Langmuir 1999, 15, 5065-5067
5065
On the Structure of Shear-Ordered Colloidal Dispersions: Bragg-Rod Intensity Distribution H. Versmold,* S. Musa, and Ch. Dux Institut fu¨ r Physikalische Chemie, RWTH-Aachen Templergraben 59, 52062 Aachen, Germany
P. Lindner Institut Laue-Langevin, Grenoble, France Received January 13, 1999. In Final Form: April 20, 1999 In this paper, scattering from Bragg rods is considered. For layered colloidal structures, the intensity along Bragg rods reflects the stacking order of the layers. It is described how the small angle neutron (SANS) scattering intensity can be used to characterize layered systems. Scattering examples are presented for colloidal layers at rest and under sheared (flowing) conditions.
Introduction It has been demonstrated by small angle neutron and light scattering that colloidal dispersions can be transferred into a layered state by shear. The scattering from such layers can be described in reciprocal space by Braggrods, labeled with two Miller indices h, k.1-5 For layers with hexagonal symmetry, two types of rods are of interest:1 Those with (h - k) ) 3n, shown in Figure 1a in a view from the top as filled black circles, and those for which (h - k) ) 3n ( 1, shown as open light circles. Here n, h, and k are all integers. After rotation by 90° about the vertical axis (1,1)-(0,0)-(1,1) a side view of the rods is obtained, which is shown in Figure 1b. For the solidly drawn (h - k) ) 3n rods the intensity concentrates to Bragg spots at l ) 0, (1, (2, etc. We call attention to the fact that this paper mainly addresses the lighter drawn (h - k) ) 3n ( 1 rods. For hexagonal layers, which are orientationally parallel aligned but translationally uncorrelated, the Bragg scattering intensity should be uniformly distributed along the (h - k) ) 3n ( 1 rods. More complicated intensity distributions are expected for layers with a well-defined stacking order.1-3 Although very helpful model calculations have been carried out for special systems,2 in general the situation is not clear at all. Therefore, measurements of the intensity distribution along the Bragg rods appears to be a first step to understand the structure of shearordered colloidal dispersions. In this paper, we first consider in which way the scattering intensity and the structure of the layered dispersion are interrelated. This clarifies how measurements along Bragg rods can be carried out for a shearaligned, ordered dispersion. The intensity distribution of the dispersion at rest will later be compared with the one of a dispersion under flow. Theoretical As usual the structure factor will be called S(Q). It depends on the Miller indices h, k, l. It should be noted * Author to whom correspondence should be addressed. (1) Guinier, A. X-Ray Diffraction; Freeman: London, 1963. (2) Loose, W.; Ackerson, B. J. Chem. Phys. 1994, 101, 7216. (3) Versmold, H. Phys. Rev. Lett. 1995, 75, 763. (4) Dux, Ch.; Versmold, H. Phys. Rev. Lett. 1997, 78, 1811. (5) Dux; Ch.; Versmold, H. Physica A: (Amsterdam) 1997, 235, 75.
that in the present case the index l is continuous and the l-axis is parallel to the Bragg rods. The structure factor S(Q), is given by
S(Q) ) |F(Q)|2 ×
1 N
m
〈
∑ exp iQ∆Rij〉
(1)
i,j)1
Here, |F(Q)|2 is the layer form factor introduced by Loose and Ackerson,2 i.e.,
|F(Q)|2 )
1
n
∑exp iQ(sp - sp′)
n p,p′
(2)
In eq 1 ∆Rij is an interlayer vector. Next, we introduce the self and the distinct part of the structure factor, eq 1. The self part accounts for the scattering of one individual layer (∆Rij ) 0 and p ) p′), whereas the interlayer interference is taken care of by the distinct part. The scattering intensity from a system of Bragg rods, eq 1, has been considered for a few kinds of stacking order of hexagonal layers1 but is unknown in general. Experimental Section Schematically, in Figure 2 our rotating disk (disk diameter ) 15 cm) shear apparatus is shown. It requires a limited amount of the highly concentrated latex material.3,8 Further, the cell is suited to perform all the necessary adjustments for a determination of the scattering intensity along the Bragg rods. Experimental results for the (h - k) ) 3n ( 1 rods as obtained with the small angle neutron diffractometer D11 at Grenoble, France, will be considered next. Right after filling, the latex material is usually disordered in the cell, even if its ionic strength is sufficiently low. To get an ordered (layered) sample we usually apply a shear rate γ˘ of about 1000 1/s for a few minutes after which time the layers are found to be parallel and stable for days. Typical scattering patterns of shear-ordered dispersions (λ ) 1.4 nm, σ ) 92 nm, polydispersity < 5%) at a particle concentration as indicated are shown in Figure 3. A few minutes after stopping the shear, an apparently relaxed layered sample (6) Chen, L. B.; Ackerson, B. J.; Zukoski, C. F. J. Rheol. 1994, 38, 193. Ackerson, B. J. Ibid. 1990, 34, 553. (7) Clarke, S. M.; Rennie, A. R.; Ottewill, R. H. Langmuir 1997, 13, 1964. (8) Dux, Ch.; Versmold, H.; Reus, V.; Zemb, Th.; Lindner, P. J. Chem. Phys. 1996, 104, 6369.
10.1021/la990034o CCC: $18.00 © 1999 American Chemical Society Published on Web 06/18/1999
5066 Langmuir, Vol. 15, No. 15, 1999
Versmold et al.
Figure 1. Hexagonal layers with Miller indices h, k. The two types of rods with (h - k) ) 3n and (h - k) ) 3n ( 1 are indicated by filled (b) and open (O) circles. (a) View from the top. (b) Side view after rotation by 90° about the axis (1,1)-(0,0)-(1,1).
Figure 2. Rotating disk shear cell (disk: diameter 15 cm; thickness 2.0 mm; material: quartz, free of boron).
was indicated by the fact that the six lowest order Bragg peaks were visible and of equal intensity at 0° scattering angle (see Figure 3a). The simultaneous visibility of the reflections indicates an extended structure in reciprocal space (Bragg rods). In order to determine the scattering intensity along the Bragg rods we recall that the scattering intensity of a Bragg reflection is given by the intersection of the reciprocal lattice with the Ewald sphere.1 In the present case, due to the short wavelength λ of the neutrons as compared with the lattice constant a of the colloidal layers, the Ewald sphere degenerates to a plane. In Figure 4 the intersection of the Bragg rods by the Ewald plane is shown. It determines the scattering intensity Ihk(l) at the height l at which the corresponding rod h, k is intersected by the Ewald plane. In order to vary l, the tilt angle R of the Ewald plane with respect to the Bragg rods can be adjusted. Finally, for ordering and/or to perform experiments at a welldefined shear rate, the angular velocity ω of the 15 cm diameter boron free quartz shear wheel can be selected. The quartz shear disk is powered by an electric precision motor. A transmission
Figure 3. SANS diffraction pattern from a shear-ordered colloidal sample at rest: (a) R ) 0° (perpendicular incidence), and (b) R ) 21°; particle concentration 46.3 vol %.
Shear-Ordered Colloidal Dispersions
Langmuir, Vol. 15, No. 15, 1999 5067
Figure 6. Intensity difference along the two types of (h - k) ) 3n ( 1 rods for shear rates γ˘ ) 503/s, 2515/s, and 5030/s as indicated.
Figure 4. Ewald plane (sphere) intersecting Bragg rods. The scattering intensity I(l) from a Bragg rod is a function of the height l at which the rod is intersected by the plane.
Figure 5. Intensity distribution I(l) along the (h - k) ) 3n ( 1 Bragg rods (a) (1,1) or (1,1), and (b) (0,1) or (1,0) or (0,1) or (1,0). The two curves are determined from the intensities (I1,1 + I1,1)/2 and (I1,0 + I0,1 + I1,0 + I0,1)/4. The curves are for a sample of volume fraction Φ ) 46.3 vol %. allows selection of the shear rate in the range 0, 0.02/s, etc. To determine the scattering at a given shear rate, the central shear wheel of the cell is set to the desired rotation speed ω.
Results Typical scattering intensity results as a function of the position l on the rods are shown in Figure 55. For the Bragg rods (1,1) and (1,1) we found within experimental uncertainty equal intensities, I1,1 and I1,1. To improve the signal to noise ratio the mean intensity (I1,1 + I1,1)/2 is given in Figure 5. Similarly, the intensity of the four rods, (1,0),(0,1),(1,0), and (0,1), was found to be equal within experimental accuracy. To improve the signal to noise ratio, again, the mean value of the intensity, (I1,0 + I0,1 + I1,0 + I0,1)/4, is shown in Figure 5. Obviously, the intensities for the two types of Bragg rods show a different l-de-
pendence. Although we would expect six-fold symmetry, for the relaxed layers at rest, there seems to be some residual symmetry breaking due to the shear applied for ordering the sample. The experimental results described so far were obtained with dispersions ordered by shear. During the scattering experiments, however, the dispersions were at rest. Next, we consider experiments which were carried out under flowing conditions. To have the system in a stationary state, the shear was turned on for several minutes before we started the scattering experiment. Since the signal to noise ratio of the scattering curves on the individual rods is poorer and more complicated to evaluate, in Figure 6 the difference of the intensities, (I1,1 + I1,1)/2 - (I1,0 + I0,1 + I1,0 + I0,1)/4, on the two types of rods is given. As Figure 6 shows the intensity distribution for the two types of rods is different and depends on the shear rate γ˘ . Less correlation of the layers is indicated at higher shear rates. Conclusion In this paper we considered the intensity distribution along the Bragg rods of a layered system. The ordering was achived by applying shear to the sample. According to our experience, that of others, and as suggested by the scattering at normal incidence, we expected a six-fold symmetry of the layers. Surprisingly, this was not confirmed by the angular scattering dependence. A residual two-fold symmetry, presumably due to the shear ordering of the sample, was observed. This investigation also shows how the stacking order of layers can be investigated via the intensity distribution along Bragg rods. Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. For this investigation, the small angle neutron facilities of the reactor in Grenoble, France, were used. LA990034O (9) Pusey, P. N. In Liquids, Freezing and the Glass Transition; Levesque, D., Hansen, J.-P., Zinn-Justin, J., Eds.; North-Holland: Amsterdam, 1991.
