NANO LETTERS
Hydrophobically-Driven Self-Assembly: A Geometric Packing Analysis
2003 Vol. 3, No. 5 623-626
Stefan Tsonchev,* George C. Schatz, and Mark A. Ratner Department of Chemistry and Center for Nanofabrication and Molecular Self-Assembly, Northwestern UniVersity, EVanston, Illinois 60208 Received January 29, 2003; Revised Manuscript Received March 4, 2003
ABSTRACT We present a new approach to the problem of finding the minimum-energy structures resulting from the self-assembly of amphiphile nanoparticles possessing a hydrophobic “tail” and a hydrophilic “head”. When the repulsive interactions between the “heads” are of hard-sphere type, the approach is rigorous and is reduced to a simple geometric problem of finding the highest density structure allowed by the nanoparticle shape. Our results show that spherical micelles always have higher fractional density for cone or truncated cone nanoparticles. This does not always agree with previous, widely used, approximate methods which have served as guides in designing new nanoscale-structured materials.
Significant progress has been made in synthesizing a variety of new materials based on the so-called “bottom-up” approach, according to which the material is built into its desired shape and properties through self-assembly of its constituent elements-usually organic macromolecules-into supramolecular structures of nanoscale dimension.1,2 Such novel functional materials are expected to be of use in a number of fields, especially medicine and biotechnology, since biological tissues, for instance, are built in a similar way, and thus, these novel nanomaterials can be designed to imitate, and possibly improve, the tissues’ characteristics through their specific functionality and three-dimensional (3D) shape. To control the functionality and shape of the nanomaterial, it is necessary to understand the interactions between the constituent macromolecules and the physical variables governing the formation of the supramolecular assembly. These forces and variables, however, are still not well understood, and experimentalists usually rely on their experience and intuition, as well as on approximate estimates of the limiting shape of the supramolecular assembly, when designing a new material with certain desired characteristics. This work was inspired by recent advances by Stupp and co-workers3-5 in the synthesis of new nanomaterials designed to mimic bone nanostructure by self-assembly of peptide amphiphiles into cylindrical fibers. Subsequently, these cylindrical structures are able to direct the mineralization of hydroxyapatite in the same alignment as observed between collagen fibrils and hydroxyapatite crystals in bone. Stupp and co-workers anticipated the formation of the cylindrical fibers based on existing knowledge of amphiphile selfassembly, according to which an amphiphile of truncatedconical shape should assemble into a cylindrical micelle (see * Corresponding author. E-mail:
[email protected] 10.1021/nl0340531 CCC: $25.00 Published on Web 03/20/2003
© 2003 American Chemical Society
the table on p 381 in ref 6). The reason behind such expectations, as expressed in the above work and also in an earlier publication,7 is based on the value of a single parameter, the so-called “critical packing parameter” (CPP), V/a0l0, where V is the volume of the nanoparticle, a0 is the surface area of its hydrophilic “head,” and l0 is its limiting length. The CPP is expected to determine the shape of the aggregate into which a given 3D amphiphile will selfassemble. For instance, if CPP e 1/3 the nanoparticles should self-assemble into spherical micelles; if 1/3 < CPP e 1/2, as in the case of truncated cones, the particles should form cylinders, etc.6 This is thought to be so because if, say, a sphere, or a cylinder, is cut into identical 3D sections which perfectly fill the corresponding structure, the CPPs of these sections are 1/3 and 1/2, respectively. However, from this it does not follow that any 3D particles with a CPP equal to, e.g. 1/3, would necessarily form spheres as their most efficiently packed structures, since the inverse correspondence between the CPP and the 3D particle shape is not unique. Thus, this approach can be viewed as approximate; when the particles are not very different in shape from the ones which perfectly fill, say, a sphere, then their CPP would be close to 1/3 and they should self-assemble into spheres. For instance, in the case of conically shaped particles the CPP is exactly 1/3, and according to this approach they should form spherical micelles, even though they cannot perfectly fit into a sphere, which is indeed the case, as we will see later on. From the same table in ref 6 it also follows that if the particles have the form of truncated cones they should assemble into cylinders. This seems counterintuitive to us, based on the particles’ shape, and also symmetry, which is the same as for the cone-shaped particles. The approach of ref 6 does not provide a quantitative measure of its inherent approximations, and thus, it is difficult to estimate its limits of validity. It also relies on a number
of additional approximations when applied to a specific system of interest, as can be seen from the example in Section 17.3 in ref 6, where it is assumed that (a) the cones perfectly fill the sphere (which is an error of 10 percent or more, as will be shown later on), (b) the volume and length of the cone-shaped particle are approximated with the volume and length of the hydrophobic tail of the macromolecule, and (c) the number of particles in the micelle is taken from experiment as an input instead of being a consequence of the theory. To avoid such approximations, which are hard to control in general, we suggest a different approach to this problem, based on the same starting physical assumptions about the particle shape, but rigorous as far as the geometric packing considerations are concerned. We assume that the hydrophilic particle “heads” interact only through short-range excluded volume interactions (in which case their repulsion is determined by their size, as assumed in the previous method), and the hydrophobic “tails” are responsible for the attractive interactions leading to self-assembly. Thus, it is clear that the lowest energy structure would be the one with the highest fractional density of particles. (Here we do not include entropic effects; therefore, strictly speaking, this would be true at zero temperature. However, we will see that ignoring the entropic effects will not prevent a successful comparison of the relative stability of different possible structures from self-assembly.) This viewpoint allows us to transform the problem of finding the minimum-energy structure into a purely geometric form. For self-assembly of conically shaped nanoparticles, the problem can be simply stated as: what is the densest structure that can be formed by cones? There are several possible candidates, and we have found that the sphere and the cylinder, as shown on p 381 in ref 6, are best. Therefore, we will show the solution for these two cases only. We will assume that our nanoparticles are cones with a spherical base, with height R and 2D angle at the vertex, R. It is easy to show that the volume of such a cone is 4 R Vcone ) πR3sin2 3 4
()
(1)
The maximum fractional density of cones in the selfassembled micelle, Frac, is equal to the number N of cones in the micelle, times the volume of the cone, divided by the volume of the structure, Vmicelle: Frac )
N × Vcone Vmicelle
(2)
We wish to find the values of Frac for two packing geometries, the micellar sphere or cylinder. For a spherical micelle, the problem is: how many cones of height R can assemble most efficiently into a sphere of radius R, which is equivalent to asking how many spherical cone bases can be placed on the surface of the sphere. We assume that the cone bases form a hexagonal lattice on the surface; this assumption will hold for small angles R up to between π/6 and π/3, which is above any reasonable limit for using cones 624
Figure 1. Plot of three neighboring cones from a spherical assembly. The dihedral angle γ is shown in the interior of the micelle, with the lines A′B and A′C perpendicular to line OA.
as models of macromolecular shapes. Now, let us consider the spherical triangle connecting the centers of the bases of three neighboring cones on the spherical surface, as seen in Figure 1. Each conical base has six neighbors; thus, each such spherical triangle contains three-sixths of a base plus one extra space between the bases. Hence, the area of each spherical triangle is one-half of the area needed for each conical base; this is all we need to solve the problem. To find this area we will use Girard’s theorem,8 a special case of a more general theorem, stating that the area of a spherical polygon of n sides is equal to the sum of all angles between the tangents at each vertex, minus (n - 2)π, multiplied by the square of the radius of the sphere. Thus, the area of our spherical triangle is A ) R2(3γ - π), where γ is the angle between the tangents at any vertex of the equilateral spherical triangle ∆ABC in Figure 1. We recognize that γ is the dihedral angle between, say, the planes OAB and OAC, which is shown in the interior of the micelle in Figure 1 as the angle ∠BA′C. From the triangles ∆OA′B, ∆OA′C, and ∆OBC, we find γ ) arccos
( ) cosR R 2cos2 2
(3)
To get the maximum number of cones in the sphere, we divide the area of the sphere of radius R by twice the area of the spherical triangle, and take the largest integer number of this ratio-as we can have only integer number of cones inside the sphere-to obtain: Nsph ) Int
(3γ2π- π)
(4)
where γ is given by eq 3. Using this in eq 2 with the help of eq 1 and the formula for the volume of a sphere, we Nano Lett., Vol. 3, No. 5, 2003
Figure 3. Fractional density of the cones in a sphere and a cylinder, from eqs 5 and 6, as a function of the 2D cone angle R expressed in radians. Figure 2. Plot of part of the hexagonal lattice of cones assembled into a cylinder.
obtain the final formula for the fraction of the cones in a sphere: Fracsph ) sin2
R 2π Int 4 3γ - π
(
)
(5)
As expected, the fraction of cones depends only on their angle R. For a cylindrical micelle, we assume that the cones point to the axis of the cylinder, while the bases form a hexagonal lattice on the side surface of the cylinder, as shown in Figure 2. Groups of cones participate in 3D circular formations resembling “fans,” each formation perpendicular to the axis of the cylinder, as seen in Figure 2, where parts of three such “fans” are shown. We have a periodic structure with an elementary cell ABCD. Thus, we need to calculate the fractional density of cones in one elementary section of height AD that contains two “fans”. The number of cones in the elementary section is easy to find: it is just the integer of twice the total circular angle, 2π, divided by the cone angle R [Ncyl ) Int(4π/R)]. To find the final density of cones, we need to know also the volume associated with the elementary section of the cylinder, for which we need to find the elementary height AD. To find AD, let us first imagine the axis of the micellar cylinder, connecting all the conical vertexes, and the side surface of the cylinder, at a distance R from its axis. Each spherical conical base touches this side surface at an arc of radius R, passing through its center-part of such a line is the line AB of the polygon ABCD. We will take the perpendicular from the edge points of each spherical conical base to the axis of the cylinder, and then will extend this line until it intersects with the side surface of the cylinder. Thus, the resulting intersections would project the spherical conical bases onto the side surface of the cylinder. We can then unfold the side surface of the cylinder onto a plane in which the projections of the conical bases would become ellipses that will have a smaller Nano Lett., Vol. 3, No. 5, 2003
diameter equal to the diameter of the flat base of the cone and a larger diameter equal to the diagonal arc length of the spherical conical base. Then, if we consider the ellipses centered at the points A and O, based on symmetry considerations, from the equation of the ellipse for the common point of these two ellipses, we can derive the elementary length AD as equal to 2x3a, where a ) R sin(R/2) is the radius of the flat base of the cone. Thus, the volume of the elementary cylindrical section is the product of the crosssection of the cylinder times the length AD, that is, Vcyl ) 2x3πR3sin(R/2). Hence, using eqs 2 and 1, we obtain the fractional density of cones in a cylinder: Fraccyl )
R 4π 1 tan Int 4 R x 3 3
() ( )
(6)
Once again, the fractional density depends only on the cone angle R. Now we can compare the fractional density of cones for the two cases considered here by plotting it as a function of the angle R up to the limit of validity of our assumption about the hexagonal lattice of cone bases on the surface of the sphere. This comparison is shown in Figure 3. We see that for the range of angles shown, a spherical micelle would always be preferable to a cylindrical one. In addition, if we consider entropic effects, the sphere would still be preferable, as it consists of a smaller number of particles. From this plot it is also easy to estimate that the packing error, based on the assumption that the cones perfectly fit into a sphere, is about 10 percent or more, for the angles R which would be appropriate for most amphiphiles. If we were to consider truncated cones as our nanoparticles, we would be following the same reasoning, starting from eq 2, and it is clear that in both cases of a spherical and a cylindrical micelle, the number of truncated cones in the micelle and the volume of the micelle would both be calculated by the exact same procedure shown above, and would therefore be unchanged. The only quantity that would differ is the volume of the truncated cone: it would be a 625
fraction of the cone volume shown in eq 1. However, in both cases of a spherical or a cylindrical micelle it would be the same fraction, and therefore, the final result would be given by eqs 5 and 6, both multiplied by that fraction. Thus, in the case of nanoparticles of truncated-conical shape, the spherical micelle would again be preferable to the cylindrical one in all cases, in accord with our expectations expressed at the beginning. In summary, we have presented a new, straightforward, approach to the problem of geometric packing of amphiphile nanoparticles and have applied it to the case of particles of conical or truncated-conical shape. When the repulsion between the hydrophilic “heads” of the particles is of hardsphere type, the approach leads to a simple, exactly solvable geometric problem, whose solution shows that in both cases of particles of conical or truncated-conical shape, the selfassembly would result in spherical micelles. Usually, the hydrophilic “heads” of amphiphile macromolecules in aqueous solution interact through long-range electrostatic forces which are not included in this model. Thus, the model can be considered as providing the solution to the limiting case when the hydrophilic “heads” are either weakly charged or are in a solution of high ionic strength where the high concentration of salt ions screens the repulsion between them to a relatively short-range interaction, which can be ap-
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proximated by their size. This is not the case with the peptide amphiphiles that inspired this work,3-5 where strong electrostatic interactions should dominate the hydrophobic ones, leading to self-assembly into cylindrical micelles. We intend to address this in another publication. Acknowledgment. S.T. is thankful to H. Frauenrath and M. Sayar for their help with the technical preparation of the figures. We are grateful to the DoD/MURI program for support of this research, and to J. Hartgerink and S. Stupp for valuable discussions. References (1) Lehn, J. M. Supramolecular Chemistry: Concepts and PerspectiVes; VCH: Weinheim, Germany, 1995. (2) Atwood, J. L.; Lehn, J. M.; Davis, J. E. D.; MacNicol, D. D.; Vogtle, F. ComprehensiVe Supramolecular Chemistry; Pergamon: New York, 1996. (3) Hartgerink, J. D.; Beniash, E.; Stupp, S. I. Science 2001, 294, 1684. (4) Hartgerink, J. D.; Beniash, E.; Stupp, S. I. PNAS 2002, 99, 5133. (5) Hartgerink, J. D.; Niece, K. L.; Stupp, S. I., submitted. (6) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (7) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (8) Berger, M. Geometry; Springer-Verlag: Berlin, New York, 1987; vol. 2, chapter 18.
NL0340531
Nano Lett., Vol. 3, No. 5, 2003
