J. Phys. Chem. B 2010, 114, 4115–4121
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Rotational Manipulation and Stacking of Nanosystems Heiner Versmold* Institut fu¨r Physikalische Chemie der RWTH, D-52052 Aachen, Germany ReceiVed: June 22, 2009; ReVised Manuscript ReceiVed: February 13, 2010
In this paper two kinds of Bragg reflections from oriented mesogenic crystals are considered. Bragg reflections behave differently for ...ABABA... and ...ABCABCA... layer stacking. We call them black and white reflections. Black reflections are characterized by the fact that they behave identical for the two kinds of stacking. Therefore, the message is the following: Black rods should not be used to identify the two kinds of stacking. By contrast, white reflections show a different behavior for ...ABABA... and ...ABCABCA... stacking and for the resulting hexagonal close-packed (hcp) and face-centered cubic (fcc) crystals. White reflections can be used to distinguish the two kinds of crystals. For an application of this method the crystals should be oriented. It will be shown that orienting by flow is possible in a Couette cell. Flow ordering occurs in all rotation cells which are often applied in rheology. The scattering presented is therefore closely related to rheology. Details and experimental results will be given in the paper. 1. Introduction Scattering experiments without sample manipulation were introduced and described some time ago by Pusey.1 They will not be repeated here. Details are given in ref 1. By contrast, we consider experiments with sample manipulation for particle alignment. The results of our various ideas have been confirmed experimentally. Experimental results will be presented later. After being introduced by Pusey,1 experiments without sample manipulation were performed by several authors.2-5 As mentioned, we consider experiments with sample manipulation. Different external fields can be applied for particle alignment: electric, magnetic, and flow fields. Since the award of the Nobel Price to Staudinger in 1953, it is known that flow is very efficient for orientational ordering. Although black and white Bragg reflections, as we call them for colloids, have been observed experimentally already by Ackerson and Hayter6 long ago, it seems that black and white Bragg reflections have received little attention until now. If a 30° rotation about the vertical z()l) axis is applied to a crystal, all reflections move to the so-called twin position. Black reflections of layered hexagonal close-packed (hcp) and cubic (fcc) crystals behave identical. Therefore, the rotation does not differentiate black reflections. Details will be given in the Discussion section. The effect is different for white reflections. Here, for hcp crystals the maxima are at l ) (1/2, whereas for cubic crystals the maxima are at l ) ( 1/3; the cubic maxima can even be shifted further.7 This behavior allows one to distinguish the two kinds of crystals. For an observation the crystals must be oriented. In order to achieve ordering of colloids, first manual rocking of the cell was applied by Ackerson.8 Later two different methods to order dispersions systematically by flow were used in our laboratory.9,10 These methods require completely different experimental setups. (1) In an older first method9 a shear wheel was used to generate on its side layers of circular flow. The shear wheel and with it the dispersion layers can be reoriented “step by step”, and their external orientation varied. As in * To whom correspondence should be addressed. E-mail: versmold@ pc.rwth-aachen.de.
Figure 1. Two possibilities of orienting an exciting beam in a small angle scattering experiment relative to the particle layers are shown: (a) with the beam normal (perpendicular) to the layers one obtains the structure in a layer; (b) with the beam tangential to the layers one obtains the stacking structure between different layers.
crystallography, the different orientations make a determination of the structure in any direction possible. This is called the “stepby-step” method in the following manuscript. (2) For our second method a previous observation of Ackerson and Hayter6 and those of Musa and Kubetzki10 were used. Although the Couette cell has been known for more than 100 years, it has almost never been used to align molecules by flow. We found the cell very useful to study molecular ordering. The following refers to our second method. A Couette cell (Figure 1) with the central part in circular motion also creates layers. However, this time it is not the side of a shear wheel, as in method 1, but the head of a rod that is used to create the flow. Two positions of the flow-generated layers in the gap are of particular importance: (a) normal, radial and (b) tangential to the beam. The two positions of the beam, a and b, are shown in Figures 1 (with a schematic drawing of a Couette cell) and 2 (with a view from the top on a Couette cell). In a Couette cell in motion there are various orientations of the beam relative to the flow-generated layers. Two important aspects of the structure are obtained with positions a and b: in position a with the beam normal, perpendicular to the layers one obtains the structure within a layer; in position b with the beam tangential to the layers one obtains the structure between the layers.
10.1021/jp905849c 2010 American Chemical Society Published on Web 03/10/2010
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Figure 2. View from the top on the orientation of layers in a Couette (or in a rotation) cell under and after flow.
Versmold
Figure 4. Tangential scattering is known from the single-crystal setting in atomic and molecular X-ray crystallography prior to application of the Weissenberg method: (a) For a tangential input beam the z* axis is adjusted vertical; (b) layer lines of different l values become visible.
Figure 5. Molecular energies for the two kinds of stacking: ...ABABA... and ...ABCA.../ACBA.... The stacking structure for ...ABAB... is hexagonal (hcp), and for ...ABCA.../ACBA... it is cubic (fcc, twins).
Figure 3. Miller indices h, k, l of an isolated layer. Indices h, k are used to describe the layer. The l direction is taken as normal (perpendicular) to the layer.
For scattering experiments the Couette cell has mainly been used in the normal, radial mode, position a. As mentioned in this normal, radial mode mainly the structure in the layers is obtainable. By contrast, in the tangential scattering geometry, position b, the stacking of the layers, i.e., the structure perpendicular to the layers, can be measured. According to Figure 2 a 90° rotation exists between the normal (radial) and the tangential layer position in a Couette or rotation cell. Thus, the indicated translation of one radius of the cell has the same effect as if the beam is rotated by 90°. Figure 2 shows that without rotation but by a translation the two aspects of structure, a and b, are measurable in one and the same cell. The refractive index of both small angle neutron scattering (SANS) and small-angle X-ray scattering (synchrotron SAXS) is very close to one, i.e., a tangential beam can enter the cell without being much deflected. However, the extension even of the well-focused synchrotron X-ray beam is finite. Thus, scattering from the cell wall (glass) and scattering from the crystalline colloidal material can be distinguished easily. Figure 3 is used to introduce Miller indices h, k, l of a hexagonal layer in reciprocal space.11,12 Visible are black (notation in terms of Miller indices; h - k ) 3n, n any integer)13-15 and white rods (h - k ) 3n ( 1) in the vertical l direction. For a single horizontal hk layer the rods are of uniform thickness. Stacking changes the thickness of the rods along the vertical l direction. For white rods this change due to stacking is different for ...ABABA... (hcp) and ...ABCABCA.../ACBACBA... (twin fcc) stacking. This different behavior of the two types of crystals is immediately visible in the scattering spectra along white rods. It is used here to distinguish mesogenic hcp from fcc crystals.
The principle of tangential scattering is known in crystallography from the single-crystal setting in atomic and molecular X-ray scattering. Prior to a measurement with the Weissenberg method the single crystal is oriented such that the l axis is vertical (h, k, l Miller indices). This is shown in many elementary textbooks of crystallography13-15 and in Figure 4a. In the case of atomic or molecular crystals the third Laue equation generates coaxial cones of different l values. As the film surrounding the camera is developed and laid flat, layer lines of different l values are visible as in Figure 4b. The determination of the distance between the layers of an atomic or a molecular crystal is a typical problem of crystallography.13-15 2. Stacking Considerations We repeat: Black and white Bragg reflections were long ago been observed in neutron scattering by Ackerson and Hayter.6 Black Bragg reflections of a crystal of hcp and cubic twins are identical. Black reflections therefore should not be used for crystal distinction.16 If a 30° rotation about the vertical z()l) axis of a crystal is performed, then all reflections move to their twin position. For black Bragg reflections this has no effect because the reflections of a hcp crystal and cubic twins behave identical. By contrast, white reflections are different10 for hcp crystals (maxima of the intensity at l ) (1/2) and cubic crystals (maxima of the intensity at l ) (1/3). Therefore, for crystal distinction white reflections should be used. As in the books of Guinier,7 we use the stacking notation of physicists, i.e., l ) 0, (1, (2, etc. For stacking problems broken values like l ) (1/2, (1/3, and (2/3 may occur. They can be avoided if a multiple of l is chosen at the beginning. There are several energy modes in the crystal cell: the acoustic, optic, and molecular branch.18 The molecular energy is modified by stacking. In Figure 5 for ...ABA... stacking (hcp) the value l ) (1/2 is degenerate and the interval A-A is split into two parts by the layer B. By contrast, for cubic (fcc) twins with ...ABCA... and ...ACBA... stacking the interval between two A layers is divided by two twin layers ...BC... or ...CB... Thus,
Rotational Manipulation and Stacking of Nanosystems
Figure 6. (a) Stacking sequence is BAB (hcp). (b) Stacking sequence is CAB (fcc). The cubic twin results from the stacking sequence BAC.
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Figure 8. Different reciprocal states of a colloidal dispersion: (a) oriented single layer, (b) oriented hcp crystal, and (c) oriented fcc crystals (in all cases the h, k layer is horizontal and the l (z*) direction is vertical).
Figure 9. Cubic twin crystals with ...ABCA... and ...ACBA... stacking and their Bragg rods. Figure 7. Equilibrium model20 spectra are shown before and after twinning with the l direction horizontal and the scattering intensity vertical. For details, see text.
for cubic stacking the interval A-A is split into three parts. This different behavior under stacking has been confirmed experimentally and will be shown in the Experimental Section (see also TOC graphic). It is well known that the two stacking sequences ...ABA... and ...ABCA... lead to different structures. Figure 6 shows both cases. In Figure 6a and 6b the hexagonal plane in the middle and the triangle at the bottom are oriented identical. Differences occur only in the upper part of the polyhedrons. In Figure 6a the triangle on top of the polyhedron has the same orientation as the triangle at the bottom. By contrast, in Figure 6b the triangle on top of the polyhedron is the mirror image of the triangle at the bottom. Thus, the stacking sequence of Figure 6a is BAB (hexagonal), whereas in Figure 6b it is CAB (cubic); a cubic twin results from the stacking sequence BAC. Oriented twins do not only have different stacking sequences but also different scattering spectra along white rods ((h - k) ) 3n ( 1, where n is an integer). For white rods the model intensity is shown in Figure 7 with the l direction horizontal and the scattering intensity vertical. For oriented monodisperse atomic and molecular crystals there are well-defined (narrow) layer lines perpendicular to the l direction as shown in Figure 4b. For mesogenic polydisperse systems such as polymer crystals the layer lines are broadened. This different behavior was observed by us experimentally some time ago with the “step-by-step” method. It will be shown at the beginning of the Experimental Section in this paper (Figures 11 and 12, see also TOC graphic). Figure 8 shows various reciprocal states of a layered system all oriented such that the l (z*) direction is vertical. In Figure 8a the reciprocal space of a single hexagonal hk layer, in Figure 8b ...ABAB... (hcp) stacking, and in Figure 8c that of ...ABCA... (fcc) stacking is shown. Although the three examples are identical concerning the hk layer at l ) 0, after stacking they show a different behavior in the l direction of white rods on which the present analysis will be based. Of interest for the experimental part of the present paper are the ...ABCA/ACBA.... cubic stacking twins in Figure 8c .
Figure 10. Twin diagonals for fcc twins ...ABCA... and ...ACBA.... For details, see text.
Before we consider Figure 8c we go to Figure 9 in which both kinds of cubic twins (...ABCA... and ...ACBA...) together with their Bragg rods are shown. The result for ...ABCA... stacking (Figure 9, left-hand side) was published more than 10 years ago.17 Figure 9 now shows both ...ABCA... and ...ACBA... cubic twins with their black and white Bragg rods. It is interesting to note that the black rods (h - k ) 3n, where n is any integer7) are intersected at l ) 0 and 1 by cubic as well as hexagonal stacking. Therefore, the spectra on black rods are not suited to distinguish the two kinds of crystals.16 By contrast, white rods (h - k ) 3n ( 1,7 shaded area in Figure 9) are alternatively intersected at l ) 1/3 and l ) 2/3 ) 1 -1/3 for cubic twins, whereas they are degenerate at l ) (1/2 for hexagonal systems. The scattering spectra along white rods are thus suited to identify both cubic twins and hexagonal crystals. We come now to Figure 8c. The gray z*/y* plane of Figure 8c includes the l direction and several black (h - k ) 3n) and white (h - k ) 3n ( 1) Bragg rods. The white rods (drawn slim) are also intersected by twin diagonals, which are explained with Figure 10. For cubic stacking (fcc) the diagonal for ...ABCA... starts at l ) 0 and ends at l ) 1. The twin diagonal (of ...ACBA... stacking) starts at l ) 1 and ends at l ) 0. We feel that it is sufficient to explain the behavior of twin diagonals and twin rods in Figure 10 for only one kind of stacking, i.e., for ...ABCA... stacking. We repeat: For ...ABCA... stacking the diagonal starts at l ) 0 and ends at l ) 1. In Figure 10 the white Bragg rod of a h,k layer (for example 1,-1) is intersected by this diagonal at l ) 1/3. The twin rod -h,-k (-1,1) is intersected at l ) 2/3. Thus, for a single twin
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Figure 11. Model and experimental (9 ) LS, low concentration) LS twin scattering intensity distribution in the l direction for step-by-step shear-disk-oriented colloid crystals. For details, see text.
Figure 12. Model and experimental (+ ) SANS, high concentration) fcc twin scattering intensity distribution in the l direction of oriented colloid crystals as determined by SANS. For details, see text.
crystal one expects one reflection from the white rod h,k (1,-1) and one reflection from the white twin rod -h,-k (-1,1). If there are two reflections on any white rod this must be due to twinning. Both cases (single crystal and twining) have been observed experimentally and will be shown in the Experimental Section. 3. Experimental Section Figure 11 shows experimental facts (9) determined by LS at rest for dilute (
