a log-normal distribution defined by Pi
- f(4
d(d) = (2a@2d2)-1/2 exp[-(ln (d/d0))2/2@2]d(d)(20)
The moments defied in eq 18a and 18b are obtained with this distribution, where the following relation exists between @ and u: exp p2 = 1
+ u2
(21) For a narrow distribution the shape is not very important, and eq 20 will always be usable. In Figures 9 and 10 we plot the experimental measured intensities against the wave vector K in a semilog plot. The drawn curve is a best simulation of the experiment by adjusting two parameters, i.e., the mean particle radius and the width of its distribution. Since the position of the minima and the sequence of relative maxima are very sensitive to the mean (number average) particle radius, it can only be varied in narrow
Hard-Sphere Colloidal Silica Dispersions 0
p-@?:y: 5
Langmuir, Vol. 4 , No. 3, 1988 673
*
I
I
*/
1
I
.5
1
>-
1.5
K2/(nm-2)
I 2
Figure 11. Plots of In S(K)against for SP23 at different concentrations. Ordinate values give the osmotic compressibility and are listed in column 10 of Table I.
limits, say 0.1-0.2 nm. The “filling in” of the minima is determined by the width of the radius distribution and the instrumental effects caused by distributions in wavelength of the radiation and acceptance angle (e.g., ref 14, Chapter 5). The drawn curves in Figures 9 and 10 are calculated without instrumental effects. Taking these into account14 smooths in particular the first minimum but has less effect on the higher extrema, which is in accordance with the experimental findings in Figures 9 and 10. Here again oscillations a t higher K values readily disappear when u is chosen too large. For SJ4 the second and third maximum in the experimental and simulated P ( K ) are clearly visible. On increasing u from 11% to 13% these details in the curve disappear. For SJ4 the mean radius determined from the P(K) extrema is 17.2 nm. It is, as expected, smaller than the rp determined from rg and from V,, since both values were found from higher moments. As one observes from Figure 9, the values at the lowest K values for SJ4 give a poor fit to the data. The experimental points are appreciably higher than the theoretical ones. This does not occur with the SP23 silica. We surmise that the 554, which is an “old” sample used for several different experiments, contains some clusters of particles. For the sole purpose of alleviating the burden put on extracting the structure factor by this inconvenience, we added a “doublet” fraction to better fit the calculation of P(K) for SJ4. This was unnecessary, however, for the silica SP23. The best fit for P ( K ) over the whole K range, see Figure 10, was obtained for r = 22.5 nm and u = 0.12. The difference with r ( r g )= 26.1 is thus accounted for by the polydispersity. %he mean particle size determines the position of the extrema while polydispersity determines other features. To conclude this section, anticipating the results of the next section, we think that fitting the form factor gives precise information on both the mean particle radius and the size distribution. S ( K =O). Osmotic Compressibility. The structure factor at K = 0 follows with eq 17 from plots of In S ( K ) against P. Such plots are very nearly linear for hard spheres and also when there is some distribution in the As an example, hard-sphere diameters (p~lydispersity).~J’ we have given here the results for SP23 in Figure 11. In Figures 12 and 13 we plot the values thus found (see column 10 of Table I) against the concentration c. For SJ4 the extrapolation is not completely unambiguous, and we
.2
0
-
.4
.6 .8 C/(g.cni3)
.10
Figure 12. Plot of In l/S(K=O) against c (columns 10 and 4 of Table I). From the slope is found a specific hard-sphere volume qHS = 0.67 cm3 g-’ (SJ4).
2
4
C(gcmi3) 6 8
Figure 13. Plot of In l/S(K=O) against c (columns 10 and 4 of Table I). From the slope is found a specific hard-sphere volume of qHS = 0.57 cm3 g-’ (SP23).
therefore give ordinate values representing extreme and mean values. For a monodisperse particle system S(K=O) is equal to the osmotic compressibility, (k!l‘)dp/da, but such a simple relation does not exist for polydisperse systems. Theoretical plots for In (l/S(K=O))may be obtained from a theory6J6 or from computer simulations results. Even for hard spheres no exact theory is known, but the Percus-Yevick theory is satisfactory in most cases, even for polydisperse systems. This was recently found from computer simulation^.'^ For a log-normal distribution the Percus-Yevick expression id6 S(K=O) = (1 - 4 ) 2
[
1-
~
64 942 l + 1 + 2d 24 I1 (1 ,2)2 .2)2 (1 24)2 (1 u2)3 l l
+
+
+
where 4 = (7r/6)Cpidi3 is the overall hard-sphere volume fraction. Some plots are shown in Figure 14 for u = 0, 0.1,
674 Langmuir, Vol. 4,No. 3, 1988
de Kruif et al.
I
/
0
2
0
6
4
- 0 Figure 14. Plots of In l/S(K-0) as a function of volume fraction for polydisperse ( u ) hard-sphere systems. The dashed lines are polydisperse hard spheres, u = 0, u = 0.1, and u = 0.2: ( A , 0) Monte Carlo simulations for u = 0.1 and u = 0.2;(0) experimental data for SP23.
t !
i
I
/ /
/ / ,
/
I
0
---?=-
U
B’ I
-Q,
.4
.6
Figure 15. Experimental structure factor at K = 0 of SJ4 as a function of volume fraction. The drawn curve is the theoretical value with u = 0.11,
and 0.2. Also some Monte Carlo results are shown. The plots of In (l/S(K=O))against r$ are surprisingly linear up to 4 0.4. The slope is found to be 7.90 and becomes smaller for polydisperse systems ( u > 0). The triangles and circles in Figure 14 represent the results of Monte Carlo simulations with u = 0.1 and c = 0.2. The least-squares slopes of the data in Figures 12 and 13 lead to a specific hard-sphere interaction volume:
-
qHS
4
6
8
K/(nm’)
*.2
2
= r$/c
where r$ is here the overall hard-sphere volume fraction and c the mass concentration of the particles. It is found that q~~ = 0.67 cm3 8-l for SJ4 and qHS = 0.57 cm3 g-’ for SP23. The difference in these values for particles which differ not much in size is noteworthy. Still we believe that this has no real physical significance. One observes for instance that the slope in Figure 12 is indeed somewhat steeper than the slope in Figure 13 but does not go exactly through the origin. A forced fit through the origin on the
Figure 16. Structure factor S(K) of SJ4 at different volume fractions. The dashed line is the calculated structure factor for a polydisperse ( u = 0.11) hard-sphere system.
other hand would lead to a smaller value of qHS. In the next section we have an independent criterion to adopt a value for q H S , i.e., by comparing the whole S ( K ) function with theory (and not only S(K=O)).There it is found that qHS = 0.62 cm3 8-l is a good choice with consistent results for both silica particles. This value is also in accordance with recent pycnometric density measurementa of SP23, leading to qp = 0.62 cm3g-l. The value qHS = 0.62 cm3g-’ is used to calculate the (hard-sphere) volume fraction of particles in the dispersion as given in column 5. Now the data of column 10, i.e., compressibility (11s(K=O)),are plotted against volume fraction and compared with (polydisperse) hard-sphere theory in Figures 14 and 15. In Figure 14 it can be seen that the S(K=O) data of SP23 (squares) do not follow hard-sphere theory completely a t high volume fractions. Two remarks can be made. Firstly, increasing polydispersity to u = 0.2 would give a good agreement. Secondly, the experimental S(K=0) is found from an extrapolation of In S(K) versus P in a region where S(K) shows a large change at high volume fractions. As indicated below this will tend to *average” the data, leading to a high value of S ( K ) (see also Figure
Langmuir, Vol. 4, No. 3, 1988 675
Hard-Sphere Colloidal Silica Dispersions
8 6
IA
I
Figure 17. Structure factor S(K) of sP23 at different volume fractions. The dashed line is the calculatedS(K) for a polydisperse (a = 0.12) hard-sphere system.
18). In general it can be remarked that extrapolations to K = 0 put a hard strain on the experimentaldata. We have
not found any such analysis in other SANS studies, e.g., on microemulsions. In accordance with previous observations on light scattering and other techniques3s4we are confirmed in our opinion that these silica particles in cyclohexane behave as hard spheres, over the whole concentration region. Deviations from this behavior are not clearly present, at least within the accuracy range of the measurements. There is, however, an independent criterion which can be found from the K dependence of the structure factor (at small K), but this will be considered next.
Figure 18. Experimentalscattering curves of 554 compared with calculated curves of R ( K ) .
S (K). Structure Function. The experimental scattering data at a given volume fraction are divided by those of a low volume fraction, see eq 17, to obtain the structure function S(K)over the whole range of K. These results are plotted in Figures 16 and 17 and represent the experimental structure factor. On the other hand we now have all parameters required to simulate the structure factors. For this we need the volume fraction of hard spheres, particle size, and size distribution. For a narrow distribution the form is not too important. The width of the distribution CT is, however, important. To calculate S ( K ) we use the (hard-sphere) volume fraction as given in Table I, column 5. The particle size and distribution are taken from the results of fitting the form factor P(K). Although the dense layer of octadecyl chains terminally grafted to the surface of the particles is virtually invisible during the scattering experiment, it does contribute to the interaction radius. Assuming a carboncarbon bond length of 1.54 A, a bond angle of 109O, and an average angle between the long axis of the parafinic chains and the tangent to the surface of 54O, the layer
de Kruif et al.
676 Langmuir, Vol. 4, No. 3, 1988
'
O
a
K-space is made. (ii) A similar effect causes the nonmonochromaticityof the neutron beam (A = 1nm; AA = 9%). (iii) Also, the divergence of the neutron beam smears the data. All three effects tend to wash out the details, as expected, but are hard to make visible for the smaller volume fractions. However, for 4 = 0.5 we calculated, e.g., that the height of the first peak should be decreased by -10% for fwhm = 0.09 and by -20% for fwhm = 0.18. The experimental data at high K values are scattered by the bad statistics of the low concentration data. For the lowest concentration examples are shown in Figures 16 and 17, but for the other concentrations points above K = 0.4 nm are deleted for clarity. There the intensity is hardly above the solvent level, although we did accumulate counts for a relatively long time. Because of the strong K dependence of the form factor, one needs a large dynamic range of the detector. The results show that the peak of the S(K) is at the right position and that the shape of the functions in front of the peak is satisfactory. It is fully consistent with our previous conclusion of a repulsive interaction of the hard-sphere type. We find no indications of an upswing in S ( K ) for K 0, which would show up in the case of attractive forces. In our opinion the comparison between calculated and experimental data is quite satisfactory. It is apparent that a polydisperse S ( K ) is needed to explain the results. This also implies that no details beyond the peak can be obtained but that on the other hand the large dip in the forward scattering, so typical of strong repulsions, is still present also in polydisperse systems. Total Scattering Intensities. In Figures 18 and 19 we plot for a few representative volume fractions the logarithm of the experimental scattering intensities against wave vector K. The drawn curves are calculated values (without instrumental effects). This type of comparison, which is often applied in microemulsion systems, is less demanding and we think also less convincing than a comparison of S ( K ) values.
r
6 4
-
c
%
2l t 0'
I
2
4
6
8
K(nm')
Figure 19. Experimental scattering curves of SP23 compared with calculated curves of R ( K ) .
thickness is 1.8 nm. This value is added to the form factor radius in order to obtain the interaction radius that should be used to calculate S(K): we take r = dHs/2 = 17.2 + 1.8 = 19.0 nm ( u = 0.11) for SJ4 and r = dHs/2 = 22.5 1.8 = 24.3 nm (a = 0.12) for SP23. With these parameters we calculated the polydisperse structure factors with the computer program according to Van Beurten and Vrij.18 We would like to emphasize that there are no further adjustable parameters used than q H S = 0.62 cm3 g-l. All other parameters were obtained from the dilute dispersion. The experimental data are somewhat blurred by three causes: (i) Due to the way the detector is set up and the data are analyzed, an averaging over a small region in
+
VI. Conclusions The experimental results presented here can adequately be described with the help of simple scattering theory and the statistical mechanical description of a polydisperse had-sphere system. The silica dispersions can be described self-consistently as a hard-sphere supramolecular fluid. This conclusion is based on the observation that parameters measured on a dilute dispersion can be used without further adjustments or supposition for the full concentration range up to 4 = 0.4-0.5. The system thus seems to be a nice model system and will be used in further studies where particular attention will be given to the (fluid) dynamic interactions, such as those present in diffusion processes, and rheological effects. Also, the effect of particle-particle attractions in poor solvents will be studied. Acknowledgment. This work was supported by the Commission of the European Communities through Grant No. STI*018. Registry No. Si02,7631-86-9.
