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Shear-Induced Structure in Concentrated Dispersions: Small Angle Synchrotron X-ray and Neutron Scattering H. Versmold,* S. Musa, and Ch. Dux Institut fu¨ r Physikalische Chemie der RWTH, D-52062 Aachen, Germany
P. Lindner ILL, 38043 Grenoble, France
V. Urban ESRF, 38043 Grenoble, France Received March 28, 2001. In Final Form: July 31, 2001
The influence of shear on the Bragg scattering of concentrated colloidal dispersions made from charged particles is considered in this paper. First, the reciprocal space of a layered sample at rest and under the influence of a disordering torque is considered. Next, the commonly used Couette cell and our disk shear cell are compared in view of the Ewald construction. An introduction of R- and β-rotation axes follows. The first experimental example is concerned with the neutron scattering from a layered colloidal sample at rest. As a second experimental example from the same sample the dependence of synchrotron X-ray scattering on the shear rate is considered. It turns out that the sample does not achieve a microcrystalline state as claimed previously but remains in an orientationally disordered layered state. Finally, the intensity distribution I(l) along the two types of Bragg rods as obtained with remarkable precision by synchrotron X-ray scattering is presented.
1. Introduction Light scattering (LS) investigations, i.e., Bragg scattering from dispersions of charged polymer particles, in which the equilibrium structure of sufficiently dilute dispersions was body centered cubic (bcc) were reported long ago.1-4 However, as one gets interested in more concentrated samples, due to multiple scattering, LS is no longer an adequate experimental method. The two color LS machines and the diffusive wave spectroscopy were invented to overcome the difficulties at higher concentrations. However, it turned out that these methods are too sophisticated for a general application, and much simpler techniques were required. Finally small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS) were found to be the methods of choice. The penetration strength of neutrons and X-rays is well-known. In Grenoble, France, the European Synchrotron Radiation Facility (ESRF) and the high flux reactor (neutron source) of the Institut Laue Langevin (ILL) are close together, and once the small angle scattering data are determined, they are usually evaluated at home. Here, we want to consider the structure of concentrated dispersions. For this the paper is organized as follows: As the reciprocal lattice is the adequate tool to discuss Bragg scattering the reciprocal space of a layered system is introduced in the next chapter. The Couette shear cell * To whom correspondence should be addressed. (1) Guiniers, A. X-ray Diffraction; Freeman: London, 1963. (2) Pusey, P. N. Colloidal Suspensions. In Liquides, Cristallisation et Transition Vitreuse, Les Houches 1989; Hansen, J. P, Levesque, D. Zinn-Justin, J., Eds.; Elsevier Science Publisher B.V.: Amsterdam, 1991; Vol. II. (3) Sirota, E. B.; Ou-Yang, H. D.; Sinha, S. K.; Chaikin, P. M.; Axe, J. D.; Fujii, Y. Phys. Rev. Lett. 1989, 62, 1524. (4) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57.
which is available at most research centers will then be introduced. We found it difficult to rotate a Couette cell about two orthogonal axes. Therefore, we have built our own disk shear cell. Experimental results obtained by neutron and synchrotron X-ray scattering with this cell will be presented. It seems to be of particular interest that with the present instrumentation structural changes are easier to identify by synchrotron X-ray than by neutron scattering. 2. The Reciprocal Space If shear is applied by one of the scattering cells mentioned above a dispersion will be ordered usually in hexagonal layers.4,5 The distance between the particles in such a layer will be called a. The reciprocal space of a hexagonal layer is a system of hexagonally arranged Bragg rods rotated, however, by 90°. In Figure 1 a reciprocal layer, i.e., the position of the rods in the (h,k,0) plane, is shown, where h and k are Miller indices. The closest distance a* between two rods is a* ) 2π/a. Also indicated in Figure 1 is the direction of flow for ordering the sample. Further, the rods h and k belong to one of the two classes1,6 (a) h - k ) 3n ( 1 and (b) h - k ) 3n, where n is a natural number. These classes are drawn in Figure 1 as open and filled circles, respectively. The (Miller) index l is used to describe the coordinate parallel to the Bragg rod. l is a continuous coordinate (along the rod), whereas for the h, k-layer, h and k assume only discrete values 0, (1, (2, (3, ..., etc. (5) Ashdown, S.; Markovic, I.; Ottewill, R. H.; Lindner, P.; Oberthu¨r, R. C.; Rennie, A. R. Langmuir, 1990, 6, 303. Versmold, H.; Lindner, P. Langmuir 1994, 10, 3043. (6) Versmold, H. Phys. Rev. Lett. 1995, 75, 763.
10.1021/la0104727 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/04/2001
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Figure 3. Couette shear cell: (a) perpendicular incidence; (b) apparent (β-) rotation about the central axis of the cell (drawn according to ref 8).
Figure 1. (a) Reciprocal layer, the direction of flow for ordering the sample, and the Miller indices h, k of the points in reciprocal space are indicated. Further, the points h, k of the layer belong to one of the two classes [1] h - k ) 3n and h - k ) 3n ( 1, where n is a natural number. These classes are drawn as filled and open circles. (b) Reciprocal space after 90° rotation about the axis (1,-1)-(0,0)-(-1,1). On the white rods (h - k ) 3n ( 1) in the upper part, the distribution for random stacking layers is drawn. For cubic crystals on the black rods (h - k ) 3n) the intensity is concentrated in real Bragg spots at l ) 0, (1, (2, (3... Figure 4. Bragg rod intersection by the Ewald plane with Rand β-rotations for the determination of the scattering intensity distribution I(l).
Figure 2. Circles with radius 1, 3, 4, 7, 9.... (h2 + k2 + hk) which represent the powder pattern of orientationally (l) disordered but parallel layers.
Next, we define the quantity N ) h2 + k2 + hk7 which assumes characteristic values for the Bragg rods belonging to a ring. For example N ) 1 for the first ring of reflections, i.e., for the rods (0, 1), (1, 0), (0, -1), (-1, 0), (1, -1), and (-1, 1). Similarly, N ) 3, 4, 7, 9, ..., for the second (2.), third (3.), fourth (4.), fifth (5.) ring....etc. After introducing Cartesian coordinates one obtains for the momentum |Q| transferred during a scattering process in the plane
Qx )
x3 a* (h + k) 2
Qy )
1 a* (h - k) 2
Thus, independent of the scattering direction, we find for the rods of a ring
Q2 ) Qx2 + Qy2 ∼ (h2 + k2 + hk)
(1)
Circles with radii 1, 3, 4, 7, 9, ... result, which represent the powder pattern of orientationally disordered but parallel layers. (7) Versmold, H.; Bongers, U.; Musa, S., in press.
Figure 5. Disk shear cell. The axes for R- and β-rotation are indicated.
3. r- and β-Rotations The Couette Cell. A shear cell which is used by almost all research centers is the Couette cell, shown in Figure 3. Commonly, the outer part is rotating whereas the inner cylinder is at rest. In the gap formed by the outer cup and the static cylinder the sample is subjected to a linear shear gradient. Under shear the sample will usually be layered. The layers being parallel to the slightly curved surface of the static inner cylinder. Depending on the scattering method, neutron or X-ray scattering, the cell will be made from quartz glass or polycarbonate or any adequate material. Little is known at present concerning the structure between the layers. This can be investigated by measuring the scattering intensity distribution along the Bragg rods. Depending on the stacking order of the layers, various distributions could exist. For a measurement of the structure of a sample in the direction of the Bragg rods, a rotation must be performed. As depicted in Figure 4, there are two directions about which the layers can be
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Versmold et al.
Figure 6. Effect of R-rotation on neutron-scattering pattern (particles of diameter σ ) 94 nm and volume fraction φ ) 34%, sample at rest): (a) R ) 0°; (b) R ) 21°; (c) R ) 32.5°; (d) R ) 47°;
Figure 7. Effect of shear rate on synchrotron X-ray scattering pattern at perpendicular incidence (sample same as in Figure 6): (a) γ˘ ) 0.0/s, (b) γ˘ ) 0.02/s, (c) γ˘ ) 20/s, and (d) γ˘ ) 1000/s.
rotated. For small angle scattering experiments, SANS and SAXS, the wavelength λ of the radiation is small as compared with the dimensions of any structure in the dispersion. The Ewald sphere with radius 2π/λ, therefore resembles a plane. This simplifies our discussion considerably. Henceforth a rotation about the direction of flow will be called a R-rotation. A β-rotation is a rotation about an axis in the layer but perpendicular to the direction of flow (Figure 4). Next, we investigate how such a rotation is performed with a Couette cell. Usually, as in Figure 3b an off-axis translation is carried out which is equivalent to a rotation. Obviously, by a translation at constant height parallel to the incident beam two symmetrically displaced layers are probed.8 This is equivalent to a rotation of the layers by an angle (β about the vertical axis. It appears to be of interest that commonly with a Couette cell only this type of β-rotation is carried out. The Disk Shear Cell. We have used a shear cell, different from the Couette cell, which is symmetrical with respect to R- and β-rotations. Our cell, shown in Figure 5, resembles the commercial plate-plate type of rheometer. With the help of Figure 4 we can see that the Ewald plane intersects the Bragg rods in a different manner for R- and β-rotations. For example, the rods (1, -1), (-1, 1) or in general (n, -n) are situated on the rotation axis for β-rotations. This means that for all orientations β these rods are intersected at l ) 0 by the Ewald plane. Only with R-rotations can these rods be intersected at variable height. This is one reason we have preferred R-rotations in our previous experimental neutron studies9 in which, (8) Ackerson, B. J.; Hayter, J. B.; Clark, N. A.; Cotter, L. J. Chem. Phys. 1986, 84, 2344. (9) Dux, Ch.; Versmold, H.; Reus, V.; Zemb, Th.; Lindner, P. J. Chem. Phys. 1996, 104, 6369. Dux, Ch.; Musa, S.; Reus, V.; Versmold, H.; Schwahn, A.; Lindner, P. J. Chem. Phys. 1998, 109, 2556. Versmold, H.; Musa, S.; Dux, Ch.; Lindner, P. Langmuir 1999, 15, 5065.
however, only the first three rings can be determined with the necessary accuracy. The situation is different for synchrotron X-ray scattering. Here, as shown in Figure 7a, (layered colloidal system, particles of diameter σ ) 94 nm and volume fraction φ ) 34%) with the present generation of instruments at least the first five rings of Bragg reflections are measurable with a very good signalto-noise ratio. 4. Small-Angle Neutron Scattering (SANS) We consider SANS first. The self-part of the layer structure factor Ls(Q) does not depend on the motion of a layer in a scattering experiment
Ls(Q) ) )
∞
1
∑0 exp(iQ‚(ri - rj))
n
at rest
(2)
∞
1
∑0 exp(iQ‚(ri + vi t - rj - vjt))
n
flowing )
)
∞
1
∑ exp(iQ‚(ri + vt - rj - vt)) n0 1
(3)
∞
∑0 exp(iQ‚(ri - rj))
n
Next, we present some scattering patterns obtained with the D11 small-angle neutron spectrometer in Grenoble, France. First, the sample must be ordered. The simplest way to do this is the application of shear. The angle dependence with the ordered sample at rest, shown in Figure 6, is typical. With R-rotations one obtains different
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Figure 8. Plot of Q2 vs N ) h2 + k2 + hk to identify the orientationally disordered layers.
signals for the reflections of the first ring. As expected four class (1, 0) and two class (1, -1) reflections each behave similarly. Evaluation shows that random stacking layers are compatible with the neutron scattering. An interesting change in the structure, first observed by Chen et al.10 can be seen as the shear rate γ˘ is increased. It will be discussed below, because details are better visible with synchrotron X-ray scattering. We would like to mention that similar series of experiments have been carried out by us at higher volume fraction φ ) 38% as well as with larger particles, σ ) 126 nm, which have confirmed the neutron-scattering results presented. 5. Small Angle Synchrotron X-ray Scattering (SAXS) With synchrotron X-ray scattering, excellent diffraction patterns can be obtained as shown in Figure 7. A variation of the shear rate γ˘ causes a retardation of the layers which is equivalent to a β-rotation. For (b) γ˘ ) 0.02/s, (c) γ˘ ) 20/s, and (d) γ˘ ) 1000/s together with (a) the scattering distribution for the sample at rest, the Bragg scattering patterns are given in Figure 7. The intensity of the three central rings has been lowered with an attenuator. There is a pronounced change in the scattering behavior as one compares Figure 7b with Figure 7a or with Figure 7c. The rings in Figure 7b require explanation. As pointed out with Figure 2 rings occur for orientationally disordered layers their radius being proportional to h2 + k2 + hk. This is exactly the observed behavior. In Figure 8 we took the data of Figure 7b, plotted Q2 vs N ) h2 + k2 + hk, and verified the linear relationship according to eq 1. Thus, our conclusion must be different from the one of Chen et al.10 who interpreted Figure 7b as the powder pattern of (10) Chen, L. B.; Zukoski, C. F.; Ackerson, B. J.; Hanley, H. J. M.; Straty, G. C.; Baker, J.; Glinka, C. J. Phys. Rev. Lett. 1992, 69, 688. Chen, L. B.; Ackerson, B. J.; Zukoski, C. F. J. Rheol. 1994, 38, 193.
Figure 9. Synchrotron X-ray scattering: Intensity distribution I(l) along the Bragg rods (a) (h, k) ) (-2, 1); (h - k ) 3n) and (b) (h, k) ) (3, -2) and (2, -3); (h - k ) 3n ( 1)
a polycrystalline state. Our experiments show that we are faced with a two-dimensional powder pattern due to a torsional distortion of the layered sample and we may speculate that the stress is released at certain layers as it becomes larger than a critical value (which is invisible in a scattering experiment). The well oriented Bragg scattering pattern is at least in part caused by the selfcontribution of the layer structure factor. By using eqs 2 and 3, we have shown that the self-part of the layer structure factor is not influenced by collective motion. At too high shear rate we expect the sample to be disordered and the scattering pattern to show rings again. It will be of interest whether these scattering pattern indicate layer-, crystal-, or fluidlike ordered structures. So far we did not evaluate the intensity of the Bragg reflections. The brightness of the spots can be used to determine the structure and stacking order of the layers.1 In Figure 9 the intensity along the two types of rods (a) (h, k) ) (-2, 1), (h - k ) 3n) and (b) (h, k) ) (2, -3) or (3, -2), (h - k ) 3n (1) is shown. Acknowledgment. Financial support from the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged. LA0104727
