Download Hydrophobically-Driven Self-Assembly: A Geometric Packing Analysis

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-
626
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