Efficiency of various lattices from hard ball to soft ball: theoretical study of thermodynamic properties of dendrimer liquid crystal from atomistic simulation.

Abstract

Self-assembled supramolecular organic liquid crystal structures at nanoscale have potential applications in molecular electronics, photonics, and porous nanomaterials. Most of these structures are formed by aggregation of soft spherical supramolecules, which have soft coronas and overlap each other in the packing process. Our main focus here is to study the possible packing mechanisms via molecular dynamics simulations at the atomistic level. We consider the relative stability of various lattices packed by the soft dendrimer balls, first synthesized and characterized by Percec et al. (J. Am. Chem. Soc. 1997, 119, 1539) with different packing methods. The dendrons, which form the soft dendrimer balls, have the character of a hard aromatic region from the point of the cone to the edge with C(12) alkane "hair". After the dendrons pack into a sphere, the core of the sphere has the hard aromatic groups, while the surface is covered with the C(12) alkane "hair". In our studies, we propose three ways to organize the hair on the balls, Smooth/Valentino balls, Sticky/Einstein balls, and Asymmetric/Punk balls, which lead to three different packing mechanisms, Slippery, Sticky, and Anisotropic, respectively. We carry out a series of molecular dynamics (MD) studies on three plausible crystal structures (A15, FCC, and BCC) as a function of density and analyze the MD based on the vibrational density of state (DoS) method to extract the enthalpy, entropy, and free energies of these systems. We find that anisotropic packed A15 is favored over FCC, BCC lattices. Our predicted X-ray intensities of the best structures are in excellent agreement with experiment. "Anisotropic ball packing" proposed here plays an intermediate role between the enthalpy-favored "disk packing" and entropy-favored "isotropic ball packing", which explains the phase transitions at different temperatures. Free energies of various lattices at different densities are essentially the same, indicating that the preferred lattice is not determined during the packing process. Both enthalpy and entropy decrease as the density increases. Free energy change with volume shows two stable phases: the condensed phase and the isolated micelle phase. The interactions between the soft dendrimer balls are found to be lattice dependent when described by a two-body potential because the soft ball self-adjusts its shape and interaction in different lattices. The shape of the free energy potential is similar to that of the "square shoulder potential". A model explaining the packing efficiency of ideal soft balls in various lattices is proposed in terms of geometrical consideration.

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Efficiency of Various Lattices from Hard Ball to Soft Ball:
Theoretical Study of Thermodynamic Properties of Dendrimer
Liquid Crystal from Atomistic Simulation
Youyong Li, Shiang-Tai Lin, and William A. Goddard III*
Contribution from the Materials and Process Simulation Center (Mail code 139-74),
DiVision of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, California 91125
Received September 19, 2003; E-mail: wag@wag.caltech.edu
Abstract: Self-assembled supramolecular organic liquid crystal structures at nanoscale have potential
applications in molecular electronics, photonics, and porous nanomaterials. Most of these structures are
formed by aggregation of soft spherical supramolecules, which have soft coronas and overlap each other
in the packing process. Our main focus here is to study the possible packing mechanisms via molecular
dynamics simulations at the atomistic level. We consider the relative stability of various lattices packed by
the soft dendrimer balls, first synthesized and characterized by Percec et al. (J. Am. Chem. Soc. 1997,
119, 1539) with different packing methods. The dendrons, which form the soft dendrimer balls, have the
character of a hard aromatic region from the point of the cone to the edge with C12alkane “hair”. After the
dendrons pack into a sphere, the core of the sphere has the hard aromatic groups, while the surface is
covered with the C12alkane “hair”. In our studies, we propose three ways to organize the hair on the balls,
Smooth/Valentino balls, Sticky/Einstein balls, and Asymmetric/Punk balls, which lead to three different
packing mechanisms, Slippery, Sticky, and Anisotropic, respectively. We carry out a series of molecular
dynamics (MD) studies on three plausible crystal structures (A15, FCC, and BCC) as a function of density
and analyze the MD based on the vibrational density of state (DoS) method to extract the enthalpy, entropy,
and free energies of these systems. We find that anisotropic packed A15 is favored over FCC, BCC lattices.
Our predicted X-ray intensities of the best structures are in excellent agreement with experiment. “Anisotropic
ball packing” proposed here plays an intermediate role between the enthalpy-favored “disk packing” and
entropy-favored “isotropic ball packing”, which explains the phase transitions at different temperatures.
Free energies of various lattices at different densities are essentially the same, indicating that the preferred
lattice is not determined during the packing process. Both enthalpy and entropy decrease as the density
increases. Free energy change with volume shows two stable phases: the condensed phase and the
isolated micelle phase. The interactions between the soft dendrimer balls are found to be lattice dependent
when described by a two-body potential because the soft ball self-adjusts its shape and interaction in different
lattices. The shape of the free energy potential is similar to that of the “square shoulder potential”. A model
explaining the packing efficiency of ideal soft balls in various lattices is proposed in terms of geometrical
consideration.
1. Introduction
The liquid crystal phase formed by self-assembled supramol-
ecules with 3D nanoscale periodicity has been researched
extensively1-13and has potential applications in molecular
electronics,2photonics,5and porous nanomaterials.6In particular,
Percec and co-workers have advanced a rational design and
synthesized monodendrons that self-assemble through various
molecular recognition mechanisms into rodlike,14cylindrical,15
and spherical16supramolecular dendrimers, which self-organize
into column lattice15or cubic lattice.1,4,16,17Wedge-shaped
(1) Ungar, G.; Liu, Y.; Zeng, X.; Percec, V.; Cho, W. D. Science 2003, 299,
1208.
(2) Percec, V.; Glodde, M.; Bera, T. K.; Miura, Y.; Shiyanovskaya, I.; Singer,
K. D.; Balagurusamy, V. S. K.; Heiney, P. A.; Schnell, I.; Rapp, A.; Spiess,
H. W.; Hudson, S. D.; Duan, H. Nature 2002, 419, 384.
(3) Percec, V.; Ahn, C.-H.; Ungar, G.; Yeardley, D. J. P.; Moller, M.; Sheiko,
S. S. Nature 1998, 391, 161.
(4) Hudson, S. D.; Jung, H.-T.; Percec, V.; Cho, W.-D.; Johansson, G.; Ungar,
G.; Balagurusamy, V. S. K. Science 1997, 278, 449.
(5) Lopes, W. A.; Jaeger, H. M. Nature 2001, 414, 735.
(6) Attard, G. S.; Goltner, C. G.; Corker, J. M.; Henke, S.; Templer, R. H.
Angew. Chem., Int. Ed. Engl. 1997, 36, 1315.
(7) Jenekhe, S. A.; Chen, X. L. Science 1999, 283, 372.
(8) Zubarev, E. R.; Pralle, M. U.; Li, L.; Stupp, S. I. Science 1999, 283, 523.
(9) Stupp, S. I.; Braun, P. V. Science 1997, 277, 1242.
(10) Stupp, S. I.; LeBonheur, V.; Walker, K.; Li, L. S.; Huggins, K. E.; Keser,
M.; Amstutz, A. Science 1997, 276, 384.
(11) Orr, G. W.; Barbour, L. J.; Atwood, J. L. Science 1999, 285, 1049.
(12) Harada, A.; Kataoka, K. Science 1999, 283, 65.
(13) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997, 277, 1225.
(14) Percec, V.; Chu, P.; Ungar, G.; Zhou, J. J. Am. Chem. Soc. 1995, 117,
11441.
(15) Percec, V.; Johansson, G.; Ungar, G.; Zhou, J. J. Am. Chem. Soc. 1996,
118, 9855.
(16) Balagurusamy, V. S. K.; Ungar, G.; Percec, V.; Johansson, G. J. Am. Chem.
Soc. 1997, 119, 1539.
(17) Yeardley, D. J. P.; Ungar, G.; Percec, V.; Holerca, M. N.; Johansson, G.
J. Am. Chem. Soc. 2000, 122, 1684.
Published on Web 01/24/2004
1872 9 J. AM. CHEM. SOC. 2004, 126, 1872-1885
10.1021/ja038617e CCC: $27.50 © 2004 American Chemical Society

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dendrons such as I, II, and III depicted in Scheme 1 have so
far been found to form either columnar or cubic phases. In the
former, dendrons assemble flat pizza-like slices into disks, which
then stack into columns, which eventually pack to form a
hexagonal array.15Dendrons with more alkyl chains are cone-
shaped and assemble into supramolecular spheres. So far, these
spherical aggregates have been known to pack on three types
of lattices, Cub Pm3 hn,16Cub Im3 hm,17and Tet P42/mnm.1The
preferred formation of the Pm3 hn lattice in thermotropic sphe-
rodic cubic mesophases is a fact which surprises, especially as
related ordered assemblies of block copolymers form the BCC
(Im3 hm) lattice. In addition, the Pm3 hn lattice was not only found
for dendritic molecules, it was also reported for the thermotropic
mesophases of amphiphilic molecules,18,19star-shaped mol-
ecules,20and amphiphiles with perfluorinated chains.21
One of the most important principles for packing hard spheres
into periodic lattice is minimization of the interstitial volume
(Figure 1), which leads to the face-centered cubic (cubic close-
packed) structure and hexagonal close-packed structure. The
packing of hard spheres (billiard balls and noble gases) into
infinite two-dimensional arrays leads to close packing with each
ball having six equally spaced neighbors in a plane with the
planes packed into periodic arrays in three dimensions that lead
to 12 nearest neighbors for each ball. Stacking the close-packed
layers as ABCABC leads to the face-centered cubic (cubic close-
packed, denoted FCC) structure, while ABABAB stacking of
close-packed layers leads to the hexagonal close-packed struc-
ture (denoted HCP). Indeed, the stable crystal structure for all
noble gases and the favored structures for most metals is FCC,
HCP, or DHCP (hexagonal with ABACABAC packing).
Another popular structure with metals is body-centered cubic
(BCC) in which atoms are at the corners and center of a cube,
leading to eight nearest neighbors. In addition, symmetric mole-
cules such as CH4and C60fullerene crystallize into structures
that are slightly distorted FCC at low temperature and fully FCC
at higher temperature. Furthermore, it has been shown that the
face-centered cubic packing maximizes the total entropy.22
Obviously, the simple principles that explain the structure
approproiate for hard balls such as noble gas atoms, CH4, and
C60 do not apply to soft spheres, which, instead of a simple
ball-ball surface contact, have flexible hair that can overlap
each other to achieve more complex packing as in Figure 2.
Almost all self-assembled supramolecular aggregates are soft
spheres and can be approximated by hard spherical cores and
soft aliphatic coronas, as shown in Figure 2.
What would be the principle for packing soft balls together?
Here, we propose three packing mechanisms of soft balls and
use classical atomistic molecular dynamic simulations to
investigate the efficiency of various lattices for soft balls from
determining their thermodynamic properties. We focus on the
three compounds I, II, and III (Scheme 1), which have been
reported to form spheres with a nearly integer number of
dendrons.16In section 2, we describe the details of our
calculations. Section 3 reports the results and the discussion.
The summary is presented in section 4.
(18) Borisch, K.; Diele, S.; Goring, P.; Tschierske, C. Chem. Commun. 1996,
2, 237.
(19) Borisch, K.; Diele, S.; Goring, P.; Muller, H.; Tschierske, C. Liq. Cryst.
1997, 22, 427.
(20) Cheng, X. H.; Diele, S.; Tschierske, C. Angew. Chem., Int. Ed. 2000, 39,
592.
(21) Cheng, X. H.; Das, M. K.; Diele, S.; Tschierske, C. Langmuir 2002, 18,
6521.
(22) Mau, S.-C.; Huse, D. A. Phys. ReV. E 1999, 59, 4396.
Scheme 1. R ) C12H25; I, G2 (12G2-AG); II, G3 (12G3-AG); III, G4 (12G4-AG)a
aWe denote these as G2, G3, and G4 through the text. Illustrations of II and III are modified from reference 16. Reference 16 denotes them as 12G2-AG,
12G3-AG, and 12G4-AG, respectively, which are 12G2-AG, 3,4,5-tris[3′,4′,5′-tris(n-dodecan-1-yloxy)benzyloxy)]benzoic acid; 12G3-AG, 3,4,5-tris{3′,4′,5′-
tris[3′′,4′′,5′′-tris(n-dodecan-1-yloxybenzyloxy)benzyloxy]benzyloxy}benzoic acid; and 12G4-AG, 3,4,5-tris(3′,4′,5′-tris{3′′,4′′,5′′-tris[3′′′,4′′′,5′′′-tris(n-dodecan-
1-yloxybenzyloxy)benzyloxy]benzyloxy}benzyloxy)benzoic acid.
Figure 1. Close-packed hard balls.
Efficiency of Various LatticesA R T I C L E S
J. AM. CHEM. SOC. 9 VOL. 126, NO. 6, 2004 1873

Page 3

2. Methods
2.1. Force Field. The methylene (CH2) and methyl (CH3) groups in
C12H25were treated as united atoms C_32 and C_33; that is, each CH2
or CH3unit was treated as a single neutral pseudoatom. The van der
Waals interaction for this coarse-grained polyethylene model is taken
from the SKS (Siepmann-Karanorni-Smit) force field,23-25which was
developed to describe the thermodynamic properties of n-alkanes. The
bond-stretching force constant, which is not presented in the original
SKS force field, is taken from the AMBER force field.26,27The torsion
potential of the SKS force field had been taken from the OPLS
(optimized potentials for liquid simulation) force field of Jorgensen.28
The parameters for all other atoms including hybrid terms with C_32,
C_33 are taken from the generic Dreiding force field.29There is no
coulomb term in the setup energy of the simulation.
The force field for the coarse-grained polyethylene model uses
valence terms of the form as shown in eq 1, with the parameters
summarized in Table 1.
2.2. The Vibrational Density of States (DoS) from the Velocity
Autocorrelation (VAC) Function. To obtain the vibrational density
of states (DoS) S(ν) as a function of the frequency ν for a given density
and temperature, we start with the mass weighted velocity autocorre-
lation (VAC) function
where miis the mass of atom i, and ciR(t) is the R component (R ) x,
y, and z) of the velocity autocorrelation of atom i
Taking the Fourier transform of C(t) then leads to the vibrational density
of states (DoS)
Integrating S(ν) gives the total degrees of freedom of the system, that
is,
Generally, MD simulations for a condensed system remove the center
of mass translations (3 degrees of freedom) because the energy must
be independent of the origin. Thus, S(ν) is renormalized such that the
integration of eq 5 gives 3N - 3.
From various test calculations, we found that recording the velocities
every 4 fs is sufficient to obtain an accurate description of the high
frequency DoS, and we found that a total time span (after equilibration)
of 20-40 ps is generally adequate to give the low frequency modes
sufficient for accurate entropies (see section 2.5).
2.3. Thermodynamic Properties from Molecular Dynamics. Given
the vibrational density of states for a given V and T, we can calculate
(23) Siepmann, J. I.; Karaborni, S.; Smit, B. Nature 1993, 365, 330.
(24) Smit, B.; Karaborni, S.; Siepmann, J. I. J. Chem. Phys. 1995, 102, 2126.
(25) Martin, M. G.; Siepmann, J. I. J. Am. Chem. Soc. 1997, 119, 8921.
(26) Weiner, S. J.; Kollman, P. A.; Case, D. A.; Singh, U. C.; Ghio, C.; Alagona,
G.; Profeta, S.; Weiner, P. J. Am. Chem. Soc. 1984, 106, 765.
(27) Weiner, S. J.; Kollman, P. A.; Nguyen, D. T.; Case, D. A. J. Comput.
Chem. 1986, 7, 230.
(28) Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. Soc. 1984,
106, 6638.
(29) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. J. Phys. Chem. 1990, 94,
8897.
Table 1. Force Field Parameters Used for Coarse-Grained Alkane Segments; The Functional Forms Are Given in Eq 1a,b(All Parameters
Are from the SKS United Atom Force Field Unless Otherwise Indicated)
EVDW(R) ) Do{(
Ebond(R)a)1
Ro
R)
12
- 2(
Ro
R)
6}
(1a)
2KR(R - Ro)2
(1b)
Eangle(θ) )1
2Kθ(θ - θo)2
(1c)
Etorsion(φ)b)∑
n
1
2
Vn[1 - dncos(nφ)](1d)
EvdW
CH2
CH3
CH2-CH2
CH2-CH3
CH2-CH2-CH2
CH2-CH2-CH3
CH2-CH2-CH2-CH2
CH2-CH2-CH2-CH3
R0c
R0c
R0c
R0c
θ0f
θ0f
V1(d1)d
V1(d1)d
4.4113
4.4113
1.54
1.54
114
114
1.4109(-1)
1.4109(-1)
D0d
D0d
Kbe
Kbe
Kθg
Kθg
V2(d2)d
V2(d2)d
0.09339
0.2265
520
520
124.19
124.19
-0.271(1)
-0.271(1)
Ebond
Eangle
Etorsion
V3(d3)d
V3(d3)d
2.787(-1)
2.787(-1)
aThe force constants for the bond-stretching potential function were introduced from the AMBER force field26,27because the original SKS force field
uses a fixed bond distance.bThe torsion potential of the SKS force field had been taken from the OPLS (optimized potentials for liquid simulation) force
field.28
Boltzmann constant.eIn kcal/(mol Å2).fIn degrees.gIn kcal/(mol rad2).
cIn Å. For vdW, SKS uses EVDW(R) ) 4?{(σ/R)12- (σ/R)6}. Thus, Ro) (2σ)1/6.dIn kcal/mol. For vdW, D0(kcal/mol) ) k?(SKS), where k is the
Figure 2. Overlapping packed soft balls.
C(t) )∑
i)1
N
∑
R)1
3
miciR(t)(2)
ciR(t) ) lim
τ-∞∫-τ
τViR(t′ + t)ViR(t′) dt′
∫-τ
τ
dt′
)
lim
τ-∞
1
2τ∫-τ
τViR(t′ + t)ViR(t′) dt′ (3)
S(ν) )2
kTlim
τ-∞∫-τ
τC(t) exp(-i2πνt) dt
(4)
∫0
∞S(ν) dν ) 3N
(5)
A R T I C L E S Li et al.
1874 J. AM. CHEM. SOC.9VOL. 126, NO. 6, 2004

Page 4

the partition function Q(V,T) by treating the continuous DoS as a
continuum of uncorrelated harmonic oscillators
where
is the partition function of a harmonic oscillator with vibrational
frequency ν. Here, ? ) 1/kT, h is Planck’s constant, and S(ν) dν is the
number of modes between frequencies ν and ν + dν. Given the partition
function, the thermodynamic properties are determined as
where
are weighting functions and Vois a reference energy. Therefore, within
the assumption that the system is ergotic within the time scale of the
calculation, all thermodynamic properties are determined. We need to
provide only the reference energy Vo and the vibrational density of
states distribution S(ν) from the molecular dynamics simulations.
We choose the reference energy such that in the classical limit (h
f 0), the energy evaluated from eq 8a is equivalent to the total energy
EMD(kinetic plus potential) obtained from molecular dynamics simula-
tion. For a system of harmonic oscillators in the classical limit, the
energy is 3NkT (the equipartition theorem), where 3N is the total degrees
of freedom of the system. Thus, we write the reference energy as
where N is the total number of atoms in the system. For condensed
systems, the total degrees of freedom of the system are 3N - 3, and
eq 10 becomes Vo) EMD- ?-1(3N - 3). It is useful to note that the
energy (and free energy) determined this way includes the zero point
energy (first term on the RHS of eq 9a). This contribution is important
when quantum effects are significant in the system.
2.4. Molecular Dynamics Simulation. Molecular dynamics software
Lammps30is used to perform NVT molecular dynamics simulations
with the time step set to 1 fs. For each structure, we start by assigning
the initial velocities (Gaussian distribution) on each atom to give a
system temperature of 20 K, and then perform MD simulations for 10
ps at 10 K to equilibrate the structure. The temperature of the system
is then increased from 10 to 277 K (recrystallization temperature)
steadily over a period of 2 ps (the temperature was increased by 267/
2000 K every time step), followed by equilibration runs at 277 K for
8 ps. A 40 ps NVT molecular dynamics is then performed with the
atomic velocities, system energy, temperature, and pressure recorded
every 4 fs. This trajectory information is later used in the velocity
autocorrelation analysis to obtain the vibrational spectrum from which
the thermodynamic properties were calculated. The use of 40 ps is found
to be sufficient for obtaining a converged free energy as shown in
section 2.5.
Molecular dynamics are performed on a Linux cluster of 80 dual-
processor Dell PowerEdge 2650s (P4 Xeon 2.2/2.4GHz, 2G Memory,
54G HD) at MSC. Each dynamics takes 1-5 days depending on the
system size (4000-25 000 atoms).
2.5. Convergence of the Free Energy Calculation. To compare
the efficiency of various lattices for dendrimer balls consistently, we
build various lattice structures from the same initial isolated dendrimer
ball (see section 3.3 below). However, there may exist deviations in
the free energy evaluated from the molecular dynamics, especially for
the entropy that is dominated by the low-frequency modes in the system.
To access the convergence of our calculated thermodynamic properties,
we run three independent molecular dynamics from the same initial
structure but assign different initial velocity distributions (same overall
system temperature). From the three different MD trajectories, we
determine the RMS deviation of free energy, enthalpy, and entropy as
shown in Figure 7 of section 3.4. To discuss how the deviation depends
on the correlation time, we select the point with a large deviation in
Figure 7A (sticky G2 Ball in BCC lattice) and calculate the deviations
in free energy using different correlation times: 5, 10, 20, and 40 ps.
Figure 3 shows that using a longer correlation time leads to a better
free energy evaluation and 40 ps is sufficient to give satisfying deviation
to compare various lattices as shown in Figure 7.
3. Results and Discussion
3.1. Three Proposed Packing Mechanisms of the Soft Balls
with Polyethylene Tails. The liquid crystal phase formed by
self-assembled supramolecules with 3D nanoscale periodicity
has been researched extensively.1-13For example, studies of
electron density profiles and histograms computed from the
X-ray diffraction data16demonstrate that compounds I, II, and
III are self-assembled in supramolecular dendrimers resembling
spherical micelles, which self-organize in a three-dimensional
cubic Pm3 hn lattice. This thermotropic liquid crystal phase is
similar to that of the lyotropic Pm3 hn phase found in some
amphiphile/water systems. These supramolecular dendrimers
contain a poly(benzyl ether) core dispersed in an aliphatic matrix
of nearly uniform density, which is made up of the melted
terminal long alkyl chains of the monodendrons.
Geometrical aspects of liquid crystal phases have been studied
in considerable detail for lyotropic systems31-33with interfacial
(30) Large-Scale Atomic/Molecular MaasiVely Parallel Simulator, Version 5.0;
CRADA collaboration, Sandia National Laboratory, USA, 1997.
(31) Sadoc, J. F.; Charvolin, J. J. Phys. France 1986, 47, 683.
(32) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic
Press: London, 1992.
Figure 3. Convergence of the free energy from VAC.
ln Q )∫0
∞dν S(ν) ln qHO(ν)(6)
qHO(ν) )
exp(-?hν/2)
1 - exp(-?hν/2)
(7)
E ) V0+ T?-1(
∂ ln Q
∂T)N,V) V0+ ?-1∫0
∂ ln Q
∂T)N,V) k∫0
∞dν S(ν)WE(ν) (8a)
S ) k ln Q + ?-1(
∞dν S(ν)WS(ν)(8b)
A ) V0- ?-1ln Q ) V0+ ?-1∫0
∞dν S(ν)WA(ν)(8c)
WE(ν) )?hν
2
+
?hν
exp(?hν) - 1
(9a)
WS(ν) )
?hν
exp(?hν) - 1- ln[1 - exp(-?hν)](9b)
WA(ν) ) ln1 - exp(?hν)
exp(-?hν/2)
(9c)
Vo) EMD- ?-13N
(10)
Efficiency of Various LatticesA R T I C L E S
J. AM. CHEM. SOC. 9 VOL. 126, NO. 6, 2004 1875

Page 5

curvature being recognized as the key factor for determining
the phase type.34,35However, it is very difficult to investigate
the interfacial curvature from experiment because of the nearly
uniform density of the aliphatic matrix.
On the basis of our simulations on the atomistic scale, we
propose three packing mechanisms of soft balls with soft coronas
composed of polyethylene as shown in Figure 4.
For the soft balls having soft coronas composed of long alkyl
chains, we considered three classes of structures:
Smooth/Valentino balls are shown in Figure 4A, which lead
to the Slippery packing mechanism. Here, the C12alkane “hair”
is slicked down on each ball. This maximizes the favorable van
der Waals attraction between intraball chains and is optimum
for separate balls. However, the best intraball interactions may
lead to an increase in the number of gauche dihedrals and hence
less favorable torsional interactions. This likely leads to a
minimum in the favorable interball interactions between the
chains.
Sticky/Einstein balls are shown in Figure 4B, which lead to
the Sticky packing mechanism. Here, the alkyl hairs stick out
perpendicular to the surface to provide maximum surface area
for adjacent balls to interact, maximizing the favorable inter-
molecular van der Waals attraction between chains of different
balls. This likely leads to a minimum in the favorable intraball
interactions between chains.
Asymmetric/Punk balls are shown in Figure 4C, which lead
to the Anisotropic packing mechanism. Here, about half of the
surface has Einstein type hair, and the other half has Valentino
hair. This was motivated by the anisotropic positions on the
faces of Pm3 hn lattice, where the balls have two close neighbor
balls and other neighbor balls at a regular distance. In this
packing mechanism, the core of the soft ball deforms to become
nonspherical. The chains are slicked down (Valentino) to provide
the most favorable by intraball VDW while accommodating the
short distances between the closest balls on the faces, and they
are extended (Einstein) to provide the best interball VDW with
the other more distant neighbors. Thus, in this Anisotropic
packing mechanism, the packing of polyethylene chains is a
balance between the intraball type and the interball type. In this
packing mechanism, the chain shape can be kept as nearly all-
trans to minimize the torsional cost while optimizing the intraball
and interball interactions.
3.2. Preparation of Dendrimer Balls of Different Shapes.
We used Cerius236to construct the three-dimensional structures
(33) Kratzat, K.; Finkelmann, H. J. Colloid Interface Sci. 1996, 181, 542.
(34) Gruner, S. M. J. Phys. Chem. 1989, 93, 7562.
(35) Tate, M. W.; Eikenberry, E. F.; Turner, D. C.; Shyamsunder, E.; Gruner,
S. M. Chem. Phys. Lipids 1991, 57, 147.
Figure 4. Three proposed packing mechanisms of soft balls with soft coronas composed of polyethylene chains.
A R T I C L E S Li et al.
1876 J. AM. CHEM. SOC.9VOL. 126, NO. 6, 2004

Page 6

including three types of dendrimer balls: Smooth/Valentino
(Figure 4a), Sticky/Einstein (Figure 4b), and Asymmetric/Punk
(Figure 4c).
From the isolated dendrons of compounds G2(I), G3(II), and
G4(III), we first constructed the Sticky/Einstein balls for G2,
G3, and G4. The analysis of the experimental X-ray result
Figure 5. Preparation of the three shapes of G2 dendrimer ball.
Figure 6. Packing dendrimer balls into A15, FCC, BCC at low density (0.2 g/cm3) followed by compression in multiply small steps until the target density
(0.99 g/cm3) is achieved.
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Page 7

Figure 7. Helmholtz free energy (A ) Emd- ZPE - TS), enthalpy (Emd) potential energy from MD including kinetic energy but not zero point energy), and entropy (S) of different shape balls in various lattices.
Each MD simulation is carried out with a fixed volume using a Nose-Hoover thermostat (denoted as NVT) at the density 0.99 g/cm3. These MD simulations are carried out at a temperature of 277 K, which is the
recrystallization temperature in ref 16. Shown here are the average and the RMS uncertainties, obtained from three independent MD simulations using independent sets of initial velocities.
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suggested that 11.3 dendrons self-assemble into the soft balls
in G2, 5.8 dendrons assemble into the soft balls in G3, and 1.9
assemble to form the balls for G4 (see Table 3 of ref 16). We
construct the G2, G3, and G4 balls to have 12, 6, and 2
dendrons, respectively. This leads to 108 C12 alkyl chains for
G2 and 162 for both G3 and G4. Figure 5 illustrates the
preparation of the G2 balls.
From the isolated G2 dendron, we first construct the “Sticky/
Einstein G2 Ball” composed of 12 dendrons, with the polyeth-
ylene chains initially all-trans, pointing outward uniformly from
the ball center. An energy minimization of 500 steps is then
performed to optimize the structure.
Next, we constructed the structure of the “Ball Frame” shown
in Figure 5. This frame is prepared using 500 C_R atoms (the
type for C in aromatic assemblies including graphite) uniformly
distributed on a sphere. The uniform distribution of C_R atoms
is obtained by first placing a point charge on each atom and
then minimizing (1000 steps) the Coulombic energy, with a
constraint of keeping the C_R atoms a fixed distance from the
center. The “Ball Frame” is a tool to be used to compress the
“Sticky/Einstein ball” to the “Smooth/Valentino ball” by starting
with a radius outside of the atoms for the sticky ball and steadily
reducing the radius, forcing the alkyl chains to deform and
eventually form the structure of the Smooth ball, as described
next.
To select an initial radius for the “Ball Frame”, we note that
the surface atoms (C_33) of the “Sticky G2 Ball” lead to a RMS
radius of 27.4 Å, whereas the closest distance of soft balls in
various lattices (A15, BCC, and FCC) at the expected density
0.99 g/cm3is 34.8 Å. Thus, we started the size of the “Ball
frame” at 27.4 Å + 4 Å ) 31.4 Å and contracted it to 34.8 Å/2
+ 4 Å ) 21.4 Å in 10 steps. Here, 4 Å is the favored distance
between C_R and C_33 atoms. In each step, the size of the
“Ball Frame” decreases 1 Å followed by 200 steps of energy
minimization on the dendrimer. The final result is the “Smooth
G2 Ball” shown in Figure 5.
By rotating a line of 12 C_R atoms 36 times, we get the
“Cone Frame” as shown in Figure 5. After compressing the
cone angle to almost 0°, we align the “Cone Frame” together
with “Sticky G2 Ball” as shown in Figure 5. We then expand
the cone angle step by step followed by 200 steps of energy
minimization unless there is no overlap between the two same
Asymmetric/Punk balls at the distance of 34.8 Å as shown in
Figure 4c.
3.3. Packing Balls into Various Lattices. From the den-
drimer balls prepared in section 3.2, except for the dendrimer
ball with Smooth shape, we first pack them together in various
lattices (A15, FCC, BCC) at low density (0.20 g/cm3) as shown
in Figure 6B,D,F. We then shrink the unit cell steadily by 1 Å
each step followed by 500 steps of minimization to allow the
polyethylene chains of neighboring balls to pack together. In
each step, the ball center is translated to be at the correct position
for the new decreased cell length, but the structure within each
ball remains unchanged. During these minimizations, we fix
the acid hydrogen atoms (denoted as atom type H___A) of the
carboxylic acid at the core of each dendron to retain the position
of the ball center. With this procedure, we get the A15, FCC,
and BCC lattices at the target density (0.99 g/cm3) as shown in
Figure 6C,E,G.
For the A15 structure, there are two crystallographically
inequivalent types of positions: the face position (2 atoms per
face, 6 per cube) and the corner/body center positions (one each
per cube), as indicated by black circles and white circles in
Figure 6. The face positions lead to the closest contacts with
neighboring balls, and hence we place the anisotropic balls
(Figure 4c) at these face positions. For the corner/body center
positions of A15, we started with sticky balls. We denote this
structure as the “Anisotropic A15” structure.
For the dendrimer ball with smooth shape, we placed the balls
on the desired positions of various lattices at the target density
(0.99 g/cm3). Our preparation procedure for smooth balls
(detailed in section 3.2) guaranteed that the balls would not touch
each other when placed into the structures at density 0.99 g/cm3.
Thus, the initial structure for the smooth balls led to significant
void volume. We then performed an energy minimization of
500 steps, followed by the molecular dynamics runs as described
in section 2.3. We find that the tail chains relax to fill in the
void volume that has been left in the initial structure during
energy minimization and molecular dynamics.
In the MD simulations, we use eight independent dendrimer
balls for A15 leading to 15 840 independent atoms per cell for
G2, 24 336 for G3, and 24 528 for G4. Note that there are no
hydrogen atoms in the alkyl chains as the united atom force
field is used. For the FCC structure, we use four independent
balls, leading to one-half the number of atoms as for A15. For
the BCC structure, we use two independent balls, leading to
one-fourth the number of atoms as for A15.
3.4. Free Energy of Various Lattices. Figure 7 shows the
free energy (A), energy (Emd), and entropy (S) of different ball
shapes packed into various lattices. The comparison of genera-
tions 2, 3, and 4 is shown in Figure 7ABC, DEF, and GHI,
respectively.
Comparing Figure 7A with 7B, 7D with 7E, and 7G with
7H, we see that free energy of various lattices follows the same
trend as the enthalpy of various lattices. The best lattice is the
“Anisotropic A15” lattice for all generations (2, 3, and 4).
For generation 2, A15, FCC, and BCC composed of sticky
balls are essentially not distinguishable. We have the same
conclusion for A15, FCC, and BCC lattices composed of smooth
G2 Balls. However, the lattices composed of sticky balls are
found to be much more stable than the lattices composed of
smooth balls.
From Figure 7C,F,I, we can see that the deviations in entropy
are pretty big as compared with those of enthalpy in Figure
7B,E,H, which makes the difference among those lattices not
distinguishable. However, we see the same trend for all
generations: Anisotropic A15 > sticky A15 > sticky FCC >
sticky BCC, indicating that A15 is slightly favorable in terms
of entropy consideration.
For generations 3 and 4, the differences among various lattices
are larger than those for generation 2. We can see that the best
lattice is still Anisotropic A15 lattice, followed by A15, BCC,
and FCC in that sequence.
In conclusion, Anisotropic A15 lattice gives the best (lowest
free energy) packing of the G2, G3, and G4 balls composed of
compounds I, II, and III. For the three packing mechanisms
proposed in section 3.1, anisotropic packing is the best. This is
(36) Accelrys_Inc. Cerius2 Modeling EnVironment, Release 4.0; Accelrys Inc:
San Diego, 1999.
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J. AM. CHEM. SOC. 9 VOL. 126, NO. 6, 2004 1879

Page 9

because it allows a good balance between interball vdW and
intraball vdW and accommodates a variety of ball-ball distances
without sacrificing the all-trans intrachain conformation of the
alkyl chains. In contrast, slippery packing with the smooth ball
shape leads to bad enthalpy, although it has the optimal intraball
vdW.
Furthermore, the Asymmetric/Punk ball shape can be viewed
as the intermediate between the disk shape and isotropic ball
shape as shown in Figure 8. Dendrons with a tapered fan
shape assemble flat pizza-like slices into disks, which then
stack into columns that eventually form a hexagonal array.15
Dendrons with more alkyl chains are cone-shaped and assemble
into supramolecular spheres. So far, these spherical aggre-
gates have been known to pack on three lattices, Cub Pm3 hn,16
Cub Im3 hm,17and Tet P42/mnm.1By tuning the number of
alkyl chains, there exists the intermediate shape between the
disk shape and isotropic ball shape as shown in Figure 8B. In
fact, from the analysis in Figure 7, this type of Asymmetric/
Punk ball shape makes the A15 structure superior to other
structures.
For a specified compound,1experiments find the following
phase sequences at different temperatures: glass < 110 °C <
Colh(Colh) hexagonal columnar) < 140 °C < Cub Pm3 hn <
153 °C < Tet P42/mnm (tetragonal distortion of cubic packing)
< 163 °C < Iso (isotropic liquid).
This indicates that the dendron exhibits a tapered fan shape
at low temperature, while it adopts a cone shape at high
temperature. In other words, the packing of disk-shaped
dendrimers in Figure 8A is enthalpically preferred and the
packing of ball-shaped dendrimers in Figure 8C is entropically
favored. The results for the Asymmetric/Punk ball shape in
Figure 8B suggest a way to understand the above phase
sequence. The Colh structure is composed of 100% disks as
shown in Figure 8A. The Cub Pm3 hn structure has 75%(6/8)
Asymmetric/Punk balls (coordination number 14) and 25%(2/
8) isotropic balls (coordination number 12). The Tet P42/mnm
structure has 67%(20/30) Asymmetric/ Punk balls (16 of them
have coordination number 14 and 4 of them have coordination
number 15) and 33%(10/30) isotropic balls (coordination
number 12).1Our assumption that tapered fans are favored
enthalpically is consistent with the observed ordering of the
phases with Colhlowest, Cub Pm3 hn next, and Tet P42/mnm at
the highest temperature.
In the Cub Pm3 hn liquid crystal phase formed by compounds
G2(I), G3(II), and G4(III), it is found that a nearly integer
number of dendrons self-assemble into a sphere. Specifically,
the packing numbers of dendrons in each sphere of G2, G3,
and G4 are 11.3, 5.9, and 1.9.16We use integer numbers 12, 6,
and 2 in the simulation. When an isolated ball formed by
aggregation of dendrons is considered, the core part of the ball
is compact involving interactions between the phenyl groups.
However, the soft corona, which is composed of polyethylene
chains, is quite loose. Thus, the packing number of dendrons
into a ball is determined by the core part of the dendrimer ball.
The average radius gyrations of the core part of the G2, G3,
G4 balls from our best Anisotropic A15 structure (equilibrated
after 60 ps dynamics) are 11.02, 13.39, and 13.88 Å, respec-
tively. Assuming that the density is uniform in the ball core
part, we derive the volume occupied by the core from the
relation of Rg ) ?3/5r. Dividing this core volume by the
volume of the phenyl group of each dendron at density 0.99
g/cm3, we get the maximum packing number of dendrons in
each G2, G3, and G4 ball: 12.48, 7.15, and 2.62. This result
indicates that the packing of dendrons into a ball is not yet
saturated. The “anisotropic packing” as shown in Figure 4C
results in the nonspherical balls and unsaturated packing. Table
2 lists the physical properties of the dendrimer balls formed by
12G2-AG, 12G3-AG, and 12G4-AG.
Figure 9 shows the density of states (power spectrums) of
various G2 lattices, which includes the Anisotropic A15 lattice
and FCC, BCC lattices composed of the sticky shape G2 balls.
Other lattices, which are not shown here, have similar power
spectrums. From Figure 9, we can see that the power spectra
of various lattices are almost indistinguishable, even in the
zoom-in of Figure 9B. Indeed, the analysis of Figure 7 shows
that there is no significant difference in the entropies of various
structures.
3.5. Comparison with Experimental X-ray. Figure 10
shows the X-ray intensities of 12G2-AG, 12G3-AG from experi-
ment16and prediction from the best “Anisotropic A15 structure”.
The predicted intensities fit the experimental intensities very
well. This confirms our predicted structure, free energy analysis
here, and the electron density profile analysis in ref 16.
3.6. Free Energy Profile over Density of Various Lattices.
To understand the packing process of the soft balls, we calculate
Figure 8. Packing of disk shape, Asymmetric/Punk ball shape, and isotropic ball shape dendrimers.
A R T I C L E SLi et al.
1880 J. AM. CHEM. SOC.9VOL. 126, NO. 6, 2004

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the free energy, enthalpy, and entropy of various lattices at
different densities as shown in Figure 11. The structures are
prepared as described in section 3.3 and Figure 6. For the
densities lower than 0.99 g/cm3, we fix the central atom H___A
in the core of each dendron to keep the ball centers at the desired
positions in the lattice during molecular dynamics.
From Figure 11B, we can see that the enthalpies of various
lattices all have a minimum at ∼0.99 g/cm3. The best structure
AA15 (Anisotropic A15) is only favored at a density above 0.90
g/cm3. The enthalpy decreases monotonically during the packing
process. The interball vdW interaction dominates the packing
energy. It decreases slowly in the early packing stage due to
less interaction among the soft balls at low density.
The entropies for all structures decrease with increasing
density, as shown in Figure 11C. That is because the overlap
between the neighboring soft balls constrains the conformations
of the chains of the balls, leading to a short-ranged repulsive
interaction. The differences among various lattices are negligible
above a density of 0.6 g/cm3. However, entropy evaluation has
a big deviation at low density as shown in Figure 11C. For
nearly isolated soft balls at low density, there is much more
flexibility, leading to the large differences.
The free energy profiles are shown in Figure 11A. The free
energy profiles of various lattices show the same trend, which
can be approximately described as three levels. The lowest level
is at the density from 0.2-0.3 g/cm3. From 0.5-0.99 g/cm3,
the free energy shows a flat stage profile as the middle level.
The free energy is the worst (highest) at the high density, 1.10
g/cm3.
The free energy profile over volume of the best Anisotropic
A15 lattice at 277 K as shown in Figure 12 indicates two stable
phases: (a) a condensed phase at the density 0.99 g/cm3; and
(b) an isolated micelle phase at the density 0.30 g/cm3. This
figure is derived from Figure 11A and essentially is an A-V
plot of the micelle from the condensed phase to the isolated
phase. The isolated phase is evaluated in a vacuum without
solvent and is unphysical. The critical pressure determined from
the two stable phases is 0.033 GPa, which is bigger than 1 atm.
This implies that in the recrystallization process, the unfavorable
solvent at 277 K is the driving force to form the condensed
phase.
In the BCC and FCC lattices, the ball interacts equally with
its neighbors. Thus, we can derive a two-body potential from
Figure 11. Figure 13 shows the two-body free energy/enthalpy
potential in BCC and FCC lattices. Obviously, the two-body
potential of the soft dendrimer balls is lattice dependent. The
reason is that the soft dendrimer balls self-adjust its shape and
interaction in different lattices. However, the two-body potential
from BCC and FCC lattices shares the same trend. The two-
body free energy potential in BCC and FCC lattices can be
explained as a so-called “square shoulder potential”.37-40In this
potential, the soft balls do not interact with each other when
they are far apart, and they interact via a constant repulsive
value at short distance. This soft shoulder arises from the entropy
repulsive interaction. The shoulder positions in BCC and FCC
are different due to the different neighbor distribution.
The two-body Emd (enthalpy) potentials in BCC and FCC
are similar to the Lennard-Jones potential. Also, the optimum
distances in BCC and FCC are different (see Figure 13B). From
the optimum point (at the density 0.99 g/cm3), we can derive
the Lennard-Jones 12 6 potentials as shown in Figure 13B (solid
points). The two-body enthalpy potential of soft dendrimer balls
has a much sharper well than the Lennard-Jones potential. The
interaction of the soft balls increases dramatically when bringing
them together.
3.7. Packing Efficiency of Ideal Soft Balls (Which Overlap
Each Other) and Some General Discussions. To investigate
the geometric difference of various lattices in the packing of
soft balls, we use the A15, FCC, and BCC lattice boxes of
generation 2 above (at the same density: 0.99 g/cm3) and put
ideal balls into desired positions. By increasing the ball radius
and letting the balls overlap each other, the void volume in the
lattice decreases as shown in Figure 14 (from numerical results).
When the ball is small enough and there is no overlap between
balls, A15, FCC, and BCC give the same void volume. By
increasing the ball size from 18 to 21 Å, the balls in A15 overlap
first, then BCC, followed by FCC. That is because FCC is the
closest packing lattice for hard balls. In the range of 18-21 Å,
FCC has the least void volume. However, if we increase the
ball size from 21 to 25 Å, FCC shows the biggest void volume.
BCC has the least void volume, but A15 is comparable. Further
increasing the soft ball size fills all of the space , and the void
volume is zero for all lattices.
The least void volume means the soft balls use the space in
the most efficient way and the overlap between the balls has
been kept to a minimum. Considering the entropy repulsive
interaction between the soft balls discussed in section 3.6, we
conclude that the less overlap in the lattice, the better the free
energy. In view of this, FCC is the lest efficient lattice for the
ideal soft ball, while BCC is the most efficient lattice.
From the analysis in the previous sections, we can see that
dendrimer balls are not ideal soft balls, which can self-adjust
their shape anisotropically to get the best packing of the soft
corona. Indeed, they adopt the intermediate shape between the
disk and isotropic ball to achieve a balance between enthalpy
and entropy, and they assemble into the A15 lattice.
(37) Blohuis, P.; Frenkel, D. J. Phys. C 1997, 9, 381.
(38) Rascon, C.; Velasco, E.; Mederos, L.; Navascues, G. J. Chem. Phys. 1997,
106, 6689.
(39) Lang, A.; Kahl, G.; Likos, C. N.; Lowen, H.; Watzlawek, M. J. Phys. C
1999, 11, 10143.
(40) Velasco, E.; Mederos, L.; Navascues, G.; Hemmer, P. C.; Stell, G. Phys.
ReV. Lett. 2000, 85, 122.
Table 2. Physical Properties of the Spherical Supramolecule
Formed by 12Gn-AG
generation µ(m odel)aN(m odel)bRg/ÅcRg(core)/Åc,dµ(predicted)eµ(exptl)f
saturation
degreeg
2
3
4
124680
7092
7116
17.75
20.24
20.97
11.02
13.39
13.88
12.48
7.15
2.62
11.3
5.8
1.9
90.5%
81.1%
72.5%
6
2
aWe use 12, 6, and 2 dendrons to form the dendrimer ball of generation
2, 3, and 4, which are close to the experimental values of 11.3, 5.8, 1.9.16
bThere are a total of 4680, 7092, and 7116 atoms in each dendrimer ball
of 12G2-AG, 12G3-AG, and 12G4-AG constructed for simulation. However,
the hydrogen atoms in the C12H25tail chains are implicit during molecular
dynamics (see section 2.1).cThe radius gyration of each ball or core of
the ball is evaluated from the best liquid crystal structure at density 0.99
g/cm3after 60 ps NVT molecular dynamics and averaged from the eight
independent balls in the unit cell.dThis is the radius gyration of the core
part of each ball (excludes the C12H25 tail chains).eOn the basis of the
relation of Rg ) ?3/5r, we determine the volume occupied by the core
part of each ball from Rg(core). We then divide it by the volume of the
core part of each dendron at the density 0.99 g/cm3, and we get the maximum
packing number of dendrons in each G2, G3, and G4 ball.fThis packing
number was determined from X-ray analysis and listed in Table 3 of ref
16.gSaturation degree is defined as µ(exptl)/µ(predicted).
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Page 11

Kamien et al.41analyzed the stability of different lattices
(A15, BCC, FCC) by considering the tension in the AB inter-
face of a diblock copolymer and the stretching of the poly-
mers. They concluded that the BCC lattice minimizes the
stretching part and the A15 lattice minimizes the tension in the
interface. They argued that the A15 lattice should be favored
as the blocks become more symmetric and corroborated this
through SCFT.
(41) Grason, G. M.; Didonna, B. A.; Kamien, R. D. Phys. ReV. Lett. 2003, 91,
art. no. 058304.
Figure 9. Power spectrum of various G2 structures using the best case for each structure (symmetric sticky balls for FCC and BCC, and asymmetric sticky
balls for A15). Each power spectrum is from one of the three MD runs. There results are normalized to one ball; separate colors (blue for A15, purple for
FCC, and green for BCC) are used for various structures, but the differences are negligible.
Figure 10. Predicted X-ray intensities of 12G2-AG, 12G3-AG dendrimers from the best structure, Anisotropic A15 structure, as compared with experiment.16
The X-ray diffraction intensities are calculated using the “Diffraction-Crystal” module in Cerius2 4.0. No polarization factor, crystal monochromator factor,
or temperature factor is applied to the intensity calculations. These intensities are from a single snapshot from the MD trajectory (after 60 ps). We apply the
factor 0.98 to the predicted intensities of 12G2-AG, which arises from power (11.3/12, 1/3); where experiment has 11.3 dendrons, our structure has 12
dendrons in each ball. The hydrogen atoms of polyethylene chains are not considered during the intensity calculations.
A R T I C L E S Li et al.
1882 J. AM. CHEM. SOC.9VOL. 126, NO. 6, 2004

Page 12

Indeed, they argued that the BCC lattice minimizes the
stretching of the polymers, which is consistent with our analysis.
We concluded that there exists an entropic repulsive interaction
between soft balls and the BCC lattice minimizes it in terms of
ideal soft balls. From an analysis of the competition between
enthalpic and entropic interactions in soft balls, we find that
the stretching of the polymers (deformation from the sphere
cell to the Voronoi/Wigner-Seitz cell) arises from the entropic
effects.
However, the tension in the AB interface analyzed by them
might not be directly applicable to the dendrimer liquid crystal
system we studied here. The authors claimed that when the
volume fraction of A-type monomers (φ) is large enough, the
AB interface takes on the shape of the Voronoi cell. On the
basis of that, they concluded that A15 minimizes the interfacial
tension.
We consider that the packing mechanism of the polyethylene
segments of the corona of the soft ball, but not the interfacial
tension between the aromatic phase and aliphatic phase, is the
key factor in determining the efficiency of various lattices.
Indeed, Percec’s group found that the density of the polyethylene
segments could be used to tune the phase of the dendrimer liquid
Figure 11. Free energy, enthalpy, and entropy as a function of density for four cases of G2 dendrimers. AA15 is the A15 structure composed of anisotropic
balls on the face and isotropic balls on the corner and body center. The other three structures, A15, FCC, and BCC, are based on isotropic balls only. MD
simulation details are the same as in Figure 7.
Efficiency of Various Lattices A R T I C L E S
J. AM. CHEM. SOC. 9 VOL. 126, NO. 6, 2004 1883

Page 13

crystal. Our simulation results indicate that the dimmer balls
on the face of A15 structure prefer a nonspherical shape, leading
to a good packing of the polyethylene segments. We believe
that the interfacial tension is less important between the aromatic
phase and the aliphatic phase, although it might be crucial for
the diblock copolymer phases.
Kamien’s group also proposed some guidelines for con-
sidering the principle of packing colloids.42,43They simpli-
fied the aliphatic phase in the dendrimer liquid crystal as a
dodecyl bilayer, which is dominated by a repulsive free energy
that scales as the inverse of the layer thickness, d-1. They argued
that A15 is the most stable structure because the entropy is
maximized through the minimization of the interfacial area in
the bilayer.
We consider the dendrimer liquid crystals to be different from
colloidal crystals, which contain solvent. The interaction among
colloidal micelles will always be repulsive, even in the crystal.
However, in the dendrimer liquid crystal, where there is no
(42) Ziherl, P.; Kamien, R. D. Phys. ReV. Lett. 2000, 85, 3528.(43) Ziherl, P.; Kamien, R. D. J. Phys. Chem. B 2001, 105, 10147.
Figure 12. Free energy profile over volume of Anisotropic A15 lattice at 277 K.
Figure 13. Two-body potential from BCC and FCC lattices.
A R T I C L E SLi et al.
1884 J. AM. CHEM. SOC.9VOL. 126, NO. 6, 2004

Page 14

solvent, the aliphatic segments of the neighboring balls can pack
each other favorably.
Furthermore, we consider that the bilayer approximation may
not be accurate enough to distinguish various lattices. Our
atomistic simulation results indicate that the various lattices
formed by the same type of dendrimer balls are not distinguish-
able in terms of free energy (e.g., Figure 7A).
4. Conclusions
Self-assembled supramolecular organic liquid crystal struc-
tures at nanoscale have potential applications in molecular
electronics, photonics, and porous nanomaterials. Most of them
are aggregated by soft spheres, which have soft coronas and
overlap each other in the packing process.
On the basis of our simulations on the atomistic scale, we
propose three packing mechanisms of soft balls, “Sticky
packing”, “Slippery packing”, and “Anisotropic packing”, and
we use the vibrational density of state (DoS) derived from
classical molecular dynamic simulations to investigate the
efficiency of various lattices for soft balls from simulation.
By focusing on the three compounds reported by Percec et
al. (J. Am. Chem. Soc. 1997, 119, 1539), which form spheres
with a nearly integer number of dendrons, we compare the
efficiency of various lattices and different packing methods. For
the soft spheres with aliphatic coronas composed of polyethylene
chains, “Sticky packing” is better than “Slippery packing”.
Anisotropic packed A15 is favored over FCC, BCC lattices.
Predicted X-ray intensities of the best structures fit the experi-
ments very well.
“Anisotropic ball packing” proposed here plays an intermedi-
ate role between the enthalpy-favored “disk packing” and
entropy-favored “isotropic ball packing”, which explains the
phase transitions at different temperatures.
Free energy profiles over the density of various lattices are
essentially the same, which indicates that the preferred lattice
is not determined during the packing process. Both enthalpy
and entropy decrease as the density increases. The free energy
profile over volume shows two stable phases: the condensed
phase and the isolated micelle phase. The two-body potential
of the soft dendrimer ball is lattice dependent, because it self-
adjusts its shape and interaction in different lattices. The shape
of the free energy potential is similar to that of the “square
shoulder potential”.
A model explaining the packing efficiency of ideal soft balls
in various lattices is proposed in terms of geometrical consid-
eration. BCC has the least void volume for the ideal soft ball,
while FCC has the biggest.
Acknowledgment. This paper is dedicated to the late Paul
Miklis, who initiated the study of these Percec systems shortly
after the first publication. In addition to the early stimulation
by Paul, we also thank Dr. Tahir Cagin and Virgil Percec for
many helpful discussions about these dendrimers, and we thank
Dr. Mario Blanco with help using Lammps. This research is
based in part on work supported by the U.S. Army Research
Laboratory and the U.S. Army Research Office under grant
number DAAG55-97-1-0126 (MURI-program officer Doug
Kiserow). The facilities of the MSC used in this research have
been supported by grants from the ARO (DURIP), ONR
(DURIP), NSF (MRI), and IBM (SUR). Other support to the
MSC is provided by the NIH, NSF, DOE, Chevron-Texaco,
General Motors, Seiko Epson, Asahi Kasei, Beckman Institute,
and Toray Corp.
JA038617E
Figure 14. Packing efficiency of ideal soft balls (which overlap each other). The curves are obtained from numerical results.
Efficiency of Various LatticesA R T I C L E S
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