J. Phys. Chem. 1995, 99, 2088-2095
2088
Structure of Disordered Droplet Domain Patterns in a Monomolecular Film Nicole Y . Morgan Department of Physics, M.I.T., Cambridge, Massachusetts 02139
Michael Seul* AT&T Bell Laboratories, Murray Hill, New Jersey 07974 Received: August 3, 1994@ Coarsening droplet patterns in a two-dimensional binary mixture of amphiphiles, confined to a monomolecular film at an air-water interface, have been investigated via fluorescence microscopy and extensive digital pattern analysis. We focus here on structural aspects of the disordered patterns formed as a result of offcritical, isothermal surface pressure quenches. Structural correlations manifest themselves in the shape of the scaled droplet radius distribution, P(R/(R)). “Packing” constraints, amplified by a substantial excluded volume effect, predominate in the densest patterns and, at an area fraction of minority phase or “coverage” 6 = 0.25, produce a virtually Gaussian shape of P(R/(R)). Patterns of lower coverage, with q5 5 0.15, tend to produce skewed shapes, as expected on the basis of droplet-droplet interactions. The initially uniform spatial distribution of droplets is observed to be unstable to the eventual formation of large-scale inhomogeneities. The possible connection of this phenomenon to the presence of electrostatic interactions is discussed.
Introduction The evolution of binary mixtures toward their equilibrium state subsequent to a temperature- or field-induced quench into a region of phase coexistence has been the subject of a great deal of experimental and theoretical interest.’ For systems with conserved order parameters, exemplified, in the simplest (scalar) case, by a binary mixture, the classic asymptotic theory for the dilute limit,2 with its prediction of a universal droplet size distribution, has provided the standard reference, as well as the point of departure for subsequent theoretical developments. In view of experimental observations of systematic departures from the single-droplet theory in a wide range of liquid and solid mixtures, theoretical efforts have aimed to extend the analysis to systems of finite volume (or area) fraction of minority phase, Theoretical progress has been made particularly for twodimensional systems in treating the effect of diffusion-mediated droplet-droplet interactions to order This analysis employs a mapping of the problem onto an electrostatic analog, with droplet charge proportional to growth rate and dropletdroplet correlations arising from the analog of a screened Coulomb interacti~n.~.~ This treatment predicts preservation of an asymptotic scaling state, characterized, as in the original theory, by a growth exponent of l13 for the mean droplet radius; however, the shape of the universal (scaled) droplet radius distributionis considerably altered from that of the single droplet the~ry.~ Recent interest has focused on the ordering kinetics of systems governed by long-range interactions. These may be of strainel as ti^,^ e1ectrostatic,l0-’* or other origin, such as the covalent binding in block copolymer^.'^ Attractive as well as repulsive interactions of long range are expected to have a significant effect not only on the scaling function but also on the growth e~p0nent.l~ In particular, the competition of long-range repulsive and short-range attractive interactions stabilizes phases with periodically modulated order parameter profiles in a wide variety of physical and chemical systems.15 This includes the system of interest here, a monomolecular film of amphiphiles subject to repulsive dipolar interactions. l6 Extensive simulations of a related model system have indicated a crossover in the ordering qi3s4
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Abstract published in Advance ACS Abstracts, February 1, 1995.
kinetics when the mean droplet radius, (R), attains the scale of the equilibrium modulation Experimental investigations of domain coarsening have relied primarily on scattering probes.’ However, direct imaging, when feasible, offers a variety of advantages and has thus been applied with increasing frequency, for example, to the study of polymer blends near surface^'^ and of thin samples of binary liquids.18 We have recently employed fluorescence microscopy to investigate the domain coarsening of a bona fide two-dimensional binary mixture, composed of amphiphiles forming a monomolecular film adsorbed at an air-water interfa~e.’~ Extensive digital pattern analysis enabled us to address the domaincoarsening kinetics following off-cxitical surface-pressure quenches and to investigate spatial correlations exhibited by disordered droplet patterns during coarsening. The latter were found to be dominated by the screening of topological charges, C n - 6 , with n denoting droplet coordination, essentially at the nearest neighbor level, in accordance with the Aboav-Weaire law of space-filling cellular structures.20 A maximum entropy analysis of such disordered droplet patterns*l was shown to account for the observed short-range screening of topological charge, as well as for the implied screening of statistical fluctuations in the droplet area. We interpret the results of this analysis as strong evidence for a prominent role of entropy maximization in the selection of configurations assumed by dense coarsening droplet patterns. It is the aim of the present article to focus in more detail on structural aspects of such patterns, prepared over a range of offcritical compositions, and of correspondingly varying (nominal) area fractions of minority phase or “coverage”. In particular, we probe the effects of structural correlations as they affect the shape of the scaled droplet radius distribution. Additional tests of the droplet radius distribution of dense patterns, whose nominal coverage, 4 1: 0.25, is amplified considerably by an excluded volume (area) effect, confirm the essentially Gaussian shape of the universal (scaling) form of the distribution. We describe evidence for a departure from this regime of predominating “packing” constraints which manifests itself in the skewed shape of the corresponding distribution at lower coverage, 4 5 0.15. Finally, we discuss the evolution of large-scale inhomogeneities in the spatial distribution of droplets which we have
0022-3654/95/2099-2088$09.0010 0 1995 American Chemical Society
Droplet Domain Patterns in a Monomolecular Film
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Figure 1. Illustration of the processing steps applied t o images of domain patterns. (A, upper left) Original, captured from video tape; the text window on the lower right of the image contains pertinent experimental parameters overlaid on the video image and saved on tape for later reference. (B, lower left) Reference image, generated by a combination of Gaussian low-pass and morphological dilation filtering. (C,upper right) Flatfielded image, resulting from division of part A by part B (followed by suitable scaling of the intensity). Insets: (top) horizontal line scan (at y = 255); (bottom) histogram of intensities, exhibiting well-defined structure: in the example, 30% of all pixels display gray values below 145. (D, lower right) Superposition of flat-fielded image and droplet contours. The horizontal dimension of each panel is 570 pm.
observed at long times in all patterns generated by off-critical surface-pressure quenches. We discuss a possible scenario to account for such an instability on the basis of electrostatic interactions. Critical mixtures were also investigated: as reported elsewhereY2'topological aspects of a stripe to droplet transition govern topology and geometry, notably the initial droplet radius distribution of the emerging droplet patterns.
Experimental Procedures Langmuir Films. Langmuir films of mixtures of dimyristoylphosphatidylcholine(DMPC) and dihydrocholesterol (dCh)23 were prepared over a range of composition corresponding to molar ratios of (88 -k 2):10 (mole fraction of dCh, XdCh = 0.1) to (68 -k 2):30 Q d C h = 0.3). The notation indicates that 2 mole % of the majority component, DMPC, carried a fluorophore, C6-NBD-PE, to enable observation of (dark) droplets of minority phase.24 Experiments employed a temperature-controlled Langmuir trough described earlier.25 Special care was taken to avoid any adhesives when manufacturing the film barrier: instead, closed loops were fabricated from polymer-coated kapton strips by a lamination process.26 Of-critical values of film composition in the range 0.1 IXdCh I0.2 were selected so as to avoid crossing the stripe phase in the vicinity of the critical mixing p ~ i n t ~ and ~ , ' ~to ensure the nucleation of circular droplet domains. In contrast, near-critical quenches, at values of XdCh
= 0.3, involved the traversal of the stripe liquid phase near the consolute point of the m i x t ~ r e ,as ~ ~described , ~ ~ elsewhere.22 Following equilibration in their uniform p h a ~ e , at~ ~ T ,=~ ~ 19 OC, films were subjected to rapid (mechanical) expansion by (manually) ramping the barrier control to rates typically in the range 10 A2/(molecule min) s dA/dt s 15 A2/(molecule min). Final values of the surface pressure were chosen to ensure formation of circular droplet^.^^**^ To monitor the coarsening dynamics at predetermined intervals for up to -90 h following layer expansion, we relied on computer-controlled laser illumination and video recording; experimental parameters such as temperature and film area were overlaid on the video signal and recorded on tape along with images of patterns. To avoid large statistical fluctuations, the number of domains in the field of view was kept (approximately) constant by compensating for domain growth by a stepwise reduction in overall magnification. To ascertain the absence of drifts in surface pressure, films were recompressed into the homogeneous phase after completion of a coarsening run. Image Analysis. Following acquisition,**images were flatfielded via division of the original by a reference image; this was generated by convolving the original with a sequence of Gaussian low-pass and dilation filters to isolate low spatial frequencies in the image intensity profile. Flat-fielded images were binarized via global thresholding. Occasionally, domains in close proximity in the original image were actually brought
Morgan et al.
2090 J. Phys. Chem., Vol. 99, No. 7, 1995
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Figure 2. Snapshots of coarsening droplet domains in a Langmuir film, composed of DMPC and dCh, XdCh = 0.2, with 2 mole ?6 of fluorescent dye; T = 19 “C. The elapsed times are indicated. To compensate for coarsening, images were rescaled by a factor of 2 (by adjusting the microscope magnification): the black label bars respectively represent 200 and 400 ,umin the left- and right-hand panels.
into contact by the binarization: the resulting “doublets” were separated manually. The sequence of images in Figure 1 serves to illustrate the various processing steps: to ascertain their fidelity, we display, in Figure ID, the superposition of droplet domain contours in the flat-fielded image and the location of contours extracted from the binarized image (not shown) by application of a Roberts edge detector.2o The binary image of droplets was scanned, and a simple filling algorithm for convex shapes was employed to evaluate domain areas and centroids. Histograms of scaled domain radii, R/(R), employing 100 bins to collect typically 350-450 data points spanning the interval [0, 2.51, were constructed to test dynamic scaling.19 Given the imperfect flat-fielding, application of a global binarization procedure has the potential of introducing systematic errors into the analysis. One of these is “binarization noise”, here predominantly in the form of small, irregular patches of pixels assigned to the “black” population. Care must be taken to reject such patches to avoid underestimating the mean of the area distribution. The second source of error is a spurious correlation between the area fraction of the minority phase, 4, and the mean droplet radius, (R): both are similarly affected by the choice of binarization threshold, a larger threshold imlying a larger 4 as well as a larger (R). We expect these effects to contribute significantly to the error (of probably 10%) in determining the mean area. The point pattern of centroids was further subjected to Voronoi analysis to probe topological statistics. To this end, the Voronoi diagram (not shown here; see ref 21) was analyzed to evaluate the distribution P(n), of droplet coordination numbers, to determine nearest neighbor (NN)distances and to construct the kth nearest neighbor (kNN) shells of a given point.
Finally, to address the interrelation of geometrical and topological properties, the joint probability distribution, P(n,A), of droplet coordination and area was evaluated by merging data sets produced in the geometrical and topological analysis of the pattern. The majority of the results of this topological analysis are reported el~ewhere.~~.*I Here, we rely on Voronoi analysis to construct the distances, S = Sd, of closest approach between the boundaries of adjacent (nearest neighbor) domains, i and j , by subtracting the radii, Ri = (Ai/n)”2 and Rj, from the nearest neighbor distance, do, between corresponding centroids, Le., S = dd - (Ri Rj). This procedure is valid only for strictly circular droplet shapes. To construct histograms, the interval [0, 2.51 was subdivided into 50 bins.
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Off-Critical Quenches Domain Radius Distribution. Off-critical quenches were performed with molar ratios in the range 90:lO to 80:20. The majority of experimental data was recorded for 8515 and 80: 20 mixtures, displaying respective area fractions of minority phase or “coverage” of 4 = 0.12 and 4 0.25. As we will see below, these nominal values of 4 are increased substantially by an excluded volume effect. In a previous report19 we presented the analysis of growth dynamics and spatial correlations in 80:20 mixtures and demonstrated dynamic scaling in a regime governed by a growth exponent of -0.28 for the mean droplet radius, (R) Figure 2 contains snapshots of patterns recorded in such a run: visual inspection of the suitably rescaled images suggests dynamic scaling. As we have shown,19 this is borne out by the analysis of the universal droplet radius distribution, P (I?@)).
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Droplet Domain Patterns in a Monomolecular Film
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QQ-Plots vs Standard Normal Distribution r
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Figure 3. Normalized, scaled droplet radius distributions, recorded during coarsening of a DMPC-dCh Langmuir film XdCh = 0.2, T = 19 “C, at (A) t = 17 min, (e)t = 357 min, (0)t = 1249 fin, and (0) t = 2946 min subsequent to an isothermal surface-pressure quench. Also shown are fits to a Gaussian (solid l i e ) and to a (2D version of
the original domain radius data (ordinates) vs the standard normal distribution (of mean zero and standard deviation unity) (abscissae) respectively corresponding to (A) t = 17 min, (e)t = 357 min, (0)t = 1249 min, and (0)t = 2946 min (see also Figure 3).
a) Lifshitz-Slyozov (dotted line) functional form. For clarity, vertical offsets of 1.5 were applied to successive plots.19
Coverage Dependence of the Radius Distribution. In contrast to the virtually symmetric shape of P(R/(R))produced by dense pattems, the shape of distributions generated by pattems of lower domain packing density is skewed.32 This is shown in Figure 5 , where we present overlays and averages involving patterns of lower coverage, 0.11 Iq5 I0.15, over a range of elapsed times from -350 to -2750 min, with samples of 200-300 data points collected into 50 bins covering the interval 0 5 N(R) 5 2.5. The superposition, in the bottom panel of Figure 5, reveals that, while the overall shape is clearly asymmetric, the magnitude of the skewness, expressed as the deviation of points from a symmetric reference shape, is of the same order as the statistical fluctuations in the individual distributions themselves. This is the source of some difficulty in making a robust determination of the skewness. Note that the magnitude of fluctuations themselves appears to display an asymmetry about R/(R) = 1. To improve the reliability of establishing a statistically significant skewness, one may appeal to more extensive averaging (over many frames) and rely on the use of QQ plots. We have not undertaken such an effort to date. We merely illustrate the effect of averaging in Figure 5A, which contains the result of averaging P(R/(R))over the first, middle, and last of the five distributions in Figure 5B. The solid line through the data points represents the result of a fit to a suitably skewed model line shape.33 We note that, as discussed in ref 21, the average over distributions obtained for patterns of higher coverage is well approximated by a Gaussian. In view of the previous observation concerning the magnitude of statistical fluctuationsin P(N(R)),it is not surprising that fits to the skewed shape (of four independent fit parameters) do not yield a statistically significant improvement over fits to a Gaussian (of only two independent fit parameters). However, the solid line does appear to reproduce the skewness satisfactorily. No extended coarsening runs were performed at lower coverage, precluding a systematic analysis for dilute pattems. However, the superposition of normalized distributionsin Figure
Surprisingly, P(R/(R)) was found to closely conform to a Gaussian shape. This may be ascertained from Figure 3, which displays four distributions, recorded at successive times of 17, 357, 1249, and 2946 min subsequent to an isothermal surfacepressure quench, along with fits to a Gaussian.lg Parametrizing the data in the form of a Gaussian has the advantage of restricting the number of requisite parameters to two while ostensibly providing a very close approximation to the actual functional form of P(R/(R)). Within the framework of a recently proposed maximum entropy analysis of disordered droplet the Gaussian is seen to arise as a close approximation to the analytic expression for P(R/(R))produced by the theory. We thus attribute the significant deviation of our virtually symmetric distribution from the Lifshitz-Slyozov mean-field solution to the predominance of global geometric (“packing”) constraints. We will see below that a decrease in the coverage, q5, affects the shape of P(R/(R)). As the result of a binning procedure, applied to data in the interval 0 5 R/(R) 5 2.5, distributions represent the outcome of an effective low-pass filtering procedure, as does any histogram. A robust and quantitative procedure to compare the distribution of the actual raw data (prior to binning) and the Gaussian distribution is afforded by quantile-quantile (QQ) plots: two distributions differing only in scale or location yield a linear QQ plot.30 Such QQ plots31of normalized droplet radii vs the standard normal distribution (of mean zero and standard deviation unity), along with linear fits, are shown in Figure 4 for the data sets in Figure 3. A linear correlation obtains over at least 11.50, with u, as usual, denoting standard deviation. In fact, when the statistical fluctuations due to the random sampling of the standard normal distribution involved in constructing the QQ plots are taken into account,30the agreement is rather close.
2092 J. Phys. Chem., Vol. 99, No. 7, 1995
Morgan et al.
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Figure 5. Effect of area fraction on the shape of the scaled droplet radius distribution, P(R/(R)). Shown are distributions produced by
patterns of low area fraction of the minority phase (see left-hand panel in Figure 6) in a DMPC-dCh mixture, with XdCh = 0.2, at T = 19 “c. (A, top) Average of first, middle, and last (in order of elapsed time) of the distributions in the bottom panel; the solid line fit is explained in the text. Also shown for reference is the asymmetric Lifshitz-Slyozov function. (B, bottom) Superposition of scaled,normalized distributions, P(R/(R)),obtained by collecting data into 50 bins spanning the interval 0 5 R/(R) 5 2.5. Elapsed times are (0)349 min, (e)1010 min, (0) 1217 min, (0)1768 min, and (0)2736 min. The vertical line at R/(R) serves to emphasize the slight skewness of the distributions,as discussed in the text. 5 does support the notion that, as long as the areafraction of domains, i.e., coverage, remains constant, so does the skewed universal shape, suggesting dynamic scaling for dilute patterns.33
Domain Boundary Separation: Excluded Volume Effect. The packing constraints just described are all the more pronounced as a result of an excluded “volume” (area) effect governing the local packing of droplets. One signature of this effect is a negative offset in the (essentially) linear dependence of the normalized droplet area, A/(A), on the normalized area, Avd(Avp), of the corresponding Voronoi cell? While the ratio of the respective mean values reproduces the nominal coverage, q5 = (A)/(AvP),the negative offset reflects an effective area of the embedded droplets which considerably exceeds, by S0.5 A/(A), the apparent droplet size.34 Mutual electrostatic repulsion between droplets, known to be relevant in the experimental system at hand,35provides a natural explanation of this phenomenon. Additional evidence for such an excluded volume effect comes from the shape of the distribution,P(S/(S)),of the distance of closest approach, S = So, between the boundaries of adjacent (nearest neighbor) domains, i andj.36 Figure 6 contains scaled distributions, P(S/(S)),corresponding to the respective scaled domain radius distributions, P(R/(R)),of Figure 3. As may be ascertained from that set of plots, P(S/(S)) also exhibits a
Figure 6. Distributions of droplet boundary separation, recorded during coarsening of a DMPC-dCh Langmuir film with XdCh = 0.2 and T = 19 “cat (A) t = 17 min, (e)t = 357 min, (0) t = 1249 min, and (0) t = 2946 min subsequent to an isothermal surface-pressure quench. Also shown are Gaussian fits (solid lines) (see also Figure 3). For clarity, offsets of 1.0 were applied to successive plots.
universal shape, with a peak position close to s S/(S) = 1, where (S) = c(R),where 2.1 5 c 5 2.5 for 4 = 0.25 and 4.5 5 c 5 5 for 4 = 0.15. Close examination of the shape reveals a small but systematic deviation from the Gaussian reference shape: for s 5 0.4, P(s) 0. If we interpret this feature as a signature of a cutoff, so that finite values of the distributions are registered only for s k 0.4, we are led37to a rough estimate 24. In view of the poor statistics, we expect this of +eff estimate to represent an upper limit of this is consistent with the more robust estimate of @eff 1.54 derived from the linear correlation of droplet and Voronoi cell areas cited above.34 We note that the values of 4 = 0.25 of the nominal coverage, typically achieved in the 80:20 mixtures, appear to be difficult to exceed. In droplet patterns of quenched connectivity, generated in near-critical k d C h = 0.3) quenches,**“pseudominority” phases appear, and even in these we observe 4 5 0.3 for the area fraction of the locally predominating minority phase.
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Inhomogeneities in the Spatial Distribution of Domains. We now turn to the issue of the stability of the disordered domain patterns. In all the extended coarsening runs performed at XdCh = 0.2, yielding an initially uniform (nominal) coverage of q5 = 0.25 in essentially the entire accessible38 film area of 25 x 25 mm2, large-scale inhomogeneities in coverage were observed at late times. That is, while the spatial distribution of domains remained homogeneous on the scale of the pertinent field of view, of maximal linear dimension 2280 pm (see above), the coverage in many randomly sampled regions of the pattern was found to decrease, leaving patterns in those regions visibly more dilute. Concomitantly, densely packed clusters of domains were observed in other regions. The two sets of patterns in parts A and B of Figure 7 illustrate the coexistence of regions of different coverage at elapsed times of respectively -1250 (top panel) and -2950 min (bottom panel). While the particular patterns compared in the top and
J. Phys. Chem., Vol. 99, No. 7, 1995 2093
Droplet Domain Patterns in a Monomolecular Film
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Figure 7. Snapshots illustrating large-scale inhomogeneity in the spatial distribution of droplets, observed at various elapsed times, t, subsequent to a surface-pressure quench in Langmuir films of DMPC and dCh with XICI, = 0.2 and T = 19 "C. (A, top) Variation in interdroplet separation, with approximately equal values of (R). Patterns were recorded at (left) t = 1217 min and (right) t = 1249 min, respectively; the horizontal dimension of the field, in each panel, is 1425 pm. (B, bottom) Variation in mean droplet radius, (R), with approximately equal interdroplet separation. Patterns were recorded at (left) t = 2736 min and (right) t = 2946 min; the horizontal dimension of the field, in each panel, is 2280 pm. The pattern on the left, with 4 = 0.15, yields a skewed distribution, similar to those in Figure 5. The pattern on the right, with 4 = 0.25, corresponds to that in the right-hand panel of Figure 2 and yields a symmetric distribution.
bottom panels of that figure were not recorded in the same film but in fact in two different coarsening runs, both runs were initiated with identical values of the nominal initial coverage. The instability in the spatial distribution of domains manifests ipelf in the>emporal evolution of the dimensionless product, Q(R); here, Q denotes the position of the first peak in the Fourier spectrum computed from the droplet patterns (not shown). Given that Q l/dd, do representing the distance between adjacent domains, i and j , and (R) @'*dij, we see that, if (minority phase) coverage remains constant, so does the product in question; in our experiments, we find deviations from a constant for elapsed times exceeding -1200 min. Figure 7A shows an example of the corresponding situation: while the respective mean droplet radii, (R), for both frames of panel A are comparable, respective values for do are seen to differ substantially. As Figure 7B demonstrates, we have also, on rare occasions, encountered cases close to the opposite extreme: in the example shown in the figure, recorded after a substantial elapsed time of -2950 min, it is the difference in respective values of (R) which primarily accounts for the difference in coverage, while values for do are comparable.
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Discussion The feature which exhibits the most significant sensitivity to a reduction in coverage is the scaled droplet radius distribution, P(R/(R)), as discussed in conjunction with Figure 5. A more prominent skewness of P (R/(R))for patterns of (locally) lower coverage may be rationalized by arguing that, while entropic "packing" constraints govern the configuration of dense patterns, these are of relatively lesser importance compared to energetic constraints at lower coverage. Energetic constraints reflect interactions between droplets known4-6.1*v39 to produce precisely the type of deviation from the Lifshitz-Slyozov form of P(R/ (R)), with remaining skewness, such as that described here in connection with Figure 5. The evolution of large-scale gradients in the spatial distribution of domains suggests the initially uniform coverage to be unstable with respect to the formation of large-scale inhomogeneities. As long as gradients remain small, coexisting regions of different coverage are separated by distances substantially exceeding the linear dimension of the pertinent field of view, and this makes it possible to track large regions which retain the initial (nominal) coverage; eventually, however, coverage
2094 J. Phys. Chem., Vol. 99, No. 7, 1995 is in fact no longer well-defined on length scales significantly exceeding the linear dimension of the field of view, and this forces the termination of coarsening runs. We note that the nature of the evolving spatial distribution is difficult to assess, requiring, in our current setup, the lateral scanning of a finite field of view across the film. Figure 7 illustrates the local droplet configurations characterizing the inhomogeneity: most commonly, we observe a spatial variation of the mean boundary separation, leaving the mean droplet size largely unaffected. To our knowledge, this phenomenon has not been previously described in Langmuir monolayers. In considering its possible origins, we must fist address the issue of surface-active impurities. Contamination of Langmuir films is notoriously difficult to control, and the slow accumulation of a suitable "fusogen" might lead to the clustering of domains, thereby generating gradients in the coverage. Drifts in the value of molecular area, ac, characterizing the transition of the film into the uniform phase, represent a sensitive indicator of the presence of surface-active impurities.@ No significant shifts in a, were obseved when films were recompressed following completion of a coarsening run, an observation which we interpret as an indication against significant contamination. Next, we turn our attention to electrostatic interactions, previously mentioned in connection with the excluded volume effect in the packing of droplet domains. The competition of short-ranged attractive interactions with long-ranged dipolar repulsion between domains is known to favor a periodically modulated ground state in the form of a droplet array."J6 The equilibrium length scale is set by the normalized ratio, Nb, of interdroplet repulsive dipolar interactions to droplet line tension, a control parameter usually referred to as the bond number.15 In the Langmuir films of interest here, Nb is determined, for given 4, by the quench depth.25 That is, following the quench, Nb remains fixed, and the lattice constant, d m , of the droplet array which represents the ideal ground state is thereby preset. In this context, we note that if competing interactions indeed govern the collective properties of droplet patterns in the Langmuir films studied here, the experimental system does appear to be overconstrained. Specifically, the quench depth, and hence Nb, as well as the coverage, 4, are set independently, while theory shows Nb to control both the equilibrium length scale, -dm,16 and the droplet size, (R)*, which signals crossover to equilibrium behavior.41 Hence, unless 4 is carefully chosen to satisfy the condition (R)* 4"*dN, number density and (mean) size cannot both be consistent with the correct (equilibrium) values, given that the overall area is also fixed. As a result, the system finds itself effectively under lateral compression or expansion. The (lamellar) stripe domain phase of systems subject to competing interactions exhibits a transverse ("buckling") instability of a finite wave vector in response to expansive ~ t r e s s . ~In~contrast, . ~ ~ a hexagonally ordered array of droplets of fixed, coverage-independent radius appears to be stable under similar conditions.4 The situation is substantially altered if the mean droplet radius is allowed to adjust itself in response to a variation in coverage, 4, in such a way that an increase of 4 favors an increase in (R).45 In particular, Timasheva and Singer have recently analyzed the case in which the instability of an ordered array of circular droplet domains manifests itself entirely in a variation of (R), maintaining constant the number of droplets and the interdroplet spacing, d,j.46 They find an instability of the uniform state in favor of a configuration of coexisting regions of differing ( R ) and correspondingly differing 4; in addition, there is a finite threshold of (R), reminiscent of the experimental observation of an onset only after finite elapsed coarsening time.
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Morgan et al. Examination of available data suggests a weak correlation of the desired type between 4 and (R), although a systematic evaluation remains to be performed; Figure 7B captures an example with the largest magnitude of such a correlation we have observed to date. Recent simulations' of coarsening droplet patterns in a model system with competing interactions indicate the preferred way of realizing the global equilibrium configuration in the form of a droplet array: domain coarsening terminates when ( R ) $l'*dm. Domain fusion facilitates the requisite adjustment of droplet size and number density, leading to the formation of a liquid of monodisperse circular droplets which subsequently orders via annihilation of topological defects. However, if domain coarsening is and if fusion is effectively suppre~sed,"~the system may favor an alternative, more expedient approach to equilibrium involving the realization of local equilibrium configurations at the expense of introducing large-scale inhomogeneities in the spatial distribution of droplet domains. The initial theoretical analysis of Timasheva and Singer suggests that electrostatic interactions are indeed capable of introducing instabilities in the uniform spatial distribution of domains, although a more general treatment may be required to capture all salient experimental features described in connection with Figure 7. On the experimental side, the clarification of the nature of the instability remains an interesting topic for further investigation.
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Acknowledgment. The work presented here has benefited from instructive conversations with R. Bruinsma, D. Dahringer, C. Mallows, and J. Marko. It is a pleasure to acknowledge S. Singer for offering a number of insightful comments regarding an earlier version of the manuscript and for providing results of a preliminary theoretical analysis. N.Y.M. acknowledges support through the Graduate Research Program for Women, sponsored by AT&T. References and Notes (1) For reviews, see: Gunton, J. D.; San Miguel, M.; Sahni, P. S . In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1983; Vol. 8. Komura, S. Phase Transitions 1988, 12, 3. Bray, A. J. Phase Transitions and Relaxation in Systems with Competing Energy Scales; NATO Advanced Study Institute Series; Geilo: Norway, 1993. (2) Lifshitz, I. M.; Slyozov, V. V. J . Phys. Chem. Solids 1961, 19, 35. Wagner, C. Z. Elecktrochem. 1961, 65, 581. (3) Voorhees, P. W. J . Stat. Phys. 1985, 38, 231. (4) Marder, M. Phys. Rev. 1987, 36, 438. (5) Marqusee, J. A.; Ross, J. J . Chem. Phys. 1984,80,536. Marqusee, J. A. J . Chem. Phys. 1984, 81, 976. (6) Zheng, Q.; Gunton, J. D. Phys. Rev. A 1989, 39, 4848. (7) Hayakawa, H.; Family, F. Physica A 1990, 163, 491. (8) Tokuyama, M.; Kawasaki, K. Physica A 1984,123, 386. Imaeda, T.; Kawasaki, K. 1990, 164, 335. (9) Kawasaki, K.; Enomoto, Y. Physica A 1988, 150, 463. (10) Roland, C.; Desai, R. C. Phys. Rev. B 1990,42, 6658. (11) Sagui, C.; Desai, R. C . Phys. Rev. Lett. 1993, 71,3995; Phys. Rev. E 1994, 49, 2225. (12) Chen, L. Q.; Khachaturyan, A. G. Phys. Rev. Lett. 1993,70, 1477. (13) Ohta, T.; Kawasaki, K. Macromolecules 1986,19,2621. Bahiana, M.; Oono, Y . Phys. Rev. A 1990, 41, 6763. (14) Hayakawa, H.; Ricz, Z.; Tsuzuki, T. Phys. Rev. E 1993,47, 1499. (15) For a review of modulated phases, see: Seul, M.; Andelman, D. Science, in press. (16) Andelman, D.; Brochard, F.; Joanny, J. F. J . Chem. Phys. 1987, 86, 3673. (17) Tanaka, H.; Hayashi, T.; Nishi, T. J . Appl. Phys. 1986, 59, 3627; 1989, 65, 4480. Tanaka, H. Phys. Rev. Lett. 1993, 70, 2770; Europhys. Lett. 1993, 24, 665. (18) Krichevsky, 0.; Stavans, J. Phys. Rev. Lett. 1993, 70, 1473. (19) Seul, M.; Morgan, N. Y.; Sire, C. Phys. Rev. Lett. 1994, 73, 2248. (20) Weaire, D.; Rivier, N. Contemp. Phys. 1984, 25, 59. Flyvbjerg, H.; Jeppesen, C. Phys. Scr. T 1991,38,49. Flyvbjerg, H. Phys. Rev. 1993, 47, 4037.
Droplet Domain Patterns in a Monomolecular Film
J. Phys. Chem., Vol. 99, No. 7, 1995 2095
(21) Sire, C.; Seul, M. J . Phys. I, in press. (37) Denoting by S,,,j,, = s,,j,(S) the lower cutoff of the distribution P(S/ (22) Seul, M. Europhys. Lett. 1994, 28, 557. (S)), we define an effective (mean) droplet radius, (R)eff (R) 0.5&,,. (23) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1992,96,6820. Given that (S) = c(R), we have S,i, = s,,c(R) and (R)eff = (1.0 (24) Hirshfeld, C. L.; Seul, M. J . Phys. (Paris) 1990, 51, 1537. 0.5sfinc)(R). With s,,,j,, = 0.4, c = 2.5, this implies, for circular droplets, (25) Seul, M. J . Phys. Chem. 1993, 97, 2941. = 26. (A),n = (1.0 + 0.5~,,,i,,c)~(A) and (26) Kapton tape of ‘/2 in width and 1 mm thickness, coated on both (38) By permitting air circulation through the interior compartment of sides with a ‘/2 mm thick layer of fluorinated ethylenepropylene (FEP), our Langmuir trough,25 slow monolayer flow could be induced. This was obtained from Fralock, Orange, CA. Lamination required heating to procedure was sometimes relied upon to visually inspect a greater portion 350 O F and exposure, for 40-60 s, to 20 psi of pressure (in a hydraulic of the film area. press kindly made available by D. Dahringer, AT&T Bell Laboratories). (39) Rogers, T. M.; Desai, R. C. Phys. Rev. B 1989, 39, 11956. (27) Seul, M.; Sammon, M. J. Phys. Rev. Left. 1990, 64, 1903. Seul, (40) In initial experiments we actually did observe upward drifts in the M. PhysicaA 1990, 168, 198. Seul, M.; Chen, V. S. Phys. Rev. Lett. 1993, surface pressure over periods of 12-24 h. These were eliminated by 70, 1658. adopting a laminating process to fabricate kapton barriers for our Langmuir (28) Seul, M.; Sammon, M. J.; Monar, L. Rev. Sci. Znstrum. 1991, 62, trough (see ref 26, in the Experimental Procedures) and were therefore 784. attributed to the slow infusion of an unknown surface-active impurity (29) Gonzalez, R. C.; Wintz, P. Digital Zmage Processing; Addisonassociated with the previously used silicone adhesive. Wesley: Reading, MA, 1992; Chapter 4. (41) Marko, J. Personal communication; the argument is cited in ref (30) Fowlkes, E. B. In A Folio ofDistributions; Fowlkes, E. B., Ed.; 19. Marcel Dekker: New York, 1987. (42) Seul, M.; Wolfe, R. Phys. Rev. Lett. 1992, 68, 2460; Phys. Rev. A (31) Becker, R. A.; Chambers, J. M.; Wilks, A. R. The New S Language; 1992, 46, 7519. Wadsworth & Brooks/Cole: Pacific Grove, CA, 1988. (32) The skewness is a dimensionless measure of the degree of (43) Singer, S. J. Phys. Rev. E 1993, 48, 2796. asymmetry of a distribution around its mean, (x). It is defined as the (44)Singer, S. J. Personal communication: as long as the droplet radius is kept fixed as coverage is allowed to vary, interdroplet interactions are normalized third moment, as follows: Sk(x1, ..., XN) = l / a z l [ ( n , found to essentially correspond to a - d ~ - repulsion. ~ ( x ) ) / u ] .As ~ usual, u denotes the standard deviation. A negative skewness (45) Singer, S. J. Personal communication: the hexagonal array of indicates an asymmetry such that the distribution falls off more sharply for interacting droplets in fact exhibits such a @-dependenceof the (equilibrium) large x than it rises for small x. droplet radius. See e.g.: Cape, J. A.; Lehman, G. W. J . Appl. Phys. 1971, (33) To test dynamic scaling, skewed distributions were in fact 42, 5732. Hurley, M. M.; Singer, S. J. J . Phys. Chem. 1992, 96, 1938. parametrized by appealing to a model line shape composed of a product of a Gaussian and an exponential with cutoff y = A exp(-(x - ( X ) ) ~ / ~ U ~ ) (46) Timasheva, A.; Singer, S. J. Unpublished. exp(-yx/(x,, - x ) ) ; this model involves four independent parameters, (n), (47) In spite of substantial Brownian motion of droplet domains u, y , andx,, (Akaiwa, N.; Voorhees, P. W. Phys. Rev. E 1994,49, 3860). immediately following the quench, domain coalescence is virtually absent. (34) As elaborated in ref 21, the plot actually involves averaging over We interpret this as a clear indication of the presence of repulsive successive values of A/(A) = x . Following such an averaging procedure, interactions between adjacent domains. Quantitative analysis of the we obtain an optimal linear fit of the form y Avd(Avp) = 1% - 0.5, dynamics may in fact lend itself to the extraction of a potential in a manner irrespective of the nominal coverage, @, in the range 0.12 5 @ 5 0.25. described for colloidal systems (see e.g.: Hurd, A. J.; Clark, N. A.; Mockler, (35) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171. R. C.; O’Sullivan, W. J. Phys. Rev. A 1982, 26, 2869. Martinelli-Kepler, McConnell, H. M.; Rice, P. A,; Benvegnu, D. J. J . Phys. Chem. 1990,94, G.; Fraden, S. Phys. Rev. Lett. 1994, 73, 356). 8965. (36) Magnasco, M. 0. Philos. Mag. B 1992, 65, 895. JP942055R
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