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Langmuir 1992,8, 1036-1038
Statistical Crystallography of Surface Micelle Spacing David A. Noever N A S A Marshall Space Flight Center, ES-76, Huntsville, Alabama 35812 Received September 23, 1991. In Final Form: February 10, 1992 The aggregation of the recently reported surface micelles of block polyelectrolyteslJ is analyzed using techniques of statistical crystallography. A polygonal lattice (Voronoi mosaic) connects center-to-center points, yielding statistical agreement with crystallographic predictions; Aboav-Weaire's law and Lewis's law are verified. This protocol supplementsthe standard analysisof surface micelles2leadingto aggregation number determination and, when compared to numerical simulations, allows further insight into the random partitioning of surface films. In particular, agreement with Lewis's law has been linked to the geometric packing requirements of fillingtwo-dimensionalspace which compete with (or balance) physical forces such as interfacial tension, electrostatic repulsion, and van der Waals attraction. Introduction Fundamental and applied interest in thin films has focused on finding and characterizing highly regular and ordered surface micelles,'P2 at both air-water interfaces and surface-deposited films. From a practical point of view, stable micelle aggregates yield monomolecular polymer films when deposited on a solid substrate and such thin films may hold some promise for designing molecular electronic device^.^^^ Physically the geometric spacing of surface micelles (Figure 1, inset) arises from a balance between hydrophobic and hydrophilic forces. In particular for block polyelectrolyte films, polystyrene blocks are strongly hydrophobic and arrange their surface aggregates to minimize waterpolystyrene contact. However, ionic side chains strongly absorb on water surfaces, such that radial, micellelike aggregates partition the surface to accommodate competing effects: (1) geometric packing requirements; (2) interfacial energies of all component groups; (3) van der Waals attraction; (4) electrostatic repulsion. As a tool to understand the first factor, geometric packing requirements, the methodology of statistical crystallography5 provides quantitative insight into space partitioning and topology. The mathematics aims to divide surfaces in such a way that neither leaves gaps nor violates physical force constraints. A limited number of (most probable) options exist for this surface coverage. In particular, the crystallographic protocol arises directly from the filling requirements for two-dimensional space. For micelles, this filling interspaces micelle centers into a polygonal grid of pentagons, hexagons, and It turns out that as a function of polygonal sidedness, definite relations or crystallographic lawsk7govern the size of these grid components (Lewis's law) and their correlation to neighboring number of polygonal sides (Aboav-Weaire law). Hence the present work takes up the question of statistical size and shape characteristics (normalized) for surface micelles formed from block polyelectrolytes. Statistical Analyses Analysis was carried out on digitized photographs featured in the work of Zhu et al.2on monomolecular films of PS-decPVP+I- block polyelectrolytes. The block (1) Zhu,J.; Eisenberg, A.; Lennox, R. B. J . Am. Chem. SOC. 1991,113, 5583. (2) Zhu, J.; Lennox, R. B.; Eisenberg, A. Langmuir 1991, 7,1579. (3) Sugi, M. J. Mol. Electron. 1985, 1, 3. (4) Burrows,P.E.; Wilson, E. G. J . Mol. Electron. 1990, 6,209. (5) Weaire, D.;Rivier, N. Contemp. Phys. 1984, 25, 73. (6) Glazier, J. A.;Gross, S. P.; Stavans, J. Phys. Reu. A . 1987, 36,306. (7)Stavans, J.; Glazier, J. A. Phys. Reu. Lett. 1989, 62, 1318.
0743-7463/92/2408-1036$03.00/0
aggregates are composed of 260 styrene units connected to 240 4-vinylpyridine units which are perquarternized with decyl iodide. When dropwise added to water surfaces, the micelle core consists of styrene chains which radiate out arms of poly(viny1pyridinium decyl iodide). From micrographs showingthe area per micelle, five independent methods2 have been put forward to measure aggregate number (the number of molecules per micelle). The latter area can be calculated by assuming that the micelles arrange themselves either on a perfect hexagonal lattice or on a random polygonal lattice. The random lattice is called by Zhu2 the individual micelle method. The technique entails connecting micelle (center-to-center) distances to form a mosaic of triangles, the bisectors of which yield a network of random polygons. The method goes by different names (Voronoi construction, mathematical mosaics, etc.) and physically arises from a democratic partitioning where each micelle center gets a territorial claim. The aim of the present work is to test this micelle spacing against predictions of statistical crystallography. The micelles show a characteristic polygonal network of connected cells with a broad distribution in cell size and shape. An example image is shown (inset to Figure 1). Boundary cells were excluded from analysis and finite size effects (246sides and vertices) were taken into account using a Student t test, which depends on sample size. The statistical analysis is introduced for micelles as a predictive and diagnostic tool, both to draw conclusions about energetic and geometric tendencies and to link experiment to numerical models of space-filling. The analysis is allied with previous work on monolayers of organic acids? but in the present case the polygonal lattice is constructed mathematically from a random array of micelles and not from interconnected monolayer boundaries directly. As lucidly reviewed el~ewhere,~ statistical crystallography can measure the balance between order (short-range physical forces) and disorder (entropy and complete spacefilling or frustration effects). In part the standard for perfect order is space-filling hexagons (honeycomb-like micelle spacing). In a variety of ~ o n t e x t sthis , ~ geometric form is taken as the lowest energy solution, minimizing the boundary tension constrained by internal resistance to pattern collapse. With this ordered reference state, the statistical physics of micelle spacing can be looked at as deviations from hexagonal order. These deviations will be called disorder in two aspects, polygonal size and shape. The probability distribution function for cell-sidedness, (8) Stine, K. J., Rauseo, S. A., Moore, B. G., Wise, J. A.; Knobler, C. M., Phys. Reu. A. 1990, 41, 6884.
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p ( n ) , and its higher moments capture the quantitative aspects of polygonal shape. The key indicator of cellular disorder is the statistical spread in the cell-side distribution, as measured by the second moment, p2. For narrow distributions (p2 < 3), some simulations5 indicate that the network disorder should show persistent behavior which evolves toward nonhexagonal cell shapes. In addition to these dynamic predictions, a set of crystallographic laws called equations of state will be tested for micelle lattices. These laws describe size-shape relations which give (1) linear relations between the number of sides for a selected cell, n, and the average number of sides for its immediate neighboring cells, m (Aboav’s law), and also (2) linear relations between the number of cell sides, n, and the average normalized area, A (Lewis’s law), and perimeter, P, of that cell. The feature of interest here is somewhat paradoxical: with little actual physical data on a random network, one can extract significant physical information (such as the dominant role of interfacial vs bulk energetic effects, the predicted direction of cell-side evolution either toward or away from greater disorder, or the relative importance of nonrandom partitioning). In fact, one powerful feature of the statistical crystallographic method is its apparent universality, owing to the strict mathematical requirements of many systems to completely fill space. In short, while there are many ways to tile a twodimensional surface, most probable arrays are contrained by a balance between entropy, physical forces, and, the quantity of greatest interest here, topology. The example addressed is block polyelectrolytemicelles compressed into a lattice network. Cell Sidedness. Figure 1 summarizes micelle data for a pattern’s form, size, and number. The mean number of polygonal sides can be tested against Euler’s space-filling ideal of h e x a g ~ n s(, ~n ) = 6. As expected for samples of finite size, ( n ) is less than six for micelle patterns owing to the preferential exclusion of large cells compared to smaller cells near the observing boundaries. The result approximates the average, pentagonal-heptagonal mix found in other random networks: soap foams,8I7 Langmuir monolayers,@theoretical vertex,1° and Pottall models. An important parameter in the selection of pattern structure is pp, the second moment of the polygon-side distribution, p ( n )
This measures the spread in cell-sidedness about the mean value and can be taken as an indicator of disorder. Larger second moments correspond to more widely disparate polygonal shapes which depart from hexagonal order. By comparing micelle spacing patterns with numerical simulations of general space-filling models, predictions of dynamic behavior can be extracted and pattern evolution either toward or away from hexagonal order can be tested. For the second moment of micelle networks, p2 is less than 3 ( p 2 = 0.5), a value which numerically corresponds to long-lived spatial disorder (e.g. spread about the mean cell-sidedness)5~7and which predicts nonasymptotic behavior (pz t ) . This finding supports a dynamic cell-side distribution consistent with formation of large wavenumbers, small-scale defects, and hence many small-area
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(9)Berge, B.; Simon, A.; Libchaber, R. Phys. Rev. A: Gen.Phys. 1990, 41, 6893.
(10) Kawasaki, K.; Nagai, T.;Nakashima, K. Philos. Mag. B 1983,48, 243. (11) Srolovitz, D.3.; Anderson, M. P.; Sahni, P. S.; Grest, G. S. Phys. Reu. Lett. 1983, 32, 263.
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Figure 1. Normalized cell-sidedistributionp(n)areadistribution p ( A ) ,and perimeter distributionp(P)a~a functionof side number n for micelle patterns. p ( n )gives the probabilityof finding a cell with n sides with polydispersity in shape given by the second moment calculated, ~ L Z= 0.5. mn = 5.45
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Figure 2. A test of Aboav’s law. The product of average neighboring sides, m, and side-numberof selected cell, n, varies linearly with n. Agreement with Aboav’s law (R> 0.99) reflects a correlation which does not extend beyond nearest neighboring cells. Aboav-Weaire coefficients: lattice parameter,a, and second moment p i (eq 2). Micelles, a = 0.825, pp = 0.5. For a < 1, the cell-sidednessevolves by a process which carries energy at the interface between different cells.
cells. In addition for increasing p2, mathematical networks have been predicted to show a trend toward greater nonhexagonal d i ~ o r d e r .Larger ~ micelle arrays and dynamic data would allow direct tests of these crystallographic predictions. Aboav-Weaire’s Law. Aboav-Weaire’s law provides a consistency test between measured values of the second moment pz (calculated from cell-side distribution, p ( n ) , Figure 1) and second moment values taken from a statistical fit to shape relations between nearest neighbors (Figure 2). Aboav-Weaire’s law is a structural or topological equation of state. In part its physical significance arises from its ability to predict localization of energyeither at cell boundaries (interfacial tension) or interior points. More broadly, it describes a statistical equilibrium, derivable from an assumption of no correlation in cell shapes beyond nearest n e i g h b o r ~ . ~Its J ~original empirical form has been adapted by Weaire to mn = ( p a + 6a) + (6 - a ) n
(2)
where the parameter a is a descriptive constant of the network and pp is defined by eq 1. Aboav-Weaire’s law fits the micelle patterns (determination coefficient, R = 0.999). Crystallographically this agreement is evidence of a correlation between angles at neighboring vertices. Within the limits of the micelle data, the values for parameters a and pp (Figure 2) correspond with the leastsquares fit to Aboav-Weaire’s law. For the lattice parameter, a < 1,numerical simulations also predict that
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(12) Peshkin, S.;Strandburg, R.; Rivier, N. Phys. Rev. Lett. 1991,67, 1803.
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F i g u r e 3. A test of Lewis’s law. The average area (A,) for n-sided polygons normalized to the average total area ( A ) for all patterns as a function of cell-side, n. Linear agreement (R = 0.99) is consistent with the topological requirements of completely filling two-dimensional space.
(in soap froths) polygonal networks evolve toward lower surface energy by minimizing distortion of cell faces.5 In micelles, an equivalent physical term might be taken as the resultant between interfacial tension and van der Waals attraction, although the long-range effects of electrostatic repulsion would considerably complicate any such analysis. Lewis’s Law. An area plot (Lewis’s law) between normalized polygonal area, AIA,,, vs number of cell sides, n, provides a good representation for micelle lattices (Figure 3, R = 0.99). This result is interesting since a maximum entropy formalism predicts that agreement with Lewis’s law indicates the presence of topological constraints on cellular structure imposed by the need to fill 2-D space.5J3 Other random structures which possess strong surface forces (such as soap forths) do not obey Lewis’s law, and the identified correspondence between area and cell-shape in surface micelles is consistent with a more topologically (and less physically) constrained arrangement. Reticulation Index. A perimeter to area relationship can be used to find the reticulation or convolveness of pattern ~e1ls.l~ This yields a regression between log Q2/4), the squared-length scale used to measure the perimeter, and log ( A ) , the area of a pattern cell. The degree of reticulation is related to the regression’s slope. As shown in Figure 4,despite diverse cell shapes, a linear fit to convolveness gives a constant slope, D 1.03. For slopes ranging between 1.0 and 2.0, higher values correspond to more convoluted boundaries; the expected slope of a random pattern is 1.5. The reliability of the present fit gives a determination coefficient, R = 0.99, across 1decade
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(13) Rivier, N. P M O S .Mag. E 1985, 52, 795. (14) Turner, M. G.; Constanza, R.; Sklar, F. Ecol. Modell. 1989,48,2.
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F i g u r e 4. Convolution index for micelle patterns. The linear (log-log) slope of squared perimeter and area gives a measure of pattern convolution with larger slopes (D)attached to more convoluted patterns. For D C 1.5, the pattern is not random but follows some short-range organizing principle. The small D = 1.033 values found for micelles correspond to less convolution and nonrandom partitioning of space. in area. Figure 4 indicates that micelle patterns deviate
from a random spatial distribution (r < 1.5) and favor a lesser degree of convolveness. The ratio of (47rAIP) can also be taken as a convenient measure of a polygons’ roundness, with a ratio of unity for circles. For the slopes shown in Figure 4,this roundness index remains constant despite a 1 decade change in area, suggesting a constant change in polygonal shape with size. In other contexts, the slope of area-to-perimeter graph is the fractal dimension,l4 although no specific self-similarity is claimed here. The reticulation slope provides a particularly convenient starting point for matching with numerical simulations and mathematical mosaic^.^ Summary The statistics of micelle lattices provide an example of random 2-D networks which are constrained by topology (the need to fill space) and physical forces. Consideration of (short-range) physical forces alone cannot account for micelle arrangements, without also taking account of the specific requirements placed on partitioning geometry. By use of similar methodology, analogous behavior can be studied and compared between patterns in soap foams! Langmuir monolayers,6 and metal grains.l5 In addition to experimental comparisons, numerical models of spacefilling allow some intuition to be built about the direction of micelle dynamics-either toward or away from cellsided disorder. This protocol yields both diagnostic and predictive information and hence adds to existing analysis using aggregation number to classify and quantify the statistical physics of micelle spacing on thin films. (15) Aboav, D.A. Metallography 1980, 13,43.
