Simulations of a Grafted Dendritic Polyelectrolyte in Electric Fields

Article pubs.acs.org/Macromolecules

Simulations of a Grafted Dendritic Polyelectrolyte in Electric Fields J. S. Kłos*,†,§ and J.-U. Sommer†,‡ †

Leibniz Institute of Polymer Research Dresden e.V., 01069 Dresden, Germany Institute for Theoretical Physics, Technische Universität Dresden, 01069 Dresden, Germany § Faculty of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland ‡

ABSTRACT: Using Monte Carlo simulations based on the bond fluctuation model, we study the behavior of a charged dendrimer grafted on one of the electrodes of a plate capacitor in an athermal, salt-free solvent. The calculations are performed for the full Coulomb potential in a wide range of parameters determining the strength of the electrostatic interactions between the charges and the magnitude of electric fields between the electrodes. For low temperatures and without an external field the dendrimer is in an osmotic regime where entropy of the trapped counterions controls the size of the molecule. Increasing the field strength leads to gradual removal of trapped counterions being localized at the opposite electrode. At a threshold value of the displacement field, D*, all counterions are stripped from the dendrimer, and the molecule is collapsed on the electrode. The collapse of the extension of the dendrimer in the direction perpendicular to the electrode is very sharp, and a bimodal distribution of monomers in the regime D < D* indicates a partial collapse of the dendrimer in this region. Here, uncompensated charges of the dendrimer are localized in substructures which are collapsed on the electrode while other parts of the molecule are still in the osmotic regime with respect to the remaining counterions. For higher temperature the collapse transition is smeared, and the osmotic effect of the counterions is less important.

I. INTRODUCTION There has been an increasing interest in studying dendrimers because of their usefulness in many applications such as multilayer thin films,1 holographic data storage,2 fluorescence turn-on biosensors,3 and gene and drug delivery.4 To name but a few, experimental investigations of dendrimers considered the thickness, morphology, and mechanical stability of monolayers of PAMAM (polyamidoamine) dendrimers made by electrostatic self-assembling from water solutions,5 the influence of ionic strength and solution pH on the molecular dimension of PAMAMs,6−8 and their structural properties at the interface between an aqueous solution and a hydrophobic or hydrophilic substrate.9 The experiments with the use of atomic force microscopy on adsorption of dendrimers to mica surfaces determined dendrimers molecular dimensions and volumes for various generations.10 The adsorption and desorption behavior of PAMAMs at the water−silica interface was studied as a function of dendrimer generation, solution pH, and ionic strength.11 A number of computer simulations were carried out to obtain the generic picture of dendrimers behavior near adsorbing surfaces. In particular, the simulations showed that dendrimers spread out and flatten down in the direction normal to the surface and expand in the direction parallel to it with increasing the strength of monomer−surface interaction.12−14 Monte Carlo simulations were used to investigate the behavior of amphiphilic dendrimers in solutions as well as in the vicinity of a planar wall. It was shown that unlike athermal dendrimers that take dense-core conformations, amphiphilic dendrimers © XXXX American Chemical Society

reveal the dense-shell structure. Near a surface depending on the wall−dendrimer interaction dendrimers display a sharp adsorption transition which is unlike the case of linear chains reminiscent to a first-order transition between a weakly localized state and a collapsed state.15,16 Adsorption of isolated charged dendrimers onto oppositely charged flat surfaces was simulated using the Debye−Hückel approximation for electrostatic interactions, and in particular the role of the electrostatic screening length on the adsorption behavior was studied.17 In this work inspired by the recent findings about polyelectrolyte brushes, we study the behavior of a grafted charged dendrimer in external electric fields. Theoretically, it was shown that for the limiting cases of low and high salt concentrations a polyelectrolyte brush with a fixed fraction of monomers bearing charges grafted on an electrode collapses under the influence of electric fields between the electrodes of a plate capacitor.18 In particular, it was demonstrated that for low salt concentrations, due to an exponential increase in the number of counterions expelled from the brush, the height of the brush reveals a sharp decrease with increasing the field’s magnitude. On the other hand, in the extreme of high concentrations of monovalent salt, instead of the steep collapse the brush height was shown to decrease nearly linearly with increasing the voltage between the electrodes. Based on the Received: November 14, 2014 Revised: January 15, 2015

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DOI: 10.1021/ma502301a Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules coarse-grained continuum model, similar field-dependent transitions from compressed to stretched conformations were found for a polymer brush grafted to a surface and bearing a charged group at its free end only.19 Monomer densities and counterion distributions were also obtained by numerical self-consistent field theory for charged polymers grafted to a flat interface in the presence of external electric fields at different charge configurations.20 Molecular dynamics simulations performed on polyelectrolyte brushes indicated that both salt concentration and chain stiffness make a tremendous impact on electroresponsive static and dynamic behavior of the brushes.21 For example, an increase in the salt concentration leads to an increase in the opposite electric field caused by the distribution of ions, which in turn resists stretching or shrinking of the brush. It was also shown that the flexible grafted chains are more sensitive to changes in salt concentration at strong positive fields. Some other molecular dynamics calculations demonstrated that in response to the applied field the brushes can reversibly change their conformations from collapsed to stretched. It was suggested that the fast response to the external electric stimuli implies that the polymer brush monofilm can be used as a high-frequency gating component for modulating electrical/fluidic impedance in nanofluidic channels.22 In this work we continue our inspection of the properties of dendrimers grafted on flat surfaces.16 In particular, we study the behavior of a single charged dendrimer accompanied by counterions without salt in the vicinity of the electrode of a plate capacitor in the presence of electric fields between the plates. Our simulations were carried out using the bond fluctuation model (BFM)23,24 of a dendrimer of generation G = 5 and spacer length S = 8 in a wide range of the electrostatic couplings between the charges present in the system and magnitude of the applied fields. In our study we focus on the conformational changes of the dendrimer caused by the two parameters and provide detailed information concerning the molecule’s shape and size as well as spatial distributions of monomers and counterions. The remaining part of the paper is organized as follows. In section II we outline the model and the simulation method. The results of our simulations are presented and discussed in section III. Finally, our conclusions and remarks are presented in section IV.

by the grafting point via the molecule’s connectivity (for a schematic representation of the studied system see Figure 1). In

Figure 1. Sketch of the simulated system.

the simulations the branching points and the terminal groups carry positive charges of valence q = 1. For each charged group one counterion is introduced to the capacitor which bears a negative charge q = −1. Thus, the system is electrically neutral as a whole. Within the applied Monte Carlo scheme configurations are sampled using the BFM including the standard Metropolis method.28 The electrostatic interaction between the charges is the total Coulomb energy defined as qiqj Uc(rij) = δc rij (2.1) where rij is the distance between the ith and jth ions with valences qi and qj δc =

e2 ϵu

(2.2)

with e standing for the elementary charge and ϵ for the permittivity of the solvent. The electrostatic energy is calculated using the standard Ewald summation method for charges in a cubic box of the size L3 = 500 × 500 × 500u3 (where u is the length unit) along with the electrostatic layer correction for Coulomb interactions in slab geometry. The calculations are performed with the optimal values of the convergence parameter κ = (5.74/L) in the real-space term with the minimum image convention and of the empty gap size in the zdirection h = 0.2L for the reciprocal-space cutoff radius Kc = 6. Periodicity is imposed on the x and y coordinates of the monomers and counterions, whereas the z coordinates range between 0 and 0.8L. The value of the convergence parameter, KLC, in the electrostatic layer correction term is set to Kc.29−33 In the simulations we assume that neither the monomers nor counterions can cross the z = 0 and z = L − h plates. The strength of the electrostatic interactions is controlled by the reduced electrostatic temperature

II. MODEL AND SIMULATION DETAILS To inspect the dendrimer’s behavior, we carry out Monte Carlo simulations using the bond fluctuation model (BFM)23,24 (see our previous work16,25). More specifically, we examine a movable, single dendrimer of generation G = 5 with the core of two bonded monomers, branching functionality f = 3, and spacer length S = 8 in an athermal solvent. Dendrimers with such architecture of their skeleton can be considered as typical in experiment and applications. For example, PAMAM dendrimers can be synthesized up to generation 10 and poly(propylene imine) dendrimers up to generation 5.26 Spacer length can also be varied in a wide range up to 20.27 Before the actual simulations, the dendrimer was generated by a divergent growth process in the ascending order of the internal generation number 0 ≤ g ≤ G starting from the core (g = 0). Additionally, one of the dendrimer’s terminal groups is immobilized and attached at the grafting electrode at z = 0. Thus, all the monomers except the grafted monomer are movable, and the motion of the dendrimer is only constrained

τc = B

kBT δc

(2.3) DOI: 10.1021/ma502301a Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules where T is the absolute temperature and kB is the Boltzmann constant. The model can be converted to real units by the inverse relation u τc = λB (2.4)

monomers so as to inspect the dendrimer in an athermal solvent. In our study the dendrimer was equilibrated for a maximum of 107 MCS (Monte Carlo steps; in one MCS on average each monomer and ion is selected to be moved in a randomly chosen, one of the six directions by a single lattice unit), whereas averages were calculated for about 104 equilibrium configurations stored every 103th MCS. An equilibrium state was considered achieved once the means of various measured quantities characterizing the molecule such as the radius of gyration revealed no systematic changes.

between τc and the Bjerrum length λB = e2/ϵkBT. Within the BFM for instance, the choice of u ≈ 2 Å corresponds to a bond length a ≈ 5 Å, which in turn concurs reasonably with real dendrimers.34 In water at room temperature one obtains λB ≈ 7 Å, and eq 2.4 yields τc ≈ 0.3. In particular, as we found in our previous works at this τc value free, charged dendrimers take swollen conformations due to the osmotic pressure exerted on them by counterions trapped inside the dendrimers.35 In the following we can consider values of τc < 1 as belonging to the osmotic regime of the dendrimer. In the Monte Carlo scheme the effect of electric fields of magnitude E between the electrodes is accounted for by the change in the system energy ΔUfi = qieEΔzu

III. RESULTS A. Contact with the Surface: Order Parameters. As we have mentioned above, our calculations involve three free parameters referred to as the reduced electrostatic, adsorption, and field temperature. The first two parameters have already been shown to have a tremendous impact on the conformation properties of dendrimers.16,35−37 However, in the rest of the paper we focus on the effects of purely electrostatic nature and drop out of those caused by the short-range attraction between the grafting electrode and monomers. The latter effect is displayed in Figure 2 which demonstrates the adsorption order

(2.5)

associated with shifting a charge of valence qi by one lattice site Δz = ±1 along the z direction. This leads to the dimensionless simulation parameter τf =

kBT eEu

(2.6)

throughout the paper referred to as the reduced field temperature. In the following we consider fields which tend to compress the dendrimer only. We use values for τf in the range of τf = 5, ..., 1000 which for the length unit u ≈ 2 Å and room temperature T ≈ 300 K refer to electric fields of magnitude 105 < E < 107 V/m. While the choice of τc and τf is natural with respect to the Monte Carlo−Metropolis scheme, in experiments one would rather control the surface charge density, i.e., the displacement field, D = ϵE, independently of variation of temperature or dielectric constant. While the latter two parameters are effectively combined in τc, the displacement field is given by τc/τf = D(u2/e). We note that e/u2 is the unit of charge density in our simulations which in the following we set to unity. Therefore, we can write τ D= c τf (2.7)

Figure 2. Fraction of adsorbed monomers M/N vs the reduced adsorption temperature, τa, for the G5 neutral dendrimer with spacers S = 8.

parameter, m = M/N (where M denotes the number of monomers in contact with the surface and N the molecular weight), as a function of the reduced adsorption temperature, τa, for the neutral G5 dendrimer. It is clearly seen that the dendrimer reveals an adsorption transition indicated by an increase in the order parameter from zero in the extreme of high adsorption temperature (nonadsorbed state) up to one at low τa values (strongly adsorbed state). For details concerning the adsorption behavior of neutral dendrimers onto a flat surface along with the scaling analysis combined with the meanfield picture, see our previous work.16 In the following we present our results obtained from simulations of the charged G5 dendrimer at various electrostatic temperature and field strength and fixed τa = 2 corresponding to the case of a dendritic skeleton in the nonadsorbed state only. Since we consider adsorption of the dendrimer in external fields, the contacts of the charged monomers with the grafting electrode should define the order parameter. Our results for the relative fraction of charged groups in contact with the electrode, nc/Nc, are displayed in Figure 3. Here Nc = 126 is the total number of charged units on the dendrimer, and nc is the number of charged groups in contact with the grafting electrode. In the upper panel (a) the order parameter as a function of the field, D (see eq 2.7), is shown. A strong transition in particular at lower electrostatic temperatures can be observed. Note that ambient conditions correspond to τc ≃ 0.3.

In a capacitor the external field is given by D = σ, where σ corresponds to the absolute value of charge density on either plate. Each dendrimer bead is also a subject to a very weak shortrange attraction exerted on it by the surface. Whenever a bead contacts the surface, the system energy decreases by δa, which leads to the adsorption temperature τa =

kBT δa

(2.8)

τa values determine the effective strength of attraction between the surface and dendritic skeleton. However, in our simulations we keep this parameter constant τa = 2, which, as we present below, practically corresponds to the case of the dendrimer’s skeleton in the nonadsorbed state.16 In other words, in the following we treat the surface as nonadsorbing for the beads. Moreover, there is no short-range attraction between the C


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charged monomers in the same molecule, much less entropy is gained in the nonadsorbed state. Therefore, the fraction of charged monomers in direct contact with the electrode is much higher. From this it becomes clear that a simple analogy between adsorption and field-induced collapse of charged polymers is not given. In the inset of Figure 3 we show the fraction, nc/M, of charged groups in contact with the grafting electrode normalized by the total number of monomers in contact with the electrode. This ratio is decreasing up to a value of D ≃ 0.01 in the limit of low temperatures and then increases again slightly. This implies that for D < D* ≈ 0.01 an increasing fraction of noncharged monomers are bound close to the surface with increasing D, indicating a cooperative effect: Loops in between adsorbed charges are more frequently in contact with the surface. The lower panel (b) in Figure 3 displays the effect of increasing the temperature or, alternatively, increasing the dielectricity of the solution. Increasing τc gradually releases the localization of the dendrimer at the surface, a process similar to the adsorption crossover. B. Monomer, Charged Groups, and Counterion Distributions. Next we consider the spatial distributions of the monomers, terminal groups and counterions relative to the grafting plate. Our analysis is done in terms of histograms, Pm, Pe, and Pc, which give the probability density of a monomer, end-monomer, and a counterion being found in the neighborhood of a certain point in the capacitor, respectively. The first physical phenomenon to point out is the occurrence of spatial separation between the dendrimer and its counterions caused by the external field. Actually, our simulations show that at sufficiently strong electrostatic fields all counterions are expelled from both the grafting electrode and the dendrimer’s volume and form an ionic layer on the other electrode (see Figure 4). Qualitatively, this effect is also well seen in the first snapshot presented in Figure 5. This can be understood by considering a virtual plane above the dendrimer. Neglecting the

Figure 3. Fraction of charged groups in contact with the grafting electrode, nc/Nc, vs (a) D at fixed τc; the inset shows the ratio, nc/M, between the number of charged groups in contact with the grafting electrode and the total number of monomers in contact with it, (b) τc at fixed D for the G5 charged dendrimer with spacers S = 8.

In order to rationalize the adsorption behavior of charged dendrimers onto an oppositely charged electrode in a capacitor, we may associate this with the behavior of a charged polymer brush. Let us consider the cross section of the dendrimer given by AD, and the number of charged units is Nc = 126. In the fully adsorbed state of the dendrimer its charge will be compensated for

D* = Nc/AD

(3.9)

At this field strength the dendrimer should be stripped from its counterions since the local field above the dendrimer is then inverted and counterions should be localized completely on the upper plate. Using the results presented later in section III.C, we obtain with Rg∥ ≃ 40 a value of about D* ≃ 0.02. Since the total field in the capacitor is not laterally homogeneous, one expects this to be an upper estimate since the field will be dominated by that of the bare capacitor above the dendrimer. Two effects take place simultaneously if the field is switched on: The number of osmotically active counterions is reduced, since only the effective charge Qeff = Nc − DAD has to be compensated by counterions. Second, the uncompensated charged groups have the tendency to adsorb at the surface where they screen the surface charges for interactions with other charged monomers. Note that collapse of polymers in a constant external field is qualitatively different from adsorption by short-range surface attraction since charged groups at any distance from the surface experience the attractive field of the surface at the same strength. The typical length scale where an isolated charge is localized above the oppositely charged surface is given by λGC ≃ kBT/Ee = τfu. Our values of τfu correspond to 5−1000 lattice units. Therefore, a strong localization of individual charges on the surface is unlikely. However, due to the connectivity of the

Figure 4. Histograms of the number of counterions (monomers), Pc(z) (Pm(z)), at fixed τc values and varying D. The blue curves refer to the monomer probability distributions. The inset shows the spatial separation between the dendrimer and its counterions which occurs at D = 0.02, D = 0.004, D = 0.0001, and τc = 0.1. D


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Figure 6. Fraction, fcin, of counterions trapped inside the dendrimer’s cloud vs D at fixed τc; the inset shows the fraction of ions, fz > 250u, at a distance z > 250u from the grafting electrode for the G5 charged dendrimer with spacers S = 8.

localization of counterions close the opposite electrode (see inset of Figure 6). The latter fraction is defined by a dividing plane at position z = 250u, thus in the range of 150 lattice units away from the upper electrode. High electrostatic temperatures cause an overlap of the counterion distribution with the dividing plane and, to a lesser extent, with the dendrimer even at D > D*. Thus, the transition is smeared in both plots for higher values of τc. We note that this is an effect of the finite width of the capacitor. In the limit of a wide capacitor, L ≫ 2λGC + Rg⊥, the counterion density inside the brush for D > D* should be zero. In Figure 4 we display the distribution of counterions and monomers at some selected values of the field strength and of the electrostatic temperature. Note that the escape of ions from the dendrimer is accompanied by considerable flattening of the molecule. The latter effect is reflected in the corresponding histogram, Pm, which becomes narrow and peaks on the plate. The phenomenon of expelling counterions by external fields was also predicted for brushes using the Poisson−Boltzmann theory18 and molecular dynamics simulations.21 For this reason we conclude that it is a general phenomenon for grafted polyelectrolytes of any architecture of their skeleton accompanied by free ions under the influence of external electric fields. As expected, for D < D* the separation between the molecule and its counterions is incomplete. This can be well observed in particular for strong electrostatic couplings through the bimodal shape of the counterion histogram, Pc. This also shows that some ions penetrate the dendrimer’s interior and the others form a layer on the opposite plate (see the inset of Figure 4a for D = 0.004 and Figure 5b). At very weak fields applied and relatively strong electrostatic couplings between the charges practically all counterions occupy the dendrimer’s volume (osmotic limit) (see Figure 5c, the inset of Figure 4a, and Figure 6). Finally, according to Figure 4c,d at high electrostatic temperature and weak fields ions are almost uniformly distributed throughout the space. In Figure 7 we display the effect of external electrostatic fields on spatial distributions of monomers, Pm, in more detail. Close to the surface the curves drop sharply to a local, narrow minimum followed by a maximum as the distance from the electrode is increased. At low electrostatic temperature where the dendrimer is in the osmotic regime (see Figure 7a,b), the distributions are considerably broader as compared with the extreme of high temperatures. In particular, a pronounced shoulder of the distribution is observed for D < D* which indicates a collapse of part of the dendrimer while the rest of

Figure 5. Snapshots of the charged G5 dendrimer with spacers S = 8 at τc = 0.1 and (a) D = 0.02, (b) D = 0.004, and (c) D = 0.0001. The red (yellow) spheres represent counterions (charged groups).

lateral inhomogeneity of the system the effective charge density/displacement field is given by Deff = Nc/AD − D. As already noted above D* = Nc/AD the field above the dendrimer is inverted and counterions are repelled rather then attracted. In order to quantify the fraction of counterions trapped in the dendrimer’s volume, we calculated the number of ions penetrating the dendrimer’s cloud visualized as a cylindrical pillbox of radius 2Rg∥ and height 1.5Rg⊥ attached with its base to the surface (for the definition of Rg∥ and Rg⊥ see the first paragraph of the next section). The resulting plots of the fraction of trapped counterions, fcin, are shown in Figure 6. It is seen in Figure 6 that the fraction of trapped ions decreases monotonously from a finite value to zero with increasing magnitude of the field. The plateau value for D = 0 depends not only on electrostatic temperature but also on the rather arbitrary height cutoff of 1.5Rg⊥ and 2Rg∥ chosen in this plot. Higher values of τc correspond to higher values of the localization length above and around the dendrimer where counterions can escape to. Thus, the ratio of the trapped counterions is always smaller than unity. The fraction of trapped counterions at low values of τc drops to zero sharply at about D* = 0.01. This behavior is accompanied by a complete E


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out of it instead. A slightly stronger localization of endmonomers occurs at lower values of the field as compared to the total density profile. To summary this section, our simulations indicate that under the influence of external electric fields the dendrimer and counterions exist in three different macrostates. In very weak fields the electrostatic attraction between the charged groups and counterions dominates the system behavior. Under such conditions ions penetrate the molecule’s volume and the dendrimer stretches out of the grafting electrode. As the field becomes stronger, for some ions the electrostatic coupling with the charged beads becomes too weak to prevent them from being expelled from the molecule. In this case a certain fraction of ions diffuse from the molecule to the other electrode, whereas the other ions remain trapped inside the molecule. In the limit of strong fields, D > D*, practically all counterions drift out of the dendrimer and form a layer on the opposite plate. In this case the dendrimer itself flattens considerably. The characteristic value for which complete stripping of counterions takes place can be estimated as D* = 0.01. This effect is smeared out for high electrostatic temperature, in particular if the Guoy−Chapman length at the electrodes becomes comparable to the width of the capacitor. Strong field-induced transitions can be seen in the osmotic limit, where the Guoy− Champan length is much smaller than the size of the dendrimer. C. Shape and Size of the Dendrimer. In this section we consider in more detail the molecule’s size and shape properties in connection with the spatial distribution of ions. Since the interface and field break the isotropy of the system, we have to distinguish between the parallel, Rg∥, and perpendicular, Rg⊥, components of the dendrimer’s radius of gyration, Rg:38

Figure 7. Linear−logarithmic histograms of the number of monomers, Pm(z), at fixed τc and varying D.

the molecule is still in a noncollapsed state (see Figure 5b). As a consequence of the partial stripping of counterions connected with an excess charge of the dendrimer, one possibility is that noncompensated charges are grouped together and adhere at the oppositely charged surface while the rest of the monomers are kept in the osmotic regime expanded by the counterion pressure. To complement our discussion of spatial distributions of monomers in Figure 8, we present histograms, Pe, of the

Rg

2

R g⊥

2

1 1 = 2N 1 = N

N

∑ ⟨(xi − xcm)2 + (yi − ycm )2 ⟩ i=1

(3.10)

N

∑ ⟨(zi − zcm)2 ⟩ i=1

(3.11)

In eqs 3.10 and 3.11 xi, yi, and zi are the coordinates of the ith monomer belonging to the dendrimer with the center of mass at xcm, ycm, and zcm, respectively. In Figure 9a,c we display the two components as a function of the applied field. In particular, for low electrostatic temperatures a collapse toward the electrode can be observed which is related with a spreading of the molecule in the lateral direction. At D* = 0.01 the lateral extension of the dendrimer reaches a plateau while the thickness of the dendrimer becomes limited by the size of the monomers. In other words, the dendrimer exists in flat conformations in the absence of counterions in its volume. In the osmotic regime, for τc < 1 this shape transition is very sharp. In particular, the collapse of the component Rg⊥ is almost steplike at low temperatures. In Figure 9b,d both components of the dendrimer’s radius of gyration are plotted versus the reduced electrostatic temperature at constant fields. In strong fields (D ≫ D*) there are no counterions inside the dendrimer; the molecule flattens considerably and expands in the lateral direction with the decreasing electrostatic temperature due to repulsion between the charged groups. For low field strengths the extension of the dendrimer displays a nonmonotonous behavior (see Figure 9b,d), which is

Figure 8. Linear−logarithmic histograms of the number of terminal groups, Pe(z), at fixed τc and varying D.

terminal groups bearing charges. The behavior of the terminal groups follows that of all the monomers discussed above. In particular, the adsorption process with increasing the field strength is clearly seen through the profiles becoming narrow and peaking on the electrode in the limit of strong fields. Note that even at the strongest fields considered not all of the terminal monomers are adhered on the substrate but stretch F


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the influence of electric fields between the electrodes of a plate capacitor. In the course of our simulations the molecule was grafted on one of the electrodes. The two parameters we varied in a wide range were the reduced electrostatic temperature, τc, and the field temperature, τf. While the electrostatic temperature corresponds to the combined effect of temperature and dielectric constant of the solution, the ratio D = τc/τf = σ corresponds to the displacement field of the capacitor and thus to the charge density on each plate. In experiments, variation of the electrostatic temperature and charging of the capacitor can be performed independently. To discuss the results, we consider the regime of low electrostatic temperature which we can associated with the osmotic regime of the dendrimer where most of the counterions are trapped inside the dendrimer and its shape is controlled by the osmotic pressure of the residual, i.e., noncondensed counterions. In our units this corresponds to τc < 1. Our main observation is that with increasing field strength counterions are gradually stripped from the dendrimer and are localized at the opposite electrode. This reduces the osmotic pressure inside the dendrimer and leads to a shrinking of its dimension perpendicular to the electrode. At the same time excess charges of the dendrimer are adsorbed on the electrode, and a nonmonotonous density profile can be observed at intermediate field strength. At a value of D* ≃ QD/AD ≃ 0.01 (QD is the total charge of the dendrimer and AD the cross section) which corresponds to the point where the total field above the dendrimer is inverted counterions are no more attracted by the molecule and are completely localized close to the opposite electrode. We observe a rather strong collapse of the dendrimer perpendicular to the electrode for values close to D*. In the lateral direction the dendrimer is flattened, which is caused by two effects: excluded volume repulsion, as for adsorption, and repulsion between charges which are locally unscreened for D > D*. These effects are expected to be most pronounced in the wide capacitor limit where the distance between the plates is much larger as compared to the height of the brush and to the localization length of counterions close to the electrodes and above the dendrimer. Overlap of these length scales will lead to crossover effects and smear out the behavior at D*. In our case this happens for higher electrostatic temperatures. Our study reveals the interplay of counterion distribution and charge effects on the behavior of dendrimers at charged substrates. This is different to the scenario of the adsorption of linear chains at oppositely charged surfaces where the assumption was made that counterions are released as a consequence of the localization at the charged substrates.44,45 For the case of grafted dendrimers the field gradually strips counterions from the dendrimer and thus induces both a reduction of osmotic pressure inside the molecule and an overall charging which leads to a strong collapse behavior as a function of the external field. It remains an interesting question how the dendrimer’s conformations are reorganized to respond to this combination of effects. Two scenarios can be possible: first, a gradual adsorption of uncompensated charges on the electrode and, second, the formation of two populations of charged strands: connected groups of uncompensated charged monomers are adhered on the oppositely charged electrode while a population of other strands are still in osmotic balance with the remaining counterions. The latter scenario leaves more extension of the strands possible for the compensated fraction

Figure 9. Perpendicular, Rg⊥, and parallel, Rg∥, components of the radius of gyration vs (a, c) D at fixed τc and (b, d) τc at fixed D for the G5 charged dendrimer with spacers S = 8.

explained by the interplay of osmotically active, condensed and free counterions as discussed in our previous work.35−37,39−41 In order to complement our analysis of the dendrimer’s shape, we consider the relative shape anisotropy, a.42,43 Within this approach the molecule is visualized as a continuous, ellipsoidal object. In particular, a = 0, a = 1/4, and a = 1 for spherical, oblate, and extremely elongated ellipsoids, respectively. In Figure 10 we plot the shape anisotropy versus the field

Figure 10. Shape anisotropy vs D at fixed τc for the G5 charged dendrimer with spacers S = 8.

strength for the considered dendrimer. The flattening of the molecule with increasing magnitude of the field is clearly seen. The dendrimer is nearly spherical in the weak field regime, passes through a variety of ellipsoidal shapes at intermediate fields, and finally flattens in the strong field regime. Note again that at higher electrostatic temperatures the shape transition is rather smooth whereas at stronger electrostatic couplings between the charges it reveals a tendency to become steep.

IV. SUMMARY AND CONCLUSIONS We applied the bond fluctuation model to investigate the behavior of a charged G5 dendrimer with spacers S = 8 under G


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Macromolecules

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of charges and thus increases the entropy of the residual counterions. Our results indicate that the latter scenario might occur because of the nonmonotonous monomer profile at intermediate values of the field. In our study we concentrated on simulations of the simplest system composed of a single charged dendrimer and counterions in a salt free, athermal solution only. Effects of salt valence, grafting density, and quality of the solvent on dendrimers in electric fields are not yet explored. Neither is the effect of varying the generation number and spacer length. Since the effective charging would be reduced by enlarging the spacers, the collapse transition could occur earlier and may become smoother. Increasing the spacer length of the dendrimer corresponds to some extent to a decrease in the grafting density in a charged brush. For this reason, though computationally very demanding, a systematic inspection of charged dendrimer brushes in various environments provides an interesting research field to take up in the forthcoming steps.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (J.S.K.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from the Deutsche Forschungsgemeinschaft (DFG) contract numbers SO-277/2-1 and KL 2470/1-1 is gratefully acknowledged. Part of the calculations were carried out at the Center for High Performance Computing (ZIH) of the TU Dresden. We thank Martin Wengenmayr for the effort he made to prepare the snapshots.

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REFERENCES

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Simulations of a Grafted Dendritic Polyelectrolyte in Electric Fields

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