Langmuir 1999, 15, 337-344
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Charge-Stabilized Liquidlike Ordered Binary Colloidal Suspensions. 1. Ultra-Small-Angle X-ray Scattering Characterization 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 Received July 3, 1998. In Final Form: October 19, 1998 In this and the following paper we present scattering studies on binary colloidal mixtures made of charge-stabilized polystyrene (PS) and perfluorinated (PFA) particles with diameters σ ) 79 and 162 nm, respectively. Both colloidal species were mixed to well-defined compositions. The total concentration was about 9 vol % fraction. By using ultra-small-angle X-ray scattering, we directly obtained the partial scattering intensities of only the PFA particles in liquidlike ordered suspensions. Furthermore, after dividing intensities by the PFA particle form factor PPFA(Q), we got the partial structure factor SPFA-PFA(Q) without any additional treatment. The experimental results are compared with theoretical predictions obtained from the pure repulsive DLVO potentials and the hypernetted chain (HNC) integral equation, as applied to charged colloidal mixtures. It is shown that all measured intensities and extracted structure factors are in good agreement with the theoretical results.
I. Introduction Recently, the general interest in colloidal suspensions has enormously increased. Colloids are ubiquitous in nature and in industrial processes. Simultaneously, considerable effort has been undertaken to understand the macroscopic and microscopic behavior of suspensions and their fundamental interactions. One of the most significant characteristics of charged colloidal particles is their ability to form short or even long range order. If sufficiently deionized, they form stable liquidlike or crystalline structures. Since polydispersity in particle size occurs in colloidal systems more or less ubiquitously, the technological relevance of polydispersity is obvious. Binary mixtures and their structural behavior represent the easiest approach to polydisperse suspensions. Therefore, binary systems have often been of interest in the literature. Several groups worked on various aspects such as liquidlike order,1-10 crystals,11-13 glass order,14 sedimentation,15-17 segregation,18,19 shearing,14,20,21 stability,16,22-25 * To whom correspondence should be addressed. (1) Krause, R.; D’Aguanno, B.; Me´ndez-Alcaraz, J. M.; Na¨gele, G.; Klein, R.; Weber, R. J. Phys. Condens. Matter 1991, 3, 4459. (2) De´rian P.-J.; Belloni, L.; Drifford, M. Europhys. Lett. 1988, 7, 243. (3) Hayter, J. B. J. Chem. Soc., Faraday Trans. 1991, 87, 403. (4) Hanley, H. J. M.; Straty, G. C.; Lindner, P. Langmuir 1994, 10, 72. (5) Ottewill, R. H.; Hanley, H. J. M.; Rennie, A. R.; Straty, G. C. Langmuir 1995, 11, 3757. (6) Me´ndez-Alcaraz, J. M.; D’Aguanno, B.; Klein, R. Langmuir 1992, 8, 2913. (7) Duits, M. H. G.; May, R. P.; Vrij, A.; de Kruif, C. G. J. Chem. Phys. 1991, 94, 4521. (8) Kline, S. R.; Kaler, E. W. J. Chem. Phys. 1996, 105, 3813. (9) Me´ndez-Alcaraz, J. M.; D’Aguanno, B.; Klein, R. Physica A 1991, 178, 421. (10) Na¨gele, G.; Zwick, Th.; Krause, R.; Klein, R. J. Colloid Interface Sci. 1993, 161, 347. (11) Bartlett, P.; Ottewill, R. H.; Pusey, P. N. Phys. Rev. Lett. 1992, 68, 3801. (12) Bartlett, P.; Ottewill, R. H. J. Chem. Phys. 1992, 96, 3306.
and dynamics10,26,27 with diverse methods such as light1,10,11,16,17,26,27 and neutron scattering2-5,7,8,12,20 or microscopy.13,18,19,23 Investigations have been carried out with different materials such as organic or inorganic polymer,1,3-5,7,8,10-14,16-23,26,27 or ferromagnetic3 particles, or micellar systems,2,8 charged 1-5,8-10,13,14,18,20,23,26,27 and hard sphere7,11,12,15-17,19,21,22,24,25 particles, or a mixture of both.6,28 In this paper we investigate a binary colloidal system of highly interacting spherical polystyrene (PS) and perfluorinated (PFA) particles stabilized by surface charges and liquidlike ordered by deionization. The behavior of binary suspensions is expected to be more complicated than that of monodisperse systems. As every particle species contributes to the microstructure we were particularly interested in partial structure factors. A size ratio of 2 should produce a significant difference in the (13) Yoshimura, S.; Hachisu, S. Prog. Colloid Polym. Sci. 1983, 68, 59. (14) Lindsay, H. M.; Chaikin, P. M. J. Chem. Phys. 1982, 76, 3774. (15) Biben, Th.; Hansen J.-P. Mol. Phys. 1993, 80, 853. (16) van Duijneveldt, J. S.; Heinen. A. W.; Lekkerkerker, H. N. W. Europhys. Lett. 1993, 21, 369. (17) Al-Naafa, M. A.; Sami Selim, M. AIChE J. 1992, 38, 1618. (18) Hachisu, S.; Kose, A.; Kobayashi, Y.; Takano, K. J. Colloid Interface Sci. 1976, 55, 499. (19) Sanyal, S.; Easwar, N.; Ramaswamy, S.; Sood, A. K. Europhys. Lett. 1992, 18, 107. (20) Hanley, H. J. M.; Pieper, J.; Straty, G. C.; Hjelm, R. P.; Seeger, P. A. Faraday Discuss. Chem. Soc. 1990, 90, 91. (21) Rodriguez, B. E.; Kaler, E. W.; Wolfe, M. S. Langmuir 1992, 8, 2382. (22) Kaplan, P. D.; Rouke, J. L.; Yodh, A. G. Phys. Rev. Lett. 1994, 72, 582. (23) Yasrebi, M.; Shih, W. Y.; Aksay, I. A. J. Colloid Interface Sci. 1991, 142, 357. (24) van Duijneveldt, J. S.; Lekkerkerker, H. N. W. Phys. Rev. Lett. 1993, 71, 4264. (25) Biben, Th.; Hansen, J.-P. Phys. Rev. Lett. 1991, 66, 2215. (26) Richtering, W.; Berend, K. Prog. Colloid Polym. Sci. 1995, 98, 79. (27) Na¨gele, G.; Medina-Noyola, M.; Arauz-Lara, J. L.; Klein, R. Prog. Colloid Interface Sci. 1987, 73, 5. (28) Vlachy, V. J. Chem. Phys. 1993, 99, 471.
10.1021/la980804g CCC: $18.00 © 1999 American Chemical Society Published on Web 12/24/1998
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particle behavior. Rather highly diluted liquidlike binary suspensions of charged polystyrene particles were already investigated by some of us with a light scattering technique,29,30 where the systems could be described as one-component macrofluids with an incoherent light scattering background. To continue our studies we turn our interest in this and the following paper to more concentrated systems of about 9% volume fractionswhere we expect that the partial structure factors finally differ from each othersand changed our investigation method to appropriate techniques: X-ray and neutron scattering. Small-angle X-ray analysis31 offers in this case a very useful tool of investigation. By means of the extended Q-region, and the, compared with light scattering, reduced multiple scattering, measurements of structure factors of concentrated and even milky suspensions become possible. Concerning latex systems with relatively large interparticle distances, ultra-small-angle X-ray scattering (USAXS) experiments of the Bonse-Hart type32 has in addition some important advantages: the construction with double crystal technique for monochromatization and analyzing leads to a significant smaller minimum momentum combined with a higher resolution in the low-Q region. This guarantees more detailed data and makes the scattering behavior for Q f 0 well accessible. Although USAXS is already established in research on colloidal systems,33-37 to our knowledge it has not been used for the investigation of binary suspensions yet. The total scattered intensity of a mixture is a superposition of all partial intensities. Due to the rather small difference of the electron densities of water and polystyrene, polystyrene in aqueous medium is a very weak X-ray scatterer. By contrast, due to the fluorine, PFA is an effective one. Therefore, by using these two in their X-ray scattering length density strongly different particle kinds, we were able to extract the PFA-PFA partial structure factor directly by USAXS. Experimental results should be compared to theoretical predictions. Colloids can be considered as supermolecular fluid particles in a continuous background. They can be described by the theory of simple liquids.38 In this paper we used the purely repulsive part of the DLVO potential39 and the hypernetted chain (HNC) closure relation38 to solve the Ornstein-Zernike equation38 for interparticle correlation. Satisfactory agreement between HNC calculations and experimental data of polydisperse suspensions has already been found elsewhere.1,29,40,41 In this work, we present to our best knowledge for the first time (29) Ha¨rtl, W.; Segschneider, C.; Versmold, H.; Linse, P. Mol. Phys. 1991, 73, 541. (30) Ha¨rtl, W.; Versmold, H. J. Chem. Phys. 1984, 80, 1387. (31) Glatter, O. In Neutron, X-Ray and Light Scattering; Lindner, P., Zemb, Th., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 33 ff. (32) Bonse, U.; Hart, M. Appl. Phys. Lett. 1965, 7, 238. (33) Reus, V.; Belloni, L.; Zemb, Th.; Lutterbach, N.; Versmold, H. J. Phys. II 1997, 7, 603. (34) Matsuoka, H.; Harada, T.; Kago, K.; Yamaoka, H. Langmuir 1996, 12, 5588. (35) Matsuoka, H.; Kakigami, K.; Ise, N.; Kobayashi, Y.; Machitani, Y.; Kikuchi, T.; Kato, T. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 6618. (36) Reus, V.; Belloni, L.; Zemb, Th.; Lutterbach, N.; Versmold, H. J. Chim. Phys. 1995, 92, 1233. (37) Yamanaka, J.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. E 1996, 53, 4314. (38) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (39) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier Publisher Company: Amsterdam, 1948. (40) De´rian, P.-J.; Belloni, L.; Drifford, M. J. Chem. Phys. 1987, 86, 5708. (41) Kunz, W.; Calmettes, P.; Jannink, G.; Belloni, L.; Cartailler, T.; Turq, P. J. Chem. Phys. 1992, 96, 7034.
Lutterbach et al.
a HNC comparison to partial scattering curves of highly liquidlike ordered binary suspensions of charged particles. For several mixtures a complete evaluation of the experimental results is shown. Total intensities I(Q), being at the same time partial intensities IPFA-PFA(Q), and extracted partial structure factors SPFA-PFA(Q) are compared to HNC based theoretical results which are received only by fixed system parameters. Reasonable agreement between the DLVO-HNC approach and the experiment will underline the validity of this pure repulsive interaction model for rather concentrated, highly interacting colloidal mixtures. II. Theory The coherent, elastic scattered intensity I(Q) of a polydisperse fluidlike sample of spherical particles is given by42
I(Q) )
∑i∑j xcicj ViVj ∆Fi ∆Fj xPi(Q) Pj(Q) Sij(Q)
(1)
Q is the scattering vector, defined as
Q)
4πn sin(θ/2) λ
(2)
θ is the scattering angle, n is the refractive index of the solvent medium for λ, the wavelength of the incident beam, ci is the number density of particles of species i, Vi is their particle volume, and ∆Fi is the effective coherent X-ray scattering length density. Here, ∆Fi denotes the difference between the X-ray scattering length density of the particle Fi and that of the solvent medium Fm (contrast). Fi denotes the electron density of the basic molecule times the scattering amplitude of one electron, whose value is 0.282 × 10-12 cm.43 Pi(Q) is the normalized form factor of particle i (Pi(0) ) 1) which is for a sphere of radius Ri given by44
Pi(Q) )
(
)
3(sin QRi - QRi cosQRi) (QRi)3
2
(3)
The spatial correlation between particles is expressed in eq 1 through a linear combination of the partial structure factors Sij(Q). For spherical, symmetrical interactions Sij(Q) is related to the Fourier transform of the partial pair distribution functions gij(r) by eq 4.38
Sij(Q) ) δij + xcicj
∫(gij(r) - 1) exp(iQB br ) drb
(4)
with δ the Kronecker symbol and r the interparticle distance. In general, the scattered intensity of a binary mixture contains three partial contributions. Thus, it is necessary to measure I(Q) for at least three different contrasts to extract the three partial structure factors Sij(Q) experimentally. In our particular case, the X-ray scattering due to the PS particles is negligible and I(Q) reduces to the PFA contribution: (42) Guinier, A.; Fournet, G. Small-Angle Scattering of X-rays; Wiley & Sons: New York, Chapman & Hall: London, 1955. (43) Stuhrmann, H. B.; Miller A. J. Appl. Crystallogr. 1978, 11, 325. (44) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley & Sons: New York, 1976.
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I(Q) ) IPFA-PFA(Q) ) cPFA ∆F2PFA V2PFA PPFA(Q) SPFA-PFA(Q) (5) For charge-stabilized colloidal suspensions the purely repulsive part of the so-called DLVO-pair potential Uij(r)39 is despite its limitations45-48 quite established. It has often been used to describe the ion-averaged potential between charged particles1,45,49-52 and can be written for binary suspensions29,53
(
)(
eff e2 Zi exp(κRi) Uij(r) ) 4π0 1 + κRi
)
Zjeff exp(κRj) exp(-κr) 1 + κRj r (6)
with r g Ri + Rj. is the dielectric constant of the suspending aqueous medium and κ denotes the screening parameter which depends on the ionic densities:54
κ2 )
e2 0kT
(
∑i ci Zieff
2
+ 2cs)
(7)
where T is the temperature (293 K). The first contribution to the ionic strength in eq 7 comes from the monovalent H+ counterions. The second term contains the added monovalent salinity cs. We use this approximate expression instead of the rigorous equation for mixtures of electrolytes,55 because the exact result would introduce unnecessary complications. As usual in this approach, the particle charge Zieff (in e units) in eq 7 is an effective value, much smaller than the structural charge. It is introduced to take the electrostatic counterion condensation into account and to correct a posteriori the Debye-Hu¨ckel linearization implicitly assumed in the DLVO expression.45,56 Despite various theoretical predictions,56,57 Zieff is usually considered for large undeformable particles as an adjustable parameter given by the fit of experimental data. Starting from the DLVO potential Uij(r), the statistical mechanical problem consists of solving the OrnsteinZernike equation (8) which relates the total hij(r) ) gij(r) - 1 and direct cij(r) correlation functions in a homogeneous fluid.38 n
hij(r) ) cij(r) +
∑ ck ∫hik(r′) ckj(|rb - br ′|) drb′
(8)
k)1
Several liquid-state models (“closure relations”) are available38,50 to calculate from this the scattering intensities (45) Belloni, L. In Neutron, X-Ray and Light Scattering; Lindner, P., Zemb, Th., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 135 ff. (46) Belloni, L. J. Chem. Phys. 1986, 85, 519. (47) Lo¨wen, H.; Kramposthuber, G. Europhys. Lett. 1993, 23, 673. (48) Lo¨wen, H.; Hansen, J.-P.; Madden, P. A. J. Chem. Phys. 1993, 98, 3275. (49) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (50) D’Aguanno, B.; Klein, R. J. Chem. Soc., Faraday. Trans. 1991, 87, 379. (51) Pusey, P. N. In Liquids, Freezing and the Glass Transition; Levesque, D., Hansen, J.-P., Zinn-Justin, J., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 763 ff. (52) Ha¨rtl, W.; Versmold, H.; Wittig, U. Langmuir 1992, 8, 2885. (53) D’Aguanno, B.; Krause, R.; Me´ndez-Alcaraz, J. M.; Klein, R. J. Phys.: Condens. Matter 1992, 4, 3077. (54) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 192. (55) Nylander, T.; Ke`kicheff, P.; Ninham, B. W. J. Colloid Interface Sci. 1994, 164, 136. (56) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (57) Belloni, L. Colloids Surf., A 1998, 140, 227.
Figure 1. Particle form factor of perfluorinated particles (PFA) measured by USAXS (O) and fitted with a mean diameter of 162 nm and a polydispersity of 9.5% (-) (see text).
and structure factors of colloidal suspensions. We decided to use the hypernetted chain (HNC) equation in view of its excellent suitability for Coulomb systems with longrange repulsive interactions.38,46 The HNC equation reads
gij(r) ) exp(-βUij(r) + hij(r) - cij(r))
(9)
where β ) 1/kT. Note that improved integral equations such as the Rogers-Young equation58 are available in the literature. Preliminary tests for our systems have shown that the corrections to the HNC results brought by such modified equations are only marginal and that the HNC equation remains correct, even quantitatively. The Ornstein-Zernike equation (8) and the HNC closure relation (9) are iteratively solved, using with the Newton-Raphson-Zerah algorithm, a classical, powerful numerical technique.59,60 Once the convergence is reached, the partial (Iij(Q)) and total (I(Q)) intensities are obtained from the calculated partial structure factors Sij(Q). III. Experimental Section Materials and Colloidal Suspensions. The colloids used were polystyrene (PS) particles and perfluorinated particles (perfluoroalkoxy-copolymer (PFA)), both dispersed in water. PS particles were prepared by radical-initiated (K2S2O8) surfactantfree emulsion copolymerization of styrene and potassium styrenesulfonate.61 PFA particles were synthesized by emulsion polymerization at high pressure (≈20 bar) with ammoniumpersulfate as initiator62 and were kindly supplied by Hoechst, Frankfurt (Germany). The latter particles consist of about 97% tetrafluoroethylene (TFE) and 3% perfluorpropylvinyl ether with perfluorooctanoic acid as surfactant. The general PFA structure is given by the following formula:
-[CF2-CF(O-CnF2n+1)]1-[CF2-CF2]mwith here n ) 3 and m = 86. Particle sizes were obtained by dynamic light scattering (DLS) and checked by fitting particle form factors measured with static light and X-ray scattering. The USAXS P(Q) of PFA is shown in Figure 1. Due to a high monochromaticity of the incident X-ray beam and a high resolution of the analyzer instrument, smearing of the P(Q) minima could be seen as polydispersity only. Fitting the PFA-P(Q) with a size distribution of 9.5% polydispersity (58) Rogers, F. J.; Young, D. A. Phys. Rev. A 1984, 30, 999. (59) Belloni, L. Chem. Phys. 1985, 99, 43. (60) Belloni, L. J. Chem. Phys. 1988, 88, 5143. (61) Shi-Der Juang, M.; Krieger, I. M. J. Polym. Sci. 1976, 14, 2089. (62) Personal communication with Dr. G. Lo¨hr, Hoechst AG, Germany.
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Figure 2. (a) Particle size distribution analysis by TEM for PS. (b) Aerosol spectroscopy size distribution for PFA (Gaussian fit was achieved with a polydispersity of 10%).
Figure 4. Titration curves of PS (O) and PFA (b). Conductivity χ versus equivalents of NaOH (concentration 10-3 mol/L). Crossing lines are indicating inflection points.
Figure 3. Atomic force microscopy (AFM) of PFA. and a mean diameter of 162 nm gives a satisfactory agreement over 4 decades in intensity. The fit was obtained by means of the zeroth order logarithmic distribution (ZOLD) function described previously.63 Other polydispersity values are 13.4% for PS verified by transmission electron microscopy (TEM) analysis and 10.4% for PFA checked by means of aerosol distribution spectroscopy. The corresponding diameter distributions are documented in Figure 2. It is obvious that there is almost no overlapping of the two particle size ranges. As shown with Figure 3, an atomic force microscopy (AFM) picture show in addition a rather narrow size distribution concerning the PFA. Both particle kinds were identified by TEM and AFM to be spherical in shape. The diameter values finally taken for the following calculations were σ ) 79 nm for PS and σ ) 162 nm for PFA (DLS). The size ratio in the binary suspensions was therefore =2. The numbers of detectable64-69 acidic surface groups, as determined by conductometric titration, were 14330e- for PS (63) Ottewill, R. H. In Colloidal Dispersions; Goodwin, J. W., Ed.; The Royal Society of Chemistry: London, 1981; p 143 ff. (64) Everett, D. H.; Gu¨ltepe, M. E.; Wilkinson, M. C. J. Colloid Interface Sci. 1979, 71, 336. (65) Yamanaka, J.; Ise, N. J. Colloid Interface Sci. 1996, 179, 324. (66) Zwetsloot, J. P. H.; Leyte, J. C. J. Colloid Interface Sci. 1994, 163, 362.
(10 500 strong SO4- and 3830 weak COO- groups61,64,66-68,70,71) and 6250e- for PFA (2525 strong COO- 70 and 3725 weak groups). Strong COO- groups result from perfluorooctanoic acid and a direct and complete hydrolysis of the sulfate-tetrafluoroethane radicals:62 ∆
TFE
2H2O
S2O82- 98 SO4•- 98 -O3SO-CF2-CF2• 98 HSO4- + HOOC-CF2• + 2HF The origin of the weak acid groups is at present unknown. We assume a presence of COO- groups without fluorine in vicinity due to defluorination processes during synthesis. Both charge numbers refer to thoroughly dialyzed and with ionic resin treated latex. Figure 4 shows the titration curves of both latex suspensions. Two two inflection points of each curve indicate the two different acid group strengths. We applied the technique of adding salt (=1 mL of 10-2 M NaCl solution to 50 mL of latex suspensions (which contained about 1 × 1019 particles/m3)) to obtain in the (67) James, R. O.; Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 65, 331. (68) Stone-Masui, J.; Watillon, A. J. Colloid Interface Sci. 1975, 52, 479. (69) Labib, M. E.; Robertson, A. A. J. Colloid Interface Sci. 1980, 77, 151. (70) Ottewill, R. H.; Rance, D. G. Colloid Polym. Sci. 1986, 264, 982. (71) Palberg, T.; Kottal, J.; Bitzer, F.; Simon, R.; Wu¨rth, M.; Leiderer P. J. Colloid Interface Sci. 1995, 69, 85.
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Table 1. Parameters of the Measured Colloidal Suspensions
a
sample no.
total volume fraction, Φ
particle number fraction, xPFA
partial volume fraction, ΦPFA
% induced polydispersitya
1 2 3 4 5 6
0.095 0.095 0.094 0.093 0.092 0.090
1 0.82 0.52 0.32 0.22 0.10
0.095 0.092 0.085 0.075 0.065 0.047
0 21.8 34.1 36.9 35.5 28.7
Calculated by considering the individual components as being monodisperse. Table 2. Electron Densities and Scattering Length Densities of the Materials Used
PFA PS H2O
electron density/nm-3
X-ray scattering length density F/1010 cm-2
618.4 341.5 334.4
17.44 9.62 9.43
minima a well-defined curve and therefore a more precise result. The pH values of all studied suspensions were determined to lie between 2.3 and 3.1. This led to a considerable reduction of weak acid group dissociation. Nevertheless the particles can be considered as highly charged. The surface strong charge densities are 0.086 C/m2 for PS (1e-/1.87 nm2) and 0.005 C/m2 (1e-/32.7 nm2) for PFA. The total volume fractions of all suspensions were kept at approximately 9%. Mixtures were prepared by adding appropriate amounts of the neat suspensions of each pure component. The concentrations of the suspensions were verified by light scattering from ordered samples.72 The parameters of the USAXS-measured systems are listed in Table 1. Complete deionization was accomplished by thorough purification with cleaned73 ionic resin. As ionic resins we used Amberlyst 15 and A27 (Fluka, Buchs, Switzerland). To avoid any loss of particles through adsorption or coagulation during purification, every resin portion was first “washed” with a small part of the respective suspension and further deionization was carried out extremely carefully. All resinfree suspensions were directly measured after preparation and purification to ensure absolutely deionized measured samples. It was taken care that during the measurements all suspensions were constantly in best experimentally possible liquidlike order and stable mixed. No crystallization or phase separations were observed. Due to deionization all particles in these samples had protons as counterions. P(Q) measurements were carried out with dilute samples with a sufficient amount of NaCl added in order to eliminate particle interactions. The X-ray scattering properties of the particles are listed in Table 2.74 Obviously, due to the larger volume occupied by the PFA particles and the contrast match for the PS particles (∆FPS = 0) the scattering is expected to be dominated by the PFA particles. All samples were dispersed in deionized water and measured at 20 ( 1 °C. USAXS and Data Treatment. Measurements were performed with the high-resolution ultra-small-angle X-ray (USAXS) camera of Bonse-Hart type32 at the Service de Chimie Mole´culaire (C.E.A., Saclay, France). As X-ray source a rotating anode with copper target was used (50 kV, 300 mA). The wavelength of the selected Cu KR line was 1.54 Å. Monochromatization was achieved by means of a triple reflecting Ge(111) channel-cut crystal. Scattered intensities were analyzed by a second Ge crystal and detected by a scintillator-photomultiplier unit. Monochromator, sample, analyzer, and detector could independently rotate. The Bonse-Hart camera was calibrated with known filters. The uncertainty left was reduced to 20% using the desmeared signal (72) Ha¨rtl, W.; Klemp, R.; Versmold, H. Phase Transitions 1990, 21, 229. (73) Vanderhoff, J. W.; Van Den Hul, H. J.; Tausk. R. J. M.; Overbeek, J. Th. G. In Clean Surfaces; Goldfinger, G., Ed., M. Dekker Inc.: New York, 1970; p 15 ff. (74) Grunder, R.; Urban, G.; Ballauff, M. Colloid Polym. Sci. 1993, 271, 563.
of 3 mm LUPOLEN polyethylene (BASF, Ludwigshafen (Germany)) as a secondary standard.75 The Q-range of the instrument was ≈ 5 × 10-4 Å-1 to 10-2 Å-1 with a resolution in Q of the order of 3 × 10-4 Å-1. Further information on the instrumental setup has previously been given elsewhere.76 All samples were measured in air-closed thoroughly cleaned borosilica capillaries of 1 mm diameter and without any contact to ionic resin. Typical count times were 1-3 h for a full Q-scan. Data were treated as follows: To obtain the absolute intensities from the raw counting rates normalization of the scattered intensities was achieved by transmission measurements. After subtraction of the capillary scattering36 and the water background, all instrument-smeared data were desmeared by the method of Strobl.77 Here, the instrumental function was independently measured and shown to be close to trapezoidal shape.76 Because of the ultra-small-angle region a correction for unpolarized incident beams78 was not necessary. Scattering was assumed to be purely elastic. Multiple and incoherent contributions were neglected.79
IV. Results and Discussion According to the X-ray scattering length densities, only the intensity due to the PFA particles will contribute and eq 5 is valid. To verify this, PS particles were proved to scatter at a concentration of 9% volume fraction and high liquidlike order 104 times less than PFA particles. Therefore, a PS contribution to the scattered intensity can be excluded. Thus, with eq 5 we were able to determine the partial structure factors SPFA-PFA(Q) of the PFA component in the mixtures directly
IPFA-PFA(Q) I(Q) ) ∝ SPFA-PFA(Q) PPFA(Q) PPFA(Q)
(10)
Figures 5-7 present the experimental results for IPFA-PFA(Q) and SPFA-PFA(Q) for our binary samples. The experimental scattered intensities are compared with HNC calculations for a two-component system. The corresponding constant system parameters are listed in Table 3. Concerning the residual salinity of the samples, we took the electrolyte concentration of pure water at 20 °C, which was always much smaller than the counterionic H+ concentration of the order of 10-5 to 10-4 mol/L in the HNC calculations. In addition, the volume fractions and particle number fractions of Table 1 completed HNC parameters. The effective charge Zieff is one of the decisive factors of the interparticle potential of charged colloids. It is usually dependent on the experimental method chosen for determination. Purely theoretical approaches such as the Poisson-Boltzmann-Cell model45 offer a possibility (75) Wignall, G. D. J. Appl. Crystallogr. 1991, 24, 479. (76) Lambard, J.; Lesieur, P.; Zemb, Th. J. Phys. I 1992, 2, 1191 and references therein. (77) Strobl, G. R. Acta Crystallogr. 1970, A26, 367. (78) Brown, C. J. In X-ray Diffraction by Polycrystalline Materials; Peiser, H. S., et al., Eds., The Institute of Physics, Chapman and Hall Ltd.: London, 1960; p 395 ff. (79) Waseda, Y. The Structure of Non-Crystalline Materials; McGrawHill Int. Book Co.: New York, 1980.
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Table 3. System Parameters Used for HNC Calculationsa T σPFA σPS ZPFAeff ZPSeff cs
293 K 80 162 nm 79 nm 260e120e10-7 mol L-1
a The effective charge of PS particles Z eff was determined by PS small-angle neutron scattering experiments (see following paper in this issue).
Figure 5. USAXS absolute intensities I(Q) of PFA pure suspension (sample 1) and PS/PFA binary mixtures (samples 2-6) in a log-log plot.
to evaluate Zieff. They predict an effective charge dependence on physical parameters such as volume fraction or salinity. Generally, agreement between the individual methods can hardly be found and corresponding results differ.1,29,36,56,57,80-83 Nevertheless, we decided here to apply the usual procedure and to consider Zieff as an adjustable value. Therefore, we carried out a fitting of the theoretical HNC curve to the experimental S(Q) values of the neat suspensions with the effective charges Zieff as the varying parameters. Because of the extremely low X-ray contrast of polystyrene in water (see Table 2) the effective charge for PS, ZPSeff, was not accessible with USAXS. Consequently, we take as fittable data the structure factor SPS(Q) measured with neutron scattering (see part 2 of this paper84). The final values are listed for each particle kind in Table 3. Note that these values ZPFAeff ) 260e- and ZPSeff ) 120e- remain in agreement with what can be expected for these colloidal systems.56,57 Despite changing partial number densities in mixtures, those charges were then left constant throughout all theoretical calculations for the whole composition range.29 Figure 5 shows the I(Q) ) IPFA-PFA(Q) values of six different suspensions with PFA number fractions xPFA ) 100%, 82%, 52%, 32%, 22%, and 10%. The strong peak of the first maximum of liquidlike order is clearly visible. However, according to the decreasing amount of PFA in the suspensions this peak gets weaker and shifts at the same time to higher Q-values. The increase of intensity at low Q-values is another significant characteristic. This will be more clearly seen regarding the S(Q) in Figure 7. (80) Ramanathan, G. V. J. Chem. Phys. 1988, 88, 3887. (81) Stevens, M. J.; Falk, M. L.; Robbins, M. O. J. Chem. Phys. 1996, 104, 5209. (82) Gisler, T.; Schulz, S. F.; Borkovec, M.; Sticher, H.; Schurtenberger, P.; D’Aguanno, B.; Klein, R. J. Chem. Phys. 1994, 101, 9924. (83) Gisler, T.; Borkovec, M.; Schurtenberger, P.; Klein, R. Prog. Colloid Polym. Sci. 1995, 98, 295. (84) Lutterbach, N.; Versmold, H.; Belloni, L.; Reus, V.; Zemb, Th.; Lindner, P. Langmuir 1999, 15, 345.
Figure 6. USAXS (b) and theoretical HNC (-) intensity curves for (a) sample 1, (b) sample 2, (c) sample 3, and (d) sample 5.
Structure Factors of Binary Suspensions by USAXS
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Figure 7. (e) USAXS (b) and theoretical HNC (-) partial structure factors SPFA-PFA(Q) for sample 6. Here, the HNC structure factor of a neat PFA system obtained at the same PFA volume fraction and screening constant than in the mixture is given for comparison (- - -).
Figure 7. (a-d) USAXS (b) and theoretical HNC (-) partial structure factors SPFA-PFA(Q). Samples as described in Figure 6.
For a PFA content of xPFA ) 10% nearly no interaction seems to be visible and the scattering curve resembles the pure form factor PPFA(Q).
Figure 6 presents the comparison of the experimental data to the calculated HNC for the PFA number fractions xPFA ) 100%, 82%, 52%, and 22%. Although after correction for instrumental constants and desmearing of the raw scattering data a fitting to the absolute intensity by X-ray scattering length densities should have been possible, we had to introduce a numerical multiplicand to align experimental and theoretical curves on an absolute scale. This factor was nearly the same for all samples (=0.8) and reflects the uncertainty in the experimental normalization of the scattered distributions. Nevertheless, the agreement between experimental data and theoretical curves is satisfactory. In Figure 7 we present the extracted structure factors SPFA-PFA(Q). Scattered intensities were divided by the PFA particle form factor (see Figure 1) which was measured at 1.0% volume fraction. After that, SPFA-PFA(Q) values were normalized to 1 in the outer Q-region and compared to the HNC curves. The agreement is quite good both for the first peak and the Q ) 0 behavior. Note that the significant rising of S(Q) for Q f 0 with decreasing PFA number density is predicted by theory for the scattering contribution of this component in the mixture and has nothing to do with incoherent sample background, as presented in refs 29 and 30. As a further result, we can see now that even for a partial number fraction xPFA of only 10% the PFA particles still have a liquidlike order inside the mixture. For the xPFA ) 100% PFA suspension, SPFA-PFA(Q) is of course identical to the usual structure factor of a monodisperse system. As the PFA composition decreases (from Figure 7a to Figure 7e), the PFA-PFA partial structure factor becomes less structured. As a reason for that, one can refer first to the decreasing PFA concentration but also to the change of counterionic concentration which, due to the exchange of PFA by PS particles, increases in parallel. Indeed, since the total volume fraction is kept constant (=9%), the replacement of large PFA particles by smaller and less charged, but more numerous PS particles increases the total ionic strength. For example, for the last case xPFA ) 0.1 (ΦPFA ) 0.047) the counterionic concentration is nearly 2.3 times larger than that for the pure PFA system. Moreover, the PFA-PFA correlation in the mixture is not simply related to the total ionic strength but is also perturbed by the presence of the PS particles. This effect is illustrated in Figure 7e where we
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in addition present the structure factor of a pure PFA system at the same PFA volume fraction (ΦPFA ) 0.047) and screening constant than those of the mixture xPFA ) 0.1. This last S(Q) is obviously much more structured than the partial SPFA-PFA(Q). This indicates that, roughly speaking, the PS particles themselves contribute to the screening of the electrostatic interactions between the PFA particles in the mixture. This behavior is more precisely studied in the accompanying paper.84 Generally, for all HNC calculations no polydispersity correction for the pure components has been considered. The agreement with experimental curves leads to the conclusion that the intrinsic polydispersity of 13.4% for PS and 10.4% for PFA (see Experimental Section) does not have any severe influence on the presented scattering results. Here, HNC could be calculated as for a binary mixture of two in theory monodisperse colloid fractions. The deviations between HNC and experimental structure factors at higher Q values should here rather be assigned to the division by the experimental quantity of P(Q) or general experimental uncertainties and not to a possible inadequacy of the DLVO potential at short range or polydispersity effects. We succeeded by means of the appropriate USAXS measurement to obtain the partial intensity of one
Lutterbach et al.
component in a binary system directly. As a main result of this paper all measured data of the studied binary mixtures of charged colloids are completely understandable and theoretically reproducible with pure repulsive interparticle interactions of Yukawa potential type and the HNC approximation. Acknowledgment. This work has been realized thanks to the PROCOPE program No 94222 between the Commissariat a` l’E Ä nergie Atomique (CEA, Saclay, France) and the Rheinisch-Westfa¨lische Technische Hochschule (RWTH, Aachen, Germany). We thank Jaques Lambard at CEA for his help in performing USAXS experiments, Elmar Jansen at the Institut fu¨r Kunststoffverarbeitung (IKV, Aachen, Germany) for assisting in making TEM photos, Dr. Birgit Severich at the Deutsches Wollforschungsinstitut (DWI, Aachen, Germany) for making AFM pictures, and Dr. Gernot Lo¨hr and Dr. Peter Fischer at HOECHST (Frankfurt, Germany) for supplying PFA particles and for analyzing them with aerosol spectroscopy. Financial support by the Deutsche Forschungs-Gemeinschaft (DFG) and the Fonds der Chemischen Insdustrie (FCI) is gratefully acknowledged. LA980804G
