Adsorption of Charged Dendrimers: A Brownian Dynamics Study

As the inverse screening length increases, the dendrimer expands in the normal direction and contracts in the parallel direction, adopting a conformat...
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J. Phys. Chem. B 2007, 111, 8728-8739

Adsorption of Charged Dendrimers: A Brownian Dynamics Study Balram Suman and Satish Kumar* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, 151 Amundson Hall, 421 Washington AVenue Southeast, Minneapolis, Minnesota 55455 ReceiVed: December 29, 2006; In Final Form: April 25, 2007

The adsorption of isolated charged dendrimers onto oppositely charged flat surfaces is studied in this work using Brownian dynamics simulations. The dendrimer is modeled as a freely jointed bead-rod chain in which excluded-volume interactions are modeled by a repulsive Lennard-Jones potential and bead-bead and beadsurface electrostatic interactions are described by screened Coulombic potentials. Adsorption behavior is studied as a function of inverse screening length, dendrimer generation, and dendrimer charge distribution. Adsorbed dendrimers adopt a disclike conformation in which they flatten in the direction normal to the surface and expand in the direction parallel to the surface. As the inverse screening length increases, the dendrimer expands in the normal direction and contracts in the parallel direction, adopting a conformation that is more stretched in the normal direction. When the inverse screening length becomes sufficiently large, the dendrimer desorbs and adopts a spherelike conformation. Bead density profiles show that adsorbed dendrimers form a two-layer structure, with one layer corresponding to adsorbed beads and a second, less dense layer corresponding to beads one rod length away from the surface. They also reveal how the distribution of monomers within the dendrimer and near the surface can be tailored by changing various problem parameters. The results presented here are expected to be helpful in providing qualitative guidance for dendrimer design in various applications.

1. Introduction Dendrimers are synthetic, highly branched, nearly monodisperse polymers in the nanometer size range.1 Since their introduction, they have received much scientific attention as they offer the control that modern drug delivery demands for the chemical nature of the drug carrier, its molecular weight and dimensions, and surface and internal structure.2,3 Drug delivery with dendrimers requires interaction with cell walls, making it essential to understand dendrimer behavior near surfaces. Apart from drug-delivery applications, Tully and Frechet4 highlight the use of dendrimers as surface functionalization agents for applications in membranes, adhesion, microelectronics, and chemical and biological sensors.5 An important class of dendrimers is those whose monomers can be electrostatically charged, as the nature of the charge distribution will govern dendrimer conformation and adsorption to adjacent surfaces. In this work, we apply Brownian dynamics (BD) simulations to explore the behavior of isolated charged dendrimers near oppositely charged flat surfaces. Numerous theoretical and experimental studies have focused on the behavior of charged and neutral dendrimers in free solution (see ref 6 and references therein). A number of experimental results concerning dendrimer adsorption onto surfaces have also been reported.4,7-16 In general, dendrimer conformations are flatter near surfaces than in free solution.7,8 For systems where electrostatic effects are important, it is found that the amount of dendrimer that adsorbs onto an oppositely charged surface can be controlled by manipulating dendrimer generation9,11 and surface charge.11,13 In some cases, nonelectrostatic interactions can also play a key role.14 There have been relatively few theoretical studies of dendrimer adsorption. Mansfield17 studied the adsorption of a * Author to whom correspondence should be addressed. Phone: (612) 625-2558. Fax: (612) 626-7246. E-mail: [email protected].

dendrimer using a lattice model in which some beads (representing dendrimer segments) are attracted to a surface by nonelectrostatic interactions. It was observed that as the strength of the dendrimer-surface attraction increases the dendrimer spreads out and flattens on the surface. Striolo and Prausnitz18 studied the effect of surface roughness and a nonelectrostatic polymer-surface attractive potential on the adsorption of dendrimers and linear polymers. When the strength of the attractive forces is weak, higher-generation dendrimers readily adsorb whereas linear polymers do not, and they reasoned that the dendrimer experiences a relatively small entropic penalty compared to that for linear polymers. When the attractive forces become sufficiently strong, both types of polymers adsorb. Ratner et al.19 used BD simulations to determine how dendrimer-surface contact area changes with the strength of a nonelectrostatic dendrimer-surface attractive potential. They observed that as the strength of the attraction increases so does the contact area. Mecke et al.20 performed atomic force microscopy experiments and atomistic molecular dynamics simulations of terminally charged and neutral poly(amidoamine) (PAMAM) dendrimers near a mica surface. They found that both terminally charged and neutral dendrimers flatten when in contact with the surface, and the degree of deformation is higher with the terminally charged dendrimers. Apart from the work of Mecke et al.,20 we are not aware of any other simulations that study the adsorption of charged dendrimers. Rather than performing atomistic molecular dynamics simulations, we carry out BD simulations in this study. By using coarse-grained models for the polymer and solvent, BD allows for simulations of phenomena that occur over relatively long time scales. Information obtained about how various properties change as key parameters are varied can give insight into what types of behavior might be expected to be independent of specific atomistic structure. Here, we examine the effect of

10.1021/jp069033c CCC: $37.00 © 2007 American Chemical Society Published on Web 07/07/2007

Adsorption of Charged Dendrimers

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8729 (Gaussian) displacement with zero mean and variance 2Doij∆t. As we are interested here in equilibrium conformations, hydrodynamic interactions are neglected, and the diffusion tensor is taken to be diagonal with nonzero elements kBT/ζ. Excluded-volume interactions are incorporated using a Lennard-Jones repulsive potential

ULJ(rij) ) 4

Figure 1. Bead-rod model of a second-generation dendrimer. The dashed circles denote the zeroth, first, and second generations of the dendrimer.

dendrimer generation, salt concentration, and different dendrimer charge distributions on dendrimer size and structure near oppositely charged flat surfaces. The latter two variables were not probed in the study of Mecke et al.20 Salt concentration is accounted for through the inverse screening length of a DebyeHu¨ckel potential, and four different types of charge distributions are considered. These types are (i) all terminal groups charged, (ii) only branching groups charged, (iii) all groups charged, and (iv) only some terminal groups charged. Each type can be synthesized and may be useful for different applications.21 The simulation model and algorithm are discussed in section 2, results and a discussion are presented in section 3, and conclusions are given in section 4. 2. Simulation Model and Algorithm A bead-rod model is used to describe the dendrimer molecules. Each bead represents a monomer and has a friction coefficient ζ. The rod length, denoted by l, represents the distance between two adjacent connected monomers. We consider dendrimers with AB2-type (trifunctional) monomers, which means that three rods are connected to every branching bead. Such a coarse-grained model has been used by Lyulin and co-workers to study various characteristic features of dendrimers, and their simulations yield results that agree well with experimental observations.22-24 The number of branching beads connecting the core bead and a peripheral bead indicates the dendrimer generation, G. A schematic of a bead-rod model of a second-generation dendrimer is shown in Figure 1 The branches emerging from the central bead are called dendrons, and there are three in Figure 1. The total number of beads, N, in a trifunctional dendrimer of generation G is given by

N ) 3(2G+1 - 1) + 1

(2.1)

Bead positions are advanced in time using the ErmakMcCammon algorithm25

ri ) roi + ∆t/kBT

∑j Doij‚Foj + Φoi

(2.2)

where the superscript “o” indicates the variables to be evaluated at the beginning of the time step, with roi and ri representing the position vectors for the ith bead before and after the time step. The time step size is denoted by ∆t, kBT is the product of Boltzmann’s constant and temperature, Doij is the diffusion tensor, and Foj is the sum of the excluded-volume and electrostatic forces on the jth bead. The term Φoi is a random

() σ rij

12

(2.3)

where rij is the distance between the ith and the jth beads. We have taken σ ) 0.8l and  ) 0.3kBT, as these were used in several previous Brownian dynamics simulations22-24,26 and proposed by Rey et al.27 to be appropriate for an athermal solvent. The force acting on bead i due to the presence of another bead j is

FLJ ij

)

{

48

()() σ rij

rˆ ij , rij < 2.5σ rij rij g 2.5σ

12

0,

(2.4)

where rˆ ij is the unit vector along rij and the cutoff radius is 2.5σ.24 The total force acting on bead i is the sum of the forces LJ due to all other beads, i.e., FLJ i ) ∑i*jFij . Bead-bead electrostatic interactions are described by a screened Coulombic (Debye-Hu¨ckel) potential. The potential energy associated with bead i is

VRi

)

kBTlBq ∑ i*j

2

e-κrij rij

(2.5)

where q is the nondimensional charge on each bead (made dimensionless by the electronic charge, e), lB ) e2/(4πkBT) is the Bjerrum length for a medium of permittivity , and κ is the inverse screening length. Increasing κ corresponds to increasing the salt concentration since electrostatic forces become more screened. The corresponding force on bead i is

FRi )

∑ i*j

( )

kBTlBq2

1

+κ

rij

e-κrij rˆ ij rij

(2.6)

The electrostatic force between a charged bead and a uniformly charged surface is given by the product of bead charge and the electric field due to the surface. The electrostatic potential energy due to an oppositely charged flat surface at the ith bead is

VSi ) -4πkBTσsq

lB 2

lκ

e-κzi

(2.7)

where σs is the nondimensional surface charge density (made dimensionless by e/l2) and zi is the z-coordinate of the ith bead with z being the direction normal to the surface, which is located at z ) 0. As the beads cannot cross the charged surface, a repulsive force is added similar to the one used in Panwar and Kumar.28 The force on the ith bead due to the surface is then

{

( )

5E rc 6 e , z e 0.25l rc 0.25l z i 5E rc 6 0.25l < zi < 0.45l FEi ) r z ez, c i ∂Vsi zi g 0.45l - ez, ∂zi

()

(2.8)

8730 J. Phys. Chem. B, Vol. 111, No. 30, 2007 where ez is the unit vector in the z-direction. In the simulations reported here, we have taken E ) 7kBT and rc ) 0.375l, which prevents beads from crossing the surface. Note that for computational ease the repulsive force is taken to be constant for zi e 0.25l and the constant force is evaluated at zi ) 0.25l. The SHAKE algorithm29,30 is employed to maintain constant rod lengths. In this algorithm, constraint equations for the rod lengths are multiplied by Lagrange multipliers, and differentiation of their sum yields an expression for the constraint force on bead i in terms of the unknown multipliers. This expression can then be used to update the position of bead i, and the updated positions (which are still in terms of the unknown multipliers) are then inserted into the constraint equations, resulting in a system of quadratic equations for the Lagrange multipliers. For computational convenience, the quadratic equations are linearized, and the multipliers are determined by assuming that the constraints are decoupled. Because of these two approximations, an iterative scheme is used to determine bead positions at a given time step, and the iterations are terminated when all constraint conditions are satisfied to within a given tolerance. For the calculations here, we have employed the condition that |(rpm/l)2 - 1| be no larger than 2 × 10-6, where rpm is the distance between two connected beads. To nondimensionalize all variables, l is used to scale length, kBT/l is used for force, kBT for energy, and ζl2/kBT for time. Henceforth, all variables are taken to be dimensionless. In our simulations, we have set q ) 1, σs ) 2, and lB ) 1 and varied κ between 0.001 and 5.0. The time step is chosen so that the maximum bead displacement in a single step does not exceed 10% of the rod length. For most of the simulations, a time step of ∆t ) 5 × 10-5 is used, but smaller time steps are needed in cases where strong electrostatic forces cause large displacements. Initial dendrimer configurations are generated using a procedure proposed by Murat and Grest.31 In this procedure, the G ) 0 generation is built by attaching three beads to the core bead in three arbitrary directions at a unit distance so that the distance between beads is not less than 0.8 to avoid beadbead overlapping. Subsequent generations are created by adding two beads at a unit distance from each bead at the free ends. A bead is considered attached if the distance between this bead and any other bead is not less than 0.8. If this is not the case, then further attempts are made to reposition the bead until this condition is fulfilled. If a new bead cannot be added in 500 iterations, then the complete dendrimer is discarded and the process is repeated again. Dendrimers up to G ) 5 were investigated in this study. For simulations near a surface, the core bead is initially placed at a distance d < 3/κ from the surface for κ < 1. At higher κ (κ > 1), the core is initially placed so that beads cannot cross the surface. Once a suitable initial configuration is generated, bead positions are evolved in time. After a steady state is reached, time-averaging is used to calculate various properties. The total number of integration steps is typically ∼107, which is much longer (>100 times) than the time it takes to reach steady state. The results presented here do not change substantially for smaller time step sizes, longer integration times, or smaller values of d. To validate our simulations, we performed runs involving dendrimers in free solution and compared the results to those in previously published studies. We found that N ∼ Rg3 for neutral dendrimers, where Rg is the radius of gyration, which is consistent with molecular dynamics simulations,31-34 Monte Carlo simulations,35 and small-angle X-ray scattering experiments.33,36,37 For the case of terminally charged dendrimers, our

Suman and Kumar results are in quantitative agreement with the radius-of-gyration values and bead density profiles presented in the Brownian dynamics simulations of ref 24. 3. Results and Discussion 3.1. Dendrimer Conformation. Dendrimer conformations are characterized in terms of the radius of gyration, Rg, and its components in the parallel (planar, x and y), Rg|, and perpendicular (z), Rgz, directions

x x x

〈

Rg )

〈

Rg| )

〈

Rgz )

N

[(xi - xcm)2 + (yi - ycm)2 + (zi - zcm)2]〉 ∑ i)1 N N

[(xi - xcm)2 + (yi - ycm)2]〉 ∑ i)1 N N

(zi - zcm)2〉 ∑ i)1 N

(3.1)

where (xi, yi, zi) is the position of the ith bead, (xcm, ycm, zcm) is the center of mass of the dendrimer, and the broken brackets denote a time average. The variation of Rg with κ is presented in Figure 2a for various dendrimer generations and charge distributions. At low κ, Rg decreases with κ since bead-bead electrostatic repulsions become weaker. As κ increases further, Rg becomes independent of κ due to complete screening of electrostatic interactions. For κ < 1, Rg is highest for a dendrimer having all of its beads charged because bead-bead electrostatic repulsion is strongest for this case. A terminally charged dendrimer has smaller Rg, followed by a dendrimer whose branching beads are charged. Since branching beads are attached to three other beads, they are more constrained compared to the terminal beads, and this leads to the lower values of Rg. (These same trends hold in free solution for κ < 1 (data not shown).) At high κ, the difference in Rg for the different charge distributions is not significant. At all κ, Rg increases with dendrimer generation. It is instructive to compare the above observations with those of Mansfield,17 who studied dendrimer adsorption using a nonelectrostatic interaction potential. In that study, the variation in Rg with the strength of dendrimer-surface interactions was not significant. In our study, a similar phenomenon can be seen by comparing values of Rg near a surface with those in free solution for a dendrimer having all of its beads charged. As seen in Figure 2a for a G ) 4 dendrimer, at a given κ there is no significant difference in Rg between the two situations. (The same is also true for the other charge distributions, but we do not show the results here.) However, for an adsorbed dendrimer the components of Rg are different compared to the free-solution values. In particular, Rg| is higher and Rgz is lower as seen in Figures 2b and 2c at low κ and in the work of Mansfield.17 We also observe that Rg| decreases with κ at low κ, then becomes constant as κ increases. The behavior of Rg| with respect to dendrimer generation and charge distribution is similar to that for Rg as seen in Figure 2a. The corresponding variation of Rgz with κ is presented in Figure 2c. At very low κ, the dendrimer adsorbs and resembles

Adsorption of Charged Dendrimers

Figure 2. Radius of gyration and its parallel and perpendicular components versus κ: (a) Rg, (b) Rg|, and (c) Rgz. A dendrimer having all of its beads charged is used for the free-solution calculation.

a thin disc having a relatively small value of Rgz and a relatively large value of Rg| (with Rgx ∼ Rgy). As κ increases, the beadsurface electrostatic interaction weakens, and as a result, Rgz

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8731 increases and Rg| decreases, leaving the dendrimer with fewer beads on the surface and more in free solution. A further increase in κ leads to a conformation in which only a few beads are on the surface and the remainder are in free solution. A maximum in Rgz is observed because at sufficiently large κ the dendrimer desorbs and adopts a less stretched and more spherical conformation with Rgx ∼ Rgy ∼ Rgz. Then, as κ continues to increase, bead-bead repulsion becomes weaker and Rgz decreases, eventually becoming independent of κ. This maximum in Rgz is not very pronounced when only the branching beads are charged since the branching beads are more constrained compared to terminal beads that are charged. With a decrease in dendrimer generation, the maximum in Rgz is less pronounced due to the reduction in the number of charged beads, and Rgz itself is smaller. We also note that at very low κ the adsorbed dendrimer never spreads onto the surface as a monolayer unlike a linear polyelectrolyte,28 where Rgz becomes independent of the chain length at low κ. This will be seen more clearly when we examine the bead density profiles. For dendrimers in free solution, Rgz monotonically decreases with κ for all charge distributions examined due to a decrease in bead-bead electrostatic repulsion. (Data for a dendrimer in which all beads are charged are shown in Figure 2c.) Finally, we discuss the behavior of the critical value of κ needed for dendrimer adsorption. The criterion used is that a dendrimer is considered absorbed if zcm fluctuates by less than 10%. For a dendrimer with G ) 3-5 and all of its beads charged, this critical value is ∼1.6, when only the terminal beads are charged, it is ∼1.4, and when only the branching beads are charged, it is ∼1.1. For G ) 1 and 2, the critical values are smaller (∼1.3, 1.2, and 0.9, respectively), and this reflects the observation of Striolo and Prausnitz18 that smaller dendrimers pay a higher entropic penalty for adsorption and the fact that the energetic benefit for adsorption is not as great for smaller dendrimers. The critical value of κ is highest when all of the beads are charged because the bead-surface electrostatic interactions are strongest in this case. As terminal beads are freer to move than branching beads, dendrimers with terminally charged beads can adsorb more easily than those that have only branching beads charged, and this is manifested in the different critical κ values. For a linear polyelectrolyte having a number of beads (all of which are charged) comparable to a G ) 2 or 3 dendrimer, the critical κ is ∼0.7,28 which is much lower than the values obtained here for dendrimers, suggesting that dendrimers adsorb more easily. This conclusion was also drawn by Striolo and Prausnitz18 in their study involving nonelectrostatic interactions alone, and a plausible reason is that the entropic penalty for dendrimer adsorption is less than that for linear polymers. 3.2. Monomer Distributions within the Dendrimer. In this subsection, we calculate the total bead density and the density of terminal beads as functions of distance from the dendrimer center of mass (r), thereby obtaining information about monomer distributions within the dendrimer. These are obtained by determining the number of beads in a spherical shell of small thickness centered around the dendrimer center of mass. The bead density inside each spherical shell is then normalized by the sum of the densities in all of the spherical shells. We first consider how the monomer distribution inside a dendrimer changes with κ. In Figure 3a, we present total bead density profiles for an adsorbed dendrimer having terminally charged groups. At low κ, the maxima and minima in the density profile are more pronounced, as the dendrons are stretched due to bead-bead electrostatic repulsion, which results in most of

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Suman and Kumar

Figure 3. Total bead density and terminal bead density profiles with respect to the center of mass for a G ) 4 terminally charged dendrimer: (a) total bead density for an adsorbed dendrimer, (b) total bead density for a dendrimer in free solution, (c) terminal bead density for an adsorbed dendrimer, and (d) terminal bead density for a dendrimer in free solution.

the beads lying on concentric circles with the dendrimer core as its center, resembling Figure 1. As κ increases, the dendrons become less stretched, and as a result, the maxima and minima become less pronounced. For κ ) 1, the density profile decreases from the center of mass and exhibits a plateau region with almost constant density before decaying to zero. The trends in the density profile for a dendrimer in free solution are similar to the ones observed for adsorbed dendrimers as seen in Figures 3a and 3b. However, the minima in the density profiles are more depressed in free solution due to the three-dimensional structure of the dendrimer, which makes backfolding of beads less likely than with adsorbed dendrimers, which have a nearly twodimensional structure. The density profile at κ ) 5 for a dendrimer in free solution has a structure similar to that for an adsorbed dendrimer at κ ) 1 and is similar to that seen for neutral dendrimers in free solution by molecular dynamics and Monte Carlo simulations31,33-35,38,40,42 and experiments.37 The terminal bead density profiles for adsorbed dendrimers having terminally charged groups do not display maxima and minima as seen in Figure 3c. At low κ, the density profile has a peak as the terminal beads are far apart due to the beadbead electrostatic repulsion, and most of the terminal beads remain at the periphery of the adsorbed dendrimer, which has a disclike shape. As κ increases, bead-bead electrostatic repulsion decreases, causing the height of the peak to decrease. Finally, at κ ) 1, the terminal groups become much more uniformly distributed throughout the dendrimer. When the

density profiles for the adsorbed dendrimers are compared with those in free solution, the general trends remain the same as those seen in Figures 3c and 3d. However, due to the two-dimensional structure of the adsorbed dendrimer at low κ, the terminal beads are more likely to move closer to the center of mass, resulting in a lower value for the peak in the density profile. For the dendrimer in free solution at κ ) 5, the distribution of the terminal groups is similar to that for an adsorbed dendrimer at κ ) 1 and resembles that seen for neutral dendrimers in free solution by experiments and simulations.22,34,41 We next examine the effect of dendrimer generation on the density profiles for terminally charged dendrimers. To more easily visualize changes in the density profile, we have normalized such that if the area under the curve for the fourthgeneration dendrimer is A, the areas under the curves are ∼(46/94)A and ∼(190/94)A for the third- and fifth-generation dendrimers, respectively. Here, 46, 94, and 190 are the numbers of beads in dendrimers of third, fourth, and fifth generation, respectively. Total bead density and terminal bead density profiles are presented in Figures 4a and 4b for various dendrimer generations. It is seen that the general shape of the density profiles is the same for each generation examined. On the basis of the values calculated before normalizing the curves, we find that the total bead densities are higher for larger generations. This can be understood by recognizing that the increase in the number of beads per generation ∼2G+1, which grows faster with

Adsorption of Charged Dendrimers

Figure 4. Total bead density and terminal bead density profiles with respect to the center of mass for terminally charged dendrimers with G ) 3, 4, and 5 at κ ) 0.1: (a) total bead density and (b) terminal bead density.

G than the volume, which ∼G3. The maxima and minima also become more pronounced as G increases. Similarly, the peak in the terminal bead density profile is found to increase with dendrimer generation, as the increase in the number of terminal beads ∼2G, which is larger than the dendrimer surface area, which ∼G2. The location of the peak is seen to shift closer to the center of mass as G decreases. The trends in the density profiles for the adsorbed dendrimers are similar to those seen for dendrimers in free solution (results not shown). In free solution, the minima in the total bead density profile are more depressed, the peak in the terminal bead density profile is higher, and the terminal bead density inside the dendrimer is less when compared to the corresponding values for adsorbed dendrimers. These differences are due to the two-dimensional disclike structure of an adsorbed dendrimer, which makes backfolding more likely than with the three-dimensional spherical structure the dendrimer has in free solution. In Figures 5a and 5b, we present total bead density profiles for different dendrimer charge distributions. The shape of the density profile is similar for all cases and resembles that seen in free solution. However, for an adsorbed dendrimer, the

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8733 maxima and minima are most pronounced when only the branching beads are charged, whereas in free solution they are most pronounced for a dendrimer in which all beads are charged. In the latter case, this is because bead-bead electrostatic repulsion is strongest when all beads are charged. In the former case, backfolding is more difficult when all beads are charged, so the densities are higher when only the branching beads are charged. Figures 5c and 5d show the corresponding terminal bead density profiles. For adsorbed dendrimers with all beads charged or only terminal beads charged, the density profile exhibits a peak due to bead-bead electrostatic repulsion. The peak is smaller when only the terminal beads are charged since the terminal beads can more easily penetrate the dendrimer interior. The peak is much less pronounced when only the branching beads are charged, with the profile being rather uniform some distance away from the center of mass. The narrow spread of terminal beads for a fully charged dendrimer implies that such dendrimers would be useful in applications where it is desired that the terminal groups contact the charged surface at the dendrimer periphery. If a uniform distribution of terminal groups is required, then dendrimers with only branching groups charged could be used. The density profiles for dendrimers having all beads charged are similar in both the adsorbed and the freesolution cases due to the difficulty of backfolding in both cases. For the other two charge distributions, backfolding is easier when the dendrimer is adsorbed and has a two-dimensional structure, so the densities are lower in the adsorbed case. 3.3. Monomer Distributions near the Surface. We now characterize how the total bead density and the terminal bead density depend on the distance from the surface. These density profiles complement the ones described in the previous section by providing information about the distribution of monomers near the charged surface. The calculation method is similar to that used previously, except that thin sheets parallel to the charged surface are used to determine bead densities. In Figure 6, density profiles for a terminally charged dendrimer are presented. From Figure 6a, we see that at low κ the beads are localized in a small region near the surface and that the size of this region increases with κ. The two peaks in the density profile are similar to those seen in the work of Mecke et al.20 The peak closer to the surface is proportional to the number of beads adsorbed to the surface while the other peak represents a layer of beads roughly one rod length away from the first peak. The height of these two peaks decreases as κ increases, and the second peak becomes considerably diminished. The density profiles for the terminal beads behave in a qualitatively similar way as seen in Figure 6b. However, the peak further from the surface is more suppressed compared to the one in Figure 6a since most of the terminal beads are attached to the surface. Note that due to the dendritic architecture and bead-bead repulsion not all of the terminal groups can attach to the surface, meaning that many beads will be exposed to free solution. In Table 1, the number of adsorbed beads is reported as a function of κ. Beads within one rod length in the direction normal to the surface are classified as adsorbed. At all κ, more terminal than branching beads are adsorbed due to the beadsurface electrostatic attraction that the terminal beads experience. As κ increases, the number of beads adsorbed decreases, scaling as κ-0.1 at low κ and κ-2.4 at higher κ. Ratner et al.19 also reported that the number of beads of a dendrimer in contact with the surface decreases as the strength of the dendrimersurface nonelectrostatic attractive potential decreases. Note that

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Suman and Kumar

Figure 5. Total bead density and terminal bead density profiles with respect to the center of mass for G ) 4 dendrimers of various charge distributions at κ ) 0.1: (a) total bead density for an adsorbed dendrimer, (b) total bead density for a dendrimer in free solution, (c) terminal bead density for an adsorbed dendrimer, and (d) terminal bead density for a dendrimer in free solution.

beads not attached to the surface are exposed to free solution, so by tailoring κ, the desired number of monomers can be placed in the solution or attached to the surface. The effects of dendrimer generation are illustrated in Figures 7a and 7b. The density profiles have been normalized and compared in the same manner as those of the previous section. As G increases, it is found that the total bead and terminal bead densities increase and that the distance over which the density profiles decay increases. The peaks also tend to be more pronounced and spaced further apart as G increases. All of these effects are due to the larger number of beads present at higher generations. The number of adsorbed beads is given in Table 2, and it seen that more beads become adsorbed at higher values of G. The number of adsorbed beads ∼G1.2 at low G and ∼G0.3 at higher G. However, the percentage of beads that adsorb decreases as G increases since the total number of beads ∼2G whereas the number of adsorbed beads grows much less quickly. A similar reduction in the percentage of adsorbed beads was also seen in the work of Mecke et al.20 Finally, in Figure 8, we explore the effect of dendrimer charge distribution. We see in Figure 8a that the highest density near the surface occurs for a dendrimer in which all beads are charged since bead-surface electrostatic attraction is strongest in this case. The densities near the surface are very similar for the terminally charged dendrimer and the dendrimer having only its branching beads charged. This is likely due to the fact that for both types of dendrimers the number of charged beads is

roughly equal and the electrostatic interactions are very strong. Thus, for applications in which a large number of monomers need to be in contact with a surface, dendrimers whose monomers are all charged should be used. We also see in Figure 8b that the smallest peak in the density profile for the terminal beads occurs for dendrimers in which only the branching beads are charged since the terminal beads do not experience any electrostatic attraction to the surface. The number of beads attached to the surface is given in Table 3. As expected, the dendrimer with all beads charged has the most beads adsorbed, followed by the dendrimer with only terminal beads charged, and then the dendrimer with only branching beads charged. In all cases, the number of adsorbed beads decreases with increasing κ. Note that when all beads are charged more branching beads than terminal beads adsorb at smaller values of κ, whereas the opposite is true at larger values of κ. For the other two charge distributions, the difference in the number of adsorbed beads is relatively small at κ ) 0.1, consistent with the behavior of the density profiles in Figure 8a. At larger values of κ, terminally charged dendrimers have more adsorbed beads as they are less constrained to move than branching charged beads and the electrostatic attraction is weaker. 3.4. Effect of Percentage of Terminal Beads Charged. In the previous sections, it was assumed that all of the beads on a terminally charged dendrimer were charged. In practice, however, it is possible to control the percentage of terminal beads that are charged,43 and this may be useful for applications in

Adsorption of Charged Dendrimers

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8735

Figure 6. Total bead density and terminal bead density profiles with respect to the surface for a terminally charged dendrimer with G ) 4: (a) total bead density and (b) terminal bead density.

TABLE 1: Number of Adsorbed Beads for a G ) 4 Terminally Charged Dendrimer κ

beads

terminal beads

branching beads

0.1 0.3 0.5 0.8 1.0 1.2 1.4

47 43 40 29 24 11 8

32 31 29 21 20 9 6

15 12 11 8 4 2 2

which is it desired to control electrostatic interactions without changing κ. Thus, in this section, we examine the effect of having only a percentage of randomly chosen terminal beads charged. To generate a dendrimer with f% of its terminal groups charged, a random number between 0 and 100 is chosen for each terminal bead. If the number is less than f, then the bead is taken to have q ) 1; else we have q ) 0. The results shown below are for representative cases. We have verified that if a different set of charged beads is chosen for the same value of f, then the values of Rg and its components vary by no more than 5% and the behavior of the density profiles is qualitatively unchanged.

Figure 7. Total bead density and terminal bead density profiles with respect to the surface for terminally charged dendrimers with G ) 3, 4, and 5 at κ ) 0.1: (a) total bead density and (b) terminal bead density.

TABLE 2: Number of Adsorbed Beads for a Terminally Charged Dendrimer with K ) 0.1 G

beads

percentage

terminal beads

branching beads

1 2 3 4 5

8 16 29 47 51

80 75 64 50 26

6 10 18 32 35

2 6 9 15 16

Figure 9 presents Rg and its components as a function of κ for different values of f. As expected, Rg decreases as f decreases at low values of κ and then becomes independent of f at higher values of κ. As was observed earlier for f ) 100, Rg for adsorbed dendrimers is similar to that in free solution even when f < 100. Similarly, Rg| is larger for adsorbed dendrimers at low κ, and its value increases with f. When κ is very small, Rgz decreases as f increases since more beads attach to the surface. As κ increases, the electrostatic attraction weakens, and Rgz becomes larger for larger f. Increasing f also accentuates the peak in Rgz. The critical value of κ for a dendrimer adsorption increases with f since bead-surface electrostatic attraction increases. For f ) 20, 50, and 80, these critical values are ∼0.2, ∼0.9, and ∼1.1, respectively.

8736 J. Phys. Chem. B, Vol. 111, No. 30, 2007

Suman and Kumar

Figure 8. Total bead density and terminal bead density profiles with respect to the surface for G ) 4 dendrimers of various charge distributions at κ ) 0.1: (a) total bead density and (b) terminal bead density.

TABLE 3: Number of Adsorbed Beads for a G ) 4 Dendrimer charge distribution

κ

beads

terminal beads

branching beads

terminally charged beads all charged beads branching charged beads terminally charged beads all charged beads branching charged beads terminally charged beads all charged beads branching charged beads

0.1 0.1 0.1 0.5 0.5 0.5 0.8 0.8 0.8

47 71 46 40 47 21 29 30 13

32 30 12 29 21 5 21 16 4

15 41 36 11 26 16 8 14 9

Density profiles with respect to the dendrimer center of mass are presented in Figure 10. It is clear that the behavior of the density profiles depends strongly on f and resembles that seen in Figure 3 when κ was varied. It is seen that even for values of f as large as 50 dendron stretching is considerably reduced, leading to fewer oscillations in the total bead density and a more even distribution of terminal groups throughout the dendrimer. The effect of dendrimer generation on the density profiles when

Figure 9. Radius of gyration and its parallel and perpendicular components versus κ of a terminally charged dendrimer: (a) Rg, (b) Rg|, and (c) Rgz.

f < 100 is similar to that seen for f ) 100 and is not presented here for brevity.

Adsorption of Charged Dendrimers

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8737

Figure 10. Total bead density and terminal bead density profiles with respect to the center of mass for G ) 4 dendrimers at κ ) 0.1: (a) total bead density and (b) terminal bead density.

TABLE 4: Number of Adsorbed Beads for a G ) 4 Dendrimera f%

beads

A

percentage of A

B

percentage of B

branching beads

100 80 50 20

47 44 29 10

32 29 18 7

66 75 88 84

1 3 2

10 12 5

15 14 8 2

a

A and B denote charged and neutral terminal beads, respectively.

Density profiles with respect to the surface are shown in Figure 11a. It is again seen that decreasing f has an effect similar to that of increasing κ (Figure 6): as f decreases, the peaks in the density profiles become smaller and the profiles become less localized. Table 4 shows the number of beads attached to the surface for different values of f. We observe that most of the charged terminal beads are attached to the surface, whereas most of the neutral terminal beads are in free solution. This is of potential use in biological sensors5 and surface functionalization, where the charged terminal groups could be used to attach the dendrimer to a surface and the neutral terminal groups

Figure 11. Total bead density and terminal bead density profiles with respect to the surface for G ) 4 dendrimers at κ ) 0.1: (a) total bead density and (b) terminal bead density.

could serve as sensor targets or modifiers of surface properties. It is also seen in Table 4 that the number of adsorbed beads decreases as f decreases, scaling as f0.1 at high f and f1.1 at low f. Consistent with Figure 11a, the number of adsorbed beads is considerably smaller when f ) 50. 4. Conclusions Motivated by the importance of understanding the behavior of isolated charged dendrimers near oppositely charged flat surfaces, we have performed Brownian dynamics simulations of dendrimer adsorption using a bead-rod model. A DebyeHu¨ckel description of electrostatic interactions is applied, and the effects of salt concentration (screening length) and charge distribution on radii of gyration and monomer (bead) density profiles are probed for a range of dendrimer generations. Adsorbed dendrimers adopt a disclike conformation in which they flatten in the direction normal to the surface and expand in the direction parallel to the surface. As the inverse screening length increases, the disc expands in the normal direction and contracts in the parallel direction, adopting a conformation that is more stretched in the normal direction. When the inverse

8738 J. Phys. Chem. B, Vol. 111, No. 30, 2007 screening length becomes sufficiently large, the dendrimer desorbs and adopts a spherelike conformation. In this situation, electrostatic interactions are strongly screened and the radii of gyration are similar for dendrimers in which (i) all terminal groups are charged, (ii) only the branching groups are charged, and (iii) all groups are charged. For adsorbed dendrimers, case iii generally has the largest radii of gyration since bead-bead electrostatic repulsion is strongest, and case ii generally has the smallest since the branching beads are more constrained relative to terminal beads. The critical inverse screening lengths at which adsorption occurs are found to be larger than that for linear polymers, reflecting the smaller entropic penalty associated with dendrimer adsorption. Increasing the dendrimer generation acts to increase the values of the radii of gyration and the salt concentration at which adsorption occurs. Total bead and terminal bead density profiles with respect to the dendrimer center of mass are qualitatively similar for adsorbed dendrimers and those in free solution, with differences due to the two-dimensional structure of adsorbed dendrimers, which makes backfolding of dendrimer segments more likely. For dendrimers in which all groups are charged, the terminal beads tend to be localized around the dendrimer periphery, whereas if only the branched groups are charged, the terminal beads are more uniformly distributed throughout the dendrimer interior. In the case of adsorbed dendrimers, density profiles with respect to the surface show that the dendrimer forms a two-layer structure, with one layer corresponding to adsorbed beads and a second, less dense layer corresponding to beads one rod length away from the surface. Dendrimers in which all groups are charged have the largest number of beads in contact with the surface at fairly low values of the inverse screening length, but this number decreases as the inverse screening length increases and becomes comparable to that for dendrimers in which only terminal groups are charged. Similarly, dendrimers with only branching groups charged and dendrimers with only terminal groups charged have a comparable number of beads adsorbed to the surface at fairly low values of the inverse screening length, but the latter type of dendrimer has more beads adsorbed at higher values of the inverse screening length. Finally, we explored the effects of having only a fraction of the terminal groups charged. When electrostatic interactions are relatively strong, increasing the fraction of charged beads increases the values of the radius of gyration and its parallel component, while decreasing the value of its perpendicular component. As electrostatic interactions weaken, the perpendicular component begins to increase as the fraction of charged beads increases since the bead-surface attraction becomes weaker. Increasing the fraction of charged beads also raises the critical salt concentration needed for adsorption. The density profiles reveal that by reducing the fraction of charged beads the terminal beads can be distributed more uniformly throughout the dendrimer. As more charged beads than neutral beads adsorb, the charged groups could be used to anchor the dendrimer to a surface while the neutral groups could be used as targets in sensors or for surface functionalization. As pointed out in the Introduction, there has been relatively little theoretical work concerning dendrimer adsorption. The present work thus fills an important gap by using a coarsegrained simulation method to explore a wide range of variables associated with the adsorption of charged dendrimers. The results obtained should be helpful in interpreting and generalizing the results of finer-grained simulation methods and in providing qualitative guidance to experimentalists designing dendrimers for various applications. The present study also

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