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Detailed Study of Diluted V2O5 Suspensions B. Vigolo,† C. Zakri,† F. Nallet,† J. Livage,‡ and C. Coulon*,† Centre de recherche Paul-Pascal, UPR CNRS 8641, avenue du Docteur-Schweitzer, 33600 Pessac, France, and Laboratoire de chimie de la matie` re condense´ e, UMR 7574 CNRS Universite´ Pierre-et-Marie-Curie, 4 place Jussieu, 75252 Paris, France Received April 25, 2002. In Final Form: July 31, 2002 Starting from gel suspensions of vanadium pentoxide (V2O5), we have prepared a series of samples by dilution with aqueous solutions of HPF6 or NH4OH to control the vanadium concentration and the pH. We have first accurately determined the limit of stability of V2O5 ribbonlike particles from visual observation and from the wavelength dependence of the absorbency of the solutions. We obtain an unexpected result as we show that flocculation is most likely observed far from the isoelectric point of these colloidal suspensions. To estimate the largest dimensions of the particles in the suspensions, we have made light and small-angle neutron scattering experiments. We show that the particle geometry depends on the preparation process. The analysis of the scattering data indicates that V2O5 ribbons behave like semiflexible objects in solution, allowing us to estimate an elastic constant. We then discuss the observed phase diagram in terms of a simple model where the stabilization of the colloidal particles is a thermodynamic process while their geometry is controlled by kinetic parameters. Finally, the flocculation of the suspension is discussed using the Derjaguin-Landau-Verwey-Overbeek model.
I. Introduction Suspensions of inorganic particles are commonly observed as the result of the precipitation of a solid phase. There are several reasons which explain why the growth of the nuclei or germs is limited to particles of finite size, instead of macroscopically sized single crystals, and the problem of the stability of such colloidal dispersions is complex.1 One important aspect is the stability of these dispersions against flocculation, which results from the balance between attractive van der Waals forces and repulsive interactions. The latter are present as soon as the particles bear surface charges. The sign and magnitude of these charges and the nature and amount of the ions present in the solution are therefore relevant parameters. As a consequence, the pH and the ionic strength of the solution are usually important to control the kinetic stability of the dispersion.2 Aqueous suspensions of vanadium pentoxide (V2O5) give an interesting example of dispersion of colloidal particles.3,4 They can be prepared by different ways, essentially leading to the same material.5 As seen by electron microscopy, the colloidal particles are long ribbons.4 They are 1 nm thick, about 25 nm wide, and several microns long. Previous light scattering measurements have shown that the largest measured dimension in solution, most likely the persistence length, is about 300-500 nm.5-8 In fact, some bending of the ribbons, apparently more difficult in * Corresponding author. Fax: +33 5 56 84 56 00. Phone: +33 5 56 84 56 50. E-mail:
[email protected]. † Centre de recherche Paul-Pascal. ‡ Universite ´ Pierre-et-Marie-Curie. (1) (a) Jolivet, J. P. De la solution a` l’oxyde; InterEditions/CNRS Editions: Paris, 1994; see also references therein. (b) Davidson, P.; Batail, P.; Gabriel, J. C.; Livage, J.; Sanchez, C.; Bourgaux, C. Prog. Polym. Sci. 1997, 22, 913. (2) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1985. (3) Ditte, A. C. R. Acad. Sci. 1885, 101, 698. (4) Livage, J. Chem. Mater. 1991, 3, 578. (5) Pelletier, O.; Davidson, P.; Bourgaux, C.; Coulon, C.; Regnault, S.; Livage, J. Langmuir 2000, 16, 5295. (6) Kerker, M.; Jones, G. L.; Reed, J. B.; Yang, C. N. P.; Schoenberg, D. J. Phys. Chem. 1954, 58, 1147. (7) Sinn, C. Eur. Phys. J. B 1999, 7, 599.
the plane of the ribbons than out of it, is visible by transmission electron microscopy.4 While concentrated samples, between 8.6 and 0.4 mol/L vanadium concentration (i.e., between 10 and 250 H2O moles per V2O5 mole), are gels, a sol is obtained by adding more water. Both gels and concentrated sols present a nematic phase. Isotropic dispersions of ribbons are found for a vanadium concentration lower than 0.2 mol/L.9 The stability of these dispersions is controlled by the pH and the vanadium concentration. The resulting “predominance diagram”10 is nothing but a phase diagram which shows the regions of stability of various vanadium species, including ribbons.4 In this paper, we give for the first time the precise limits of the ribbon domain, focusing our study in the dilute regime. We show that analyzing the absorption spectra of the different samples in the wavelength range of 230500 nm is an accurate technique to discriminate between different species, and therefore to confirm the location of the ribbon domain. From the pH dependence of its limits, we deduce an estimation of the surface charge of the ribbons on each side of the isoelectric point (located at pH ≈ 2.6). As the pH is controlled by adding a given amount of HPF6 (to decrease the pH) or NH4OH (to increase the pH), the ionic strength of the solution is necessarily modified and flocculation is sometimes observed. We show that flocculation is found far from the isoelectric point, suggesting a dominant role of the ionic strength. Next, a systematic light scattering study and small-angle neutron scattering (SANS) data on selected samples are presented. As a very high dilution of the dispersions is reached, we measure the form factor of individual ribbons and therefore determine the geometric parameters of these objects. In particular, we estimate their length and width and discuss the influence of the preparation conditions on particle geometry. Finally, we discuss the results and show that (8) Pelletier, O.; Bourgaux, C.; Diat, O.; Davidson, P.; Livage, J. Eur. Phys. J. E 2000, 2, 191. (9) Davidson, D.; Bourgaux, C.; Schoutteten, L.; Sergot, P.; Williams, C.; Livage, J. J. Phys. II France 1995, 5, 1577. (10) Baes, C. F.; Mesmer, R. E. The hydrolysis of cations; John Wiley and Sons: New York, 1976.
10.1021/la020387e CCC: $22.00 © 2002 American Chemical Society Published on Web 11/02/2002
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they are a consequence of a subtle interplay between thermodynamic and kinetic arguments. II. Experimental Section II.1. Samples. The synthesis of V2O5 gels was performed by the ion exchange method.5 A solution of a sodium methavanadate (NaVO3) precursor of concentration 0.1 mol/L is passed through a proton exchange resin (DOWEX-50W-hydrogen, strongly acidic cation, 2% cross-linking 50-100 mesh). As acidification occurs, the pH of the effluent solution decreases from 7 to 2.6, the optimum pH for the polymerization of the precursor. After a few hours, a dark red gel of ribbons of V2O5 appears. The vanadium weight fraction in this gel is determined by preparing a dry extract and is about a few weight percents. To prepare different samples with lower vanadium concentrations11 (typically between 0.1 and 1 × 10-4 mol/L) and with a pH between 1 and 5, we have diluted the initial gel with solutions of controlled acidity. For pHs ranging from 1 to 2.8, we have used solutions of hexafluorophosphate acid (HPF6). To go from pH ) 2.8 to pH ) 5, we have used solutions of ammonium hydroxide (NH4OH). The counterions (PF6- and NH4+, respectively) were chosen for their lower tendency to induce flocculation compared with smaller ions such as Cl- or Na+. The stability of the samples, specially regarding pH, was systematically checked by allowing them to settle for at least a few weeks before performing the experiments. To study the influence of sample preparation, two series of samples were made. The first series, labeled as the “new series” (N) in the following, was prepared 1 week after the synthesis of the gel. The physical properties of these samples remain stable during a period of time of at least 2 years. Another series was prepared, labeled as the “old series” (O), by dilution of a gel prepared 7 months ago (for the SANS study) or 3 years ago (for light scattering measurements). The results presented here were obtained between 1 week and 2 months after the preparation of the samples. II.2. Characterization Techniques. The sample absorption was recorded with a UNICAM spectrometer. We swept a domain of wavelength ranging from 230 to 500 nm. Quartz cells were used to avoid cell absorption in the UV range. Thin cells were chosen (optical path, 1 mm) to limit absorption from the samples. The absorption was measured relative to air (indeed, the pure solvent was reactive toward cells and could not be used). For light scattering experiments, the incident monochromatic beam (vertically polarized) is produced by a Coherent I90-K laser. The wavelength (λ ) 6471 Å) is chosen to minimize the absorption from the samples. We used cylindrical glass cells (diameter, 7 mm) immersed in a bath of Decalin (decahydronaphthalene) necessary for temperature control (all experiments were performed at 25 °C) and index matching. The explored wave vector range is 5 × 10-4 to 3 × 10-3 Å-1. Small-angle neutron scattering for selected samples in the ribbon domain was also recorded. These experiments were performed at Laboratoire Le´on-Brillouin (Laboratoire mixte CEACNRS) (Saclay, France), on the spectrometer PACE. The neutron wavelength (λ ) 10 Å) was set with a mechanical velocity selector, and the incident beam was collimated with circular apertures of diameter 12, 16, and 7.6 mm at a distance of 5, 2.5, and 0 m, respectively, from the sample. The sample-to-detector distance is 4.57 m, yielding a wave vector in the range of 4 × 10-3 to 4 × 10-2 Å-1. The samples are held in flat quartz cells with a rectangular cross section and optical paths of 2 or 5 mm. The spectra are corrected for solvent and cell scattering and for background noise.
III. Results III.1. Determination of the Phase Diagram. Samples were prepared as described above. A first identification of the stable species in the solution is possible from the (11) There are different but equivalent possibilities to define the vanadium concentration in the V2O5 solutions: the number n of H2O molecules per V2O5 unit, the volume fraction or weight fraction in vanadium, or the vanadium concentration; n ) 20 000 corresponds to a volume fraction of 1.5 10-4 and to a weight fraction of 5 × 10-4. The corresponding vanadium concentration is 5 × 10-3 mol/L.
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color of the sample. In the investigated domain of pH and vanadium concentration, three different species are encountered. Figure 1 shows a series of N samples at various pHs and a given vanadium concentration of 1 × 10-2 mol/L. The VO2+ solutions, which exist for low pH (pH < 2, Figure 1a), are faintly yellow. Solutions containing V2O5 ribbons are red-brown (Figure 1b-f), while the decavanadate (V10O28H24-) solutions, stable at high pH (pH > 3, Figure 1g), are dark yellow. In the ribbon domain, the samples are either dispersed (Figure 1d) or flocculated (aggregated) (Figure 1b,c,e,f). In this latter case, the samples are inhomogeneous and present a dilute solution of either VO2+ ions (at low pH) or decavanadate ions (at high pH) floating above the denser flocculate. Moreover, the flocculate is gel-like at lower pH (Figure 1b,c). By variation of pH and vanadium concentration, a more systematic study has been made. Figure 2 summarizes as a phase diagram, displaying, in particular, VO2+ (V), ribbon (R), and decavanadate (D) domains, the results obtained with the N series. The first conclusion, consistent with previously published data,5,10 is that ribbons are not stable at very high dilution. The optimum stability limit (i.e., maximum possible dilution) is obtained at a pH of about 2.6 for a vanadium concentration of about C*2.6 ) 2 × 10-3 mol/L. More generally, the continuous line in Figure 2 gives the critical concentration where V2O5 ribbons disappear when decreasing the vanadium concentration. Note that this transition is reversible, in the sense that evaporating some water from a sample taken slightly below C*2.6 induces the ribbon formation. In the ribbon domain, one obtains either a colloidal dispersion (R) or a flocculated sample (F). As shown in Figure 2, the flocculation occurs on both sides, low pH or high pH, of the ribbon domain. The limits of these flocculated domains are given by the dashed line. At a pH lower than 2.6, the addition of a small amount of ammonium hydroxide to a flocculated sample gives rise to dispersed ribbons. On the other hand, the flocculation is irreversible at a pH higher than 2.6. Note that some irreversibility of the sample aspect is also observed at C*2.6 when crossing the precipitation line. Starting from a dispersed sample, one obtains a flocculated system after the following cycle: dilution followed by evaporation of some water. This shows a subtle interplay of thermodynamic and kinetic effects that shall be discussed in more detail in the last part of the paper. Similar studies have been made for the O series. Within the experimental error, the limits of the ribbon domain are the same. However, the aspect of the samples is different, as shown in Figure 1h-l, which presents a series of O samples at a given concentration (5 × 10-3 mol/L), 2 years after their preparation. Even close to the optimum pH, the samples show a slow sedimentation (Figure 1ik). At a lower pH (Figure 1h), samples have the same aspect as the N tubes of the previously labeled F domain. The situation seems more complex at higher pH, as there is no clear transition in the tube aspect (Figure 1k,l). To ascertain quantitatively the limits of the ribbon domain, we have performed absorption experiments in the range of 230-500 nm. We present first the results concerning samples of the N series. Figure 3 shows the typical absorption spectrum of a VO2+ solution (Figure 3a), a ribbon dispersion (Figure 3b), and a decavanadate solution (Figure 3c). VO2+ solutions exhibit an absorption hump around 280 nm and a very weak shoulder around 340 nm. In the ribbon domain, there are two peaks around 270 and 380 nm. These two features are still present for decavanadate solutions, as humps around 265 and 400 nm. This reveals characteristic features of the different
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Figure 1. Photograph of samples taken 2 years after their preparation. Samples of the N series along a line at constant concentration (C ) 1 × 10-2 mol/L): (a) pH ) 1.5 in the VO2+ domain, (b) pH ) 1.55, (c) pH ) 1.7, (d) pH ) 2.6, (e) pH ) 3.5, (f) pH ) 4 in the ribbon domain, and (g) pH ) 4.5 in the decavanadate ion domain. Samples for the O series for a constant concentration (C ) 5 × 10-3 mol/L) in the ribbon domain: (h) pH ) 1.8, (i) pH ) 2.1, (j) pH ) 2.5, (k) pH ) 2.8, and (l) pH ) 3.6.
Figure 2. Phase diagram in the pH/C (vanadium concentration) plane. The continuous thick line gives the limit of the ribbon domain. The dashed line indicates the limit between dispersed (R) and flocculated (F) ribbon samples. V and D stand for VO2+ and decavanadates. White circles are samples obtained by dilution with pure water (CI ) 0); the dashed line linking circles is a guide for the eyes.
species, most likely electronic transitions. To characterize quantitatively these transitions, we have estimated their intensity and characteristic wavelength. The intensity was obtained by integrating the absorption after subtraction of a straight baseline as shown in Figure 3b. The absorption wavelength is then taken at the maximum of the resulting peak. Figure 4 presents the variations of intensity and absorption wavelength with pH for a given vanadium
concentration of 4.4 × 10-3 mol/L. As flocculated samples are inhomogeneous, only V2O5 dispersions were measured in the ribbon domain. Each plot exhibits three parts, corresponding to the three different homogeneous domains. The dashed zones give the limits of the aggregated ribbons. Both electronic transitions have a maximum intensity in the ribbon domain (Figure 4a,b). The lower wavelength peak has a comparable intensity in the VO2+ and decavanadate domains, while the intensity of the higher wavelength absorption is very weak at low pH, in agreement with Figure 3a. In the same way, the position of the peaks is not the same in the different domains (Figure 4c,d). An important change of the wavelength is observed at the border between the VO2+ and ribbon domains. A smaller but significant variation is found at the border between the ribbon and decavanadate solutions. We also measured the absorbency along a line at constant pH ) 2.6 for decreasing vanadium concentrations. This value of the pH corresponds to the maximum of stability of the ribbon phase under dilution. As it is also the pH of the initial gel, one simply obtains the different samples by diluting the gel with a solution of HPF6 at pH ) 2.6. As suggested by the predominance diagram,10 around a concentration of about 2 × 10-3 mol/L, we should observe the frontier between the ribbon and VO2+ domains. There is no variation of the peak wavelengths as a function
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comment, we shall note that this absorption is the most sensitive to the object dimensions: it is very weak for the molecular VO2+ species, somewhat stronger for the decavanadates, and even stronger for the ribbons. The present result can be taken as an indication that bigger objects are present in ribbon dispersions of the O series as compared to the N series, which is consistent with the observed sedimentation in the former samples. III.2. Scattering Experiments. As we have prepared very diluted samples, the scattering intensity at a given scattering wave vector q is proportional to the form factor P(q) (normalized to 1 in the limit q f 0) of an individual object and written
I(q) ) KCRVoP(q)
Figure 3. Typical absorption spectra in the explored domain, from 200 to 520 nm: (a) in the VO2+ domain (C ) 4.4 × 10-3 mol/L, pH ) 1.4), (b) in the ribbon domain (C ) 4.4 × 10-3 mol/L and pH ) 2.7), and (c) in the decavanadate domain (C ) 4.4 × 10-3 mol/L and pH ) 4.6).
of vanadium concentration; we therefore merely display in Figure 5a the peak intensities. The two peaks have about the same intensity, which decreases approximately linearly with the vanadium concentration to nearly vanish when the critical concentration at pH ) 2.6, C*2.6 ) 2 × 10-3 mol/L, is reached. This is consistent with Figure 2: below C*2.6 one enters into the VO2+ domain, where the vanadium concentration is too weak to observe any absorption. Qualitatively similar data are found for the samples of the O series. Figure 5b gives the variation of the intensity of the two peaks with the vanadium concentration at the optimum pH ) 2.6. As already mentioned, the critical concentration C*2.6 is nearly the same. In comparison with the data of the N series (Figure 5a), we notice that the higher wavelength peak is now stronger. As a first
(1)
where K is a constant depending on the experimental setup, more precisely defined below, CR is the concentration of the vanadium which enters into the ribbon phase,12 and Vo is the volume of an individual object. In eq 1 and for light scattering, K results from the normalization of the raw scattering data to a benzene standard. Neutron scattering data, corrected for sample transmission and scaled to a 1 mm optical path, are normalized to the scattering of the 1 mm H2O standard, which yields a different K for neutron scattering. In a very simple geometric description, a ribbon will be modeled as a rigid parallelepiped with dimensions δ (thickness), l (width), and L (length). Note that we neglect any polydispersity of the ribbons, which should in fact exist, at least as far as L and l are concerned. It is expected that the thickness is constant, as it should correspond to two layers of distorted V2O5 octahedrons.13 The volume of each ribbon is thus Vo ) Llδ. As we shall see in the discussion, the length L should be identified with the persistence length in a more realistic semiflexible description. First, we present scattering results of the N series (restricted to homogeneous samples). We have measured the SANS spectra on some selected samples along the line of optimum pH, for vanadium concentrations between 1.8 × 10-2 and 6.2 × 10-3 mol/L. Significant scattering intensities were obtained despite the extreme dilution of the samples, making long exposures (about 7 h for the most dilute solution). However, scattering conditions were not good enough to measure intensity at large q (for qδ ≈ 1). In the investigated range (4 × 10-3 Å-1 < q < 3 × 10-2 Å-1), qδ , 1 and the approximate expression of the ribbon form factor reads14
P(q) )
∫0π dθ sin θ∫02π ×
1 4π
(
dφ
)
qL cos θ ql sin θ sin φ sin 2 2 qL cos θ ql sin θ sin φ 2 2
sin
2
(2)
The form factor approaches 2π/(Llq2) for ql . 1. A crossover is found toward a 1/q regime when q becomes smaller than 1/l, the inverse of the ribbon width. Finally, a saturation at the value 1 is obtained when the wave vector becomes smaller than 1/L. We show in the Appendix that expression 2 can be reduced to a factorized approximation, (12) As we shall see in the discussion, in the ribbon domain a fraction of the vanadium goes into molecular species. In fact, CR varies like C - C* (C is the total vanadium concentration in the solution). (13) Yao, T.; Oka, Y.; Yamamoto, N. Mater. Res. Bull. 1984, 52, 1433. (14) Berne, B. J.; Pecora, R. Dynamic Light Scattering; John Wiley and Sons: New York, 1975.
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Figure 4. Intensity and wavelength position of the absorption peaks of the N series at a fixed vanadium concentration (4.44 × 10-3 mol/L), varying the pH from 1 to 5: (a,b) intensity for the two characteristic absorptions (around 270 and 380 nm, respectively); (c,d) absorption peak position. Dashed zones indicate the flocculated ribbon domains.
namely,
P(q) ) F1(qL)F2(ql)
(3)
Expression 3 has been used to fit the data using the numerical approximations also described in the Appendix. We give in Figure 6a the SANS intensity versus q. In the investigated range, we observe a clear departure at small angles from the expected asymptotic q-2 behavior (continuous line), emphasized in Figure 6b. These data are used to estimate the ribbon width l. In Figure 6b, the continuous line is built from the theoretical form factor, using a simplified asymptotic form of F1 (cf. eq A9), F1 ∝ 1/q, and keeping the complete expression for F2 (cf. expression A10). The best agreement is obtained with l ) 320 ( 50 Å. In the 1/q2 regime, the scattered intensity is proportional to CRδ. From absorption data, we know that CR varies as C - C*2.6. We have therefore plotted in Figure 6c the value of I/Iw (at q* ) 10-2 Å-1 ) as a function of C. The concentration dependence is consistent with a variation linear in C - C*2.6, that is, with a constant value of δ upon dilution.13 Let us now consider the light scattering data. Results are shown in Figure 7a. The straight line is a guide for the eyes corresponding to a q-1 law. As expected, we observe a crossover toward a saturation at small angles. This crossover is emphasized by the plot Iq versus q, in Figure 7b. The continuous line results from a fit using expression 3, from which I(0)/Ib (the normalized intensity at q ) 0) and L can be extracted, as well as an order of magnitude of l in the single case of the most concentrated sample. The obtained value for l, namely, 400 ( 150 Å, is consistent with the SANS data.
In the ribbon domain, we have systematically measured the light scattering curve at different concentrations and pHs. For several fixed concentrations, no significant evolution of L and I(0)/Ib with the pH was observed. On the other hand, fixing the pH at the optimum value, that is, 2.6, we found that I(0)/Ib vanishes below the critical vanadium concentration C*2.6 (2 × 10-3 mol/L) and there is a nonlinear increase of the scattered intensity for C above C*2.6 (Figure 8a). The characteristic size L also increases with the vanadium concentration (Figure 8b). Note that the most significant variation is observed close to C*2.6. As I(0)/Ib should scale like CR(Llδ), that is, like (C - C*2.6)(Llδ), the ratio I(0)/(C - C*2.6)L should be proportional to l (the thickness δ being constant). We have used the most concentrated sample (Figure 7b), for which the fitted value of l is the most reliable, to estimate the proportionality constant and therefore deduce the variation of l with the vanadium concentration. Figure 8c shows that the two characteristic lengths, l and L, are in fact proportional. We now discuss the results of the O series. Let us first recall that a slow sedimentation was observed for these samples. Therefore, the scattering data are expected to be time dependent and the experiments described below capture the behavior of the samples at a given time. We give in Figure 6a the SANS intensity versus q for a selected sample (C ) 1.8 × 10-2 mol/L), 2 weeks after its preparation, when it still appears homogeneous. These data are already qualitatively different from those of the N samples, however. A different power law is observed with an exponent near -1.4 (dashed line). Light scattering was also measured for a series of samples at the optimum pH (Figure 9). To avoid dust contamination, measurements
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Figure 5. Intensity of the absorption peaks versus vanadium concentration at a fixed pH ) 2.6: (a) for the N series and (b) for the O series; (b) 270 nm absorption and (4) 380 nm absorption.
were made 2 months after sample preparation when the sedimentation still remains marginal. As for the N series, the scattered intensity decreases when the vanadium concentration decreases to vanish at C*2.6. However, the saturation at low angles is hardly observed which prevents accurate determinations of either I(0)/Ib or a characteristic length. At higher angles, a power law behavior is still observed, but with an exponent which varies with the vanadium concentration. For the most concentrated samples, the exponent value (-1.4, continuous line) is consistent with the available SANS data, that is, the same power law is found in the whole range between q ) 1.5 × 10-3 and 3 × 10-2 Å-1 . This behavior is not consistent with an isolated ribbon form factor. It is rather the signature of ribbon aggregates with presumably fractal structure.15 The most dilute samples in Figure 9 show a different power law with an exponent close to 1 (dotted line), reminiscent of the ribbon form factor. This may suggest a slower aggregation kinetics for dilute samples, as the scattered light intensity profiles are expected to change as aggregation proceeds.15 In any case, the observed sedimentation prevents any detailed study of this kinetics. IV. Discussion Two different aspects appear in the experimental results presented in the previous parts of this paper. On one hand, some characteristics such as the limit of the V2O5 domain (continuous line in Figure 2) are independent of the sample preparation. This suggests that their analysis relies on thermodynamic arguments. On the other hand, other (15) Schaefer, D. W.; Martin, J. E.; Wiltzius, P.; Cannell, D. S. Phys. Rev. Lett. 1984, 52, 2371.
Figure 6. Small-angle neutron scattering data. (a) Variation of I/IW (normalized by water intensity IW) versus scattering wave vector q: (4) N sample (C ) 1.8 × 10-2 mol/L) and ([) O sample (C ) 1.8 × 10-2 mol/L). The continuous (respectively dashed) line is a guide for the eyes of a q-2 (respectively q-1.4) law. (b) We have plotted Iq2 versus q: (4) experimental data for the N sample. The continuous line is a fit from expressions F1 (eq A9) and F2 (eq A10). (c) ([) Variation of intensity at q* ) 1 × 10-2 Å-1 of N samples versus concentration.
results such as the limit of the flocculated domain (dashed line in Figure 2) or the ribbon geometry are sample or preparation dependent and are therefore probably the consequence of a kinetic process. The complexity of the present system is certainly in a large part due to this interplay between kinetic and thermodynamic aspects. Let us first introduce the thermodynamic arguments. Their illustration is essentially evidenced by the phase diagram summarized in Figure 2. The first important feature here is the existence of a stability limit for the V2O5 colloidal particles. It shows that the colloidal particles are no longer stable under very high dilution. The transformation is reversible in the sense that the pre-
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Figure 7. Static light scattering data. (a) Variation of I/Ib (normalized by benzene intensity) versus q for N samples at pH ) 2.6: (O) C ) 1.56 × 10-2 mol/L, (b) C ) 1.04 × 10-2 mol/L, (0) C ) 9.34 × 10-3 mol/L, (9) C ) 8.41 × 10-3 mol/L, (]) C ) 7.35 × 10-3 mol/L, ([) C ) 6.31 × 10-3 mol/L, (4) C ) 4.47 × 10-3 mol/L, and (2) C ) 3.70 × 10-3 mol/L. The continuous line is a guide for the eyes of a q-1 law. (b) (4) Experimental data as qI vs q of an N sample (C ) 1.56 × 10-2 mol/L). The line results from a fit using expression 3.
cipitation of the particles can be obtained by concentrating a solution which contains only molecular species (V10O28H24- or VO2+ ions). Although this process is reminiscent of the well-known dissolution/precipitation of a solid in solution, it is important to note that the precipitation here does not yield to either amorphous or crystalline bulk (i.e., 3D) solid. Moreover, colloidal particles are also obtained when dry, solid V2O5 is introduced into water.5 At the molecular scale, this transformation may be understood as the result of the intercalation of water molecules into the solid structure. We may first imagine a layer separation resulting from the breaking of the weakest interlayer bonds in the solid. The second step is most likely the breaking of the weaker intralayer interactions, eventually leading to the formation of V2O5 ribbons. In this context, the colloid formation is closer to the swelling process of a pre-existing structure than to a salt dissolution, and the quality of the crystalline order of the starting material is critical. Indeed, the colloid formation is found to be much easier starting from amorphous V2O5 and would be extremely slow for highly crystallized samples. As explained in more detail below, the protonation of the oxide surface plays an important role in the colloid stabilization, the latter being understood as the result of a stabilizing effect of hydration and partial ionization of the ribbon surfaces due to acid-base reactions. According to this picture, the continuous line in Figure 2 gives the solubility limit for species dispersed at the molecular scale.
Figure 8. Light scattering parameters deduced from the fit for the N series: (a) variation of the intensity extrapolated at q ) 0, I(0) with vanadium concentration; (b) evolution of the persistence length L with vanadium concentration; (c) plot of l versus L to show that these two quantities are proportional. Continuous lines are guides for the eyes.
Beyond this limit, V2O5 ribbons should precipitate as hydrated crystals of a two-dimensional solid. Indeed, the structure of these ribbons implies that each vanadium atom is hydrated and should be considered as in contact with the solvent. This is independent of the preparation process since the thickness δ is always the same. However, the larger dimensions L and l of the particles do depend on the preparation, which means that the growth of the 2D crystals is also controlled kinetically. As shown by experiments, parameters such as the particle size or the nature of the colloidal suspensions (dispersed or flocculated) are not important to describe the precipitation of the ribbons. In other words, notwithstanding kinetic aspects, we can rely on simple thermodynamic arguments
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(2.6), the amount of either decavanadate or VO2+ ions is negligible, and one equilibrium only has to be considered. We first discuss the lower pH side (pH < 2.6) where VO2+ species are dominant. Expression 7 should be taken with a2 ) c2 ≡ c* where c ) C/C0 is a dimensionless concentration and c* is the dimensionless vanadium concentration along the precipitation line. This gives
log(c*) ) log K - (1 - σ) pH
Figure 9. Variation of I/Ib versus q for the O series at pH ) 2.6: (b) C ) 1.54 × 10-2 mol/L, (0) C ) 1.10 × 10-2 mol/L, ([) C ) 6.25 × 10-3 mol/L, (×) C ) 5.49 × 10-3 mol/L, (1) C ) 4.38 × 10-3 mol/L, (4) 3.29 × 10-3 mol/L, and (9) C ) 2.74 × 10-3 mol/L. The continuous line represents a -1.4 power law and the dashed line a -1 power law.
to compare the relative stability of ribbons and species dispersed at the molecular scale. The competition between colloidal suspensions and species dispersed at the molecular scale will be simply discussed starting from the following chemical equilibriums:
VO2+ + (2.5 - σ)H2O a (1 - σ)H3O+ + VO3.5H2+σσ (4) and
0.1[V10O28H2]4- + (0.3 - σ)H2O + (0.4 + σ)H3O+ a VO3.5H2+σσ (5) In these expressions, VO3.5H2+σσ corresponds to an actual “monomer” in the ribbon after hydration and partial ionization (σ is the charge per vanadium; it can be either positive or negative, depending on the pH). Formally, a ribbon can be described as an association of a large number of such elementary units. Each of them can be obtained from V2O5 through the reaction
0.5V2O5 + (1 - σ)H2O + σH3O+ a VO3.5H2+σσ
(6)
The resulting thermal equilibrium can be described by writing the law of mass action for chemical reactions 4 and 5. A general expression is obtained in terms of the activity of each chemical species. As the activity of the ribbons (described as a 2D, solid phase, not to be confused with the bulk V2O5) is equal to unity, we obtain from (4)
a2 ) K(aH)1-σ
(7)
a1 ) K′(aH)-(4+10σ)
(8)
and from (5)
where aH (respectively a1, a2) is the activity of H3O+ ions (respectively decavanadate, VO2+ ions). Since aH ) 10-pH, the activity of the vanadium-containing ions is fixed by the pH when ribbons are present. We use expressions 7 and 8 to discuss the location of the continuous line in Figure 2. The ribbon concentration vanishes along this line, and as the solution is diluted, electrostatic effects can be neglected. The activities can then be identified to the ion concentrations (normalized to C0 ) 1 mol/L). Moreover, far from the optimum pH
(9)
As the experimental result (continuous line in Figure 2) is indeed a straight line with a slope close to 0.8, the charge σ is approximately constant in this part of the phase diagram and equal to 0.2. In the same way, the analysis above pH ) 3 can be done using expression 8. Now a1 ) c1 ≡ c*/10 (with a factor of 10 because c* represents the vanadium rather that the decavanadate concentration). We now obtain
log(c*) ) 1 + log K′ + (4 + 10σ) pH
(10)
As we find experimentally a straight line with a slope close to 0.4, we also conclude to a constant charge per vanadium, now equal to -0.36. This value is in agreement with previous estimates of the surface charge in concentrated samples.4 On both sides of the optimum pH, the absolute value of σ is smaller than l. This is consistent with the so-called Manning condensation which implies that the surface charge on a colloid should saturate owing to electrostatic effects.16 Since the charge σ changes sign in the vicinity of the optimum pH, the isoelectric point of the suspensions should be found near pH ) 2.6. This is also suggested by the shape of the line, shown as a dotted line in Figure 2, which corresponds to a dilution with pure water. In the absence of externally added ions, electroneutrality reads
4C1 - CH ) C2 + σCR
(11)
the amount of OH- ions being obviously negligible in the relevant pH range. Far from the precipitation limit, CR is much larger than C1 and C2 and an approximate form of eq 11 is
CH ) -σCR
(12)
This can be used to roughly estimate the charge σ. Neglecting here the salt exclusion effect, to be described in detail below (cf. eq 16), we take CH ≈ 10-pH, whatever the ribbon concentration. Since an asymptotically constant value of pH ) 2.6 is reached as the vanadium concentration increases, σ should become smaller and smaller. For C ≈ CR ) 10-1 mol/L, we estimate σ ≈ -0.025. Let us now discuss the ribbon dimensions deduced from scattering data. This part of the discussion essentially concerns the results obtained with the N series from which the geometric parameters of a single ribbon have been studied. As already mentioned, the total length of the ribbons measured with electron microscopy is larger than the characteristic size L deduced from our scattering experiments. This suggests that L has to be identified with the persistence length of the ribbons. To support such an assumption, let us consider again the data shown in Figure 8c: length L and width l are proportional. This would be a surprising coincidence in a purely geometric description, since it seems hard to imagine a ribbon dissolution mechanism affecting proportionally the length and the width. The observation becomes easy to under(16) Manning, G. S. J. Chem. Phys. 1969, 51, 924.
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stand, however, if L is a persistence length. For onedimensional elastic objects, semiflexible polymers for instance, the persistence length reads
L)
κ kT
(13)
where κ is the elastic constant. Taking profit of the theory of elastic plates, one knows the expression of κ as a function of l and δ:17
κ ) Rδ3l
(14)
where R is proportional to Young’s modulus E of the material. As a consequence, L is proportional to l, δ being constant.18 A similar conclusion would arise from a simple microscopic description of the ribbon elasticity, ribbons being described as bilayers of vanadium atoms. Considering that these atoms are linked by springs of force constant C and that all distances between them are of the order of δ, Young’s modulus should scale like C/δ while the elastic constant κ will be proportional to C lδ2. Thus, expression 14 would also be obtained. Our experimental determination of the persistence length gives an estimation of R of the order of 4 × 107 N/m2. This value should be compared with Young’s modulus of macroscopic crystals, usually of the order of 1011 N/m2.19 Although a numerical factor dependent on the model should be present between R and E, we do not expect such a large discrepancy. We therefore conclude to a large reduction of the stiffness for the ribbon nanoparticles compared to the bulk material. We have presently no specific explanation for this result. However, elastic behavior of nanoparticles may diverge from bulk behavior when a characteristic length becomes in the order of atomic scales.20 In our case, each vanadium atom should be considered at the surface of the ribbon and the elastic properties may be strongly influenced by hydration or ionization at the surface. Finally, we note that the present study gives a very simple method to determine the elasticity of nanoparticles, important information not easily obtained by other techniques.21 The last point which should be discussed is the occurrence of flocculated samples in some parts of the phase diagram (see Figures 1 and 2). This flocculation is found on both sides of the ribbon domain, that is, far from the isoelectric point, independently of the sample preparation. This is an unexpected result, as we expect the dispersions to be stabilized by repulsive electrostatic interactions. In fact, a maximum of the sol stability near the isoelectric point remains the exception, as for example in silicate dispersions where the gelation kinetics is much slower near this point.1a,22 Moreover, the flocculation also (17) Landau, L.; Lifshitz, E. Theory of elasticity, 3rd ed.; Mir: Moscow, 1986. (18) There are in fact two persistence lengths as one should consider bending in and out of the ribbon plane. From the theory of plate elasticity, the corresponding bending elastic constants read κ1 ) Rl3δ and κ2 ) Rlδ3, respectively. As δ , l, the second mode has a much lower energy and out-of-plane bending is favored as already mentioned in ref 8. For this reason, bending in the ribbon plane has been ignored in our discussion. (19) Kittel, C. Introduction to Solid State Physics; John Wiley and Sons: New York, 1996. (20) Baker, P. B.; Vinci, R. P.; Arias, T. Mater. Res. Soc. 2002, 27, 26. (21) Wong, E. W.; Sheehan, P. E.; Libert, C. M. Science 1997, 277, 1971. (22) Iler, R. J. The Chemistry of Silica; John Wiley and Sons: New York, 1979.
Figure 10. DLVO interaction potential for Γ ) 10. The interaction potential V (normalized to B) is plotted as a function of x ) r/lD. The parameters Γ and B are defined in the text.
occurs when the ribbon concentration decreases at a given pH. We shall first discuss the results found for the N series and then generalize the discussion to the O series. We have presently no deep argument to explain the optimum stabilization of the dispersion close to pH ) 2.6, that is, when the averaged surface charge is expected to vanish. However, because of different chemical bonding of the vanadium atoms, a distinction should be made between the two different faces of a ribbon. Exposed to the solvent, these faces may bear different charge densities, and electrostatic repulsion may still play a role at the isoelectric point. At the same time, the optimum dispersion also corresponds to the minimum of C* and to small amounts of counterions (PF6-), that is, to a solution where the total concentration of molecular species is small. This implies a weak ionic strength and a better efficiency for the electrostatic interactions. Let us now discuss the effect of adding HPF6 or NH4OH on flocculation. Since counterions (NH4+ or PF6-) are introduced, the ionic strength of the solution is necessarily modified.23 The pH of the solution being always below 5, we shall neglect the presence of OH- ions. Quite generally, flocculation of colloidal dispersions is described using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory which introduces a competition between attractive van der Waals and repulsive electrostatic interactions. For large interparticles distances (i.e., diluted colloidal systems), the van der Waals interaction varies like A/r6 (r is the distance between two colloidal particles) while the electrostatic term is dominated by an exponential term B exp(-r/lD) (lD is the Debye length). A and B are prefactors depending on the size and charge of the particles, respectively.2 To describe the competition between these two terms, the reduced parameter Γ ) AlD-6/B can be introduced. For values of Γ smaller than Γ* ) 125.16, the total interaction has the dependence shown in Figure 10. At short distances, a deep primary minimum is found. For larger values of r, a weaker secondary minimum is also observed. When the particles fall in the primary minimum, irreversible flocculation is usually observed. Oppositely, the reversible formation of gels is generally found when only the secondary minimum is reached.1a,24 In fact, gelation in V2O5 suspensions has already been considered as the first sign of attractive forces which become important when electrostatic repulsions are screened.25 To quantify this screening effect, the important (23) The pKa for (NH4+/NH3) being equal to 9.25, NH3 species can be neglected at low pH. (24) Wierenga, A.; Philipse, A. P.; Lekkerkerker, H. N. W.; Boger, D. V. Langmuir 1998, 14, 55. (25) Pelletier, O.; Davidson, P.; Bourgaux, C.; Livage, J. Prog. Colloid Polym. Sci. 1999, 112, 121.
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occurs. Putting together these two scaling laws gives
quantity is the Debye length which reads
∑R qR2CR
lD-2 ) 4πlB
(15)
where the sum runs over all the ions present in solution and lB is the Bjerrum length (about 7 Å for an aqueous solution at room temperature). To estimate the different ionic concentrations in the ribbon domain, one should consider the so-called “salt exclusion” or Donnan effect originating in the electrostatic (ion-colloid) interactions. The first-order correction to the ideally dilute situation is proportional to the colloidal concentration CR. In other words, an activity coefficient γi defined by ci ) ai/γi should be introduced for an ion i, with26
γi-1 ) 1 + bicR
(16)
where bi is proportional to the charge of the ion, to the potential at the surface of a colloidal particle, and (in the limit of a weak interaction) to the square of the Debye length, lD2. This means that for a given value of its activity, the concentration of an ion is reduced if it bears a charge of the same sign as the colloid, enhanced in the opposite case. Let us now discuss separately the two sides of the phase diagram, beginning by the lower pH domain. In that case, the flocculation (or gel formation) is an experimentally reversible process. At lower pH, the dominant species are H3O+, VO2+, and PF6- ions, with dimensionless concentrations cH, c2, and cion, respectively. As the two cationic species bear the same charge, we have
γH-1 ) γ2-1 ) 1 - |b|cR
(17)
The electroneutrality of the solution (generalization of expression 11, introducing Cion) then implies
cion ) σcR + (1 - |b|cR)(a2 + aH)
(18)
The ionic strength (proportional to lD-2) is obtained as
I)
∑R qR2cR ) 2(aH + a2) + [σ - 2|b|(aH + a2)]cR
(19)
Equation 19 shows that, considering the salt exclusion effect, the ionic strength may increase when the ribbon concentration decreases. Thus, the observed flocculation may be consistent with the conclusions of the DLVO theory. To go further into the analysis of the reversible flocculation occurring at low pH, let us present simple scaling arguments. We shall assume that the secondary minimum shown in Figure 10 is reached at the flocculation threshold. The value of the reduced variable x ) r/lD at this secondary minimum, xmin, changes only weakly with the reduced parameter Γ previously introduced (for example, increasing Γ from 0.01 to 0.1 only implies a 15% change of xmin). Thus, the mean distance between two colloidal particles at the flocculation threshold, rm, is roughly proportional to lD. At the same time rm scales like (CR*)-1/3, where CR* is the ribbon concentration when flocculation (26) Stigter, D.; Terrell, L. H. J. Phys. Chem. 1959, 63, 551.
lD3CR* ) const
(20)
To compare with the experimental results (dashed line, lower pH side in Figure 2), two further simplifications can be done. First, we can note that because of the salt exclusion effect the VO2+ concentration should be very small at the flocculation threshold (at least, far below the optimum pH ) 2.6). Thus, in expression 20 we can identify CR* as the total vanadium concentration, Cf, at the flocculation. Moreover, in expression 19 the activity aH is the dominant term (aH is also equal to the H3O+ dimensionless concentration on the dilution line; it corresponds to the largest ion concentration) and lD-2 ∝ aH. Finally, expression 20 becomes
aH-3/2Cf ) const
(21)
In other words, we expect a straight line with a slope of -3/2 in the semilog plot given in Figure 2 as pH ) -log(aH):
3 log(Cf) ) - pH + const 2
(22)
This is exactly the experimental result found at lower pH. The same analysis can be done at higher pH (above the optimum pH ) 2.6). The charge of the ribbons is now negative, and decavanadate ions are the dominant molecular species to consider in order to estimate the Debye length. With similar arguments as above, we now obtain lD-2 ∝ a1, where a1 is the activity of the decavanadate ions given by the law of mass action (expression 8). The limit for a reversible flocculation would then be given by aH-(6+15σ)Cf ) const, or (with σ ) -0.36)
log(Cf) ) 0.6 pH + const
(23)
Such a behavior is not observed: the flocculation on the right side of the ribbon domain is irreversible and roughly occurs at a given pH (close to 3.5). This may be explained if one assumes that the primary minimum is reached. The criterion for the flocculation threshold is now that the energy barrier to reach the minimum should be below a critical value. As the height of this barrier is essentially controlled by the Debye length, a simple version of this criterion reads lD ) const, or (considering the relations between lD, a1, and aH given above) pH ) const. Thus, our simple analysis is also consistent with the experimental result found at higher pH. To conclude this discussion, let us briefly discuss the results observed in the O series. Although the limits of the ribbon domain are the same as in the N series, we find a higher tendency to flocculation for dilute samples. On both sides of the optimum pH, the aspect of the tubes is reminiscent of the one found in the N series (see Figure 1h,k). The main difference is found close to the isoelectric point (Figure 1i,j), as these samples seem to indicate a coexistence between dispersed and flocculated ribbons. Moreover, we have found a slow evolution of these samples with time, and the presence of presumably fractal aggregates is revealed by light or neutron scattering experiments. These results suggest a larger polydispersity in the O series, where a fraction of the ribbons would remain dispersed as in the N series while another part of the dispersion would flocculate under dilution even at the optimum pH. This subtle difference is certainly due to a different nature of the ribbons in the two series of samples.
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This confirms that their growth is controlled by a kinetic process rather than by thermodynamics. More systematic studies would be useful to clarify this point.
With x ) (qL cos θ)/2, one obtains
P(q) )
V. Conclusion We have described in this paper a detailed experimental study of dilute dispersions of vanadium pentoxide. The topology of the phase diagram has been first established from visual observations and absorption study. The limit of the ribbon domain is independent of the preparation conditions, indicating that the precipitation of V2O5 ribbons is a thermodynamically controlled process. The study of the pH dependence of the limits of the ribbon domain gives an estimate of the surface charge of the aggregates. Inside the ribbon domains, it is possible to measure the form factor of isolated ribbons by light and neutron scattering when dilute samples are prepared from a recently synthesized concentrated gel (N series). These experiments give the characteristics sizes l and L of the dispersed nanoparticles. The simple proportionality between the two lengths shows that L is the persistence length of the ribbons, which should therefore be described as semiflexible objects. It is interesting to note that such techniques give here a simple way to determine the elasticity of nanoparticles. Finally, we interpret the observed flocculation on both sides of the isoelectric point using simple arguments derived from the DLVO theory. The difference between the two series of samples confirms that the flocculation can be strongly sensitive to small details such as the chemical characteristics of the surfaces or the size of the colloidal particles. Therefore, the sample preparation becomes important at this point of the discussion. The main conclusion of the study is a subtle interplay between thermodynamic and kinetic aspects. On one hand, the precipitation of V2O5 ribbons, described as single crystals of a two-dimensional solid, is a reversible thermodynamical process. On the other hand, the nature and dimensions of the colloidal particles are kinetically controlled. The comparison of the two series of samples indicates that a slow kinetic process is still active several weeks and even several months after the preparation of the concentrated gel. As expected, the characteristics of the colloidal particles are important to describe the reversible or irreversible flocculation of the dispersions. Acknowledgment. It is a pleasure to thank A. Perrot and M. Maugey for their participation in the experiments and R. Cle´rac for technical help. We also thank J.-F. Joanny, R. Bruinsma, P. Davidson, O. Pelletier, and P. Poulin for fruitful discussions. We gratefully acknowledge Laboratoire Le´on-Brillouin for allocating neutron beam time. Appendix: Approximate Expression of the Form Factor
(x
sin2
2
∫0qL/2 sinx2 x dx∫0π/2 dφ
4 πqL
ql 2
1-
(x ql 2
4x2 cos φ q2L2
)
)
2
1-
4x2 cos φ q2L2 (A1)
The first integral to estimate is
(u2 cos φ) (u2 cos φ)
sin2
∫0
2 F2(u) ) π
π/2
dφ
2
(A2)
with
u ) ql
x
1-
4x2 l ) xq2L2 - 4x2 2 2 L qL
Then the form factor reads
P(q) )
2
∫0qL/2 sinx2 xF2(Ll xq2L2 - 4x2) dx
2 qL
(A3)
For small values of l/L, the function F2[u(x)] is varying slowly with x compared to (sin2 x)/x2. Then an expansion of F2 around x ) 0 (where (sin2 x)/x2 is maximum) can be made to estimate P(q). One gets
F2(u) ) F2(ql) -
l x2 F′2(ql) L2 q
(A4)
where F′2(u) is the first derivative of F2(u). Introducing this expression in eq A2 gives the first two terms of an expansion in powers of l/L:
P(q) )
2
∫0qL/2 sinx2 x dx -
2 F (ql) qL 2
∫0qL/2 sin2 x dx
(A5)
)
(A6)
2l F′2(ql) q2L3 or
P(q) ) F1(qL) F2(ql) -
(
sin(qL) l 1F′2(ql) 2 qL 4qL
where
F1(qL) )
2
∫0qL/2 sinx2 xdx
2 qL
(A7)
is the form factor when l ) 0. At first order, an even simpler expression is obtained:
Let us start from expression 2:
P(q) ) F1(qL) F2(ql)
P(q) ) qL cos θ ql sin θ cos φ sin2 sin2 2π 2 2 1 π sin θ dθ 0 dφ 4π 0 qL cos θ 2 ql sin θ cos φ 2 2 2
∫
∫
(
) (
)
(A8)
Both contributions to eq A8 can be estimated:27
F1(v) )
2
∫0v/2 sinx2 x dx ) 2vSi(v) - v42 sin2(2v)
2 v
(A9)
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where Si is the Sine Integral function28 and
g(v) )
2 F2(u) ) 2J0(u) - J1(u) - π[J0(u)H1(u) - J1(u)H0(u)] u (A10) where Jn and Hn are the Bessel and the Struve function of order n, respectively.28 The functions F1(v) and F2(u), tabulated in ref 28, can be estimated. When F2(u) is known, its derivative can be deduced numerically. Note that one may use
Si(v) )
π - f(v) cos v - g(v) sin v 2
(A11)
(
)
1 v4 + 7.241163v2 + 2.463936 v v4 + 9.068580 + 7.157433
(27) These integrals were calculated with Maple 6.01, Waterloo Maple Inc. (2000).
)
(A12)
We now discuss the practical relevance of the more complete expression A6 as compared to the simpler estimate, eq A8. For l/L ) 0.2, we have checked that the correction is always less than 2% of the complete expression. We have therefore used the simpler form (eq A8) to fit the data. In the investigated range of q, either with light or neutron scattering, the approximations for v > 1 (i.e., eqs A11 and A12) can be used to estimate F1(v). In the case of F2(u), an approximate expression valid for u < 2.5 reads
F2(u) )
and the rational approximation of f(v) and g(v) valid for v > 1 found in ref 28:
f(v) )
(
1 v4 + 7.547478v2 + 1.564072 v2 v4 + 12.723684v2 + 15.723606
(
)
1 - 0.05886u 1 - 0.05776u + 0.03869u2
(A13)
This has been used to fit the light scattering data. To analyze the neutron scattering data, F2(u) has been computed numerically. LA020387E (28) See for example: Handbook of mathematical functions; Abramowitz, M., Stegun, I., Eds.; Dover: New York, 1970.
