J. Phys. Chem. B 2007, 111, 2081-2089
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Self-Assembly of T-Structures in Molecular Fluids Amar B. Pawar and Ilona Kretzschmar* Department of Chemical Engineering, City College of New York, 140th Street & ConVent AVenue, New York City, New York 10031
Gregory Aranovich and Marc D. Donohue Department of Chemical and Biomolecular Engineering, Johns Hopkins UniVersity, Baltimore, Maryland 21201 ReceiVed: July 21, 2006; In Final Form: December 20, 2006
A lattice density functional approach is used to describe the equilibrium assembly of three types of anisotropic patchy particles into a T-structure. The T-structure is comprised of one three-patch, three two-patch, and three one-patch particles. All patches are positioned orthogonal to each other. Temperature, particle concentration, and interaction energy ranges are determined that lead to T-structure formation. T-structure formation is investigated for two types of two-patch particles: Case 1 uses two identical patches and Case 2 employs two differing patches. Sets of parameters leading to T-structure assembly are determined for both cases. We find that in Case 1 the symmetric two-patch particle enforces T-structure formation, while the asymmetric two-patch particle in Case 2 leads to formation of chains, dimers, and incorrect and extended T-structures in addition to the T-structure. Synthetic strategies for both cases are discussed and reveal that Case 2 presents the more straightforward synthetic route.
Introduction Self-assembly as a “bottom-up” strategy has great potential to overcome the size limitations and processing restrictions of present-day, “top-down” manufacturing processes such as photolithography.1-7 Depending on the interactions between building blocks and the thermodynamic conditions in the system, various supramolecular entities can be formed by virtue of selfassembly.8 In the context of self-assembling target structures, anisotropic particles play an important role as building blocks that induce directionality in the self-assembly process. Sitespecific functionalization of the surface of a spherical particle is one way to introduce anisotropy leading to surface-anisotropic particles.9-16 The anisotropic nature of the particle surface enables a certain degree of control over the assembly process, since the specific interactions between the anisotropic particles can provide a driving force for the assembly of a target structure. The controlled assembly of anisotropic particles also requires knowledge of the effect of parameters such as temperature, interaction energies, and particle concentrations on the selfassembly process. While it is time-consuming to test every possible condition experimentally, the theoretical study of selfassembling anisotropic particles presents an efficient tool for the prediction of process parameters and conditions for selfassembly. Recently, several computational studies on the self-assembly of anisotropically interacting particles have been reported.8,17-23 Glotzer et al.21 used Monte Carlo simulations to investigate the biomolecule-directed self-assembly of nanoscale building blocks. Zhang et al.8 predicted the formation of various complex structures such as ring, chain, and staircase structures by anisotropic, weakly interacting “patchy” particles using Brown* Address correspondence to this author. E-mail: kretzschmar@ccny. cuny.edu.
ian dynamics. They further showed that diamond-like structures can be formed by a system of hard spheres with sticky patches.17 Malescio and Pellicane18 predict the formation of stripe patterns caused by repulsive interactions in a two-dimensional system of core/shell particles. Vanakaras20 presented computer simulations in which he studied the equilibrium self-organization of Janus particles. De Michele et al.24 used Monte Carlo and event driven molecular dynamics simulations to describe the dynamics of a system in the presence of attractive patchy interactions. Most of the reports concerning the self-assembly of anisotropic particles are directed toward (i) understanding the principles of self-assembly or (ii) the order-disorder phase transitions in a system of particles. Our study addresses the effect of experimentally controllable parameters such as temperature, attractive and repulsive interactions, and particle concentrations on the thermodynamic stability of an equilibrium structure using a lattice density functional approach. Lattice density functional theory has been used extensively for the description of hard-sphere solids,25 phase transitions in fluids of anisotropic monomers,26 molecular diffusion,27 adsorption energy distribution in an N2 isotherm,28 pore-shape effects on adsorption,29 and phase diagrams of binary mixtures.30 More specifically, Tarazona25 introduced a free-energy density functional for a system of hard spheres. He showed that the functional was able to reproduce the thermodynamics and direct correlation function of a homogeneous fluid and successfully tested it for the description of a highly inhomogeneous hardsphere solid. Further, Aranovich and Donohue (AD)26 developed a lattice density functional model describing the phase transitions in a fluid comprised of anisotropic monomers. The basic principle behind the lattice density functional approach is to consider the Helmholtz free energy, F, as a functional of the density distribution, F(r). The free energy functional, F[F(r)],
10.1021/jp0646372 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/07/2007
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Figure 1. T-structures formed with (a) symmetrically (B1) and (b) asymmetrically (B2) modified two-patch particles for Case 1 and Case 2, respectively.
is then minimized at equilibrium resulting in the density distributions at equilibrium, eq 1.26
δF[F(r)] )0 δF(r)
(1)
Here, we report our computational findings on the selfassembly of anisotropic (“patchy”) particles into a T-structure. The T-structure is chosen because it presents an interesting model for plasmonic waveguides,31,32 three-way interconnects,33 three-way connectors,34 spin filters,35 or a transistor structure with leads,36-38 where the center particle has the properties of a transistor. Figure 1 shows the targeted T-structure. We employ an Ising lattice model along with principles of lattice density functional theory to describe the equilibrium self-assembly of anisotropic particles as a function of interaction energies, the concentration of the anisotropic particles, and temperature. The AD lattice density functional model26 is extended from one to three types of anisotropic particles. We compare two specific systems of particles leading to the T-structure. The two systems differ from each other in the modification of the two-patch particle, which is symmetric in Case 1 (Figure 1a) and asymmetric in Case 2 (Figure 1b). Computational Details In this section, we present the theoretical model used for our calculations and our rationale for choosing the parameter space investigated. The system of particles used in the calculations consists of three types of particles. The number of patches on the particles is differentiated by a letter code (A-C) and is used to indicate the interactions between the particles: A ) center particle with three identical patches, Bn ) middle particle with two patches (n ) 1 for symmetric patches and n ) 2 for asymmetric patches), and C ) end particle with one patch. The particles used for the T-structure in the calculations are shown in their respective positions in Figure 1. For Case 1 (Figure 1a) the B1 particle has isotropic interactions on both poles since it has identical patches, while for Case 2 (Figure 1b) the anisotropic interactions on opposite poles arise because the B2 particle has two different patches. For simplification of the calculations, the system of particles is assumed on a two-dimensional planar square lattice with 21×21 lattice sites.39,40 The particles can orient themselves in any of four orientations on a lattice site (i,j). The four poles are the patch positions on the particles (see the Supporting Information). The lattice sites are occupied either by the particles or by solvent molecules. The probability of a lattice site (i,j) to be occupied by a specific particle (A, Bn, or C) is derived from the contributions of the four orientations of the specific particle type at the site (i,j) and is hereafter referred to as the total density distribution of the particle. The interaction energies between different poles (patches) are used to simulate anisotropic interactions between the particles. Equilibrium density distributions of A, Bn, and C particles are studied around a particle with three patches (A particle) located at the center (0,0) of the lattice.
Note that the density distribution, F(i,j), around the particle at (0,0) does not represent the complete statistics. It is the conditional probability of finding certain particles as neighbors of an A particle at the site (0,0). The complete picture of all possible structures would require the consideration of the density distributions for all types of particles (A, B, and C) at the (0,0) position. Model. The model used for the calculations is an Ising lattice model employing a lattice density functional approach with anisotropic interactions as developed by Aranovich and Donohue (AD).26 In brief, the AD model considers a single type of particle with four poles interacting with other particles of its own kind on a square lattice. Different values for the interaction energies between the different poles account for anisotropic interactions between the particles. The Helmholtz free energy, F, is obtained by using
F ) U - TS
(2)
where U is the configurational internal energy, T is the absolute temperature, and S is the total entropy summed over all lattice sites. The configurational internal energy for the system is calculated based on the interaction energy between particles sitting on two neighboring sites applying the mean-field approximation. The equilibrium density distribution of the particles is obtained by minimizing the Helmholtz free energy of the system. For more details, the reader is encouraged to consult ref 26. The T-structure discussed here requires three types of particles (A, Bn, and C). To extend the AD model from a single type of particle to a mixture of three types of particles, the interactions between the same kind of particle and different types of particles have to be considered. The internal energy per particle resulting for two neighboring lattice sites then is given as
E)
() 1
2
particle 3
∑
particle 3
∑
orientation 4
∑
orientation 4
∑
a)particle 1 b)particle 1 m)orientation 1 n)orientation 1
(abmnp(a,m)p(b,n)) (3) where abmn is the interaction energy between the ath and bth type of particle (a, b ) 1-3) with contact points m and n (n,m ) 1-4). p(a,m) and p(b,n) are the respective probabilities of finding the ath and bth type of particle on neighboring sites to have a contact point m on the ath type of particle and a contact point n on the bth type of particle. The configurational internal energy and the entropy (per site) for the system of three types of particles are developed in an analogous manner to the single type particle system. The Helmholtz free energy, F, for the system is obtained according to eq 2. It should be noted that F is the functional of the order parameters xt,k,i,j, which are the density distributions for the different types of particles (t ) 1 to 3) with varying orientations (k ) 1 to 4) on the planer lattice (i ) -10 to 10 and j ) -10 to 10). Minimization of the free energy functional will give the density distributions at equilibrium. The bulk densities of the particles, x∞A, x∞Bn, and x∞C referred to as the concentrations of A, Bn, and C particles, respectively, are themselves solutions for the equilibrium density distribution equations due to the self-consistent nature of the lattice density functional equations. To obtain nonuniform density profiles, the system is perturbed by placing an A particle with a particular orientation at the center point (0,0) of the lattice. The equilibrium density distributions of A, Bn, and C particles then are studied around the A particle at (0,0) and are calculated based on the equality of the chemical potentials which result
Self-Assembly of T-Structures in Molecular Fluids
J. Phys. Chem. B, Vol. 111, No. 8, 2007 2083
TABLE 1: Range of Parameters Explored in Calculations Case 1 (n ) 1)a
parameter temp [K] interactions [eV] specific interactions in the T-structure nonspecific interactions concentration A Bn C a
Case 2 (n ) 2)b
275 - 375
300
strong attractive: 1.0-8.0 weak attractive: 0.5-7.5 rest all attractive: 0-0.047 rest all repulsive: n.a.c
strong attractive: 2.5-3.5 weak attractive: n.a.c weak attractive: 1.0-2.0 rest all repulsive: 0.3-2.0
0.001 0.003 0.003
0.0001-0.0005 0-0.0031 0-0.016
T-structure with symmetric B1 particle. b T-structure with asymmetric B2 particle. c Not applicable.
from classical Ono-Kondo theory.41 The nonlinear simultaneous lattice density functional equations for the equilibrium density distribution are solved by using the method of successive substitution with an initial guess of a random density distribution at all lattice sites. The initial random density distributions at all lattice sites represent the mathematical starting point for the successive substitution calculations and are not related to the density distributions in the real system evolving toward equilibrium. The latter are needed for dynamic calculations. Details of the equations used are given in the Supporting Information. Choice of Parameters (Table 1). A system of three types of particles with four poles gives rise to 82 independent parameters in three sets: i.e., concentration (3), interparticle interactions including only distinguishable pairs (78), and temperature (1). Temperature is a very important experimental parameter as it determines the feasibility of experiments. Of the remaining two sets, variations in relative concentrations have a greater effect than relative interaction energies when more than one type of particle competes for a specific binding site. Interaction energies can be used to tune the specific patch interactions and thereby control particle assembly. The interparticle interactions can be categorized as (i) the interactions that lead to the formation of the T-structure, including A-Bn and Bn-C particle patch interactions, and (ii) the interactions that do not take part in the formation of the T-structure and are responsible for uncontrolled particle condensation such as A-C particle patch interactions and nonspecific interactions between patches and bare particle surfaces. The large number of parameters (82) would require numerous calculations. Thus, we have reduced the parameter space to 8 parameters using the arguments given in the following. The parameter space investigated is summarized in Table 1. The two T-structures displayed in Figure 1 are comprised of the same number of A, Bn, and C particles; however, the Bn particle is different for each of the structures (n ) 1, 2). Case 1 (Figure 1a, n ) 1) presents an approach where the interactions between patches are defined in such a way that they only lead to formation of the T-structure, while in Case 2 (Figure 1b, n ) 2) structures different from the desired T-structure are possible. As a result, a different parameter space is chosen for each case. Case 1, B1: Symmetric Two-Patch Particle. To form the T-structure shown in Figure 1a, both the A-B1 and B1-C particle patch interactions must be attractive. The attractive A-B1 particle patch interaction assures the binding of a B1 particle to the A particle in a T-shaped precursor. Since B1 particles have identical patches on the two equatorial poles and both A and C particle patches have attractive interactions with B1 particle patches, the T-shape precursor attracts A and C particles at its terminal positions. To ensure the binding of C particles to B1 particles in the T-shape precursor, the B1-C
particle patch interaction should not only be attractive but has to be more attractive than a certain threshold to avoid formation of an A and B1 particle network. All other nonspecific interactions between nonbinding patches and bare surfaces are summarized under the term rest all interactions. Since rest all interactions account for attractive physical interactions, e.g., van der Waals interactions (0.02-1.3 eV),42 we calculate the maximum attractive rest all interaction that the system can tolerate without nonspecific condensation of particles. The concentration ratio of A:B1:C is chosen as 1:3:3 matching the ratio of particles in the targeted T-structure. Calculations with increased relative concentrations of B1 and C particles yield nonspecific condensation at lower attractive rest all interactions (data not shown). Decreased B1 and C particle concentrations enable the system to tolerate larger attractive rest all interactions; however, they will also lead to a smaller yield of T-structures formed (data not shown). Absolute particle concentrations of A of 0.001, B1 of 0.003, and C of 0.003 lead to maximum attractive rest all interactions in the range of feasible attractive forces from 0.01 to 0.1 eV. An increase of the absolute concentrations decreases the maximum attractive rest all interactions exponentially, while a decrease leads to higher maximum attractive rest all interactions (data not shown). Thus, for Case 1 we define the B1-C particle patch interaction as a strong attractive interaction, the A-B1 particle patch interaction as a weak attractive interaction, and all remaining nonspecific interactions as attractive rest all interactions. The concentration ratio of A:B1:C is kept at 0.001:0.003:0.003. The strong and weak attractive interactions are varied from 1.0 to 8.0 eV and 0.5-7.5 eV, respectively, such that the strong interaction is always at least 0.5 eV higher than the weak interaction. The temperature range from 275 to 375 K is investigated, because of the freezing and boiling point of water, which is a likely solvent for the assembly. Case 2, B2: Asymmetric Two-Patch Particle. In Case 2 (Figure 1b), attractive interactions between identical patches are used to form the T-structure. Therefore, B2 particles should have at least one patch identical with the patches of an A particle, binding B2 particles to the A particle in a T-shaped precursor, and another patch identical with the C particle patch, enabling the C particles to bind to the B2 particle ends of the T-shaped precursor, completing the T-structure. The asymmetry of the B2 particle patches ensures that the A particles will not bind to the B2 particles in the formed T-shaped precursor by attractive interactions between identical patches; therefore both the A-B2 and B2-C particle patch interactions are defined as strong attractive interactions. Note the strong attractive interactions between identical patches not only allow binding of A-B2 and B2-C particles but also allow interaction of same-type particles (A-A, B2-B2, and C-C) leading to various structures such as dimers, trimers, and chains. Thus, the relative concentrations
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TABLE 2: Carbon and Oxygen Binding Energies43 C-C bond
BDE [eV]
O-O bond
BDE [eV]
C-O bond
BDE [eV]
C2 C2H6 C4H10 C8H18
6.059 3.888 3.768 3.590
O2 H2O2 CH3O2CH3 C2H5O2C2H5
5.116 2.219 1.658 5.275
CO CH3OCH3 C2H5OC2H5 CH3OC2H5
11.109 3.594 5.490 3.652
TABLE 3: Metal Oxygen and Sulfur Binding Energies for Neutral Diatomic Species and Other Noncovalent Bonding Interactions BDE BDE M-O bonda [eV] M-S bonda [eV] Au-O Ti-Od Pt-O
2.30 6.94 4.06
Au-S Ti-Sd Pt-S
bonding
BDE [eV]
4.33 electrostatic 1 kTb,c 4.40 van der Waals 0.02-1.30c 2.43 H-bond 0.1-0.41c
Taken from ref 44. b At separations of ∼56 nm. Electrostatic interaction range (Debye length) is proportional to 1/xC0, C0 ) ionic concentration, k ) Boltzman constant, and T ) temperature. c Taken from ref 42. d Taken from ref 45. a
of A, B2, and C particles play an important role in the formation of the T-structure. The equilibrium density distributions around the A particle at (0,0) are studied with varying concentrations of A (0.0001-0.0005), B2 (0-0.0031), and C (0-0.016) particles. In contrast to Case 1, we define two groups of nonspecific interactions: (i) weak attractive interactions between non-identical patches and (ii) rest all interactions between patch/ bare and bare/bare surfaces. The reasoning behind the weak attractive interactions between non-identical patches is that there is always a finite probability that a functional group on a molecule of one patch interacts with a patch of another particle. The relative ratio of interaction energies influences the distribution between the different structures and is varied accordingly. Strong attractive (2.5-3.5 eV), weak attractive (1.0-2.0 eV), and repulsive rest all interactions (0.3-2.0 eV) for Case 2 are chosen such that they lead to a maximum phase space for T-structure formation. The temperature is kept at 300 K for all calculations. The attractive weak and strong interactions in both cases are in the range of covalent bonding energies (1.0-7.0 eV), while the attractive rest all interactions fall into the van der Waals interaction range (0.02-1.3 eV).42 Binding energies43 for C-C, O-O, and C-O bonds ranging from 1.66 to 11.11 eV are summarized in Table 2 and those for M-O and M-S bonds44,45 with M ) Au, Ti, and Pt ranging from 2.30 to 6.94 eV are summarized in Table 3. The third column in Table 3 lists energies for electrostatic, van der Waals, and H-bonding interactions. Identification of Phase Transitions. Finite structures (e.g., T-structure) are expected to form at the low concentrations of particles (
