Charge-Stabilized Liquidlike Ordered Binary Colloidal Suspensions. 2

IPFA-PFA(Q), IPFA-PS(Q), and IPS-PS(Q) and partial structure factors SPFA-PFA(Q), SPFA-PS(Q), .... and (2) is expressed in cm-1, the partial quantitie...
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Langmuir 1999, 15, 345-352

345

Charge-Stabilized Liquidlike Ordered Binary Colloidal Suspensions. 2. Partial Structure Factors Determined by Small-Angle Neutron Scattering N. Lutterbach and H. Versmold* Institut fu¨ r Physikalische Chemie, Templergraben 59, RWTH Aachen, 52062 Aachen, Germany

V. Reus, L. Belloni, and Th. Zemb CEA/Saclay, Service de Chimie Mole´ culaire, 91191 Gif sur Yvette Cedex, France

P. Lindner Institut Laue-Langevin, 38042 Grenoble, France Received July 3, 1998. In Final Form: October 19, 1998 In this paper we present small-angle neutron scattering (SANS) studies on liquidlike ordered binary colloidal suspensions. Using perfluorinated (σPFA ) 162 nm) and polystyrene (σPS ) 79 nm) particles, we prepared samples made of the same colloids as in the preceding paper. All system parameters of the mixtures were as before. Via neutron contrast variation we directly obtained all three partial intensities IPFA-PFA(Q), IPFA-PS(Q), and IPS-PS(Q) and partial structure factors SPFA-PFA(Q), SPFA-PS(Q), and SPS-PS(Q). The experimental results are compared with theoretical predictions for binary colloidal mixtures, based on the pure repulsive part of the DLVO potential. Within the hypernetted chain (HNC) closure an accurate agreement is achieved. The particle correlation in the mixtures is clarified by means of the partial distribution eff eff (r) and UPS-PS (r). From this we functions gij(r) and the one-component effective pair potentials UPFA-PFA deduce for the examined suspensions a partial clustering or local demixing by attractive interaction contributions of depletion origin.

I. Introduction Great technical interest in colloidal behavior resulted in recent years in the extension of investigation from monodisperse to polydisperse colloids as model substances. With this paper, we present the continuation of our work on binary suspensions of highly charged polystyrene (PS) and perfluorinated (PFA) particles in the liquidlike ordered state. In the preceding paper, part 1,1 we examined concentrated binary systems by ultra-small-angle X-ray scattering (USAXS). Due to this technique and the different relative electron densities of the materials used, we were able to obtain the partial structure factor SPFA-PFA(Q) of the PFA component1 only. Here, we investigate the same particles and identical systems by means of small-angle neutron scattering (SANS),2,3 which combines the advantages of an extended available Q-range and of multiple scattering control. Therefore, it becomes suitable for structural investigations on concentrated colloidal systems. Due to the different neutron scattering length densities of the particles, we were able to control the scattering properties of each compound in the binary latex mixtures by an appropriate H2O/D2O composition of the suspending medium. This neutron contrast variation technique permitted us to determine now all three partial structure factors SPFA-PFA(Q), SPS-PS(Q), and the cross term SPFA-PS(Q). Binary colloidal mixtures of charged polymer particles * To whom correspondence should be address. (1) Lutterbach, N.; Versmold, H.; Reus, V.; Belloni, L.; Zemb, Th. Langmuir 1999, 15, 337. (2) Feigin, L. A.; Svergun, D. I. Structure Analysis by Small Angle X-Ray and Neutron Scattering; Plenum Press: New York-London, 1987. (3) Ottewill, R. H. In Colloidal Dispersions; Goodwin, J. W., Ed.; The Royal Society of Chemistry: London, 1981; p 143 ff.

have already been subject of investigation. Hanley et al.4 measured 10 vol % suspensions with SANS and extracted partial structure factors. Although obviously liquidlike, their mixtures were treated as having a crystalline structure. Structure factors were then explained with a more or less stringent lattice model. Ottewill et al.5 worked on much more diluted binary suspensions containing salt. Using SANS, they received partial structure factors. Unfortunately, their experimental curves were not compared to any simple-liquid-theory approach. In our paper, a rather concentrated (≈9% volume fraction) binary system of highly interacting charged particles is brought to high liquidlike order by complete deionization. We present for a size ratio of ≈2 and for three compositions the partial intensities Iij(Q) and structure factors Sij(Q). These experimentally extracted quantities allow us to investigate the influence of bidispersity on the internal microstructure. To better understand the local organization of such mixtures, we compare all experimental data to theoretical simple-fluid-model results. We use the powerful and most established twocomponent approach based on the binary OrnsteinZernike and hypernetted chain (HNC) equations6 and on the purely repulsive electrostatic DLVO potentials.7 This model reproduces very well the experimental Q-dependence of scattering. To elucidate the various particle correlations more physically, we then focus on the pair (4) Hanley, H. J. M.; Straty, G. C.; Lindner, P. Langmuir 1994, 10, 72. (5) Ottewill, R.; Hanley, H. J. M.; Rennie, A. R.; Straty, G. C. Langmuir 1995, 11, 3757. (6) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (7) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier Publisher Co.: Amsterdam, 1948.

10.1021/la980822y CCC: $18.00 © 1999 American Chemical Society Published on Web 12/23/1998

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distribution functions and the one-component effective potentials which illustrate the interactions between two like colloid particles averaged over the configuration of all particles of the opposite type. These functions are well suited to highlight the importance of so-called depletion effects inside our mixtures. Depletion is in the literature a well-known effect in asymmetrical binary systems and generally denotes an indirect attraction between like particles induced by a direct repulsion between unlike particles. This effect is an expression of volume exclusion from one to the other particle kind and could be considered therefore as an osmotic effect.8-10 So, to our best knowledge we present in the following for the first time a complete treatment of highly liquidlike ordered charged binary systems: (a) SANS measurements at different contrasts for three different mixing ratios; (b) extraction of all partial scattering intensities Iij(Q) and partial structure factors Sij(Q); (c) testing of the reliability of the two-component theoretical approach by comparing all experimental data to calculated results; (d) explanation of scattering behavior and particle correlation by means of radial pair distribution functions and effective pair potentials. II. Theory In the following some basic equations of scattering and interaction related to binary colloidal suspensions are listed. The total scattered coherent, elastic intensity I(Q) of spherical particles of a polydisperse suspension is given by11

I(Q) )

∑i∑jxcicjViVj ∆Fi ∆FjxPi(Q) Pj(Q) Sij(Q)

(1)

ci and Vi represent the number density and volume of particle species i, respectively. ∆Fi ) Fi - Fm denotes the difference between the coherent neutron scattering length density3,12 of particle i (Fi) and that of the suspending medium (Fm), which gives the contrast of the sample in SANS. Pi(Q) and Sij(Q) are the form factors for spherical particles and the partial structure factors, respectively. For our binary PFA/PS system we can theoretically express the intensity I(Q) in eq 1 as a linear combination of the three partial quantities Iij(Q):

binary Ornstein-Zernike and HNC integral equations.6 For Uij(r) we choose the well-known repulsive DLVO form:7

Uij(r) )

)

κ denotes the Debye-Hu¨ckel screening constant which depends on the counterion concentration ∑iciZieff (no added salt), Ri values are the particle radii, and r represents the interparticle distance. As usual, the effective charges Zieff are considered as adjustable parameters determined by the fit of experimental data. By use of this potential (3), the binary Ornstein-Zernike equation is solved with help of the binary HNC closure and by means of standard numerical iterative techniques.13 Once the convergence is reached, the obtained partial structure factors Sij(Q) allow calculation of the partial quantities Iij(Q) ( eq 2) and the total intensity I(Q) ( eq 1). Although the Sij(Q) contain in a hidden form the same information about pair correlation, the pair distribution functions gij(r)6 are more suitable to elucidate the physical behavior of the various components inside the mixture. One further step in the understanding of the different cross correlations consists of calculating the one-compoeff eff (r) and UPS-PS (r). The nent effective potentials UPFA-PFA idea is to map the two-component PFA/PS system on a one-component system made of PFA (respectively PS) particles which interact through the effective pair potential eff eff UPFA-PFA (r) (respectively UPS-PS (r)). The precise definieff tion for UPFA-PFA(r) is: the potential which leads within the one-component model to the same pair distribution function gPFA-PFA(r) than that found in the original binary model, with the PFA concentration being the same in both approaches. Using the one-component Ornstein-Zernike eff (r) is given by14,15 and HNC equations, UPFA-PFA eff UPFA-PFA (r) eff ) γPFA-PFA (r) - ln gPFA-PFA(r) kT

(4)

where

I(Q) ) (FPFA - Fm) IPFA-PFA(Q) + (FPFA - Fm)(FPS -

eff γˆ PFA-PFA (Q)

Fm)IPFA-PS(Q) + (FPS - Fm)2IPS-PS(Q) (2)

(8) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (9) Mondain-Monval, O.; Leal-Canderon, F.; Philip, J.; Bibette, J. Phys. Rev. Lett. 1995, 75, 3364. (10) Ro¨hm, E. J.; Ho¨rner, K. D.; Ballauff, M. Colloid Polym. Sci. 1996, 274, 732. (11) Guinier, A.; Fournet, G. Small-Angle Scattering of X-Rays; Wiley & Sons: New York, Chapman & Hall: London, 1955. (12) Windsor, C. G. In Chemical Applications of Thermal Neutron Scattering; Willis, B. T. M., Ed.; Oxford University Press: Oxford, 1973; p 1 ff.

)(

with r g Ri + Rj

2

Note that while the scattered total intensity I(Q) in eq 1 and (2) is expressed in cm-1, the partial quantities Iij(Q) in eq 2 are expressed in cm3 and are not, strictly speaking, intensities by unit. They are contrast independent quantities which keep the same values for all H2O/D2O solvent compositions. Equation 2 is the basic equation for the use of the contrast variation technique in neutron scattering. Details are given in the Experimental Section. The partial structure factors Sij(Q) can be calculated from the three ion-averaged colloid-colloid pair potentials Uij(r) using the statistical mechanical model based on the

(

eff eff e2 Zi exp(κRi) Zj exp(κRj) exp(-κr) 4π0 1 + κRi 1 + κRi r (3)

)

[SPFA-PFA(Q) - 1]2 SPFA-PFA(Q)

(5)

and eff eff (r) ) hPFA-PFA(r) - cPFA-PFA (r) γPFA-PFA

(6)

with h(r) the total and c(r) the direct correlation function.6 eff UPS-PS (r) is calculated similarly. While UPFA-PFA(r) is a solvent- and ion-averaged potential between PFA particles eff (r) must be understood as an in the mixture, UPFA-PFA averaged potential, with the average being taken also on eff (r) must the configuration of the PS particles. (UPFA-PFA distinguished from the potential of mean force, -kT ln gPFA-PFA(r), which averages over all particles, including the PFA neighbors.) Note again that, while eff UPFA-PFA (r) contains in principle no more information than gPFA-PFA(r), it is well suited to elucidate the various (13) Belloni, L. J. Chem. Phys. 1988, 88, 5143. (14) Belloni, L. J. Chem. Phys. 1986, 85, 519. (15) Belloni, L In Neutron, X-Ray and Light Scattering; Lindner, P.; Zemb, Th., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 135 ff.

Structure Factors of Binary Suspensions by SANS

couplings in the mixtures. In particular, it will clearly illustrate the screening and possible depletion effects of the PS particles onto the effective PFA-PFA interactions (and vice-versa). III. Experimental Section Materials and Colloidal Suspensions. The colloid particles used for these studies were polystyrene spheres (PS) of diameter σ ) 79 nm and perfluorinated spheres (PFA) of diameter σ ) 162 nm. The size ratio σPFA/σPS was =2. Both particle species were stabilized by negative surface charges and dispersed in water or mixtures of H2O (F ) -0.56 × 1010 cm-2)3 and D2O (F ) 6.40 × 1010 cm-2)3 with different ratios for contrast matching. Water was deionized completely and D2O (99.8% deuteration (Riedelde Hae¨n, Germany)) was used without any further purification. In comparison to pure H2O, the chemical and physical properties of the colloidal systems in H2O/D2O mixtures are taken as unchanged. Pure one-component suspensions were enriched with D2O by dialysis in purified regenerated cellulose tubing of type VISKING. The H2O/D2O ratio was controlled by Raman spectroscopy. By mixing appropriate volumes of these and in pure H2O dispersed mother suspensions, we obtained the mixtures with distinct neutron scattering length densities and particle fractions. The binary mixtures and the neat suspensions were of nearly the same overall volume fraction (=9%). Details are given in Table 1. Before every measurement the suspensions were deionized with mixed-bed ion-exchange resin of corresponding medium neutron scattering length density to obtain a highly liquidlike order. Dilute (1 vol %), disordered samples were used to obtain the P(Q) of the two particle species. The temperature was 20 ( 1 °C. Additional information about materials, preparation aspects, and sample handling is given in the preceding paper.1 SANS and Data Treatment. All measurements were made on the D11 instrument at the ILL in Grenoble, France.16,17 The source of neutrons was a 57 MW high flux reactor with a liquid deuterium cold source. The D11 is a small angle diffraction spectrometer with a resolution dependent on the sample-detector distance L. We measured at L ) 5, 20, and 35.7 m with a collimation of 10.5, 20.5, and 40.5 m. Wavelength divergence was approximately 10% (fwhm). With λ ) 14 Å the Q-range extended from 7 × 10-4 to 3 × 10-2 Å-1. The detector was a two-dimensional BF3 detector with 64 × 64 elements, each element 10 × 10 mm2. All suspensions were measured in closed, carefully cleaned rectangular quartz cells from HELLMA (Mu¨llheim, Germany). They offered various thicknesses of either 1 mm or 0.3 mm as sample path length and contained no resin. Measuring times were of the order of 10 min up to 2 h. Since we measured in principle spatial symmetric intensity distributions, the data files could be radially averaged around the previously masked beam center. Before that, care was taken that the scattered intensities were uniform and without any peaks. All scattered intensities were corrected for electronic and spherical background and quartz-cell scattering. Solvent correction appeared not to be necessary. To convert raw data in the absolute intensities, we normalized all data by means of transmission values and instrument calibration by water scattering data at L ) 5 m. Intensities measured at L ) 20 and 35.7 m were correspondingly aligned.18,19 Multiple scattering was avoided by using thin sample cells and contrast variation. No further mathematical treatment was carried out. (16) Lindner, P.; May, R. P.; Timmens, P. A Physica B 1992, 180 & 181, 967 and references therein. (17) Ibel, K., Ed. Guide to Neutron Research Facilities at the ILL; Institut Laue-Langevin: Grenoble, France, 1994. (18) Cotton, J. P. In Neutron, X-Ray and Light Scattering; Lindner, P., Zemb, Th., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 19 ff. (19) Ghosh, R. E. A Computing Guide for Small Angle Neutron Scattering; Institut Laue-Langevin: Grenoble, France, 1989.

Langmuir, Vol. 15, No. 2, 1999 347 Complete data evaluation was made under the assumption of pure elastic and coherent sample scattering. Corrections for 1H (and 2H) were disregarded and no Placzek correction20,21 was applied. Extraction of the Partial Quantities. One of our goals in the present study is to experimentally separate the different partial contributions Iij(Q) from the total intensities I(Q). The neutron scattering length densities Fi calculated3 for the particles are FPS ) 1.42 × 1010 cm-2 and FPFA ) 4.6 × 1010 cm-2. These values were confirmed by intensity measurements on pure individual suspensions with different solvent compositions. The strategy used to extract the three Iij(Q) for a given PFA/PS mixture was as follows: the SANS intensity is recorded at the PFA contrast match Fm ) FPFA ) 4.6 × 1010 cm-2. The intensity ( eq 2) reduces to the PS-PS contribution, I(Q) ≡ I0(Q) ) (FPS FPFA)2IPS-PS(Q), which directly leads to IPS-PS(Q). In principle, IPFA-PFA(Q) could similarly be extracted at the PS contrast match, Fm ) FPS. But, due to the relatively large volume occupied by PFA particles and their extremely good scattering ability, multiscattering effects arise when Fm is too far away from the PFA matching point FPFA. (It turned out that such effects were decisively smaller for the smaller PS particles and could therefore be neglected.) Thus, in general we limited our Fm range to the vicinity of FPFA (from 3.9 to 4.4 × 1010 cm-2, see Table 1) where the multiscattering is negligible. For each of the these Fm values the ratio (I(Q) I0(Q))/(Fm - FPFA) was calculated and plotted as a function of (Fm - FPFA). We verified that the plot was linear within the experimental uncertainties and therefore in agreement with the general expression eq 2. For each Q-value, a least-squares linear fit allowed IPFA-PS(Q) to be extracted from the intercept (Fm FPFA f 0) and IPFA-PFA(Q) to be extracted from the slope. A more detailed listing of the calculation is given in the Appendix. Note that, due to the large asymmetry in scattering power between the two components, the experimental uncertainty (using any extraction technique) is maximum for the cross partial intensity IPFA-PS(Q). Finally, we verified that the intensity measured at the PS contrast match, Fm ) FPS, is much smoother than the above correctly extracted IPFA-PFA(Q), revealing the presence of multiscattering. Additionally, the intensity has been recorded Fm ) FPS for a thinner sample cell (0.3 mm sample thickness) where the multiscattering should be significantly reduced. As expected, the so directly measured intensities IPFA-PFA(Q) got in almost perfect agreement with the previously extracted IPFA-PFA(Q). With the described strategy we are sure that the intensities measured at the different contrasts Fm correspond all to the same three partial quantities Iij(Q) (eq 2).

IV. Results and Discussion As already mentioned, one aim of this work is to extrude all three partial structure factors Sij(Q) of a binary colloidal suspension. Since we succeeded in measuring one partial intensity by USAXS directly (see preceding paper1), we chose here the same system with its same parameters. In Figure 1 the measured SANS particle form factors PPS(Q) and PPFA(Q) are shown. The best theoretical fits to spherical shape give particle diameters of 78 nm (PS) and 162 nm (PFA), respectively. These values are in good agreement with USAXS and static and dynamic light scattering measurements.1 In contrast to the USAXS experiments1 a fitting to polydispersity was not reasonable because finite size of detector elements, polychromaticity of incident wavelength, and possible multiple scattering prevented exact analysis of the I(Q) minima. Moreover, the second maximum of the SPFA-PFA(Q) lies in the region of the first minimum of PPFA(Q) and therefore showed a rather high sensibility to the division of IPFA-PFA(Q) by PPFA(Q). Thus, a good comparability of I(Q) and P(Q) in (20) Blum, L.; Rovere, M.; Narten, A. H. J. Chem. Phys. 1982, 77, 2647. (21) Page, D. I. In Chemical Applications of Thermal Neutron Scattering; Willis, B. T. M., Ed.; Oxford University Press: Oxford, 1973; p 173 ff.

348 Langmuir, Vol. 15, No. 2, 1999

Figure 1. SANS particle form factors of polystyrene particles (PS) (Fm ) -0.56 × 1010 cm-2) (O) fitted with a diameter of 78 nm (s) and of perfluorinated particles (PFA) (Fm ) 4.2 × 1010 cm-2) (b) fitted with a diameter of 162 nm (s). In addition, the PFA form factor measured at Fm ) -0.56 × 1010 cm-2 is shown (9) (see text) (for clarity intensities are arbitrarily multiplied).

Figure 2. SANS structure factors of the neat suspensions (samples 1 and 5) and their corresponding one-component HNC fits (s). For comparison, a HNC PS structure factor with 13.5% particle polydispersity is shown (- - -) (see text).

their I(Q)-over-Q-dependence is necessary to obtain by division an accurate S(Q). Concentration and contrast dependent multiple scattering effects prevent often such a comparability.18 We achieved the best result for the division using a particle form factor PPFA(Q) measured with 1% volume fraction and Fm ) -0.56 × 1010 cm-2 (pure H2O solvent). Note that, using this method, we were sure to have not significantly influenced height and position of the first S(Q) peak. In the following, this PFA form factor and the PS form factor presented in Figure 1 will serve as normalization to extract the partial structure factors from the partial intensities. Figure 2 presents the structure factors SPS(Q) and SPFA(Q) of the neat suspensions. Fitting these curves with the classical theoretical one-component HNC approach gives for the effective charge parameter values Zeff PFA ) 260e- (in agreement with the USAXS fit) and Zeff PFA) 120e-. We have chosen to keep these charge values fixed for all mixtures. This strong constraint choice will underline the agreement between theoretical and experimental results. The main results of our work for the Iij(Q) and the Sij(Q) are compiled in Figures 3-5. In Figure 3 the dependence of the SANS intensity on the scattering length density of

Lutterbach et al.

Figure 3. SANS intensity distribution of the binary mixture sample 2 depending on Fm. Full lines are two-component HNC fits. For the sake of more clarity two intensities were multiplied and the intensity axis was broken.

the suspending medium is illustrated for the first mixture (sample 2, PFA number fraction xPFA ) 82%). Here, the influence of contrast variation is clearly visible. At the PFA contrast match, Fm ) FPFA ) 4.6 × 1010 cm-2, only the low and weakly structured PS-PS contribution is visible. Slight variation of Fm away from FPFA leads to a fast increase of the PFA contribution and reveals a more and more developing peak in I(Q). Next, we consider the direct determination of the partial structure factors. In Figure 4 we present the three partial intensities Iij(Q) of the liquidlike ordered binary systems with a number fraction of xPFA ) 82%, 52%, and 22% PFA. IPS-PS(Q) was directly measured by elimination of the scattering of the PFA particles by exact contrast matching (Fm ) 4.6 × 1010 cm-2). The IPFA-PFA(Q) and IPFA-PS(Q) were determined from the scattering obtained at Fm ) 3.9-4.4 × 1010 cm-2. In addition, we compare the received IPFA-PFA(Q) with those IPFA-PFA(Q) directly obtained by measuring at the contrast match point of PS (Fm ) 1.42 × 1010 cm-2) and with reduced sample cell thickness to avoid multiscattering. The agreement between both results illustrates the quality of the arithmetical method used and the experimental coherence of all measured distributions. The curves in Figure 4 indicate that the PFA-PFA contribution to I(Q) is much larger than that of the PS-PS, due to the larger size of the PFA particles (VPFA = 8.5VPS). As the number fraction xPFA decreases, IPFA-PFA(Q) weakens while IPFA-PS(Q) and IPS-PS(Q) increase and become more structured. All these scattering curves cannot be identified as those of monodisperse systems and are typical of binary mixtures. Figure 5 displays the partial structure factors of the three examined mixtures. Those were obtained according to eqs 1 and 2 by dividing the Iij(Q) of Figure 4 by (2 - δij)xcicjViVjxPi(Q)Pj(Q). The SPFA-PFA(Q) shown were extracted from the IPFA-PFA(Q) measured with reduced sample thickness. Since the scattered intensities have been transformed to absolute units, the Sij(Q) should in principle be correctly “normalized” without further correction. In practice, we noted that the SPFA-PFA(Q) and SPS-PS(Q) tend in the large Q-region to a constant K different from 1 (K = 1.8 and roughly keeps the same value for all studied cases). This reflects the uncertainty in the SANS normalization process, i.e., due to not detectable instrument constants. Thus, all Sij(Q) have further been divided by K to get the correct absolute quantities with Sii(Q f ∞) ) 1. Note that in case of

Structure Factors of Binary Suspensions by SANS

Langmuir, Vol. 15, No. 2, 1999 349

Figure 4. SANS partial contributions Iij(Q) (full symbols) for sample 2 (a), sample 3 (b), and sample 4 (c), extracted by calculation as described in the text. The open circles correspond to IPFA-PFA(Q) at the PS contrast match. The lines represent the two-component HNC fits. Magnification factors are introduced for clarity.

Figure 5. SANS partial structure factors Sij(Q) (symbols) deduced from Iij(Q) in Figure 4 and two-component HNC fits (s) for binary mixtures sample 2 (a), sample 3 (b), and sample 4 (c).

SPFA-PS(Q) no additional adjustment to zero for high Q has been done. As before, it is obvious in Figure 5 that the Sij(Q) behavior, especially at the Q f 0 limit, is for all partial structure factors typical of mixtures and significantly different from those of one-component systems. For one given mixture the three Sij(Q) crucially differ even in shape and refer to unlike interparticle interactions. To understand the physical information contained in the experimental results in Figures 3-5 more quantitatively, we now use our two-component theoretical approach. HNC calculations were made with the values given

in Tables 1 and 2, without adjustable parameters. Note that the residual salinity of the medium, with regard to the thorough deionization, considered to be the one of pure water at 20 °C, is much lower than the counterion concentration in the HNC calculations (a few 10-5 mol/L) and can be neglected in the total ionic strength. The theoretical intensities I(Q) and Iij(Q) in Figures 3 and 4 have been calculated according to eqs 1 and 2 and further multiplied by the previously determined constant K to allow for an absolute comparison. Due to a 10% polychromaticity of the neutron wavelength λ, HNC intensities

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Table 1. Parameters of the Measured Suspensions and Scattering Length Densities of the Solvents Used sample no.

total volume fraction, Φ

particle number fraction, xPFA

partial volume fraction, ΦPFA

% induced polydispersitya

neutron scattering length densities of medium Fm/1010 cm-2

1 2 3 4 5

0.095 0.095 0.094 0.092 0.086

1 0.82 0.52 0.22 0

0.095 0.092 0.085 0.065 0

0 21.8 34.1 35.5 0

4.2 1.42/3.9/4.2/4.4/4.6 1.42/3.9/4.2/4.4/4.6 1.42/3.9/4.2/4.4/4.6 4.6

a

Calculated considering the individual components as being monodisperse.

Table 2. System Parameters Used for HNC Calculations T  σPFA σPS

293 K 80 162 nm 79 nm

Zeff PFA Zeff PS residual salinity cs

260 e120 e10-7 mol L-1

and the structures factors as well have been smoothed to mimic this smearing effect. The width ∆Q of the Gaussian convolution function is given by

∆Q ) Q∆λ/λ

(7)

with ∆λ/λ ) 0.1. No additional correction for intrinsic polydispersity was made. As one can see in Figures 3-5 the agreement between the experimental and calculated data is satisfactory. The model is able to reproduce all features of the total intensities as well as of the three partial quantities and structure factors. Within the experimental accuracy and considering the nonnegligible uncertainties in the experimental partial quantities resulting from the extraction process, the agreement is quantitative as well. All oscillations, peak (minima) positions, and heights (depths) in the I(Q), Iij(Q), and Sij(Q) are correctly fitted. Although all information concerning the pair correlation is in principle contained in the Sij(Q), we also analyze our results in real space. In Figure 6 the three HNC pair distribution functions gij(r) for the three mixtures are shown. The general behavior of the curves is typical of charged colloidal mixtures.22-24 The three gij(r) present a main peak at different locations. No matter what the amount of PS particles is, the peak for the smaller particles (PS-PS) is always located at smaller distances r than that for the larger particles (PFA-PFA) (the peak for the cross PFA-PS function being at intermediate distances). If all particles were identical from the interaction point of view (as in the substitutional model25), the three gij(r) would be identical and coincide with the one-component function gm(r) obtained for the total concentration ctot ) cPFA + cPS. For highly deionized charged suspensions as ours, gm(r) would have a main peak located at the mean h ) 2820, distance between first neighbors d h ≈ ctot-1/3, i.e., d 2475, and 2025 Å for xPFA ) 82%, 52%, and 22%, respectively. The differences between the three gij(r) for a given mixture composition (Figure 6) thus reveal the polydispersity in interaction in the suspension and indicate that PFA and PS particles cannot be considered as randomly distributed on a common network. Note that the DLVO pair potentials (eq 3) essentially differ through the charges Zieff and not through the sizes Ri, since κRi < 1. Thus, for a given interparticle distance, the PFA-PFA (22) Ha¨rtl, W.; Segschneider, C.; Versmold, H.; Linse, P. Mol. Phys. 1991, 73, 541. (23) Me´ndez-Alcaraz, J. M.; D’Aguanno, B.; Klein. R. Physica A 1991, 178, 421. (24) Wagner, N. J.; Klein, R. Colloid Polym. Sci. 1991, 269, 295. (25) Faber, T. E.; Ziman, J. M. Philos. Mag. 1965, 11, 153.

Figure 6. HNC partial pair distribution functions gij(r) of the binary mixtures sample 2 (a), sample 3 (b), and sample 4 (c).

electrostatic repulsion is roughly five times stronger than the PS-PS one. Let us focus on the composition with xPFA ) 82%, whose gij(r) values are presented in Figure 6a. The gPFA-PFA(r)

Structure Factors of Binary Suspensions by SANS

has a main peak at 2940 Å which is larger than d h ) 2820 Å. If the PFA particles were alone in the suspension without the PS particles (cPFA and κ keeping the same values as in the mixture), the peak of gPFA-PFA(r) would be at 3030 Å (≈cPFA-1/3). On the other hand, the position of the gPS-PS(r) peak, being at 2130 Å, has nothing to do with d h or with the mean distance between PS colloids (≈cPS-1/3). The mixture can be viewed as a network of PFA colloids, slightly disturbed by the presence of the less numerous and less charged PS colloids. These are located in the interstitial regions between the large PFA particles, the interstices being of electrostatic origin. As a consequence, the PS colloids are forced to be close to each other resulting in a gPS-PS(r) peak at short distance (attractive depletion effect). As xPFA decreases (at constant total volume fraction), PFA particles are replaced by more numerous PS ones and the mean distance decreases. At xPFA ) 22% (Figure 6c) the PS particles dominate but the PFA ones still occupy a nonnegligible electrostatic volume h) fraction. The peak in gPFA-PFA(r) is located at 2550 Å (>d while, if the PFA particles were alone, it would be at 3315 Å. In addition, the corresponding values for gPS-PS(r) are 1830 Å (