THE JOURNAL OF CHEMICAL PHYSICS 139, 164906 (2013)

The escape of a charged dendrimer from an oppositely charged planar surface P. M. Welch Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545-0001, USA

(Received 2 August 2013; accepted 9 October 2013; published online 30 October 2013) Many of the envisioned applications of dendrimers revolve around placing these molecules at and removing them from charged interfaces. Herein, we provide a prescription for the conditions needed to release a charged dendrimer from an oppositely charged flat substrate. Identifying an effective segment step length that reflects the intramolecular repulsions due to excluded volume and electrostatics, as well as the dendrimer’s branching, provides the essential concept leading to an analytical prediction for the boundary between captured and free molecules. We find that this effective step length obeys trends similar to those predicted for linear chains, but is modified by the dendrimer’s connectivity. Moreover, the boundary predicted for the capture of linear chains holds for dendrimers once this effective step length is employed. Monte Carlo computer simulations of coarse-grained model dendrimers escaping from charged surfaces validate these findings. The simulations consider generations 2 through 6 with a range of lengths between the branch points, as well as a range of solution ionic strengths and surface charge densities. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4826575] I. INTRODUCTION

Dendrimers hold promise for a number of applications in which their behavior at interfaces is key to their successful use, including roles in therapeutic agent delivery1–6 and medical imaging.7–9 These applications typically involve the partitioning of dendrimers to charged interfaces from solution, driven by electrostatic attraction between the dendrimer and surfaces such as cell membranes, organic films, and metal substrates. Numerous experimental studies illuminate the adsorption behavior of dendrimers on these surfaces, especially detailing the relationship between the amount taken up by the surface and the environmental parameters. However, much remains unknown about how changes in those parameters affect the stability of the complexed system and when such changes may result in the dendrimer’s escape from the interface. We address this question here and present a prescription for the conditions needed for an isolated charged, flexible dendrimer to release from an oppositely charged planar surface. Literature studies clearly demonstrate that electrostatic attraction dominates the behavior of dendrimers near charged surfaces. Amine terminated molecules such as the much studied poly(amido amine) (PAMAM) and poly(propylene imine) (PPI) dendrimers present positive charges at pH values as high as 10.10, 11 Many studies indicate that the resulting attraction between positively charged dendrimers and negatively charged surfaces results in a flattening of the molecule. This has been observed by Mecke et al.12 and shown computationally by Suman and Kumar,13 as well as suggested by Lewis and Ganesan.14 Similar flattening was also noted in Mansfield’s15 computational study of non-charged dendrimers at adsorbing interfaces, in the computational study of amphiphilic dendrimers by Lenz et al.,16 and in a study2 0021-9606/2013/139(16)/164906/8/$30.00

of dendrimer-polyelectrolyte complexes adsorbing to charged surfaces. This maximization of the contact between the surface and the dendrimer, in addition to the exponential relationship between the number of charged sites and the generation of growth G, leads to a strong dependency of the behavior at the interface on G, as observed by Esumi and Goino.17 van Duijvenbode et al.18 also noted that the solution ionic strength plays an important role in determining the surface coverage, since the charge-charge repulsion is mediated by the presence of other ions in solution. Many surfaces change with variations in solution pH, resulting in modification of the surface charge density and attendant adsorption characteristics. A subtle effect may also couple to these changes in environmental and molecular parameters; adding charge to dendrimers via variation in solution pH or through more direct chemical modification may produce a molecule whose conformation and mass distribution varies with solution ionic strength. Although the literature contains contradicting experimental19–23 and computational24, 25 findings, many computational26–36 and experimental22, 37–42 reports support this idea. Thus, to provide the most generally applicable theory for dendrimer interactions with charged surfaces, we need a treatment that captures this phenomenon without explicitly requiring it. The solution to the analogous problem for linear chains exists in the literature. Wiegel43 first solved a version of this problem, considering an ideal linear chain with uniform charge density in the presence of a flat, oppositely charged plane. He found that a closed form expression relating the solution ionic strength, the linear charge density on the chain, the surface charge density, and the bare step length for the walk composing the chain exists at the boundary between the free and adsorbed states. However, his model did not include the relevant intra-chain interactions that lead to chain swelling

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due to excluded volume and charge-charge repulsions. Rather, his solution applies only to the ideal phantom chain. Later, Muthukumar44 expanded Wiegel’s43 treatment by introducing an effective model for the charged linear chain by reparameterizing the ideal phantom chain with a step length that accounted for both the excluded volume and electrostatic repulsion. The use of an effective step length to account for intramolecular interactions can be traced back to the work of Kuhn,45 as embodied in the well-known Kuhn model. This concept was put into a modern formulation by the work of Edwards and Singh.46 Those authors extracted the full crossover behavior of the effective step length, characterizing how it increases as the excluded volume interactions become more important. Muthukumar’s extension to charged chains with excluded volume yields insight into the dependence of the effective step length l1 on the solution ionic strength (as reflected in the Debye length κ −1 ) and the contour length of the chain L. In particular, he found that l1 ∝ L1/5 κ −4/5 in the limit of high solution ionic strength and l1 ∝ L in the low ionic strength regime. Combining that result with Wiegel’s43 findings thus provides a simple inequality, discussed in detail below, relating the environmental and molecular parameters at the boundary between adsorption and desorption. Kong and Muthukumar47 later validated this inequality with computer simulations. More recently, Cherstvy and Winkler48–50 have applied the WKB (Wentzel-Kramers-Brillouin) method and approximations to the Debye–Hückel potential to study polyelectrolyte chains near flat and curved surfaces. Their findings closely agree with the solution found by Wiegel43 for the planar case, as well as allows for consideration of induced image charges near the interface that result from dielectric mismatch. We follow the approach of Wiegel and Muthukumar as a guide for this investigation and arrive at the surprising conclusion that the same boundary exists for dendrimers, once the effective step length that captures both the repulsive interactions and the branching is identified. In particular, we find that an adsorbtion boundary exists as defined by Eq. (1) and depends on the Debye length κ −1 , the surface charge density σ , the linear charge density q, the Bjerrum length lB , and the effective step length for the dendrimer l1 : |σ q|lB 0.12 > . 3 κ l1 π

(1)

counting for excluded volume and electrostatics. Note that the first two terms dominate for large L and G. Section II details the theoretical development that leads to these results. In Sec. III, we describe the bond fluctuation Monte Carlo method that we employ to validate these predictions. The simulations consider values of G falling between 2 and 6 with a range of contour lengths L, as well as a range of surface charge densities and ionic strengths. Section IV provides a detailed comparison between the theory and the simulation. Finally, Sec. V provides a summary of these results and a discussion of where we believe they will prove most applicable. II. THEORY

Following Muthukumar’s investigation of charged linear chains with excluded volume, one can construct an effective model for dendrimers in which the excluded volume and charge are accounted for by reparameterizing an ideal model that does not explicitly include these intramolecular interactions. The key to building this effective model lies in finding an effective step length l1 that captures the swelling effect of the intramolecular interactions while preserving the underlying molecular structure and contour length of the molecule. Consider a uniformly charged dendrimer of generation G with linear charge density q and bare step length l0 with P steps between branch points, resulting in a contour length between these junctions of L = Pl0 , and trifunctionality at each branch junction. Figure 1 illustrates this model and the intramolecular interactions, as well as the model surface that is introduced below. The average squared radius of gyration 0 Rg2  of an ideal dendrimer with the pictured structure yields to exact analytical evaluation and is given by Eq. (3):51, 52  2 l0 0 2  Rg  = [3P 3 (−1 + 6 ∗ 2G + [3G − 5]22G ) N −

P (P − 1) G (2 − 1)(9 ∗ 2G P − 7P + 2)]. 2

(3)

Here, N is the total number of segments in the dendrimer and is given by N = 3P(2G − 1) + 1. Analogous to the well-known procedure for mapping a linear chain to an effective Gaussian model, the physical invariants that constrain

The effective step length is given by the cubic root of Eq. (2), connecting the structural parameters and the observed squared radius of gyration of the dendrimer Rg2  to l1 : Rg 2  = l1

M1 (L, G) M2 (L, G) + l12 (3L[2G − 1])2 (3L[2G − 1])2

+ l13

M3 (L, G) . (3L[2G − 1])2

(2)

The three functions of contour length between branch points L and generation of growth G are given by M1 (L, G) = L3 (3{−1 + 6∗2G + [3G − 5]22G } − 4.5∗2G (2G − 1) + 3.5(2G − 1)) , M2 (L, G) = L2 (−(2G − 1) + 4.5∗2G (2G − 1) − 3.5(2G − 1)), and M3 (L, G) = L(2G − 1). Given a measured value of Rg2 , one can calculate the effective step length l1 for that specific dendrimer (viz. fixed L and G), ac-

FIG. 1. A cartoon sketch of the system under consideration in this study. The molecule is a monocentric charged dendrimer with contour length L between the tri-functional branch points. The surface is a flat homogeneous, oppositely charged substrate.

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the value of l1 must be identified to perform the reparameterization of Eq. (3). In the linear case, the contour length of the entire chain provides a sufficient invariant. In the dendritic case, however, care must also be taken to preserve the branching connectivity of the molecule. Thus, we must hold both G and the curvilinear distance between branch points L constant. From Eq. (3) and this set of constraints (identifying terms of L = Pl0 and simplifying), one finds that l1 approximately obeys the cubic equation, Eq. (2), assuming that l1  3L[2G − 1]. We are interested in the general behavior of l1 , since it informs us about the change in the dendrimer with varying solvent and pH conditions. Thus, we need to know how l1 varies, not only as the excluded volume begins to become important, but also when intramolecular charge repulsions cause the molecule to expand. We obtain the dependence of l1 upon the Debye length κ −1 and L in the limit of high salt concentration from a simple scaling analysis. Following Sheng53 and co-workers’ study of non-charged dendrimers, one finds the dependence of Rg2  on κ −1 and L for a charged dendrimer with excluded volume in the limit of high salt concentration from a simple Flory argument. Consider the dimensionless free energy den0 Rg 2 F N N 2 ∝ ( 0Rg )2 + ( Rg ) + w2 GP Rg sity kT 3 + w3 GP ( Rg 3 ) . The Rg first two terms correspond to the entropic penalty for stretching or compressing the dendrimer. The latter two represent the contributions due to the intramolecular interactions. The B ) in two-body penalty factor w2 is proportional to (W + 4πl κ2 26 the high salt limit. Here, W is the excluded volume parameter and lB is the Bjerrum length. The excluded volume penalty may be estimated from the underlying short-ranged (non-Coulombic) potential Up (ri, j ) by Eq. (4):54

({e Rγ ,b ; L, G})    NB G  = dR0 D[Rγ ,b (s)]d J Rγ ,b exp(−H (Rγ ,b )) γ =1 b=1

× δ(Rγ ,b (P ) −J Rγ ,b )δ(Rγ ,b (0) −J Rγ −1,h(c) ) ×

d ri,j [1 − exp(−Up (ri,j )/kT )].

(4)

The Bjerrum length reflects the dielectric strength of the media and is approximately 7 Å in water at 25 ◦ C. Minimization of the F/kT with respect to Rg leads to Eq. (5):  1/5  4π lB 0 2 N GP  Rg  . Rg ∝ W + 2 κ

(5)

Equation (5) can be simplified by noting that  Rg  ∝ B l0 L at large G and P. When electrostatics dominate, 4πl κ2 W , and we recover Rg ∝ κ −2/5 L3/5 . From this result and the linear term of Eq. (2) above, we find that l1 ∝ κ −4/5 L1/5 in the high salt limit for large G and P. Similarly, one expects that Rg ∝ L when the dendrimer is fully stretched in the low salt limit, leading to l1 ∝ L. Thus, l1 for the dendrimer behaves similarly to the linear analog in these limits with respect to contour length L and Debye length κ −1 . The full crossover behavior and dependencies of l1 on G may be probed by generalizing Muthukumar’s approach for linear44 and star-branched55 polyelectrolytes. Unfortunately, the approximation required to make such a model analytically tractable loses much of the connectivity information for dendrimers. Combining Eqs. (2), (3), and (5) yields insight into the full behavior of l1 in the limit of high salt, the region most interesting here. This is discussed in detail in Sec. IV. 0

2

G−1 3∗2 

δ(e Rγ ,j −J Rγ ,j ).

(6)

j =1

Each string composed of the steps between junction points is labeled by the generation at which it falls γ and an index for the specific string within that generation, b. Note that there are NB = 3∗2γ −1 strings in each generation. Equation (6) also introduces the junction points J Rγ , b and integrates over all possible positions. The subscript denoted h(c) is given as the integer of the two choices c = b/2 and c = (b − 1)/2 + 1. Define J R0, h(c) = R0 to be the position of the central junction point. The first two delta functions maintain connectivity between generations. The final set of delta functions enforce that the terminal groups fall within the set {e RG, b }. Equation (7) gives the Edwards Hamiltonian H(Rγ , b ) that captures the energetics of the charged dendrimer: H =

 W =

With an effective model for a charged dendrimer in hand, we may now find the conditions needed for it to be captured by an oppositely charged flat planar surface. The probability function for mass distribution ({e Rγ , b }: L, G) of a dendrimer whose terminal units fall at {e Rγ , b } can be described by the Edwards path integral Eq. (6):

G NB  L dRγ ,b 2 3  ds 2l0 γ =1 b=1 0 ds +

NB  L  L G  G NB  W  ds ds δ(Rγ ,b (s) − Rγ ,b (s )) 2l02 γ =1 b=1 γ =1 b =1 0 0

+

 L NB  L G NB  G  wc   ds ds exp[−κ|Rγ ,b (s) 2 γ =1 b=1 γ =1 b =1 0 0

− Rγ ,b (s )|]/|Rγ ,b (s) − Rγ ,b (s )|.

(7)

The first term represents the connectivity of the molecule, summing over all of the strings. The second captures the pair-wise excluded volume interactions. The third represents the charge-charge repulsive interactions modeled within the Debye–Hückel approximation. Here, wc = q 2 /kT , and  is the dielectric constant of the media. The Hamiltonian for the isolated dendrimer dramatically simplifies to a simple connectivity term when the effective model is used because l1 captures all of the intramolecular contributions due to electrostatics and excluded volume. Thus, the Hamiltonian for the effective model dendrimer in an external potential V (Rγ ,b (s)) is given by Eq. (8): 

 2  L G NB L dR 3  1 γ ,b + He = ds dsV (Rγ ,b (s)) . 2l1 γ =1 b=1 0 ds kT 0 (8)

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Examining Eqs. (6) and (8) reveals that there are no cross-terms involving different portions of the dendrimer. Within this approximation, all of the complications due to intramolecular interactions are now embodied in l1 . The connectivity plays a role only in the requirement that the strings join in the appropriate fashion. Thus, we can factor Eq. (6) into a product of propagators for each string within the dendrimer, as in Eq. (9). This decoupling of the strings is similar to the approach that Lewis and Ganesan56 recently employed in their field theoretical study of charged dendrimers, in which the strings evolve in a mean-field but are required to join at the proper locations along the molecule: 0 ({e RG,b }; L, G)    NB G  d J Rγ ,b L (J Rγ −1,h(c) −J Rγ ,b ; L) = dR0 γ =1 b=1

×

G−1 3∗2 

δ(e RG,j −J RG,j ).

(9)

j =1

The Green’s function for each linear segment  L is the solution to differential equation, Eq. (10):   ∂ l1 2 1 − ∇R + V (R) L (R − R0 ; L) = δ(R − R0 )δ(L). ∂L 6 kT (10) Equation (10) obeys the boundary conditions that  L vanishes when R either goes to infinity or when it hits the wall at z = 0. Note that this expression assumes that l1 is not impacted by the presence of the external potential. Now we may consider a single chain’s behavior in the presence of a charged planar surface. The potential V (Rγ ,b (s)) for a charged plane with surface charge density σ located at the origin and normal to the z-axis is given by Eq. (11), the planar-averaged Debye–Hückel potential. Note that the definition used here assumes that σ accounts for all charge in the vicinity of the surface. This varies by a factor of two from the definition used in other works in the literature where a distinction is drawn between charges above and below the plane defining the surface:48, 57 exp(−z RG,b (s)κ) . (11) κ Here, z RG, b (s) is the z-component of the distance of chain segment away from the origin. We seek to find the conditions under which a single chain is bound by the surface potential V . This corresponds to treating Eq. (10) as an eigen function problem and noting the set of parameters that lead to a non-oscillatory solution for  L . Identifying the precise conditions for this to occur and for the surface to capture the chain is not trivial. However, Wiegel43 and Muthukumar44 interrogated this model and concluded that the chain becomes bound when the condition given in Eq. (1) is satisfied. Since we can factor ({e Rγ , b }; L, G) for the entire dendrimer into products of  L and since  L reflects a bound state under the condition of Eq. (1), one may expect that the conditions needed for a charged surface to capture a flexible charged dendrimer are approximately identical to those for a V = −2π kT lB |σ q|

linear chain. The connectivity of the molecule does play an important role in determining the release conditions. However, all of the molecular information that pertains to that transition resides entirely within the effective step length l1 . III. SIMULATION MODEL AND ALGORITHM

To test these predictions, we probed the behavior of a simple bead-spring model dendrimer in the presence of a flat planar charged surface. Specifically, a uniformly charged monocentric dendrimer of generation G with P steps between tri-functional branching points was placed in the presence of an oppositely charged plane, as illustrated in Figure 2. This coarse-grain model aims to capture the behavior of the general class of dendrimers that are nearly homogeneous in their composition, with each bead representing a group of several atoms. The beads are only distinguished by their branching functionality (1, 2, and 3). Thus, without capturing the local details of the chemistry, the model approximates the general behavior of dendrimers similar to the amine-terminated PAMAM or PPI. The connectivity of the dendrimer is maintained by a harmonic spring potential Us = 3K(l − l0 )2 . Here, l is the distance between two bound beads, K is the spring constant, and l0 is the equilibrium bond length. The Morse potential captures the pair-wise interactions between all non-bonded beads with Up = u0 Exp[−2α(ri, j − rmin )] − 2 Exp[α(ri, j − rmin )]. Here, ri, j is the distance between two non-bonded beads, u0 is the well depth of the potential, α is the range parameter, and rmin is the location of the minimum. The Debye– Hückel screened electrostatic potential represents the chargecharge repulsion between the beads in the model dendrimer with UDH = kTlB ψ 2 exp [−rij κ]/ri, j . Here, kT is the thermal energy, lB is the Bjerrum length, and ψ is the charge per bead. on the solution ionic strength The Debye length κ −1 depends such that κ −1 = (4π lB ci zi2 )−1/2 with ci the concentration and zi the valence of the ith ion species. Similarly, the planar averaged Debye–Hückel potential given by Eq. (11) above dictates the interaction between the charged plane and the dendrimer’s segments. The Bjerrum length lB sets the length scale for the simulations and typically falls to 7.1 Å in water at 25 ◦ C. The thermal energy determines the energy scale for the simulations and is set to 0.85. To facilitate rapid equilibration,

FIG. 2. A diagram of the simulation model used to challenge the theoretical predictions. The Morse potential captures the short-ranged excluded volume interactions. Connectivity is maintained by a simple harmonic spring potential. All charges are represented within the Debye–Hückel approximation. A G = 2, P = 2 model is shown.

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K/ lB2 = 20. The ratio of the equilibrium bond length to the Bjerrum length is set to l0 /lB = 0.7, corresponding to a roughly 5 Å step length in a typical chemistry. To prevent bond crossing, a maximum bond length is set such that lmax /lB = 1, the Morse potential range is set to α/lB = 24, and the minimum of the Morse potential was rmin /lB = 0.8. The Morse potential well depth divided by the thermal energy was set to uo /kT = 1.18 to ensure that the simulations were carried out in the good solvent regime. The charge per bead ψ was set to +1. The Debye length κ −1 was varied to reflect changes in the solution salt concentration, from high to low, with κ −1 /lB = 0.5 to κ −1 /lB = 10. This corresponds to solution salt concentrations ranging from 1.9 mM to 0.77 M in a real system. The surface charge density of the plane varied from σ lB2 = −0.1 to σ lB2 = −1. Finally, a number of different dendrimers were examined with G spanning from 3 to 6 and P ranging from 2 to 10. Bond fluctuation Monte Carlo was employed to determine the effective step length for this model dendrimer in the absence of the charged plane, as well as the conditions needed for these molecules to escape from the plane as a function of the solution salt concentration and surface charge density. In this scheme, beads are chosen at random and moved by a random displacement along the x-, y-, and z-axis ranging from zero to ±0.25lB . Trial moves are accepted with the Metropolis criteria, rejecting any moves that position a bead into the plane (passing through the origin with a normal parallel to the z-axis) or exceeding lmax . Estimates of l1 were constructed from simulations spanning 100 000–10 × 106 Monte Carlo steps (each step defined as N attempted perturbations to the molecule) with samples taken every 10 000 Monte Carlo steps. Simulations evaluating whether a given molecule at a specific value of κ −1 and σ would escape the surface are begun with the dendrimer equilibrated on the surface at a high value of σ . This approach ensures that the molecule does not simply diffuse away without interacting with the charged plane as may happen if the dendrimer were simply placed nearby. The dendrimer is judged to have been released if, after 5 × 106 Monte Carlo steps, the attractive potential exerted by the surface on a bead is greater than −0.02kT per bead.

J. Chem. Phys. 139, 164906 (2013)

FIG. 3. The root-mean-squared radius of gyration plotted as a function of the predicted scaling dependencies of the structural and environmental parameters in the high salt concentration regime, as given by Eq. (5). Data shown for a range of P and G values with κ −1 /lB = 0.5 and κ −1 /lB = 1.0. Different colors are used for each value of P.

above. Note that this scaling prediction differs slightly from that given in Ref. 26 and reflects the more recent findings of Sheng53 and Kröger58 and their respective co-workers. Similarly, the simulations probed Rg2 1/2 /lB for a range of structural parameters P and G in the lowest salt regime of the simulation data, κ −1 /lB = 10. Figure 4 presents those values as a function of the contour length from the branching

IV. RESULTS AND DISCUSSIONS

The dendrimers were first equilibrated under conditions of different solution salt concentrations represented by the Debye length κ −1 . This permits us to identify the high and low salt regimes of the simulation model by comparison of the simulated root-mean-square radius of gyration Rg2 1/2 to the scaling law proposed in Eq. (5). Figure 3 presents Rg2 1/2 divided by lB as a function of 0 Rg2 1/2 , P, G, N, lB , κ −1 , and the excluded volume parameter W . The latter was evaluated by numerical integration of Eq. (4) above, leading to W/ lB3 = 0.56. Results for dendrimers with P in the range of 2 through 10 and G falling between 2 and 6 are shown for the high salt limit. Two different values for the Debye length are represented, κ −1 /lB = 0.5 and κ −1 /lB = 1.0. The solid line corresponds to the best fit to a power law expression with an exponent that falls to roughly 1/5, as predicted in Eq. (5)

FIG. 4. The root-mean-squared radius of gyration plotted as a function of the predicted scaling dependencies of the structural parameters in the low salt concentration limit. Data shown for a range of P and G values with κ −1 /lB = 10. Different colors are used for each value of P.

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FIG. 7. Typical snapshots from the simulations of dendrimers in their equilibrium position relative to the surface. Images from simulation of a G = 5, P = 10 dendrimer with κ −1 /lB = 1.0 shown. From left to right, the surface charge densities are σ lB2 = −0.05, σ lB2 = −0.1, and σ lB2 = −0.3.

FIG. 5. The effective step length divided by the bare step length l1 /l0 plotted as a function of the number of steps between branch points P and the generation of growth G in the high salt limit with κ −1 /lB = 0.5.

center of the dendrimers divided by the Bjerrum length, GL/lB , over the same range of P and G values as those in Fig. 3. The solid line represents the best fit to a power law with an exponent of roughly unity, as predicted in Sec. II above. However, note that the data do not collapse upon the master trend as expected if this simple scaling argument were accurate. This suggests that the truly rigid, fully extended state is not achieved in the simulations. Rather, the data fall in the cross-over region between the high and low salt limits where we have no clear understanding of the dependence of the size of the dendrimer on κ −1 , G, and P. This result mirrors Kong and Muthukumar’s47 findings for linear chains; they, too, observed that the fully rigid-rod scaling could not be recovered in their simulations. Solving the cubic expression given in Eq. (2) in combination with the simulation estimates of Rg2  yields only one physically meaningful (positive) value for l1 in each of the scenarios examined here. However, combining Eqs. (2), (3), and (5) yields clearer insight into the dependence of l1 on the structural and environmental parameters studied than a simple presentation of those numerical results. Figures 5 and 6 were generated to illustrate the trends using the values of lB , l0 , and W relevant to the simulation model, in addition to the missing constant of proportionality implied in Eq. (5) (≈1/2, as determined by examining the simulation predictions for Rg2 ).

FIG. 6. The effective step length divided by the bare step length l1 /l0 plotted as a function of the Debye length divided by the Bjerrum length κ −1 /lB and the generation of growth G with P = 2.

Note that the numerical values obtained directly from solving Eq. (2) with the simulated values of Rg2  fall on these surfaces, as one would expect. Figure 5 specifically presents the ratio of the effective step length to the bare step length l1 /l0 as a function of both the number of steps between branch points P and the generation of growth of the dendrimer G in the high salt limit with κ −1 /lB = 0.5. From the plot, one sees that G more greatly impacts the value of l1 than P. Also note that the effective step length may assume values as high as 15 times that of the bare step length in the limit of high G and P. However, for smaller values of P and G, the ratio approaches 1 to 2. Figure 6 explores the impact of changing κ −1 in the limit of high salt concentration. Fixing P = 2, the figure presents the dependence of l1 /l0 on κ −1 /lB and G. One observes that both parameters strongly influence the value of l1 in this regime. Thus informed, now we may evaluate the applicability of Eq. (1) to dendrimers. In addition to the structural parameters, two variables dictate the behavior of the dendrimers on the charged surface. As noted in the referenced experimental studies, the surface charge density and the solution ionic strength play major roles in mediating the interaction between the surface and the dendrimer. Figure 7 presents typical snapshots from the simulations of a G = 5 and P = 10 dendrimer with κ −1 /lB = 1 with varying surface charge density σ . From left to right, σ lB2 = −0.05, σ lB2 = −0.1, and σ lB2 = −0.3. At relatively low values of |σ | the dendrimer is released. However, as |σ | increases, the dendrimer tends to flatten, first exhibiting a droplet-like conformation and ultimately laying down completely flat to maximize the contact with the surface. Similarly, increasing the solution ionic strength via changes to κ −1 modifies the extent of the dendrimer’s contact with the surface. Figure 8 presents typical snapshots from simulations of a G = 5 and P = 10 dendrimer with

FIG. 8. Typical snapshots from the simulations of dendrimers in their equilibrium position relative to the surface. Images from simulation of a G = 5, P = 10 dendrimer with σ lB2 = −0.015 shown. From left to right, the solution ionic strengths are κ −1 /lB = 10, κ −1 /lB = 5, and κ −1 /lB = 1.0.

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line should produce captured dendrimers. In the figure, the open black circles indicate simulations that resulted in the escape of the molecule; the red squares, simulations where the dendrimer remained trapped by the surface. All of the parameters used in constructing the figure were input variables except l1 , which was estimated from the simulations of the isolated dendrimers. The predicted boundary between escaped and captured molecules predicted by Eq. (1) clearly proves to be highly accurate. The failures may be attributed to inaccuracies in estimating l1 , a computationally expensive procedure for larger dendrimers, as well as the difficulties associated with accurately identifying such transitions from computer simulation.

V. CONCLUSIONS

FIG. 9. A map of the conditions for releasing versus capturing the dendrimer, as predicted by Eq. (1). Data for a range of molecular and environmental parameters are shown. The green line is the theoretical boundary, while the points are results from the simulations. Red squares indicate instances in which the dendrimer remained trapped by the surface. The black circles indicate simulations in which the dendrimer escaped.

σ lB2 = −0.015 and varying κ −1 . From left to right, κ −1 /lB = 10, κ −1 /lB = 5, and κ −1 /lB = 1, corresponding to an increasing solution ionic strength or salt concentration. In the low ionic strength limit, the dendrimer’s conformation balances the attraction between the surface and the molecule against the intra-molecular repulsion to produce a cactus-like structure with a few segments bound to the surface. As κ −1 decreases, the structure begins to relax (with a decreasing value in l1 ) while maintaining contact with the surface. Ultimately, as κ −1 decreases to the high salt concentration limit, the conformational entropy of the dendrimer appears to overcome the attraction to the surface and the molecule escapes. Equation (1) captures these changes and the resulting release of the dendrimer. Clearly, lowering the magnitude of σ and κ −1 drives the left-hand side of Eq. (1) to smaller values. Similarly, lowering l1 drives the ratio to higher values, albeit at a slower rate of change than that affected by modifications to κ −1 . Figure 9 illustrates the accuracy of Eq. (1) in predicting this release transition. Shown are data from simulations of dendrimers with structural parameters in the range of G = 2 to 6 and P = 2 to 10 in media with κ −1 /lB ranging from 0.5 to 10 in the field of a flat surface with σ lB2 spanning from −0.015 to −0.9. A dimensionless product of the surface and solution characteristics |σ |κ −2 comprise the abscissa. The product of the right-hand side of Eq. (1) and the dimensionless ratio of the molecular properties and solution ionic strength 0.12 κl1 make up the ordinate. As prescribed by Eq. (1), simπ |q|lB ulations with conditions falling above the line where the abscissa equals the ordinate, represented by the green line in the figure, should result in freed dendrimers. Those simulations whose parameters map them to the region below the green

We have presented a prescription for constructing an effective ideal model for flexible, charged dendrimers. The central element of this method is a rescaled effective step length l1 that captures all of the intra-molecular repulsions and is analogous to the Kuhn statistical step length. The procedure relies upon knowing the analytical expression for radius of gyration of the ideal (non-interacting) molecule and understanding which structural features must be preserved in rescaling the fundamental step length in the model. Thus, it should prove portable not only to other dendritic structures (di-centric, etc.), but also to other branched polymers. Using a scaling argument informs us about the dependence of this effective step length on the generation of growth and contour length of the molecule, as well as how it should change with variations in solution ionic strength. These predictions were tested using Monte Carlo simulation of a beadspring model of a charged dendrimer. While the simulation model does oversimplify the system compared to the real material, the theoretical framework for estimating the effective step length should be independent of these shortcomings. Most notably, the simulation model does not take into account explicit counter ions, a potential source of discrepancy between the simulations and actual experiments.24 However, the procedure for estimating the effective step length should not be impacted by this approximation. With a description of the effective step length in hand, we are able to construct an effective model for the dendrimer in an external field by appealing to the separability assumption that the model for the effective step length does not change by imposing the external field due to the charged surface. The path integral description for this effective dendrimer model factors into terms of the Green’s function for simple linear chains. Thus, we are able to directly apply the results from analogous problems already solved for linear chains to approximate solutions for flexible dendrimers. This study specifically examines the scenario of a dendrimer escaping from a charged, flat planar surface. Here the solution for the linear chain analog is well-known.43, 44 The effective step length provides the essential connection between the dendrimer and that result. This finding was also tested using Monte Carlo computer simulation of bead-spring models of charged dendrimers. That study showed that, within the

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limits of its applicability, the linear chain solution accurately predicts the boundary between captured and free dendrimers. Several important caveats bear noting: (i) As with the simulation model, the theory for the linear chain model does not include such important elements as explicit counter ions or the possibility of a dynamically varying surface as the dendrimer approaches (as may be especially important in biological membranes,59–61 for example). (ii) Relatively long contour lengths are needed since l1 is assumed to be much smaller than 3L[2G − 1] and the ground state dominance approximation is applied in the original theories of Wiegel43 and Muthukumar.44 (iii) Naturally, open and flexible dendrimers are assumed rather than dense colloid-like structures in which the internal degrees of freedom are less important. (iv) The planar-averaged Debye–Hückel potential does not account for mismatch between the dielectric of the medium and surface, as addressed by Chertsvy and Winkler.50 (v) The model should not work for strongly charged surfaces where the separability assumption (l1 is independent of the external potential) may fail due to perturbations to the ion cloud associated with the dendrimer. However, the findings presented herein should provide a guide for the most commonly studied dendrimers today and be relevant to physiological conditions. ACKNOWLEDGMENTS

This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. Financial support provided by the U.S. Department of Energy Office of Biological and Environmental Research under Proposal SCFY081004 and the Los Alamos National Laboratory Directed Research and Development program. Computing time was provided by Los Alamos National Laboratory’s Institutional Computing. We thank Kim Rasmussen and Boian Alexandrov for many insightful discussions of the issues addressed herein. 1 O.

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