Langmuir 1994,10, 3043-3045
3043
Reinterpretation of Small-Angle Neutron-Scattering Studies on Ordered Colloid Dispersions Heiner Versmold” Institute of Physical Chemistry, RWTH, 52062 Aachen, F.R. Germany
Peter Lindner Institut Laue Langevin, Grenoble, France Received May 4,1994. I n Final Form: June 14,1994@ It is shown that the original interpretation of the small-angle neutron-scattering data presented by Ashdown et al. in ref 1, i.e. that the particles are arranged in a fcc array with a (111)face oriented against the surface ofthe Couette cell, suffersfrom several deficiencies. As an alternative interpretation, a hexagonal layer structure is proposed here, which is in perfect agreement with the experimentally observed Bragg diffraction pattern.
Introduction Very interesting small-angle neutron-scattering studies have been reported by Ashdown, Markovic, Ottewill, Lindner, Oberthur, and Rennie in this journal.’ Ionexchanged polystyrene lattices of particle diameter cr = 205 nm have been examined by small-angle neutronscattering in a Couette cell, both at rest and under sheared conditions. After the shear is turned off, a well-defined spot pattern was obtained, indicating a highly ordered system. In ref 1for such a system at rest, the diffraction pattern is compared with the ones expected for fcc, bcc, and hcp lattices with the conclusion that the particles are arranged in a fcc array with a (111)face oriented against the surface of the Couette cell. Although good correspondence between the experimentally determined Bragg spot pattern and the fcc assignment is claimed in ref 1, there remains considerable doubt concerning the correct assignment because several fcc reflections are missing. In view of this difficulty, a reinterpretation of the data of ref 1is carried out in this paper. First, we present a detailed comparison of the experiental data of ref 1with the expected diffraction pattern for a fcc lattice and show that a satisfactory description of the experimental data cannot be achieved. Next, a hexagonal layer structure is proposed, and it is shown that this structure prefectly accounts for the experimentally observed Bragg spot pattern.
Experimental Procedure For the discussion below, it is important to recall how the neutron-scatteringexperiments were carried out. According to ref 1,immediately after the Couette cell was filled, a pattern of rings (Figure 6a of ref 1)was observed which, as stated, either suggests a “powder”of small randomly oriented crystallites or a liquid-likestructure. On the other hand, a considerable point of interestwith the system was that after the system was sheared and then the shear rate reduced from 10 000 s-l to zero within a few minutes, the scattering pattern turned to sharp Bragg peaks (Figures4a and 6 of ref 1). Thus, shear was necessary to initiate the growth of the structure for which the Bragg diffraction spots (Figure4a of ref 1)occurred. Since the spots were observed in transmission with a wide beam, the result implies that there are large ordered domains in the sample. Further,the hexagonal
* To whom correspondence should be addressed. Abstract published inAdvance ACSAbstracts, August 15,1994. (1)Ashdown, S.; MarkoviC,1.;Ottewill, R. H.; Lindner, P.; Oberthtir, R. C.; Rennie, A. R. Langmuir 1990,6 , 303. @
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Figure 1. Indexing of the diffraction data for fcc crystal structure as proposed in ref 1. The reflections N = (h2+ k2 P)= 4 (200),N = 12 (222),N = 16 (4001,and N = 19 (331)are missing. For details see text.
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structureof the pattern indicatesstrongorientationalcorrelation between the domains ifthere existed more than a singledomain.
Results and Discussion For cubic lattices, diffraction from a plane with Miller indices h,k,l occurs for scatteringvectors Qhkl given by the Bragg equation2
~i~~= (Fsin ehkz)2 = (F)z (h2+ k2 + 1’) 2
(1)
Here, 1 is the wavelength of the incident neutrons, 8 is the scattering angle, and a is the lattice constant. According to eq 1, a plot of QhkL2 versus N = h2 k2 l2 should give a straight line with slope ( 2 n l ~ ) For ~ . fcc lattices, certain reflections are missing; i.e. the following rule for the Miller indices holds: If h,k,l are all even or all odd, the corresponding reflection is allowed; otherwise it is forbidden. Thus, reflections should occur for N = 3, 4,8,11,12,16,19,20, ... In Figure 1 a standard technique of indexing a cubic diffraction pattern when the unit cell dimension is not known is a p ~ l i e d . ~Horizontal .~ lines are drawn for the observed Qhk12 values which are taken from Figure 2b ofref 1. These should intersect the vertical
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(2) Kittel, C. Introduction to Solid State Physics; Wiley: New York, 1966. (3) McKie, D.; McKie, C. Essentials of Crystallography; Blackwell: Oxford, 1986. (4) Segschneider, C.; Versmold, H. J . Chem. Educ. 1990,67, 967.
0 1994 American Chemical Society
3044 Langmuir, Vol. 10, No. 9, 1994
Versmold and Lindner
Figure 2. Stereo picture of fcc unit cell with two hexagonal (111)layers perpendicular t o the (111)direction.
Figure 3. Stereo picture of reciprocal bcc lattice with the proposed (111)direction of the incident neutrons.’ The Ewald sphere, which becomes a plane for small-angle neutron scattering, passes through the origin. None of the (111)corners is in the Ewald plane; i.e. no (111)Bragg reflection occurs with the proposed crystal orientation.
lines drawn a t N = 3,4,8,11,12,16,19,and 20 such that the intersections define a straight line through the origin. If, according to ref 1,the lowest reflection is indexed as (111)N= 3, then, asFigure 1shows, onlytheN= S(2201, the N = 11(311), and the N = 20 (420) reflections fall on the predicted straight line through the origin. The reflections N = 4 (200), N = 12 (222), N = 16 (4001, and N = 19 (331) are missing. Thus, the attempted fcc powder pattern assignment is by no means convincing. A second point of interest is that it was not necessary to vary the orientation of the Couette cell in order to obtain the various Bragg peaks. Thus the question must be raised whether the large number of reflections observed under this condition is consistent with the proposed fcc structure with a (111)face oriented against the surface ofthe Couette cell. In Figure 2 a fcc unit cell with two hexagonal (111) layers perpendicular to the 111 direction is shown. The diffraction pattern from three-dimensional structures is conveniently discussed in terms of the reciprocal lattice and the Ewald sphere of radius 2 ~ 1 1 .The ~ reciprocal lattice of fcc is bcc. In Figure 3 the part of interest of the reciprocal lattice (bcc) is shown; the proposed (111) direction of the incident neutrons is included. With a neutron wavelength of 1 = 9.96 x 10-lo m and a lattice constant a x 1 x m, the radius of the Ewald sphere 2 d is about 3 orders of magnitude larger than the Qhkl of interest, and thus on the scale of the reciprocal lattice, the Ewald sphere is a plane. Thus, if we take the center of the cell as the origin of the reciprocal lattice, then the Ewald plane is perpendicular to the (111)direction and contains the origin as shown in Figure 3. Obviously, none of the lower reciprocal lattice points &hkl is located on the Ewald plane, and thus with this orientation of a fcc crystal no (111)Bragg reflection occurs a t all. In order to fulfill the Bragg condition, the crystal must be rotated until one
ofthe reciprocal lattice points moves into the Ewald plane. This, however, is in contradiction to the experiment reported in ref 1. We are thus forced to conclude that the interpretation of the diffraction pattern given in ref 1 cannot be correct. Next, a simple plausible reinterpretation of the data of ref 1 is given. Let us assume that, due to the applied shear, hexagonal layers are formed parallel to the surface of the Couette cell. At a low shear rate these planes slide over each other. As the shear stops, the equilibrium stacking structure of the layers (ABCABC... for fcc or ABAB... for hcp) is not achieved on the time scale of the experiment, but laterally uncorrelated stacking remains frozen in the structure. For such a system of orientationally ordered but laterally uncorrelated layers, one expects the diffraction pattern of independent hexagonal layers. Agood argument in favor of our uncorrelated layer description is the fact that a t low shear rates, where the layers are uncorrelated due to the slip motion, no noticeable change of the diffraction pattern as compared with the system at rest is observed (see Figure 4b of ref 1). Let us next consider the diffraction pattern for such a system of hexagonal layers. The reciprocal lattice of a two-dimensional hexagonal layer is again a hexagonal layer. The lattice constant a* of the reciprocal lattice is given by a* = (2/1/3)2~la.The scattering situation is depicted in Figure 4. Since there is no constraint concerning the z* component of Q for a two-dimensional system, the reciprocal lattice consists of “Bragg Each intercept of the Ewald plane with a rod defines a scattering vector Q h k . (Note that k,and k f a r e not drawn t o scale.) It is obvious that for an ideal lattice all Q h k lead to observable Bragg peaks and that the scattering pattern ( 5 ) Guinier, A. X-Ray Diffraction; Freeman: London, 1963.
Langmuir, Vol. 10, No. 9, 1994 3045
Small-Angle Neutron-Scattering Studies 25
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N Figure 5. Indexing of the diffraction data of ref 1,according to eq 2, the Bragg conditionfor a hexagonal layered structure. For details see text.
Figure 4. Diffractiongeometry for scattering from hexagonal layers. The lengths of the wavevectors k,and kfarenot drawn t o scale.
will be hexagonal. Further, a slight tilt of the sample does not cause vanishing or appearance of reflections. All this is in good agreement with the experimental observations. For a quantitative evaluation the Bragg condition for a hexagonal layer can be written as
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According to eq 2, now a plot of &hk2 versus N' = h2 k2 hk should give a straight line with slope S = (4n/n/J3a)2. Since eq 2 is derived for a primitive hexagonal twodimensional lattice, all h and k values are allowed.
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Nevertheless, not all N' values but only N' = 1 , 3 , 4 , 7 , 9 , 12,13,16,19,21,etc. occur. In Figure 5 we present a plot similar to that in Figure 1 but now with the hexagonal two-dimensional lattice assignment of eq 2 for the data of ref 1. Now the first five reflections convincingly follow a straight line through the origin. There are no extra peaks, and no peak is missing. From the slope of the line in Figure 5 , S = 2.5 x A-2, the value a = 460 nm for the lattice constant of the hexagonal layers is obtained. In conclusion, we have shown that the original interpretation of the small-angle neutron-scattering data presented in ref 1 suffers from several severe deficiencies. As an alternative interpretation, a hexagonal layer structure is proposed here, which is in perfect agreement with the experimentally observed Bragg diffraction pattern.
Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is gratefully acknowledged.
