J. Phys. Chem. B 2007, 111, 5819-5828
5819
Molecular Dynamics Simulations of Dendritic Polyelectrolytes with Flexible Spacers in Salt Free Solution Yong Lin, Qi Liao,* and Xigao Jin State Key Laboratory of Polymer Physics and Chemistry, Joint Laboratory of Polymer Science and Materials, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China ReceiVed: January 21, 2007; In Final Form: March 23, 2007
We present the results of molecular dynamics simulations of dendritic polyelectrolytes in dilute salt-free solutions. The dendritic polyelectrolytes are modeled as an ensemble of regular-branched bead-spring chains of neutral and charged Lennard-Jones particles with explicit counterions. A wide range of molecular variables of the dendritic polyelectrolytes such as generation number, spacer length, and charge density were considered in the simulations. The effect of dendrimer size on relaxation time, the conformation of spacers, and the size dependence of the dendrimer on molecular variables are discussed and compared with a Flory type theory. The osmotic coefficients of the dilute dendritic polyelectrolyte solutions, as well as the profiles of monomers and counterions, are calculated directly from the simulations. Our simulation results show that the inner spacers of the dendrimers are extensively stretched, and the size dependence on the molecular weight deviates from the scaling prediction that assumes a Gaussian elasticity of the spacer.
1. Introduction Dendrimers are regularly branched macromolecules with a large number of terminal groups. The well-defined dendritic structure with monodisperse molecular weight may be fabricated by the recursive synthesis, either by the divergent approach,1 sequentially filling out generations from an initial core, or by the convergent approach,2 attaching branched arms to a core. The unique molecular architecture and dimensionality of a dendrimer may alter the local chemical environment of the molecule significantly in contrast to the traditional linear polymers; for example, the intrinsic viscosity of dendrimers exhibits a maximum with increasing molecular weight while that of linear polymers shows a linear dependence on molecular weight.3,4 These unique properties of dendrimers suggest a wide range of potential applications as novel alternatives of traditional polymeric materials in chemical catalysis,5-7 drug delivery,8 and viscosity modification.9 During the past two decades, significant efforts have been made theoretically and experimentally to understand the solution and bulk behaviors of neutral dendrimers. The main progress is the confirmation of the dense-core structure10 of neutral dendrimers in solution by many theoretical and experimental works. The prediction of the structure of neutral dendrimers in various solvents by Flory type theory based on the dense-core structure is in perfect agreement with the results of selfconsistent field calculation,10 molecular dynamics (MD),11 and Monte Carlo simulations.12 A summary of the related topic based on neutral dendrimers is found in ref 4. Considerable attentions have been paid to dendritic polyelectrolytes with ionizable groups that can dissociate into charged monomers and counterions in aqueous solution. These charged dendrimers are used for different applications, especially in host-guest systems because the inner space of the molecular structure has been predicted to undergo significant changes in * To whom correspondence should be addressed. E-mail: qiliao@ iccas.ac.cn.
size and density profile with a change in molecular variables and local environment. For example, the charge density can be changed by adjusting the pH value and the salt concentration of the solution.13,14 However, some experimental results reported that the structural flexibility of the dendrimer does not change with varying the pH value of the solution because the spacers contain only a few monomers, which should be considered as rigid spacers.15 We expect that the flexibility of the chain will significantly affect the static and dynamic properties of the dendrimer in the potential applications and, therefore, should be investigated and understood quantitatively. To our knowledge, there are only a few reports of simulations on dendrimers with long flexible spacers. Borisov and Zhulina16 studied weakly charged starburst polyelectrolytes in good solvents using scaling and self-consistent field theory. They assumed all spacers of the charged dendrimers were Gaussian chains and based the theory on this premise. But as we know, topological constraints on dendrimers require the charged monomers to be located in a confined space, which may mean that the spacers are strongly stretched. Up to now, the majority of simulation studies of dendritic polyelectrolytes have not yet considered the effect of counterions explicitly. The counterion interaction was implicated by the screened Coulomb interaction in the precious simulations.13,14,17 In understanding the counterion effect, two regimes must be considered. The first is known as the osmotic regime where the dendritic polyelectrolytes tend to absorb most counterions into their occupied space. In contrast, the counterions may distribute in all of the space because of the entropy consideration in the polyelectrolyte regime. As the dendrimer generation increases from low to high, the crossover from the polyelectrolyte regime to the osmotic regime must be considered, which is significantly complicated for theoretical prediction. Theoretical and simulation works have been devoted to the dynamics of neutral and charged dendrimers. Lyulin et al.17 presented some results on the dynamic properties of dendrimers with short spacers by molecular dynamic simulations. For the
10.1021/jp070514l CCC: $37.00 © 2007 American Chemical Society Published on Web 05/08/2007
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charged dendrimers, they considered only the case of terminal charged groups. Their results were not quantitative because few data points were considered. However, qualitative results from their simulations show that the theoretical prediction of dynamics based on the Rouse model of a flexible spacer could not be adequate to describe the dynamic behavior of the dendrimers.18 The aim of the present paper is to analyze and verify theoretical prediction of the equilibrium conformational structure of dendritic polyelectrolytes with long spacers in dilute solutions by performing MD simulations. A Flory type theory of strongly stretched dendrimers in dilute solutions is also proposed, and the predictions of this model are tested by the simulations. The simulations also give us chances to compare our results with the theoretical prediction of dynamic properties of dendrimers in which the Rouse model for spacers connecting two consequent branching points has been used, especially for the case of neutral dendrimers. This work with explicit counterions is a direct way to overcome the theoretical difficulties of handling the crossover from the polyelectrolyte regime to the osmotic regime and can help us to understand the effect of counterions on the molecular structure. In the osmotic regime, the clouds of the counterions could be dragged by the movement of the dendrimers. The dynamics of the charged dendrimers will deviate from the standard Rouse model because the interactions between monomers could be transferred by the absorbed counterions even in the simulation model without explicit solvents and an algorithm of hydrodynamic interaction. Our simulation results may be helpful in further understanding the dynamics of charged dendrimers. The rest of the paper is organized as follows. The simulation model and the algorithms are described in section 2. We provide a detailed account of the simulation results for spacer length, molecule size, and form factors in dilute solutions, as well as compare the simulation results with the prediction of the strongly stretched dendritic polyelectrolyte model, in section 3. Finally, in section 4, a summary of the results is presented. A strongly stretched dendritic polyelectrolyte model, which is based on the Flory type mean field theory, is introduced in the appendix. 2. Model and Methodology A dendritic polyelectrolyte of g generation is represented in our simulation by an ensemble of spacer chains each consisting of n Lennard-Jones (LJ) monomers with a charge fraction f. The total number of monomers of one dendrimer is N ) n(2g+1 - 3) + 1 (Figure 1a). We have considered dendritic polyelectrolytes with a series of charge fractions ranging from f ) 0 (neutral chain) to f ) 1/2 (every second monomer charged). All charged particles are monovalent; therefore, the total number of charged monomers is equal to the number of counterions Nc ) fN, and counterions are confined in a cubic simulation box of size LC with periodic boundary conditions (Figure 1b). The main simulation results in good solvent are summarized in Table 1. Excluded volume interactions between every pair of monomers are modeled by the truncated shifted LJ potential (or called Week-Chandler-Andersen potential19) set to zero at the cutoff,
ULJ(r) )
{
[(σr ) - (σr ) - (rσ) + (rσ) ] r e r
4LJ
12
6
12
c
0
6
c
c
r > rc
(1)
where the cutoff distance is equal to rc ) 21/6σ for good solvent, which means that interaction between monomers is purely repulsive. The parameter LJ controls the strength of the shortrange interactions, and the simulations for the good solvent were
Figure 1. Schematic picture of a four generation dendrimer in our simulation model (a). Snapshot from the simulation of a four generation dendrimer in solution with concentration c ) 1.0 × 10-4 σ-3, number of monomers in spacer n ) 30 and charge fraction f ) 1/5. Counterions, charged, and neutral monomers are represented by blue, red, and green spheres, respectively (b).
carried out for LJ ) 1.5kBT (kB is the Boltzmann constant and T is the absolute temperature). θ point is usually defined by the temperature at which the second virial coefficient between the chains vanishes. The attractive and repulsive interactions cancel for the polymer at θ point, and the polymer shows ideal chain behavior. For comparison, we also carried out the simulations of the dendritic polyelectrolyte in the θ solvent by choosing LJ ) 0.34kBT and rc ) 2.5σ, which is the θ point corresponding to the uncharged system.20 The connectivity of monomers in the chains is maintained by the finite extension nonlinear elastic (FENE) potential
( )
1 r2 UFENE(r) ) - kR02 ln 1 - 2 2 R
(2)
0
where k ) 7LJ/σ2 is the spring constant and R0 ) 2σ is the maximum bond length at which the elastic energy of the bond becomes infinite. The FENE potential gives only the attractive
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TABLE 1: Radii of Gyration of Dendritic Polyelectrolyte of Different Generation, Spacer Lengths, and Charge Fraction in Good Solvent Rg/σ f
n
g)3
g)4
g)5
0 0 0 0 1/30 1/15 1/10 1/10 1/10 1/10 1/6 1/5 1/5 1/5 1/5 1/3 1/2 1/2
10 20 30 40 30 30 10 20 30 40 30 10 20 30 40 30 20 30
6.98 ( 0.82 10.3 ( 1.19 13.0 ( 1.13 16.4 ( 1.35
9.44 ( 0.78 14.8 ( 1.21 18.0 ( 1.12 21.5 ( 1.38 20.2 ( 1.47 24.9 ( 1.59 11.0 ( 0.70 19.7 ( 1.21 28.8 ( 1.40 36.7 ( 1.48 34.2 ( 1.17 13.3 ( 0.63 24.9 ( 0.94 36.3 ( 1.28 46.9 ( 1.31
12.2 ( 0.71 19.2 ( 1.12 22.1 ( 1.01 27.9 ( 1.24 28.3 ( 1.57 36.6 ( 1.50 15.6 ( 0.81 29.1 ( 1.04 41.8 ( 1.43 55.6 ( 1.43 49.3 ( 1.08 19.0 ( 0.62 35.8 ( 0.94 51.8 ( 1.04 66.9 ( 1.30
7.38 ( 0.69 12.7 ( 1.11 17.7 ( 1.44 23.5 ( 1.86 8.54 ( 0.62 15.9 ( 1.01 23.2 ( 1.29 30.8 ( 1.74 21.5 ( 0.76 31.3 ( 0.95
g)6
14.8 ( 0.76 25.3 ( 1.44 30.7 ( 1.03 38.6 ( 1.18 41.0 ( 1.22 51.2 ( 1.41 21.3 ( 0.64 39.9 ( 1.01 57.9 ( 1.26 75.6 ( 1.56 66.5 ( 1.03 25.5 ( 0.57 48.1 ( 0.85 69.7 ( 1.08 90.9 ( 1.06 78.6 ( 0.91 32.2 ( 0.80 44.4 ( 0.72 58.3 ( 0.65 47.0 ( 1.05 64.7 ( 0.85 85.1 ( 0.76
g)7 19.3 ( 0.72 29.4 ( 0.56 36.6 ( 0.74 42.4 ( 1.09 28.2 ( 0.59 53.2 ( 0.96 77.2 ( 1.14 32.5 ( 0.52 62.2 ( 0.73 90.9 ( 0.80 74.1 ( 0.60 108 ( 0.78
part of the bond potential. The repulsive part of the bond potential is provided by the LJ interaction (eq 1 above). The solvent is represented by a continuum with the dielectric constant . In such effective medium representation of the solvent, all charged particles interact with each other via the unscreened Coulomb potential
UCoul(r) ) kBT
lBqi qj r
(3)
where qi is the charge valence of the ith particle equals to +1 for a positive charge and -1 for a negative charge. The Bjerrum length lB ) e2/(kBT) determines the strength of the electrostatic interactions. The electrostatic interactions between all charges in the simulation box and all their periodic images were computed by the smoothed particle mesh Ewald (SPME)21 algorithm implemented in the DL_POLY version 2.14 software package.22 The MD simulations were performed by the following procedure. The initial conformation of one polyelectrolyte dendrimer in the cubic cell with periodic boundary conditions was generated as a set of self-avoiding walks. The counterions were placed randomly in the unoccupied space of the simulation box. Both theta and good solutions of dendrimers with generation g varying from 2 to 7 and Bjerrum length lB ) 1.0σ were studied over charge fractions f ranging from 0 to 1/2. The monomer number concentration is 0.0001σ-3, which ensures the dendrimer in the dilute regime far below the overlap concentrations in all simulations. The simulations were carried out at a constant temperature using the Langevin thermostat.23 A velocity Verlet algorithm was used to integrate the equations of motion with a time step equal to ∆t ) 0.012τLJ, where τLJ )
xmσ2/kBT is the characteristic LJ time. The numbers of MD
steps chosen are large enough to allow the mean square endto-end distance and the mean square radius of gyration to relax to their equilibrium values. This requirement led to a simulation range between 500 000 and 4 000 000 MD steps (10-100 relaxation times of squared radius of gyration) depending on the generation number and charge fraction of the dendrimer. Further details and discussion of the relaxation time will be presented in section 4. Table 1 also presents the squared root of the mean square of gyration radii of the dendrimers averaged after the simulated solution had equilibrated.
3. Results and Discussion A. Relaxation Time. The dynamic properties of dendrimers in solution are of great interests because of their unique viscosity behavior. For example, the intrinsic viscosity has a nonmonotonic dependence on the molecular weight. However, simulations of the dynamic properties were limited by the algorithm of the hydrodynamic interaction and the computer capacity. Only the noise dependence of relaxation time on generation (or molecular weight)11 was obtained, or few sets of dendrimer systems were investigated17 in previous studies; therefore, it is difficult to draw quantitative conclusions from their simulation results. We have two motives in discussing the dynamics of dendrimers in the first sections of our results and conclusions, in particular, the relaxation time as it relates to dendrimer size. First, we want to ensure that the simulations are long enough to give sufficient averaging of the static properties. From the simulation point of view, an equilibrium state is crucial to the statistics of any static properties of the dendrimers in solution. In our simulation model there is only one dendrimer in a large periodic box, so we may monitor the fluctuation of the dendrimer size to check whether the internal structure has equilibrated. The more important reason is to investigate the idea of the well-known electroviscous effect due to the counterions of the charged dendrimer system and to compare it with the theoretical prediction of the dynamics of the Rouse model and Zimm model. The solvents in our model are treated as a viscous continuum interacting with all particles by a random force, so initially, the Rouse model of dynamics may be expected. However, the explicit counterions in our simulation may transfer the interaction between the monomers as the hydrodynamic interaction of the solvent molecules because most of the counterions will be absorbed into the charged dendrimer in the osmotic regime. Thus, even we did not take into account explicitly the hydrodynamic interaction in our algorithm; the dynamics of the Zimm model could not be ruled out when we analyzed the relaxation time of dendritic polyelectrolytes with explicit counterions. The relaxation of the internal structure of the dendrimer can be monitored by the fluctuation of dendrimer size, or the squared radius of gyration. The dynamics is characterized by the autocorrelation function CRg2(t) for the squared radius of gyration in most of the literature11,17, which is evaluated by the expression:
CRg2(t) )
〈(Rg2(t) - 〈Rg2〉)(Rg2(0) - 〈Rg2〉)〉 〈Rg4〉 - 〈Rg2〉2
(4)
The generation dependence of CRg2(t) for a neutral dendrimer with spacer monomer number n ) 30 is shown in Figure 2a, and that for charged dendrimers with f ) 1/5 is shown in Figure 2b. The estimated relaxation time of the function either from the long-time slope or from CRg2(t) ) 1/e gives noisy results with a large error bar. To obtain reliable relaxation times, we fit CRg2(t) to the stretched exponential function C(t) ) exp[-(t/R)β], and then the relaxation time is estimated by the time integral of the fitting function C(t).24 The relaxation times τ estimated in the manner described above for neutral and charged dendrimers are shown in Figure 3. The longest relaxation time is about 200 000τ LJ for neutral dendrimers of seven generations, and the run time of the sample lasts about 2 000 000τ LJ in our simulation. We keep the run time in the range of 10-100 relaxation times of the squared radius of gyration for all samples to guarantee the equilibrium state of the systems. The relaxation time of the neutral dendrimer is
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Figure 2. Autocorrelation function of Rg2 for neutral (a) and charged (b) dendrimers with n ) 30 and f ) 1/5.
much longer than that of the corresponding charged one because the electrostatic repulsion increases the internal strain of the dendrimer backbone, which leads to a faster fluctuation of the internal structure. Cai and Chen18,25 have developed a theoretical model to predict the relaxation time of a dendrimer with flexible spacers at different time scales. In their prediction, the relaxation time of internal elastic motion based on Rouse dynamics for a large generation dendrimer is
τ ) 2g+1 ζ/K ≈ Nζ/ω
(5)
where ζ is the friction coefficient for the single monomer, ω ) 3kBT/b2, and K ) 3kBT/a2 with the average spacer length a ) n1/2b, and b is the Kuhn length for the flexible spacer. Our simulated relaxation time of squared gyration radius is a function of the generation number and the molecular weight of the dendrimers as shown in Figure 3a,b, respectively. Our results for the neutral dendrimer confirm eq 5 and show a very good prediction of the size fluctuation dynamics of the dendrimer with flexible spacers free from the hydrodynamic interaction proposed by Cai and Chen.18,25 The simulation of dendrimers with short spacers and no hydrodynamic interaction17 shows that the dynamics of the size fluctuation differs significantly from Cai and Chen’s theoretical prediction. Their simulation results, which show a weak dependence of N, are not observed in our simulation of dendrimers with long flexible spacers. As proposed by Lyulin
Figure 3. Relaxation time of squared gyration radius as a function of generation number (a) and molecular weight (b) with different charge densities. The solid lines are fitting results by eq 5, and the dash lines are guide for eyes (see the text for details).
et al.,17 the difference between the simulation results of dendrimers with short spacers and the theoretical prediction is due to the spacer flexibility, which is clearly supported by our simulation results of dendrimers with long spacers. The relaxation time of size fluctuation of charged dendrimers in our simulation shows a weak dependence on N with increasing charge density of the dendritic polyelectrolytes (Figure 3a,b), which deviates from the Rouse model prediction. This weak dependence is in agreement with the simulation results of the dynamics behavior of the dendrimers with hydrodynamic interaction by Lyulin et al.17 but deviates from the prediction of Cai and Chen,18,25 which shows a linear dependence of relaxation time on the generation number.18,25 As suggest by Lyulin et al.,17 the large generation dendrimer with hydrodynamic interaction may be considered as an impenetrable sphere in a viscous medium, and the relaxation time of size fluctuation should be independent of the size of the dendrimer. Our simulation results of the weak dependence on generations or molecular weight show the crossover from the Rouse model (τR ∝ N, solid lines in Figure 3a,b) to the Zimm model (τR ∝ N1/2, dash lines in Figure 3a,b) of a flexible dendrimer. We conclude that the behavior of the Zimm model is caused by the counterions absorbed into the dendrimers and this effect would be intensified by increasing the charge density and molecule weight of the dendrimer.
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Figure 4. Spacer lengths at different generations. The solid line gives the fitting result of eq 9, ri ) 42.0 - 11.9 exp(i/7.97), i ) 8 - i′.
B. Spacer Length. A snapshot of the charged dendrimer for g ) 4 and f ) 1/5 are shown in Figure 1b. The snapshot clearly shows that the spacers of the first and second generations are strongly stretched. In contrary to the case of neutral dendrimers, the assumption of Gaussian spacers in the previous scaling theory26 is invalid for the charged dendrimer with large charge density and high molecular weight. A weak size dependence on the charge density may be expected because of the finite extension of the spacer. We develop a scaling theory to analyze the conformation character of the strongly stretched dendritic polyelectrolyte model, in Appendix. The lengths of the stretched freely joined linear spacers are (see Appendix for details)
[
ri ) nb coth(β1/2i-1) -
]
2i-1 β1
where i ) 1, 2, ..., g (6)
where β1 ) f1b/kBT, f1 is the force acting on the spacer of the first generation, and b is the bond length or Kuhn length of the order of σ for our flexible dendrimer. It is possible to get simple relationships for the spacer length dependence on force for two limiting cases. For small relative elongations (ri , Rmax ) nb), the spacer length dependence on the force is approximately linear and Hooke’s law is followed by a Gaussian chain:
ri ) r1/2i-1 )
β1 3‚2
i-1
nb
(
)
2i-1 nb β1
observed that the most outside spacers have the same length and are almost independent of the generation number of the dendrimers. The spacer length of the first generation (r1) increases with the generation number of the dendrimers, as shown in Figure 5a. First generation r1 is related to the molecular variables by the following equation (see Appendix for details)
2(Rmax - Rg) r1 1 )1- )1) nb β1 Nb
(7)
For large extensional force βi ) β1/2i-1 . 1, the extension has the simple form
ri ) 1 -
Figure 5. Spacer length of the first generation for dendrimers of different generation (a), dependence of r1 on the Rg/N for the charged dendrimer (b). The charge density is f ) 1/5, and the spacer monomer number is n ) 30.
(8)
This means that the spacer lengths in the same dendrimer exponentially decrease with an increasing generation number of the spacers given the force acting along them is large enough. Figure 4 shows our simulation results of the spacer length dependence on the generation number in the same dendrimers. The solid fitting curve gives the exponential dependence on the ith generation. This exponential fit, which means that the spacers are strongly stretched, is in good agreement with the observation from the snapshot, as well as the prediction of eq 8. We also
1-
(
lBbφ2f 2N 2 Rg2
)
(1 - φ)fNb + Rg
-1
(9)
where φ is the effective charge fraction of the dendrimer. It is also possible to get the simple relationship for the spacer length dependence of force for two limiting cases, corresponding to the polyelectrolyte (φ ) 1) and osmotic (φ ) 0) regimes:
{
Rg2
1, polyelectrolyte regime r1 lBbf 2N 2 ) nb Rg 1, osmotic regime fNb
(10)
Figure 5b shows the dependence of r1 on the Rg/N for the charged dendrimer with f ) 1/5 and n ) 30. We suppose that the deviation from the linear dependence is because of the crossover from the polyelectrolyte regime to the osmotic regime.
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Figure 6. Size of neutral (a) and charged (b) dendrimers. Figure 7. Density profiles of monomers (a) and counterions (b) of dendritic polyelectrolytes with n ) 30, g ) 6 in a dilute solution.
C. Radius of Gyration. The properties of neutral dendrimers in solvent of variant quality have been extensively studied both theoretically and experimentally.18,25,27 For a neutral dendrimer in good solvent,
( ) ( )
Rg 5 Rg 3 νN = 3 Rg0 Rg0 b (ng)1/2
where Rg0 ≈ b(ng)1/2 (11)
If Rg/Rg0 . 1, then the second term of the left side of eq 11 can be ignored, giving
Rg = ν b N (ng) 1/5 2/5 1/5
2/5
(12)
Figure 6a shows our simulation results of Rg for the neutral chain. The error bars are at the order of the point size. All data points with different spacer lengths and generation numbers can be collapsed into a line with slope of 2/5, which means the prediction of eq 12 is in good agreement with our simulation results of neutral dendrimers. The electrostatic interaction between charges leads to the complicated and rich behavior of polyelectrolyte solutions in contrast to those of neutral polymers. The most important difference is that the osmotic pressure caused by the counterions in the inner space of charged dendrimers plays a key role in the intramolecular repulsion, while the excluded volume interaction between monomers is the main contribution for the nonionic dendrimers. A scaling theory of dendritic polyelectrolytes with
relatively low generation and sufficiently long spacers was developed by Wolterink et al.26, which gives the Rg ∼ f1/2 dependence typical in the osmotic regime for all types of branched polyelectrolytes with Gaussian elasticity. Figure 6b shows our simulation results of Rg for the charged dendrimers. All data points with different charge densities and sufficiently long spacers can be collapsed into a universal line,
( )
Rg f 2N ) Rg,max - Rg n
1/4
(13)
The above empirical function deviates from the simple predictions for two limiting cases, corresponding to the polyelectrolyte (φ ) 1) and osmotic (φ ) 0) regimes. We argue that the scaling prediction of Wolterink et al. systematically overestimates the size of dendritic polyelectrolytes because of the assumption of Gaussian elasticity of the spacers. We consider the finite extension of the spacers and a revised version of the theoretical prediction for the size of dendritic polyelectrolytes presented in the Appendix. The effect of charge renormalization and the localization of counterions inside the macromolecule make a comparison of the predicted and simulated results quantitatively impossible for relatively low generations. The crossover from the polyelectrolyte to the osmotic regime happens at a critical generation
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J. Phys. Chem. B, Vol. 111, No. 21, 2007 5825
Figure 8. Density profiles of monomers (a) and counterions (b) of dendritic polyelectrolytes with n ) 30, f ) 1/5 in a dilute solution.
Figure 9. Osmotic coefficients of dilute solutions of dendritic polyelectrolytes (a) and the phase diagram of the two zone model (b). The solid line is γR ) γ0 (see the text for details).
g*, which depends on the charge densities. We cannot simulate charged dendrimers with very high generations because of the computer capacity; thus, the simulations cover only a range of the transition from low g to high g. D. Density Profiles. The density distribution function of the monomers and counterions from the dendrimer center may give important structure information. The density distribution function of the monomers and counterions, called density profile in the following discussion, is defined as g(r) ) 〈n(r)〉/Vs(r), where 〈n(r)〉 is the ensemble average number of monomers or counterions in the shell at distance r from the mass center of dendrimer and Vs(r) is the volumes of the shells. The density profiles of monomers and counterions for dendrimers with different charge densities in good solvent are shown in Figure 7a,b, respectively. The oscillations become more pronounced with increasing the charge density of the dendrimers. The position of the minima corresponds to the branching points of different generations. The monomer density profiles with more pronounced minima clearly confirm the extensive stretching of the inner spacers. The monomer density is inversely proportional to the square of the distance in the range between the neighboring generations (the dashed lines in Figure 7a with a slope of -2). Therefore, the dependence was attributed to the rod-like spacers in the inner generations because of extensive
stretching. The radii of gyration are marked by the arrows. The peaks around the radii of gyration indicate that there is a shell in the charged dendrimer, which is contrary to the monotonically decreasing profile for a neutral dendrimer. The counterion density profiles shown in Figure 7b suggest that the counterions almost homogeneously distribute in the inner space of the dendrimer. A non-monotonic profile could be observed in the most strongly charged dendrimers. Figure 8 shows the density profiles of monomers (a) and counterions (b) for dendrimers of different generations in good solvent. The absorption of the counterions into the inner space of the dendrimer indicates a crossover from the polyelectrolyte regime to the osmotic regime (Figure 8b). The absorbed counterions screen the elastic interaction and stretch the inner generation spacer; therefore, the cascade structure of different generation spacers is observed in monomer density profiles. E. Osmotic Coefficient. Figure 9a presents the generation number dependence of the osmotic coefficient of dendritic polyelectrolytes, defined as the ratio of the solution osmotic pressure π to the ideal mixture osmotic pressure kBTcf of all counterions. The osmotic coefficient φ decreases with increasing charged density and molecular weight. This behavior of the osmotic coefficient is similar to that reported for polyelectrolytes in the θ solvent and poor solvent.
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In our previous work,28,29 we have shown that the osmotic coefficient φ of the dilute polyelectrolyte solution can be approximated by an empirical function
(
)
γR cRe(c)3 φdilute ≈ exp Be γ0 N
(14)
where Re is the root-mean-square end-to-end vector of flexible chains, γ0 ) QlB/Rg2 is the reduced bare charge density of the macroion, γR ) QRlB/Rg2 is the reduced effective charge density of macroion with radius Rg, and B is the numerical coefficient on the order of unity. Figure 9b shows the dependence of osmotic coefficients on the parameter γR/γ0, which indicates the effective charge density at the surface of the dendrimers. F. Form Factors. Dendrimer conformations are studied by various scattering experiments. Small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS) techniques can be used to analyze the intramolecular structure of dendrimers.30-32 In dilute solutions, the scattering is dominated by the contribution of a single dendrimer, and both techniques measure the single molecule form factor
Pintra(q) )
1 N2
〈
∑ i,j
exp(iq b‚b r ij)〉 )
1 N2
〈∑ 〉 sin(qrij) qrij
i,j
(15)
where b q is the wave vector, b rij is the vector between the ith and jth monomers of the dendrimer and brackets denote ensemble average over all possible dendrimer conformations. Figure 10 shows the form factors of dendrimers with n ) 30 obtained from our simulations for different generations (Figure 10a) and different charge densities (Figure 10b). The peaks in the form factor at large q (around 6.6 σ-1) correspond to length scales on the order of a bond length and are due to correlations between positions of neighboring monomers along the spacer backbone. As can be seen in these figures, there are local minimums, which become significant with an increase in the generation number and the charge density. The first minimum may contribute to the overall size of the dendrimer in the previous experimental works.15,31 The form factor of a uniform sphere of radius R is
P(q) ) 9
[
]
sin(qR) - qR cos(qR) (qR)3
Figure 10. Form factors of dendritic polyelectrolytes with n ) 30 and f ) 1/10 in a dilute solution at different generations (a). Form factors of dendritic polyelectrolytes with n ) 30 and g ) 6 in a dilute solution at different charge densities (b).
2
(16)
which gives the first minimum of the form factor at qminR ) 4.49, or qminRg ) 3.48. More general is the form factor of the shell of radius R with a hollow core of radius r
P(q) ) 9
[
]
sin(qR) - qR cos(qR) - sin(qr) + qr cos(qr) q (R - r ) 3
3
3
2
(17)
The structural information for a single dendrimer could be extracted in the q range from the first minimum to the peak position, which corresponds to the spacers. The first minimum becomes more significant, and its position moves toward lower q values as generation and charge density increase. The generation and charge density dependence of the maximum indicate that it can be associated with the dendrimer size. To verify the correlation between the minimum and the overall dendrimer size, we show the value of qminRg for dendrimers with a different charge fraction in Figure 11. All data points
Figure 11. Relationship between qmin and Rg for dendritic polyelectrolytes with n ) 30 and g ) 6.
fall into the line qminRg ) 3.48 within the range of the error bars, which is indicative of a sphere structure of the charged dendrimer. To display the conformation evolution of a dendrimer with increasing the charge density, lines with a slope of -5/3 corresponding to the neutral spacer in good solvent and -1 corresponding to the rod-like structure are put in Figure 9b as a guide. In the case of the strongly charged dendrimers,
MD Simulations of Dendritic Polyelectrolytes additional oscillations are noted. These oscillations are associated with the correlations between the spacer locations along the dendrimer backbone. With increasing the generation and charge density, the magnitude of this peak decreases a little while its position shifts toward lower q values. This shift is a result of the spacer extension, which leads to an increase in the length. The significant dependence of size on the charge density has not been verified yet experimentally for a dendrimer with relatively short spacers, for example, poly(amidoamine),15 or for most of the commonly available dendrimers today. Our simulation results suggest that flexibility of the long spacer plays a key role in the more pronounced sensitivity to the charge density, which relates to the change of solution pH value and ionic strength; thus, the dendritic polyelectrolyte with flexible spacers may provide more controlled release of drug molecules though volume and density changes of dendrimers. 4. Conclusions We have performed MD simulations on neutral and charged dendrimers with long spacers in dilute solutions. The dendrimers were modeled as an ensemble of Lennard-Jones particles, and explicit counterions were included for the charged systems. No algorithm of hydrodynamics interaction was considered in the simulations. It is observed that the relaxation time of neutral dendrimers with long spacers agrees well with the prediction of the Rose model, while that of the charged dendrimer shows a weak dependence on generations or molecular weight with increasing the charge density. The simulation results reveal that the inner spacers of charged dendrimers are strongly stretched and the most outside spacers always have the same length for dendrimers with the same n value. The spacers in high generation charged dendrimers are strongly stretched for their exponential dependence on the ith generation, and the spacer length of the first generation (r1) increases with the generation number of the dendrimer. Our simulation results of Rg for a neutral dendrimer are in good agreement with theoretical prediction. For a charged dendrimer, the simulated results of Rg could be described by an empirical function, deviating from our theoretical prediction on the two limiting cases, that is, polyelectrolyte and osmotic regimes. The monomer profile of the charged dendrimer also indicates that there is a shell in the charged dendrimer, and the minima in the profile confirm the extensive stretching of the inner spacers. The counterions distribute homogeneously in the inner space of the charged dendrimer except for the most strongly charged one, which presents a non-monotonic profile of counterion distribution. The osmotic coefficient φ decreases with increasing charged density and molecular weight, similar to the results reported for polyelectrolytes in the θ and poor solvent. The simulation results reveal that local minima in the form factors become significant with an increase of the generation number and charge density, and all data of qminRg fall into the line qminRg ) 3.48, which unambiguously demonstrates the spherical structure of the charged dendrimer. Acknowledgment. This work was financially supported by NFSC Grants 20474075 and 200674091, Innovation Funding No. KJCX2-SW-H07 from the Chinese Academy of Sciences, and 973 Grant 2003CB615604 from the Chinese Ministry of Science and Technology. Appendix Flory Type Theory of Strongly Stretched Dendrimers. a. Polyelectrolyte Regime Without Confinement Entropy. If coun-
J. Phys. Chem. B, Vol. 111, No. 21, 2007 5827 terions of charged dendrimers are distributed evenly throughout the whole system, this is known as the polyelectrolyte regime. In the polyelectrolyte regime, the free energy of the charged dendrimer comprises two terms, Fconf (conformational entropy for the extension of all of the spacers) and Fcoulomb (accounting for unscreened Coulomb repulsion).
F ) Fconf + Fcoulomb
(18)
Fcoulomb ) kTlB f 2N 2/Rg
(19)
The total radius of the dendrimer can be written as: g
Rg = r1/2 +
ri ∑ i)2
(20)
We assume all spacers of the dendrimer are strongly stretched, so
(
r1 ) 1 -
(
ri ) 1 -
) )
( (
)
1 1 R ) 1nb β1 max β1
)
1 i-1 1 2 Rmax ) 1 - 2i-1 nb β1 β1
(21)
where β1 ) f1b/kT, f1 is the force acting on the spacer of the first generation, b is the bond length, k is the Boltzmann factor, and T is the absolute temperature. From eqs 20 and 21, we get
β1 )
(2g+1 - 3)nb Nb ) (2g - 1)nb - 2Rg 2(Rg,max - Rg)
(22)
Conformational entropy of the first generation Fcon,1 can be rewritten as17
(
Fcon,1 ) -kTn ln 1 -
)
r1 ) kTn ln β1 nb
(23)
and conformational entropy of the ith generation Fcon,i is
(
Fcon,i ) -kTn ln 1 -
)
ri β1 ) kTn ln i-1 nb 2
(24)
The total conformational entropy of the dendrimer is the sum of all contributions of spacers, that is Fcon ) Fcon,1 + g 2iFcon,i Ignoring the terms not depending on Rg, we get ∑i)2
Fcon ) kTn(2g+1 - 3) ln β1 ≈ kTN ln
Nb (25) Rg,max - Rg
Substituting eqs 19 and 25 into eq 18 and minimizing the total free energy, we get
Rg2 ) lB f 2N Rg,max - Rg
(26)
b. Osmotic Regime Without Confinement Entropy. If the Coulomb interaction inside a charged dendrimer is screened predominantly by counterions trapped within the dendrimer volume, this is referred to as osmotic regime. In this regime, swelling of the charged dendrimer is ensured by counterions trapped in the pervaded volume of the dendrimer. When the intramolecular Coulomb interactions between charged monomers are partially screened by the ions of a low molecular weight salt, the repulsive Coulomb interaction inside the dendritic
5828 J. Phys. Chem. B, Vol. 111, No. 21, 2007
Lin et al.
polyelectrolyte can be described as a short-range binary repulsion with the effective second virial coefficient VDH = f 2/cs, where cs is the salt concentration.16,26 The contribution of the screened Coulomb interaction is13
(
)
( )
f 2lB N2 f 2 N2 FCoulomb ) kT ν + 2 ≈ kT ν + cs R 3 κ Rg3 g
(27)
where κ2 ) 8πlBcs. Combining eqs 18, 25, and 27 and minimizing the total free energy, we get
( )
Rg4 f2 )N ν+ Rg,max - Rg cs
(28)
In the osmotic regime of a salt-free solution, cs is roughly equal to the concentration of counterions (cs = fN/Rg3) in the pervaded volume of the charged dendrimer. Ignoring the excluded volume of monomers, we get
Rg )f Rg,max - Rg
(29)
The above results could also be derived from the force balance point of view. The elastic force acted on the first generation spacer is balanced by the electrostatic repulsion of the effective charges in the dendrimer and the force caused by the osmotic pressure; that is,
lBφ2f 2N 2 (1 - φ)fN f 1 β1 N + ) ) ) kT b 2(Rg,max - Rg) Rg R2 g
(30)
where φ is the effective charge fraction of the dendrimer. The first term on the right side in eq 30 comes from the electrostatic repulsion of the effective charges, and the second term comes from the osmotic pressure, which may be estimated by the total pressure acting on the surface of the dendrimer. We may set φ ) 0 to get eq 29 in the osmotic regime or set φ ) 1 to get eq 26 for a dendrimer at low generation, which is in the polyelectrolyte regime. In reality, there is a crossover from the polyelectrolyte regime to the osmotic regime as generation and
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