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Highly Ordered Size-Dispersive Packings of Polydisperse Microgel Spheres Markus Antonietti,* Ju¨rgen Hartmann, Martin Neese, and Udo Seifert Max-Planck-Institute of Colloids and Interfaces, Research Campus Golm, D-14424 Potsdam, Germany Received April 1, 2000. In Final Form: June 22, 2000 Dilute suspensions of polydisperse polystyrene microgels in the size range of 5-50 nm in organic solvents were evaporated under controlled conditions, and the resulting two-dimensional packing structures were investigated. The particles form highly ordered structures, as revealed by electron microscopy. The ordering process is understood as a unique self-organization process, in which differences in the particle size and the polydispersity of the particles are a main driving force. The structures follow the rules of hyperbolic geometry and are named “Zenon”-packing, indicating that the two-dimensional structures without translational invariance can still be understood as a special form of a regular lattice. A systematic variation of the solvent quality and the surface tension of the suspension shows that the ordering essentially is based upon size-dispersive van der Waals attractions between the microgels.
1. Introduction Crystallization, ordering, and self-assembly of colloidal objects are key topics of modern material science. These phenomena are interesting from a fundamental point of view to analyze and understand structure formation of matter on the intermediate length scale by using colloidal models which are more easy to observe by standard techniques for structural or dynamic analysis. Examples for work on nanocrystalline superlattices in three dimensions is given by the work of Pusey1 (principles) or Weller2 (materials). More easy to analyze is the structure formation of nanoparticles in two dimensions, which was recently reviewed by Wang3 (layers made by the selfassembly technique); the interested reader is also pointed to the work of Fendler4 and Heath et al.5 (layers made by the Langmuir-Blodgett technique). In the case of monodisperse nanoparticles, colloidal crystallization in a cubic lattice occurs; the more complicated case of bimodal particle size distributions responds to the size ratio and results either in demixing or in rather well ordered binary alloys.6,7 The spontaneous ordering of bimodal mixtures in two dimensions toward regular patterns was just recently described.8 For polydisperse systems, it was generally believed that crystallization is suppressed and just disordered systems can be obtained.9 More recent work however had shown that this cannot be the complete truth. A so-called “Apollonian” packing was predicted and numerically “verified” for systems where a hierarchy of smaller particles fits into the interstitial sites of the larger objects.10,11 More complex structure patterns generated (1) Pusey, P. N.; Poon, W. C. K; Ilett, S. M.; Bartlett, P. J. Phys.: Condens. Matter 1994, 6, 29. (2) Weller, H. Curr. Opin. Colloid Interface Sci. 1998, 3, 194. (3) Wang, Z. L. Adv. Mater. 1998, 10, 13. (4) Fendler, J. H. Curr. Opin. Colloid Interface Sci. 1996, 1, 202. (5) Heath, J. R.; Knobler, C. M.; Leff, D. V. J. Phys. Chem. B 1997, 101, 189. (6) Bartlett, P.; Ottewill, R. H.; Pusey, P. N. Phys. Rev. Lett. 1992, 68, 3801. (7) Eldridge, M. D.; Madden, P. A.; Frenkel, D. Nature 1993, 365, 35. (8) Kiely, C. J.; Fink, J.; Brust, M.; Bethell, D.; Schiffrin, D. J. Nature 1998, 396, 444. (9) Pusey, P. N. J. Phys. (Paris) 1987, 48, 709. (10) Anishchik, S. V.; Medvedev, N. N. Phys. Rev. Lett. 1995, 75, 4314.
from polydisperse nanoparticles were described by Ohara et al.12 and Meldrum and Fendler.13 In these cases, the particles order in a size-specific way, where the transition between the regions with different size is not very sharp and defect poor. A similar effect was indirectly observed during examination of densely packed polydisperse polymer spheres where changes in optical density could be produced with intense photon illumination.14 The purpose of the present paper is the analysis of the latter order phenomenon in colloidal model fluids with controlled polydispersity. It will be shown that the previously described effects are due to a special form of ordering or self-assembly where polydispersity is an essential element by focusing the structural analysis on electron microscopy of two-dimensional arrays of such particles obtained by a controlled evaporation process. As model particles we select polystyrene microgels, a recently developed model system which turned out to be the ideal candidate for a variety of physicochemical model examinations.14-16 These microgels can be made in the size range between 5 and 250 nm, are uncharged, dissolve in organic solvents, and interact purely via van der Waals forces. Illustratively spoken: they can be described as soft, rubbery spheres with weak interactions. Due to analytical reasons, the order phenomena found are characterized just in the two-dimensional plane, but are expected to occur in three-dimensional systems too. 2. Experimental Section Highly cross-linked polystyrene microgels with spherical morphology are synthesized by polymerization in microemulsions, where size and cross-linking density can be easily varied. The (11) Brilliantov, N. V.; Adrienko, Y. A.; Krapivsky, P. L.; Kurths, J. Phys. Rev. Lett. 1996, 76, 4058. (12) Ohara, P. C.; Leff, D. V.; Heath, J. R.; Gelbart, W. M. Phys. Rev. Lett. 1995, 75, 3466. (13) Meldrum, F. C.; Fendler, J. H. In Biomimetic Materials Chemistry; Mann, S., Ed.; VCH: Weinheim, 1996. (14) Bartsch, E.; Antonietti, M.; Schupp, W.; Sillescu, H. J. Chem. Phys. 1992, 97, 3950. (15) Antonietti, M.; Pakula, T.; Bremser, W. Macromolecules 1995, 28, 4227. (16) Antonietti, M.; Briel, A.; Fo¨rster, S. J. Chem. Phys. 1995, 105, 7795.
10.1021/la000508f CCC: $19.00 © 2000 American Chemical Society Published on Web 08/31/2000
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Table 1. Samples Used in This Examinationa sample
RH (latex) (nm)
std dev (%)
Mw (LS)
M13 M16 M20 M27
13.2 16.2 20.4 27.2
33.0 31.0 26.0 23.0
2.65 × 106 5.82 × 106 1.35 × 107 3.66 × 107
a The hydrodynamic radius R H and the polydispersity as expressed as the width of a gaussian distribution are determined from the latex prior to purification and reswelling in organic solvents, where aggregation phenomena occur. The molecular weight of the microgels was determined by static light scattering in THF by extrapolating the form factor of the single microgel.
Figure 2. Higher relution TEM picture of single Zenon clusters: (a) pure sample M27; (b) blend of M27, M20, M16, and M13.
Figure 1. Overview of the dried microgel patterns as revealed by TEM: (a) pure sample M27; (b) blend of M27, M20, M16, and M13.
Table 1 summarizes the properties of the microgel samples used in this examination. Structure formation or ordering was induced by dissolving the microgels in organic solvents, such as THF, toluene, or DMF, and placing those solutions on a carbon-coated copper grid for electron microscopy. To control the evaporation rate of the solvent and to increase the available time for structure formation, the grid was placed in a pure solvent atmosphere. With exchange of the solvent atmosphere continuously with air, the evaporation of the horizontally placed droplet can be easily controlled. After the visible disappearance of the droplet, the grid is dried under vacuum to remove any residual solvent. The electron micrographs were taken with a EM 912 OMEGA transmission electron microscope (Zeiss). The samples show sufficient contrast to be depicted without staining.
synthesis of the samples as well as purification and isolation was performed according to the literature.17,18 The microgels used for the present examinations vary between 5 and 50 nm in radius, with a cross-linking density of one cross-linker (divinylbenzene) per 10 linear (styrene) units. The particle size distribution of such single samples, based on electron microscopy, is usually well described with a Gaussian distribution and a standard deviation of 30%. For some of the experiments microgels were blended to ensure a bimodal distribution or even wider particle size distributions with a controlled shape. After synthesis and purification, the microgels are pure apolar polymers and can be viewed as mesoscopic subunits of a statistical network with a spherical shape. The density profile of these spherical particles is nearly homogeneous with a sharp decrease at the surface, the thickness of which corresponds to half the mesh size.19
Although microgels show some minor tendency toward aggregation already in solution,19,20 structure formation essentially takes place during evaporation of the solvent and the resulting high microgel concentrations. In addition, there is a solvent selectivity which is discussed below. Figure 1 depicts two typical structures of the microgel clusters found after controlled evaporation of THF as a solvent (here 30 min) from sample M27 (a) and a blend with controlled polydispersity of the samples M27, M20, M16, and M13.
(17) Antonietti, M.; Basten, R.; Lohmann, S. Makromol. Chem. Phys. 1995, 196, 441. (18) Antonietti, M.; Hentze, H.-P. Adv. Mater. 1996, 8, 840.
(19) Neese, M. Doctoral Thesis, Potsdam, 1998. (20) Davankov, V. A.; Ilyin, M. M.; Tsyurupa, M. P.; Timofeeva, G. I.; Dubrovina, L. V. Macromolecules 1996, 29, 8398-8403.
3. Results
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Figure 3. Schematic representations illustrating the structure of the cluster shown in Figure 2b: (a) sorting by size; (b) lattice plot; (c) schematic lattice lines; (d) the Coxeter diagram as an ordered hyperbolic model structure.
The samples are composed of larger aggregates, in coexistence with some isolated microgels, which obviously did not find their way to condense into the clusters. It is found that the size of the aggregates is dependent on the rate of evaporation and the polydispersity of the samples: the smaller the polydispersity, the larger and the more anisometric the aggregates. Already in these low-magnification overview pictures, it is seen that the aggregates are composed in a typical way where the large microgels are on the inside while smaller ones are located on the surface. We name these structures, analyzed in more detail below, “Zenon” clusters, following the famous fable of Achilles and the turtle that introduced the concept of hyperbolic borderlines into philosophy and science. The structure of the single aggregate is better judged in a higher magnification, shown in Figures 2. Both clusters contain on the order of 1000 particles and have an overall size in the micrometer range. In both cases, the single microgels show spherical shapes with defined borders to neighboring particles. The speciality of the twodimensional aggregates is the size-selective ordering of the structure. Starting in the center, one finds the biggest microgels with R ) 26 nm, while to the outside the particles decrease continuously to smaller sizes. The size decrease can be more easily seen in Figure 3a, where an outline plot marks the different sizes with varying gray tones. In this highly ordered aggregate one can identify about 400 microgels in a characteristic radial
position due to their distance from the center. The smallest distinguishable microgels of a size of about 6-7 nm are found at the rim, followed by some other, nonresolved species, which can be even smaller spheres, but also impurities within the sample. The found sizes and the broad polydispersity in this single cluster arise from the probed sample, which is a mixture of four different-sized microgel samples with overlapping particle size distributions with mean sizes of 10, 12.5, 17, and 25 nm. Looking at the local packings in the aggregate, one finds mainly quenched hexagonal arrangements (lattice plot in Figure 3b), with some defects where microgels only have five neighbors. Due to this good preservation of the coordination number 6, the aggregate is essentially ordered, although the layer lines are bent. One tends to describe this long-range order as a lattice, but because of the varying size of the constituting elements, there is no translational invariance. The microgels in Figure 3c are marked to underline some representative lattice lines, which show the typical bending that is due to the difference in particle size. In Figure 3c, it is also seen that the system follows a hexagonal coordination as much as possible; i.e., one observes a fan-shaped substructure where the whole cluster is composed of three fans. It is also seen that the center of this specific cluster is formed by the seven biggest microgels, which presumably represent the nucleation center of this cluster. This pattern is comparable to a hexagonal lattice in a hyperbolic space, where the degree
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of hyperbolic distortion is directly coupled to polydispersity. An idealized example for such a hyperbolic geometry, the well-known Coxeter diagram, is plotted in Figure 3d to illustrate the similarity between these finite hyperbolic lattices. As the size decrease in the Coxeter diagram would correspond to one very special particle size distribution (which is practically impossible to adjust) packing defects have to be expected. As compared to standard crystallization (where all particles are exchangeable), this process relies on a far more delicate ordering phenomena and the coupled entropy loss. The cluster analyzed in Figure 3 represents one selected species, but as already seen in the overview micrograph (Figure 1b), all clusters of a specific sample with its specific polydispersity are quite similar and are characterized by the typical geometric rules delineated above. In principle, it is obvious that the hyperbolicity and the coupled size of the clusters depends on polydispersity: the higher the polydispersity, the smaller the clusters. The structure of a typical aggregate of an “unblended” microgel fraction with a Gaussian polydispersity of about 15% was already shown in Figure 2a. Here the aggregates grow larger, but usually also possess an anisometric shape. Also for these structures, we observe the typical bent layer lines with the small particles in the outer zones of the cluster, but preferentially in the caps of the structures. This deviation from spherical shape might be due the kinetics which becomes increasingly more demanding for the larger structures. Another explanation is that the shape essentially follows the envelope of the particle size distribution: the area covered with smaller particles increases with the radius of the cluster as R2; i.e., for Gaussian distributions there are not enough small particles available to cover the rim of a spherical object. This is why we blended different microgel fractions and tried to realize distinct envelopes of the particle size distribution, and the close-to-spherical structures shown in Figures 1b and 2b have been already the product of such an optimization procedure. Increasing the polydispersity above a critical value results in a new type of cluster structure, as shown in Figure 4. In is clearly seen that in addition to the hyperbolic packing, smaller particles occupy the interstitial sites of the larger lattice, which is the so-called Apollonian packing.10,11 This is a well-known effect that is always found for a mutually attracting system and just corresponds to the “principle of maximum density”. Besides its beauty, one can learn for the present context that the effective particle potential is attractive and that the existence of the Zenon structures is limited toward larger polydispersities by the onset of the Apollonian packing. We also performed experiments with different evaporation rates and different microgel concentrations. For the rate experiments, it is a trivial outcome that the slower the evaporation, the more ordered the structures. The structure formation obviously relies on reorganization processes which take place on the time scale of minutes; faster evaporation leads to essentially disordered arrangements. Higher concentrations of microgels in the evaporating solvent lead to larger structures (note that the surface coverage is directly proportional to concentration, which start to get more than one center with large particles; later, the Zenon structure is transformed in large scale size fluctuations where big and small particles “demix” in separated regions. In this case, the characteristic length of the fluctuation again depends on polydispersity and is on the micrometer scale, i.e., 20-100 particle diameters.
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Figure 4. Simultaneous occurrence of the Zenon and the Apollonian packing: two different blends of M13 and M27.
Due to technical details, the micrographs in this region are however not good enough to enable a quantitative evaluation. We also tried to generalize this Zenon packing to other polymer colloids. Since it is known that sterically stabilized latex particles possess a shallow secondary minimum in the interaction potential,21 which is also size specific, we treated a sample of polystyrene latex stabilized with a nonionic stabilizer (Lutensol AT25), which was made by microemulsion polymerization (experimental details described in22) in a similar fashion, that is, we diluted the latex to an intermediate concentration and evaporated the water as slow as possible. As shown in Figure 5, nicely organized Zenon clusters are obtained in this case, too. For demonstration purposes, we depict a cluster with two centers, that is the beginning of a size fluctuation. 4. Discussion Searching for an explanation of this novel structure formation, we have to discuss both bulk interactions between the microgel spheres and surface tension effects due to the evaporating solvent as well as quasi-equilibrium and kinetic effects. As an experimental handle, we have varied the solvent and checked for the occurrence of ordered structures. The results of these experiments are summarized in Table 1. (21) Evans, D. F.; Wennerstrom, H. The colloidal domain; VCH: Weinheim, 1994. (22) Antonietti, M.; Bremser, W.; Schmidt, M. Macromolecules 1990, 23, 3796.
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Table 2. Viscosity, η, Surface Tension, γ, and Refractive Index, n, of the Solvents Used in This Study (T ) 20 °C)a THF DMF toluene ethylnaphthaline
η, mPa s
bp, °C
1.560
65 153 111 251
0.588 2.020
γ, mN/m
n
A × 10-20 J
Zenon crystals
29.3 32.3
1.407 1.430 1.497 1.599
4.8 3.7 1.3 0.2
very good good none none
a The Hamaker constant A is calculated according to the scheme given in: Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991.
Figure 5. Zenon structure as formed by a different chemical system: a polydisperse polymer latex, sterically stabilized by a nonionic surfactant.
Concerning bulk effects, the van der Waals interaction V(D) between two spherical particles of radius RA and RB separated at the surfaces by a distance D reads:23
V(D) )
[
2RARB A + 2 6 D + 2D(R + R ) A B 2RARB D2 + 2D(RA + RB) + 4RARB ln
(
+
D2 + 2D(RA + RB)
)]
D2 + 2D(RA + RB) + 4RARB
(1)
The scale of this interaction is set by the Hamaker constant A tabulated for different solvents in Table 2. In Figure 6a, we plot V(D) for D ) 2 nm as a function of R ) RA ) RB. In Figure 6b, we show V(D) for D ) 2 nm as a function of R ) RB for RA ) 10 nm fixed. In each case, we present data for the three solvents THF, DMF, and toluene; the forth solvent, ethylnaphthalene, was chosen to be isorefractive, which according to Lifshitz theory should result in practically no van der Waals attraction at all. The results show that for toluene the interaction never exceeds kBT, whereas for DMF and THF, larger particles experience an attraction larger than kBT. As shown in Table 1, we find Zenon packing only in the latter cases. This comparison between experiment and calculation suggests that Zenon packing requires a bulk interaction between the particles of at least kBT. (23) Hiemenz, P. C. Principles of colloid and surface chemistry, 2nd ed.; Marcel Dekker: New York, 1986.
Figure 6. van der Waals interaction (1) at a separation D ) 2 nm between microgel spheres. (a) The potential V as a function of the radius R for equal size spheres in different solvents. (b) The potential V as a function of the radius RB of one sphere for a fixed size RA ) 10 nm of the other sphere. The different symbols describe different solvents: (9) THF; (2) DMF; (0) toluene.
The larger the particles, the stronger their interaction. From an equilibrium point of view, this effect supports the formation of clusters in which on average the larger particles form the center whereas the smaller ones are at the edge. Since V(D) is only a few kBT, particles in equilibrium do not stick to each other permanently. The larger the concentration of microgel spheres, the more probable are bound configurations. As more and more solvent evaporates, the concentration increases and aggregation clusters become more likely. From a theoretical perspective, another driving mechanism for the Zenon packing could be due to surface tension of the evaporating droplet. As Figure 7 shows, if the receding droplet leaves a microgel sphere behind it, the total surface area will increases. This would amount to an increase in surface energy of about 4πR2γ ≈ 104kBT using R ) 10 nm and a surface tension of γ ≈ 40 mJ/m2 as appropriate for toluene and THF.24 Such a large energy is unlikely provided thermally as long as other arrange-
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Figure 7. Sketch of a receding droplet (a) and (c) dragging along a microgel sphere. The hypothetical intermediate state (b) has an energetically costly larger total surface area than state (c).
ments are possible. Hence the receding droplet drags along the enclosed microgel spheres thus avoiding an extension of surface area. Note that the same energy consideration favors putting the large particles into the middle and the small ones at the rim of the droplet, since the droplet has its maximum height in the middle. The latter argument could suggest that surface tension is the main driving force for Zenon packing. However, our results on the dependence of the type of solvent show that this is not the case. The absence of Zenon ordering in toluene which has roughly the same surface tension as THF rules out surface tension as the single dominant mechanism. In principle, one could also imagine depletion forces to provide a mechanism for the ordering (due to the presence of very small particles). However, the polydispersity in our samples seems not to be strong enough to contribute substantially via such an effect. Moreover, our results on the strong influence of the type of solvent exclude this possibility. Clearly, the picture just presented, which relies on energy minimization and, thus, on equilibrium properties, demands that there is enough time for the reorganization of configurations. This requirement puts a limit on the evaporation rate. A typical time needed to diffuse a distance of the size of a final cluster (ca. 100 nm) is 1 ms using the diffusion coefficient in dilute solution. Even though this crude estimate neglects the fact that with increasing evaporation the diffusion becomes slower, it indicates that at our drying rate of about 1 nm/ms (or 1 mm/1000 s) there should be enough time for rearrangements in order to reach the minimum energy configuration described above. 5. Conclusion We have shown that polydisperse colloidal particles with weak interactions under appropriate conditions can form finite, regular lattices which are characterized by a hyperbolic architecture. Larger particles are located on the inside where the small ones are on the outside of a cluster. Due to the preservation of a rather uniform coordination number, the local arrangement is not liquidlike, and curved layer lines can be observed. On the other hand, the structures do not show translational invariance; i.e., they are “scattering-amorphous” and not crystalline per definition, but well ordered. The resulting “Zenon structures” are, to the best of our knowledge, a form of self-organized structure which is not classically (24) Riddick, J. A.; et al. Organic Solvents: Physical Properties and Methods of Purification, 4th ed.; Wiley-Interscience: New York, 1986.
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treated and hard to file between the classical differentiations “crystalline” or “liquid-crystalline”. The underlining principle “sorting-by-size” was already described for polydisperse gold and silver colloids,12,13 but for the generation of larger and more marked structures described here, careful adjustment of intermolecular forces (by solvent variation) and sufficiently slow formation rates have to be applied. In the slow evaporation limit, the overall size of these structures is related to the hyperbolic bending or the particle polydispersity: polydisperse particles form rather small aggregates whereas species with a narrower particle size distribution form larger and more extended clusters. The larger clusters also adopt an irregular outer shape, which is either caused by kinetic effects or by the particle size distribution and the availability of appropriate particles for every site. In the limiting case of near monodisperse particles, classical colloidal crystals are expected, which can possess macroscopic dimensions. Exceeding the polydispersity above a critical value results in an overlay of the Zenon packing with an Apollonian packing: in addition to the hyperbolic arrangement of the particles, small particles are built in the interstitial sites of the larger particles. This shows thatsanalogous to classical crystallizationsmaximization of the mass density is a main driving force, and the ordering is driven by size-selective van der Waals attraction. This interpretation is supported by experiments where the condensation rate and the Hamaker attraction (via refractive index of the solvent) were varied: the most ordered structures were obtained in the case of slow evaporation and sufficiently strong particle-particle interactions, while surface tension only has played a minor role. It is obviously very important that the mutual interactions are of the order of kBT and, in the intermediate range, the particles have to explore the whole configurational space and have to react also to minor differences of local packing energies. On the other hand, the attractions have to be strong enough to compensate the entropy loss coupled with this rather demanding type of ordering. As a consequence, this type of structure is only found within a certain range of particle sizes and interactions. It is an open question if such structures also exist in the three-dimensional bulk state, although first observations such as cluster formation in dilute solution19,20 or an anomalous high scattering power at low scattering angles in concentrated systems14 already hint at the universality of the phenomenon. In the very end, it is stimulating to state that the described observations, along with similar experiments, break the paradigm that structure formation relies on uniformity. In energetically balanced cases, it is more that the polydispersity adds new effects and additional length scales; i.e., finite, “multicellular” objects are obtained. It is a vision that such effects can also be used for a materials aspect where small and large particles carry different properties, e.g., conductivity or magnetism. However, these speculations have to await experiments which look for similar structures in three dimensions. LA000508F
