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Molecular Modeling of Dendrimers for Nanoscale Applications

by
Tahir Cagin*, Guofeng Wang, Ryan Martin, and William A. Goddard III

aCalifornia Institute of Technology, Materials and Process Simulation Center and Division of Chemistry and Chemical Engineering
Pasadena, CA 91125 USA

*tahir@wag.caltech.edu
http://www.wag.caltech.edu/home/tahir/

This is a draft paper for the
Seventh Foresight Conference on Molecular Nanotechnology.
The final version has been submitted
for publication in the special Conference issue of
Nanotechnology.


Abstract

Dendrimers and hyperbranched polymers represent a novel class of structurally controlled macromolecules derived from a branches-upon-branches structural motif. Dendrimers are well defined, highly branched macromolecules that radiate from a central core and are synthesized through a stepwise, repetitive reaction sequence that guarantees complete shells for each generation, leading to polymers that are monodisperse. The synthetic procedures developed for dendrimer preparation permit nearly complete control over the critical molecular design parameters, such as size, shape, surface/interior chemistry, flexibility, and topology. Synthetic techniques proved effective include the Starburst divergent strategy, the convergent growth strategy, and the self-assembly strategy. These methods have proved effective in generating macromolecules with a unique combination of properties. Recent results suggest that dendritic polymers, a new class of macromolecules developed over the last decade may provide the key to developing reliable and economical fabrication and manufacturing of functional nanoscale materials that would have unique properties (electronic, optical, opto-electronic, magnetic, chemical, or biological) that could be the basis of new nanoscale technology and devices. In this paper, we describe some of the dendrimers obtained from each route and study their energetic and structural properties using CCBB to build the structures and Molecular dynamics and minimization to anneal these molecular level representations.

Introduction

Dendrimers are a new class of three-dimensional, man-made molecules produced by an unusual synthetic route which incorporates repetitive branching sequences to create a unique novel architecture. Exceptional features of the dendritic architecture include a high degree of structural symmetry, a density gradient displaying an intra-molecular minimum value and a well defined number of terminal groups which may be chemically different from the interior. The combination of these features creates an environment within the dendrimer molecule facilitates an avenue to developing reliable and economical fabrication and manufacturing of functional nanoscale materials that would have unique properties (electronic, optical, opto-electronic, magnetic, chemical, or biological) that could be the basis of new nanoscale technology and devices.

Dendrimers and hyperbranched polymers represent a novel class of structurally controlled macromolecules derived from a branches-upon-branches structural motif. [1] Dendrimers are well defined, highly branched macromolecules that radiate from a central core and are synthesized through a stepwise, repetitive reaction sequence that guarantees complete shells for each generation, leading to polymers that are monodisperse. [2] The synthetic procedures developed for dendrimer preparation permit nearly complete control over the critical molecular design parameters, such as size, shape, surface/interior chemistry, flexibility, and topology. [1,2] Synthetic techniques proved effective include the Starburst divergent strategy (Tomalia and coworkers [1,2] ), the convergent growth strategy (Fréchet and coworkers [3]), and the self-assembly strategy (Zimmerman and coworkers [4]). These methods have proved effective in generating macromolecules with a unique combination of properties. [5,6]

The geometric characterization of dendrimer structure has lagged this rapid progress in synthesis and design. [1] The problem is that these molecules possess an enormous number of energetically permissible conformations, and in solution there is rapid interchange between them. Thus diffraction techniques yield little structure information. Also a number of generations involve the same monomers, making it difficult to extract precise information about the local structure from infrared or NMR experiments. Thus the most precise experimental data about overall structure comes from size exclusion chromatography (SEC). [1,2] The main experimental data about the geometric character of particular sites has come from NMR relaxation times for molecules able to partially penetrate into the dendrimer.[7]

A particular advantage of using theory is that the properties of new materials can be predicted in advance of experiments. This allows the system to be adjusted and refined (designed) so as to obtain the optimal properties before the arduous experimental task of synthesis and characterization. However, there are significant challenges in using theory to predict accurate properties for functional dendritic materials.

Predicting the structure, dynamics, and properties of dendrimers at the nanoscale and microscale regimes requires substantial improvements in theory (FF and simulation methodologies) and the software (the algorithms and their implementation in order to do the calculations). Below we describe some recent developments in the area of dendrimers and molecular modeling applications to a list of dendritic polymers, PAMAM, Stimuli responsive polymers, Colloidal crystals of self assembled dendrimers, dendritic architectures for molecular imprinting.

Results

The Continuous Configurational Boltzmann Biased Direct Monte Carlo Method for Polymers

To predict the properties of polymers it is necessary to determine an ensemble of conformations highly populated at the temperature and pressure of interest. The most efficient method for predicting these conformations is by Monte Carlo (MC) sampling. However, for polymers with molecular weights of interest in polymer applications, such MC is too slow. The CCBB, an improved Monte Carlo method, was developed to solve this difficulty. CCBB combines continuous configurational biased (CCB) direct sampling method with Boltzmann factor biased (BFB) enrichment. CCBB is 300,000 times faster than simple sampling Monte Carlo in generating the free energy properties of polymer chains [8] CCBB has been implemented in a program compatible with the MPSim MD software. Here we have applied it to generate initial structures for branched, hyperbranched and hybrid polymers to be used atomistic simulation and structural and energetic characterization of dendrimers.

PAMAM: Change in structure as a function of generation

Recently, dendritic polymers have been used as soluble templates/unimolecular reactors from which nano-clusters of inorganic compounds or elements can be synthesized. The basic concept involves using dendrimers as hosts to preorganize small molecules or metal ions, followed by a simple in situ reaction which will immobilize and stabilize domains of atomic or molecular guest components (inorganic compounds as well as elemental metals). In one of these examples poly(amidoamine) (PAMAM) dendrimers have been used, to attract copper(II) ions inside the macromolecules where they are subsequently reacted with solubilized H2S to form metal sulfides. These organic/inorganic, dendrimer-based hybrid species have been termed 'nanocomposites' and display unusual properties. For example, solubility of the nanocomposites is determined by the properties of the host dendrimer molecules. This allows for solubilization of the inorganic guest compounds in environments in which they are inherently insoluble. Since it has been established that there is no covalent bond between host and guest, these observations suggest that the inorganics are physically and spatially restricted by the dendrimer shell. However, this structure has not been verified. Here we use structure building and molecular mechanics and dynamics techniques to investigate the structural characteristics of PAMAM dendrimers.

We carried out molecular dynamics simulations at room temperature to investigatethe structure of the PAMAM dendrimers up to generation 7. We have used two different initiators, ethylenediamine (EDA) and ammonia (NH3). The core and the monomers determines the number of atoms for each generation by

c + c*m + ... + c * m n - 1 = c * (mn - 1)/(m - 1)

Thus, the number of atoms increases exponentially with the generation number leading to steric overlaps after some generation. [12] Figure 1 shows the structure of PAMAM-EDA generation 7 dendrimer.


FIgure 1.

In the molecular model building process, we used an annealing schemes which uses successive steps of energy minimization and molecular dynamics runs. After constructing initial structures molecular dynamics simulations are carried out at T = 300 K using DREIDING FF [9]. Each generation (including the half generation simulations were carried out over 200 ps (with a time step of 1 fs). Initial 100 ps of the run is treated as equilibration. The structural analysis carried over the last 100 ps of each simulation. In Table 1 we list the calculated radius of gyration for generations 1 through 7th.

Table 1. Variation of radius of gyration at 300 K as a function of generation and with amine and ethylene di-amine cores.

 G      Rg ( Å )      Rg ( Å )
 1      4.971490      3.774253 
 2      7.034312      6.030406 
 3      9.774769      8.409326 
 4     13.016935     11.155713 
 5     16.355944     16.001749 
 6     21.668465     20.598940 
 7     27.624895     26.454330 

Self-Assembly of dendrimers leading to colloidal crystals and discotic liquid crystals

In 1989, it was predicted that a change in dendritic shape to a nearly spherical one should occur upon increasing the generation number [1] The absence of long-range order required for X-ray analysis allowed only indirect evidence to be provided for this concept. Recently Percec and coworkers [11] reported the synthesis of three generations of self-assembling monodendrons based on die AB(3) building block methyl 3,4,5-trihydroxybenzoate. The first 3,4,5-tris[p-(n-dodecan-1-yloxy)benzyloxy]benzoic acid and the second-generation methyl 3,4,5-tris{3',4',5'-tris[p-(n-dodecan-1-yloxy)benzyloxy]benzyloxy}benzoate monodendrons self-assemble into cylindrical supramolecular dendrimers that self-organize in a two-dimensional p6mm lattice. The third-generation monodendron 3,4,5-tris(3',4',5'-tris{3 ",4 ",5 "-tris[p-(n-dodecan-1-yloxy)benzyloxy]benzyloxy}benzyloxy)- benzoate self-assembles in a spherical dendrimer that self-organizes in a three-dimensional cubic Pm3-n lattice. Structural analysis of these lattices by X-ray diffraction provided the first direct demonstration of the supramolecular dendrimer shape change from cylindrical to spherical and indirect determination of the average shape change of the monodendron from a quarter of a disk to a half of a disk and to a sixth of a sphere as a function of generation number. These results have demonstrated the concept of monodendron and supramolecular dendrimer shape control by generation number.

As depicted in Figure 2, dendrimers can be designed to aggregate to form cylinders or spheres depending upon the nature of the fundamental building unit. One class of these systems leads to cone type dendrimers that organize into spheres (2 to 12 per sphere depending on generation) which then pack into the unusual Pm3-n (A-15) type cubic crystal structure having spheres at the corners and body center of the cube plus two on each face.


Figure 2.

In CCBB technique [8] starting from monomer, we have built structures for generations 2 through 4 dendrimers. We then carried out successive minimization and dynamics simulations to anneal the structures to obtain stable structures. Figure 3a, displays the generation 2 dendrimer, Figure 3b is generation 3 and finally Figure 3c is the generation 4. In Table 2, we have listed the physical properties of these dendrimers.


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Figure 3a
Generation 2
Figure 3b
Generation 3
Figure 3c
Generation 4

Table 2. Theoretical number of mondendrons, molecular weight, number of atoms and number of end groups for (3,4,5)n12Gn-COOH spherical supramolecule

n	u 	Mw 	NA	NC	NH       NO      NE

2      12    25192.6   4680    1632    2880     168     108
3 	6    38234.1   7092    2490    4356     246     162
4       2    38382.3   7116    2504    4368     244     162

u = Number of mondendrons in a spherical supramolecular dendrimer
Mw = Molecular weight
NA = Number of atoms in supramolecule
NC, NH, NO = Number of C, H and O atoms in supramolecule
NE  = Number of end groups in supramolecule

Table 3. Average radius of gyration, average radii and average lattice parameters of (3,4,5)n12Gn-COOH spherical supramolecule


n         Rg (A)             Rv (A)            a =4Rv (A)
     CCBB    Theo.       CCBB    Theo.     CCBB    Theo.   Exper.

2    13.84   17.19       16.91   18.77     67.66   75.07   68.3 
3    16.84   19.82       19.80   21.56     79.21   86.22   79.2
4    18.28   20.55       20.32   21.59     80.91   86.35   84.0

Rg = Radius of gyration
Rv = Derived from 3/4 pi Rv3 =V, V is volume of supramolecule

We also have analyzed the variation of density as a function of distance from the center of mass of the dendrimer, Figure 4.


FIgure 4

We then carried out large-scale MD calculations on this system to determine the equilibrium structure at room temperature. These are large systems with 37000 atoms (excluding hydrogen) per unit cell of pm3-n structure. The equilibrated structures led to xray diffraction intensities in good agreement with experiment, Figure 5.


FIgure 5

This self assembly of this system represents a model problem for a coarsening approach starting from classical atomic level simulation and averaging over atoms to obtain a supramolecular representation. To do this, we performed molecular dynamics simulation on isolated single spherical assembly at 300 K. We also performed successive simulations with two spherical supramolecular assemblies as the distance between the sphere centers varied until they overlap. From these simulations we determined interaction enegetics of a pair of spherical assemblies. The interaction energy can be approximated by a flat bottom Morse potential. The inner core and dispersion parts have the same depth, however different minima. Preliminary studies using the interaction potential gave rise to stable Pm3-n structure. [11]

Stimuli-responsive macromolecules

Fréchet [3] has prepared new stimuli responsive macromolecules based on linear, star and dendritic blocks. A novel amphiphilic hybrid macromolecule has been synthesized by Fréchet et al. [3] This macromolecule has hydrophobic dendritic groups at the periphery of a hydrophilic polyethylene glycol (PEG) star. Light scattering experiments suggest that changing the solvent from THF (tetrahydrofuran) to methanol leads to large changes in structure. To study the response of these macromolecules to such variations in environment, we used molecular dynamics (MD) to predict structures and properties for the macromolecule in methanol and THF. These calculations used explicit solvent and periodic boundary conditions. The Fréchet macromolecule has 2761 atoms and we included also 26,892 methanol molecules or 13,140 THF molecules. We used the MPSim MD program to carry out 700ps of NVT dynamic simulations at 300K. These results show that that in THF the Frechet hybrid macromolecule has a somewhat compact PEG core with the dendrimer extending outward into the solvent while in methanol the PEG tends to wrap around the dendrimer to bury it away from the solvent. These results validate the interpretations by Fréchet et al.

We have first carried out large-scale MD calculations on this system in two different solvents (THF and H3COH) to determine the equilibrium structure as a function of solvent and temperature. The structures of the slightly extended and coiled forms of dendrimers are shown in Figure 6. The variation of radius of gyration vs time is given in Figure 7.


Figure 6. Last structures obtained from MD simulations with explicit solvents 84 monomer coil (left) and 113 monomer coil (right).


Figure 7

Dendrimers for Unimolecular Imprinting

The development of unimolecular imprinting is critical in many nanoscale applications ranging from the development of new chemical sensors, electronic, chemical sensing, nano- and molecular-electronics and shape and functional group selective binding especially for nanoscale catalytic reactors.

Dendrimers due to their very well defined structures are the most likely candidates for this purpose. Recently, Zimmerman has proposed dendrimers with homoallyl groups on their periphery which could be linked through a ring closing methathesis reaction. These dendrimers contain three cleavable ester bonds at their core with robust ether linkages throughout the remaining structure.

The determination and study of the three dimensional molecular struture of these supramolecular assemblies is a challenging problem. Using Continuous Configurational Biased Monte Carlo Method we have developed candidate 3-D molecular structures, after annealing these structures using molecular dynamics methods we have generated the final supramolecular dendrimer structures for molecular imprinting.

Below we present the results of these structural, energetic and functional properties of these dendrimers for nanoscale applications. These structures are analyzed using their final energy as criteria. Energy distribution for structures for the final structures is given in Figure 8. Average Radius of Gyration for the ensemble of 90 molecules, is 15.93 Å. The standard deviation of the distribution is only 0.42 Å. Figure 9, displays the distribution of radius of gyration for the ensemble of structures generated through the CCBB procedure described in this section. We also measured the principal axes lengths of the generated ensemble. In Table 4 we list the values of Principal axis components of the moment of inertia tensor.


Figure 8.

 


Figure 9.

Table 4. Principle moments of inertia

Axis	Moment	standard deviation
x	9.20		0.93
y	9.38		0.84
z	8.89		0.81

The distribution of x-, y- and z-principal axis components of radius of gyration tensor is displayed in Figure 10 a through 10 c.


larger image

larger image

larger image
Figure 10. Principal axis components of radius of gyration tensor calculated over ensemble of 90 structures:
a) x-axis b) y-axis c) z-axis

In Figure 11, we display a self assembled monolayer of these structures to demonstrate the use to create a very regular imprinting pattern.


Figure 11.

Discussions

To conclude we would like to address some of the connections between dendrimers and technological applications especially in the nanotechnology.

Recent results suggest that dendritic polymers, a new class of macromolecules developed over the last decade [1-7, 13] may provide the key to developing reliable and economical fabrication and manufacturing of functional nanoscale materials that would have unique properties (electronic, optical, opto-electronic, magnetic, chemical, or biological) that could be the basis of new nanoscale technology and devices.

Dendrimer technology has been established to obtain cone shaped, spherical, or disk like shapes that are mono disperse with sizes in the range of 2 to 12 nm. These structures can be designed to be containers for organic dye molecules or for metal or semiconductor clusters, with exteriors that dissolve them in appropriate media or stick them onto appropriate surfaces. The sequestered metal clusters of Fe and Co show magnetic properties and the sequestered semiconductor probably behave as spherical quantum dots. [14]

Probably the metals could be prepared pure and the surfaces oxidized to provide protection or to provide a barrier for electronic purposes. The cones, spheres, or disks can be covalently attached at regular spaces to polymer backbones to form linear necklaces. Thus one can imagine a regular array of metals or quantum dots whose spacing and diameter can be controlled exactly. These linear necklaces that can be directed to associate in bundles which could then be cross-linked to provide stable 2 dimensional networks of metals or quantum dots. Used as a mask this could be the basis for nanolithography with lines of -2 to 5 nm. Used as a detector it could provide a nano channel plate of magnificent resolution. Used as an active electronic element it could provide for entirely new types of devices.

The current magnetic clusters are magnetically soft, which could be useful for some applications. Placing a cap on the dendrimer that interacts strongly with the metal surface could increase their coercive force. Using appropriate dendrimers the shape of the metal could be changed to disk-like or cylinder, providing the possibility of using as a magnetic storage medium. The dendrimers can be designed so that the interior or exterior is hydrophobic or hydrophilic and rigid or flexible and small pore or large pore. This provides enormous opportunities for chemical sensors.

Acknowledgements

The research projects reported in this paper are supported by grants ARO-MURI, ARO-AASERT and ARO-DURIP. The facilities of MSC is also supported by funds from NSF (CHE 95-22179), DOE-ASCI, NASA/Ames, Avery Dennison, BP Chemical, Beckman Institute, Chevron Petroleum Technology Co., Chevron Chemical Co., Exxon, Owens Corning and Seiko-Epson.

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