Article pubs.acs.org/Langmuir
Dendritic Brushes under Good Solvent Conditions: A Simulation Study Leonidas N. Gergidis,† Andreas Kalogirou,† and Costas Vlahos*,‡ †
Department of Materials Science and Engineering, University of Ioannina, 45110 Ioannina, Greece Department of Chemistry, University of Ioannina, 45110 Ioannina, Greece
‡
ABSTRACT: The structural properties of polymer brushes, formed by dendron polymers up to the third generation, were studied by means of Brownian dynamics simulations for the macroscopic state of good solvent. The distributions of polymer units, of the free ends, of the dendrons centers of mass, and of the units of every dendritic generation and the radii of gyration necessary for the understanding of the internal stratification of brushes were calculated. Previous self-consistent field theory numerical simulations of firstgeneration dendritic brushes suggested that at high grafting densities two kinds of populations are evident, one of short dendrons having weakly extended spacers and another with tall dendrons having strongly stretched spacers. These Brownian dynamics calculations provided a more complicated picture of dendritic brushes, revealing different populations of short, tall, and in some cases intermediate height dendrons, depending on the dendron generation and spacer length. The scaling dependence of the height and the span of the dendritic brush on the grafting density and other parameters were found to be in good agreement with existing theoretical results for good solvents.
1. INTRODUCTION End-grafted polymer chains have important applications in the stabilization of colloidal suspensions, surface modification, reduction of viscosity compared to that of the bare surface, drug delivery systems, and biosensors.1−6 With regard to grafting density, two limiting regimes can be distinguished. At low grafting densities, polymer chains are isolated, occupying a half-sphere with a radius comparable to its radius of gyration, often called the mushroom regime. At high grafting densities, the polymer chains stretch away from the surface to avoid the unfavorable excluded volume interactions, leading to the formation of the polymer brush. Experimentally, the preparation of grafted polymers on surfaces follows two approaches, grafting-from and grafting-to techniques.7 The grafting-from technique is preferentially used when high polymer grafted densities are required. Theoretical models, computer simulations, and experiments have been employed by many research groups to calculate the relevant physical parameters such as density profiles, thickness, etc., of the polymeric brushes under different solvent conditions and grafting densities.8−12 To date, polymer brush studies, almost exclusively, focused on linear polymer chains. A few results regarding brushes composed of star, comb, and dendritic polymers are reported in the literature13−16 despite the synthesis of these architectures and the technology to attach polymers on surfaces being available. It is expected that brushes from dendritic polymers with a large number of free ends and excess polymer density at the periphery of the brush (Figure 1) may introduce novel desired features. © 2012 American Chemical Society
Figure 1. Cartoon representation of a dendritic G2F3S3 brush formed by a second-generation trifunctional dendron with a spacer length S of 3. The longest path (N) equals 9, and the total molecular weight (M) equals 21.
Polotsky et al.14 combined the numerical Scheutjens−Fleer17 self-consistent field theory approach with a Flory type analysis, based on the simple Alexander−de Gennes box model.18 The purpose was to study the structural organization of polymer brushes formed by dendritic molecules (dendrons) under good Received: October 8, 2012 Revised: November 5, 2012 Published: November 7, 2012 17176
dx.doi.org/10.1021/la3039957 | Langmuir 2012, 28, 17176−17185
Langmuir
Article
⎧ 2⎤ ⎡ ⎪−0.5kR 2 ln⎢1 − ⎛⎜ rij ⎞⎟ ⎥ , r ≤ R ⎪ 0 ij 0 ⎢⎣ ⎝ R 0 ⎠ ⎥⎦ Ubond(rij) = ⎨ ⎪ ⎪∞ , rij > R 0 ⎩
solvent conditions, which are grafted by the terminal unit of the root spacer to a planar surface. They found that at high grafting densities the packing of dendrons, in a manner independent of dendron generation and functionality, leads to a relatively uniform plateaulike distribution of monomer unit density. This implies the existence of an intrabrush segregation of dendrons having contributions that belong to different dendron subpopulations. For brushes formed by first-generation dendrons, two types of dendrons were found. The planar surface is enriched by units of dendrons with weakly extended spacers, while the other population of dendrons that is characterized by completely stretched root spacer and branches was found predominately at the outer surface of the brush. The effects of solvent quality on the span of dendritic brush, determined by the end-to-end distance of the strand joining the root to a terminal unit, were studied by Kröger et al.13 A Flory type theory of dendritic brushes yielded scaling rules, state diagrams, and information about the collapse transition. They have assumed that spacer units, junctions, and free end (terminal) units are identical in shape and interactions. The span of dendritic brushes for flexible spacers and unity Kuhn length scales as n1/3G2/3Sτ1/3δ−2/3, n1/2G1/2Sδ−1, and nS|τ|−1δ−2 for the good solvent, θ, and bad solvent, respectively, where n is the number of branching points (junctions), G the dendron generation, S the spacer length, δ the distance between dendrons, and τ the reduced temperature of the system. Their results suggest a continuous collapse in dendritic brushes. The scaling predictions of the dendritic brushes under various solvent conditions can be checked using numerical simulations. We employed Brownian dynamics simulations to elucidate the effects of the dendron architecture, such as spacer length, dendron generation, functionality of branching points, and dendron grafting density on the brush organization in good solvents. The properties of interest are the distributions of the dendron center of mass, dendron free ends, and units, the mean distance of units from the surface (directly proportional to brush thickness), the mean square radii of gyration components and of the whole dendron, and the end-to-end distance between the grafting point and the terminal units. In addition, the back folding probabilities of spacers were calculated. Our results are compared with existing theoretical findings from the literature.
(1)
where rij is the distance between units i and j, k = 25Tε/σ2, and R0 is the maximal extension of the bond (R0 = 1.5σ). These parameters19 prevent chain crossing by ensuring an average bond length of 0.97σ. Monomer−monomer interactions were calculated by means of a truncated and shifted Lennard-Jones potential: ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎛ ⎞12 ⎛ ⎞6 ⎤ ⎪ ⎢⎜ σ ⎟ σ σ σ ⎥ ⎪ 4ε⎢⎜ r ⎟ − ⎜⎜ r ⎟⎟ − ⎜⎜ r ⎟⎟ + ⎜⎜ r ⎟⎟ ⎥ ⎪ ⎝ ij ⎠ ⎝ cij ⎠ ⎦ ⎝ cij ⎠ ⎝ ij ⎠ ULJ(rij) = ⎨ ⎣ ⎪ + ε , rij ≤ rcij ⎪ ⎪ 0, rij > rcij ⎩
(2)
where ε is the well depth and rcij is the cutoff radius. Between polymeric units and the substrate wall there are no interactions. The surface is considered as a reflecting wall. The solvent molecules are considered implicitly. The Brownian dynamics simulation method allows the statistical treatment of the solvent, incorporating its influence on the polymer by a combination of random forces and frictional terms. The friction coefficient and the random force couple the simulated system to a heat bath, and therefore, the simulation has canonical ensemble (NVT) constraints. The equation of motion of each unit i of mass m in the simulation box follows the Langevin equation: mi rϊ (t ) = −∇ ∑ [ULJ(rij) + Ubond(rij)] − miξ ri̇ (t ) + Fi(t ) j
(3)
where mi, ri, and ξ are the mass, the position vector, and the friction coefficient of unit i, respectively. The friction coefficient is equal to ξ = 0.5τ−1, where τ = σ(m/ε)1/2. The random force vector Fi is assumed to be Gaussian, with zero mean, and satisfies the equation ⟨Fi(t ) ·Fj(t ′)⟩ = 6kBTmξδijδ(t − t ′)
(4)
where kB is the Boltzmann constant and T is the temperature. Grafted dendritic polymers of first (G1), second (G2), or third (G3) generation with branching point functionality three (F3) and four (F4) and spacer length varying from ten (S10) to fifty two (S52) units are simulated. Except from the total molecular weight M = S[(FG+1 − 1)/(F − 2)], we also introduce a quantity N describing the monomer units along the longest path. This path includes one spacer of each generation N = S(G + 1). The Brownian dynamics simulations were performed using the open-source massive parallel simulator LAMMPS,20,21 extensively used in our previous studies of self-assembly of dendritic-star, H-shaped polymers.22−24 Different cutoff distances in the Lennard-Jones potential were used to describe the interactions between polymer units under good solvent conditions. Under good solvent conditions, where the interactions are repulsive, we have set cutoff radii rcij equal to 21/6σ. For the sake of simplicity, all units were considered to have unity mass (m = 1) and diameter (σ = 1).
2. MODEL The substrate surface for our dendritic polymer brushes is taken to be a piece of the x−y plane with linear dimensions L × L and periodic boundary conditions applied in the x- and y-directions. We consider np the number of polymeric dendrons, of equal molecular weight M grafted onto the plane by the end unit of the root spacer. The grafting separation distance between dendrons is set to δ in the x- and y-directions as shown in Figure 1. The surface area per dendron s is calculated as s= δ2, and therefore, the grafting density of dendrons is specified as d = 1/s = 1/δ2. We employed a coarse-grained model to study the dendritic polymers. A group of atoms was modeled as a unit (with diameter σ), while different units were connected with flexible finitely extended elastic bonds (FENE). The FENE potential is expressed as 17177
dx.doi.org/10.1021/la3039957 | Langmuir 2012, 28, 17176−17185
Langmuir
Article
In this work, dendritic brushes with 196 polymeric dendrons and varying dendron grafting densities d ranging from 1/(3σ2) to 1/(36σ2) were simulated. The integration time step (Δt) was set to 0.012τ. We performed 1 million time steps with all cutoff radii set (rcij) equal to 21/6σ. The system then was allowed to equilibrate for 1 million steps. According to the autocorrelation functions of the radius of gyration and the span of the dendron, the aforementioned equilibration period was adequate to eliminate any bias introduced from the initial conformation. The production phase of the simulation was subsequently conducted for 50 million steps. The properties of interest were calculated from 2000 snapshots of the simulated system.
3. RESULTS AND DISCUSSION 3.1. Internal Stratification of Dendritic Brushes. 3.1.1. Dendritic Brushes Formed by Trifunctional Dendrons. We start with the analysis of dendritic brushes formed by dendrons up to third generation with functionality F = 3 and spacer length S = 10 (G1F3S10, G2F3S10, and G3F3S10). The smallest grafted areas per dendron (s) we were able to achieve were 3σ2, 4σ2, and 9σ2 for the G1F3S10, G2F3S10, and G3F3S10 dendrons, respectively. In Figure 2, the probabilities of finding a dendron unit, the dendron’s free ends, and the dendron's centers of mass at a distance z perpendicular to the substrate surface are plotted for various grafting densities, namely, 1/ (3σ2), 1/(6σ2), and 1/(12σ2) for G1F3S10 brushes, 1/(4σ2), 1/ (8σ2), and 1/(16σ2) for G2F3S10 brushes, and 1/(9σ2), 1/ (18σ2), and 1/(36σ2) for G3F3S10 brushes, respectively. For the G1F3S10 brushes (panel a) with the highest grafting density [1/ (3σ2)], the probability of finding any unit at height z from the surface reveals a plateau region for a wide range of z values indicating a constant density of the brush. Close to the surface there is a very small depletion regime. For high values of z, the probability distribution decays abruptly and vanishes at z = 18, very close to the dendritic longest path N = 20, indicating very stretched branches for both generations. The probability distribution of the dendron's centers of mass is 2-fold, indicating the existence of two kinds of populations of dendrons. We calculate the total mean square radius of gyration of the dendron, ⟨Rg2⟩, and to improve our understanding of the differences in the conformations of the two populations of grafted dendrons, we additionally calculate its components 2 ⟨Rgxy2⟩ and ⟨Rgz2⟩ defined as ⟨Rgxy2⟩ = (1/M)∑M i=1[(xi − xcm) + 2 (yi − ycm)2] and ⟨Rgz2⟩ = (1/M)∑M i=1[(zi − zcm) ], where M is the total number of dendron units and xcm, ycm, and zcm are the coordinates of the dendron's center of mass. The ⟨Rgxy2⟩ and ⟨Rgz2⟩ components of the mean-squared radius of gyration, ⟨Rg2⟩, describe the size of the dendron projections on the x−y plane and on the z-axis, respectively. The probability distributions of these quantities are presented in Figure 3a. It is observed that for the half of the dendrons, ⟨Rgxy2⟩ takes small values of ∼1.5. Also, dendrons having ⟨Rgxy2⟩ values of 6 are evident. For the purpose of comparison, we also simulated the G1F3S10 brush with an infinitesimal grafted density using a single grafted dendron in a very large surface (mushroom regime). The respective value of ⟨Rgxy2⟩ is 6.18, indicating 2.5fold lateral compression of dendrons in the brush with respect to the mushroom regime dendrons. The ⟨Rgz2⟩ distribution is 2fold and takes much higher values than ⟨Rgxy2⟩ because the
Figure 2. Probability density distributions of z for units, free ends, and the dendron's centers of mass for dendron brushes (a) G1F3S10, (b) G2F3S10, and (c) G3F3S10 for different grafting densities. The uncertainty in the simulated quantities is
