The PoissonâBoltzmannâFlory Approach to Charged Dendrimers

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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The Poisson−Boltzmann−Flory Approach to Charged Dendrimers: Effect of Generation and Spacer Length J. S. Kłos*

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Faculty of Physics, A. Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland ABSTRACT: Using the Poisson−Boltzmann−Flory approach and a cell model, we examine the properties of dendritic polyelectrolytes and counterions in a broad range of molecular weights under variation of dendrimer generation and spacer length. The theory predicts that for the assumed cell size the lower generation dendrimers exist in the polyelectrolyte regime characterized by low and nearly uniform counterion densities. When the dendrimer generation is increased beyond G7, a crossover to the osmotic regime occurs where pronounced condensation of counterions onto the dendrimers takes place. In both regimes the molecules are swollen as related to their neutral state. For a given spacer length the expansion factor changes nonmonotonously with dendrimer generation and is maximal at the crossover. For fixed generation in the polyelectrolyte regime, it is the greatest for shorter spacer chains. The dendrimer effective charge grows monotonously with increasing generation number and, for a given generation in the osmotic regime, is subtly affected by spacer length. The theory also reveals an effect of both parameters of the dendritic architecture on the mean electrostatic potential, the electric field, and the ζ potential.

I. INTRODUCTION Besides generation, spacer length is another important parameter adjusted in the synthesis that influences the properties of neutral and charged dendrimers. The question of conformations of isolated neutral dendrimers, the scaling behavior of their size, and the role of spacers in it were handled by theoretical and simulation approaches. As a result, a number of scaling laws for the radius of gyration, R, incorporating molecular weight, N, and generation number, G, were proposed and verified numerically. Among them the formula R ∼ N1/5(GS)2/5, including spacer length S, explicitly was proved most appropriate on the mean-field level.1,2 For charged dendrimers, it was observed in molecular dynamics simulations that the inner spacers are strongly stretched and the most outside spacers always have the same length. The relaxation time of neutral dendrimers with long spacers was found to agree well with the prediction of the Rose model, while that of charged dendrimers to show a weak dependence on generations or molecular weight with increasing the charge density.3 The role of spacers and their local relaxation behavior were also considered in the study of the relaxation process of a single dendrimer based on the assumption that the molecule relaxes hierarchically from the outermost generation to the first one. As a result, scaling laws for various relaxation times were derived and supported by the results of Brownian dynamics simulations.4 Furthermore, according to some Monte Carlo simulations, swelling of charged dendrimers appears whose degree is affected by both the reduced temperature and spacer length. In particular, for longer spacers the maximum swelling was found to shift to lower temperatures.5,6 Experimentally, the shape and the structure of starlike dendrimers under variation of generation and spacer length in a good solvent were measured with small-angle neutron scattering and neutron spin−echo spectroscopy.7 It was inferred that dendrimers © XXXX American Chemical Society

become more compact with increasing generation and spacer length and that deviations from Gaussian behavior are larger with decreasing spacer length. It was also concluded that the dendrons in dendrimers with short spacers stretch out to gain volume, whereas dendrimers with longer spacers exist in a more space filling shape most likely by backfolding of the dendrons to the center. Last, but not least, understanding the influence of the parameters characterizing the dendritic architecture on the structural properties of dendrimers is of importance in such fields as nanomedicine and pharmaceutical science.8 In this respect the question of the effect of spacer length on the tumor-targeting potential of dendrimers serves as the key example.9 Motivated by the aforementioned findings, in the present work we examine dendritic polyelectrolytes in salt-free solutions using the Poisson−Boltzmann−Flory theory and a cell model.10 In continuation of our previous study which narrowed down to a variety of dendrimers under the constraint of fixed spacer length,11 we inspect here dendritic polyelectrolytes in a broad range of molecular weights under variation of both generation number and spacer length. This enables us to provide further insight into the relation between the branched architecture of charged dendrimers and their conformational properties in equilibrium. Our approach follows the current trends in the field of theoretical methods of studying various polyelectrolyte systems. 12−19 It is complementary to Monte Carlo and molecular dynamics simulations2,3,5,6,20−23 and to investigations based on smallangle scattering of X-rays and neutrons, optical probe Received: March 5, 2019 Revised: April 17, 2019

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DOI: 10.1021/acs.macromol.9b00446 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules Ω = Ωel + Ωexcl + Ωele

techniques, and transmission electron microscopy as well as diffusion NMR.24−34 The paper is organized as follows. In sections II and III we recap the model along with the numerical method for solving the Poisson−Boltzmann equation. The results of our calculations are presented and discussed in section IV. Section V contains our conclusions and remarks.

where Ωel = kBT

(2.1)

Nt = 2G + 1

(2.2)

3N R2b−2 2(GS)2

(2.4)

is the elasticity term and

II. MODEL We inspect charged dendrimers with the core of two bonded monomers, branching functionality, f b = 3, immersed in an implicit solvent with a constant electric permittivity, ϵ. The molecular weight, N, and the number of terminal groups, Nt, in the molecules are N = 2 + 4S(2G − 1)

(2.3)

Ωexcl = kBTνb3

3N 2 8πR3

(2.5)

represents the mean excluded volume interactions. Here kBT is the thermal energy, b is the Kuhn statistical length, and ν is the excluded volume parameter. In the following we consider charged dendrimers in a good solvent for their molecular skeleton, in which case ν > 0. The equilibrium size of the dendrimers without charges, R0, is derived by the minimization of the sum Ωel + Ωexcl, with respect to R, which yields6,35 i 3 y R 0 = bjjj ν zzz k 8π {

1/5

where G is dendrimer generation and S is spacer length. Our calculations are performed using a Wigner−Seitz cell model of molecular solutions14 (see Figure 1). We study a single

N1/5(GS)2/5

(2.6) ÄÅ ÉÑ ÅÅ 1 i ∂ϕ y2 ÑÑ r 2ÅÅÅÅ ϵjjj zzz + kBTne βϕÑÑÑÑ dr ÅÅ 2 k ∂r { ÑÑ ÅÇ ÑÖ (2.7)

The third term on the right-hand side of eq 2.3 Ωele = 4π

∫0

L

r 2ρext ϕ dr − 4π

∫0

L

is the mean-field Poisson−Boltzmann grand potential; see refs 19, 36, and 37 as well as the Appendix. In the formula ϕ is the mean electrostatic potential, n is the counterion density at the e reference point where ϕ = 0, and β = k T with e standing for B

elementary charge. Here, the external charge density, ρext, due to the bare positive charge of the dendrimers is assumed as ρext =

Θ(R − r )

(2.8)

3fN d2Φ 2 dΦ + = − 3 Θ(Y − y) + me Φ 2 y dy dy Y

(2.9)

where for convenience we introduce dimensionless Φ = βϕ, y = r/lB, Y = R/lB, m = 4πlB3n, and lB =

e2 4πϵkBT

is the Bjerrum

length. Thus, by combining the Poisson−Boltzmann approximation for electrostatics and the Flory theory for conformation of polymers with the excluded volume interaction between monomers, we examine dendritic polyelectrolytes on the standard mean-field level without including the electrostatic fluctuations and correlations. As indicated by a number of studies, to take into account effects of this kind, more sophisticated methods must be employed.38−40 Furthermore, the counterion-polymer steric interaction is also neglected. The equilibrium size of the charged dendrimers is obtained by the minimization of the thermodynamic potential with respect to R14,19,37

dendrimer with the radius of gyration R placed in the center of a spherical cell of radius L > R. The concentration of the dendrimers in the solution, ρd, is related to the radius of the −1

( 43 πL3)

4πR3

with Θ(x) standing for the unit step function and r for the radial distance from the coordinate origin.11,14,37 The mean electrostatic potential, ϕ, is related to the volume charge density through the Poisson−Boltzmann equation19,36

Figure 1. Sketch of a charged dendrimer of radius R and counterions in a Wigner−Seitz cell of radius L. The red circle represents the dendrimer−bulk interface separating the molecule from the bulk. The Wigner−Seitz cell is sketched with the blue circle. The charged groups of the dendrimer (counterions) are shown with the black (white) circles. Because of the spherical symmetry, the electric field vanishes in the center of the dendrimer. Because of the spherical symmetry and charge neutrality inside the cell, the electric field is zero at the surface of the cell.

cell through ρd =

3fNe

. In the following we consider salt-

free solutions and assume that a fraction f of the monomers bear monovalent positive charges that are accompanied by f N pointlike monovalent negative counterions due to the requirement of electroneutrality inside the cell. It is assumed that with the center of the dendrimer in the coordinate origin all the properties of the system are spherically symmetric and depend on the distance from the coordinate origin only. The grand potential of a charged dendrimer and counterions has the form

∂Ω =0 ∂R

(2.10)

which in the reduced units yields B


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2 9fN Δ 3NY −3 9νN − − =0 l 4 2 8πY Y4 (GS)

tration, m, is unknown. Therefore, we employ here a version of the shooting method47 in which the reference ionic concentration is the guess-value parameter: For a given Y we integrate eq 2.9 a number of times with various m values and, using the aforementioned interpolation procedure, seek mr such that |∂Φ/∂y(D)| is on the order of 10−14 or less. Once mr has been found, we accept the corresponding electrostatic potential as the correct solution to the boundary value problem.

(2.11)

where Δ=

∫0

Y

y 2 [Φ − Φ(Y )] dy

(2.12)

l = lB/b is the reduced Bjerrum length, and Φ fulfills the Poisson−Boltzmann equation (eq 2.9). Equation 2.11 expresses the balance between pressures of different origin within the dendrimers in equilibrium. The first term refers to the elastic stress of the molecules; the second and the third one refer to the pressures due to the steric repulsion between the monomers and the electrostatic interaction, respectively. For the sake of numerical calculations, in the following we assume that ν = 0.5, D = L/lB = 3000, and l = 1.4, which refers to the Bjerrum length in water at room temperature, lB ≈ 7 Å, to a typical Kuhn length b ≈ 5 Å,41 and to the cell size L = 2 μm. In order that the model corresponds better to terminally charged dendrimers such as PAMAMs (polyamidoamine) at neutral pH,24,42−46 we set the protonation fraction to f = Nt/N. Note that according to eqs 2.1 and 2.2, this means that upon varying spacer length the bare charge of the molecules is kept constant for given dendrimer generation. In the following we present our results obtained for dendrimers with generations between G3 and G9 and with S8, S12, and S16 spacers.

IV. RESULTS A. Ionic Spatial Distribution. In Figure 2 we display the reduced counterion density

Figure 2. A lin-log plot of the reduced counterion density, ρc, versus the reduced distance from the center of the cell for the dendrimers with S12 spacer. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

III. NUMERICAL SCHEME To derive the equilibrium features of the charged dendrimers, the set of eqs 2.9 and 2.11 is solved numerically. We first solve eq 2.9 with the boundary conditions (3.13a)−(3.13c) with a number of Y’s picked from the proximity of Y0 = R0/lB through trial and error and a visual check such that ∂Ω/∂R(Y) changes its sign. Then using the set of pairs, A = {(Y, ∂Ω/∂R(Y))}, interpolation is applied, the root, Yr, of the interpolating function is found, and the boundary value problem with Y = Yr in the Poisson−Boltzmann equation is solved again. Should |∂Ω/∂R(Yr)| not be small enough the pair (Yr, ∂Ω/∂R(Yr)) is attached to A, and the procedure is repeated. Applying this scheme leads to Φ and Y such that |∂Ω/∂R(Y)| is on the order of 10−5 and less, which we accept as the solution to eq 2.11. As mentioned above, solutions to eq 2.9 must be found at each stage of solving eq 2.11. Because no analytical solutions to eq 2.9 are known, we seek numerical ones which fulfill the boundary conditions: ∂Φ = 0 at y = 0 ∂y (3.13a) ∂Φ =0 ∂y

Φ=0

at y = D

at y = 0

ρc = me Φ

(4.14)

versus the dimensionless radial distance from the center of the cell for various dendrimer generations and S12 spacer. It is seen that the Poisson−Boltzmann−Flory theory predicts a tendency of counterions to accumulate inside the molecules and to decrease their density sharply across the interface and in the bulk. As compared with the lower-generation dendrimers, the higher-generation ones are characterized by a significantly higher counterion density, especially in their interior. Such an effect we attribute to stronger Coulomb attraction between the higher-generation dendrimers and counterions due to a larger bare charge of the former and a larger number of the latter. Thus, the uptake of counterions by the higher-generation molecules becomes apparent and results in inhomogeneous spatial distributions of counterions throughout the cell. This kind of counterion condensation onto the dendritic polyelectrolytes signals that the molecules exist in the osmotic regime. In turn, the lower-generation dendrimers remain in the polyelectrolyte regime because of minor counterion condensation and nearly even ionic spatial distributions.3,11 To localize the crossover between both regimes more precisely, in Figure 3 we display the fraction

(3.13b)

fcin = (fN )−1

(3.13c)

Equations 3.13a and 3.13b establish the condition for a vanishing electric field in the center and on the boundary of the cell due to the spherical symmetry and charge neutrality inside the cell, respectively. Equation 3.13c refers to the reference point of the electrostatic potential, which without a loss of generality we set to zero at the coordinate origin. Knowing Φ and ∂Φ/∂y at y = 0 makes it possible to integrate eq 2.9 using the Runge−Kutta method. However, for our system the integration cannot be performed straightforwardly because the numerical value of the reference ionic concen-

∫0

Y

y 2 ρc dy

(4.15)

of the condensed counterions versus generation number for various spacer lengths. For the dendrimers with generations lower than G7 the fraction remains nearly zero for each spacer length, indicating that the molecules are in the polyelectrolyte regime. For G7−G9 dendrimers it is finite and increases with dendrimer generation, which signals that the osmotic regime is explored. Thus, the crossover between the two regimes is found for G7 dendrimers whose bare charge is large enough to attract a considerable fraction of the counterions into their C


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chains are more compact molecules with a somewhat improved capability to uptake counterions in the osmotic regime. It is worth noting that qualitatively similar tendencies of ionic density profiles for dendritic polyelectrolytes with spacers of different lengths were found in Monte Carlo simulations as well.5,6 In relation to the phenomenon of counterion condensation, Figure 5 presents the cumulative fraction of counterions Figure 3. Fraction of the condensed counterions, fcin, versus generation number for various spacers.

fc = (fN )−1

∫0

y

y′2 ρc dy′

(4.16)

interior. Note that in the osmotic regime the fraction of condensed counterions reveals a spacer-length dependence such that the molecules with shorter spacer chains absorb a larger amount of counterions and that the crossover takes place for G7 generation no matter spacer length. The latter observation we attribute to the fact that, in line with PAMAMs at neutral pH, the bare charge of the considered dendrimers is set down by generation number only (see eq 2.2). Thus, the Poisson−Boltzmann−Flory theory in its current form predicts that for dendrimers with the bare charge independent of spacer length the crossover between both regimes is not clearly affected by spacer length either. We complement the presentation of the reduced counterion density profiles with Figure 4, in which we display them for

Figure 5. A log-lin plot of the cumulative fraction of counterions, fc, versus the reduced distance from the center of the cell for fixed dendrimer generations and various spacers. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

versus the radial distance from the center of the cell for fixed dendrimer generations. For the dendrimers in the polyelectrolyte regime the fraction increases sharply to one at the boundary of the cell due to small ionic densities in the interior region. For the molecules in the osmotic regime where a significant ionic penetration of the molecules takes place, the profiles increase abruptly already in the vicinity of the interface, saturate in the bulk, and peak to one as the boundary of the cell is approached. Unlike the polyelectrolyte regime, here the effect of spacers becomes noticeable since the fraction is greater for a smaller spacer length. To show the effect of dendrimer generation on the cumulative fraction of counterions, in Figure 6 we display the fraction for varying generations and S12 spacer. In this case the transition between the polyelectrolyte and the osmotic regime is clearly signaled by the change in the appearance of the profiles in going from low to high generations. In particular, a significant increase in the fraction is seen in the bulk region as a result of increasing dendrimer generation. It is also informative to examine the spatial distribution of the charge present in the system. In Figure 7 we display the reduced overall charge density

Figure 4. A lin-log plot of the reduced counterion density, ρc, versus the reduced distance from the center of the cell for fixed dendrimer generations and various spacers. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

fixed dendrimer generation and varying spacer length. It is seen that the dendrimers with shorter spacers are characterized by higher reduced counterion densities in their interior, which corresponds with the aforementioned observation of a larger condensed ionic charge in this case. A contrary tendency of the counterion densities become apparent in the bulk, especially for the dendrimers in the osmotic regime (see Figure 4d−f). On the basis of the observation that the radius of gyration of the molecules decreases with decreasing spacer length, we conjecture that the charged dendrimers with shorter spacer D


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which results in a monotonously increasing reduced overall charge density in the interior region. Like in the polyelectrolyte regime, the size of the molecules increases with increasing spacer length, which brings about a considerable reduction of the reduced overall charge density as well. Furthermore, because of low ionic concentrations in the bulk, the reduced overall charge density there is nearly zero as compared with that in the dendrimer domain. In Figure 8, we present the Figure 6. A log-lin plot of the cumulative fraction of counterions, fc, versus the reduced distance from the center of the cell for the dendrimers with S12 spacer. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

Figure 8. Reduced overall charge density, ρch, versus the reduced distance from the center of the cell for the dendrimers with S12 spacer. The vertical jumps occur at the dendrimer−bulk interface.

profiles for fixed spacer length and increasing generation number. Here, the crossover between the polyelectrolyte and the osmotic regime for G7 generation is detected by the change in the profile shape. Knowing the overall charge density enables us to calculate the normalized reduced cumulative charge Q nc = (fN )−1

3fN Y3

Θ(Y − y) − ρc

y

y′2 ρch dy′

(4.18)

as a function of the radial distance from the center of the cell under the constraint of fixed generation number and fixed spacer length (see Figures 9 and 10). This quantity goes up sharply to a finite value at the dendrimer−bulk interface, saturates in the bulk, and subsequently drops to zero at the

Figure 7. Reduced overall charge density, ρch, versus the reduced distance from the center of the cell for fixed dendrimer generations and various spacers. The vertical jumps occur at the dendrimer−bulk interface.

ρch =

∫0

(4.17)

versus the reduced radial distance from the center of the cell for fixed dendrimer generations and various spacers. For the molecules in the polyelectrolyte regime the profiles are steplike, which shows that the reduced dendrimer bare charge density itself contributes most to the reduced overall charge density in the interior region (see Figure 7a−c). The jump in the function at the dendrimer−bulk interface is a consequence of the assumption that the dendrimer bare charge is distributed uniformly over the volume of the molecules and drops to zero discontinuously across the interface. In comparison, simulations, for which such an assumption need not be made, also predict a sharp, but continuous, decrease in the overall charge density on the periphery of charged dendrimers.5,6,22,48 Note that the reduced radius of gyration of the dendrimers increases with increasing spacer length, which leads to a reduction of the reduced overall charge density due to the constraint of a fixed dendrimer bare charge in this case. As displayed in Figure 7d− f, the overall picture of the spatial charge distribution changes significantly for the molecules existing in the osmotic regime. Here, the dendrimer positive charge is considerably neutralized in the vicinity of the center by the condensed counterions,

Figure 9. A log-lin plot of the normalized reduced cumulative charge, Qnc, versus the reduced distance from the center of the cell for fixed dendrimer generations and various spacers. The red vertical lines indicate the approximate location of the dendrimer−bulk interface. E


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Figure 10. A log-lin plot of the normalized reduced cumulative charge Qnc, versus the reduced distance from the center of the cell for the dendrimers with S12 spacer. The red vertical lines indicate the approximate location of the dendrimer−bulk interface.

Figure 12. Ratio, Qeff/f N, between the reduced dendrimer effective charge and the reduced dendrimer bare charge versus dendrimer generations for various spacers.

Q eff /fN = 1 − fcin

is nearly one in the polyelectrolyte regime because of slight counterion condensation and drops below one in the osmotic regime because of the condensation effect being apparent. Note that as dendrimer generation is increased, the ratio between both quantities decreases, which indicates that the difference between them grows. It is worth mentioning that similar tendencies of the behavior of the dendrimer effective charge upon increasing generation number were found in small-angle neutron scattering measurements and a combination of diffusion and electrophoresis NMR of PAMAM dendrimers27,49 as well as in molecular dynamics simulations.50 B. Electrostatic Potential and Electric Field. In Figure 13, by the example of G4 dendrimers composed of spacers of

boundary of the cell according to the condition of charge neutrality inside the cell. For the molecules in the polyelectrolyte regime the effect of different spacer lengths is hardly seen, whereas for those in the osmotic regime the profiles become distinguishable in the bulk (see Figure 9). In particular, a slight reduction of the normalized reduced cumulative charge is observed for shorter spacers. As presented in Figure 10, in reaction to increasing dendrimer generation the normalized reduced cumulative charge decreases both at the periphery of the molecules and in the bulk as a consequence of counterion condensation. The uptake of counterions by the dendrimers has an influence on the total charge enclosed by the dendrimer−bulk

Figure 13. Reduced mean electrostatic potential, Φ, versus the reduced distance from the center of the cell for G4 dendrimer and various spacers. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

Figure 11. Reduced dendrimer effective charge, Qeff (black), the absolute reduced charge of the condensed counterions, Qcin (blue), the reduced dendrimer bare charge, f N (red), versus dendrimer generations for various spacers.

various length, we display the mean electrostatic potential, Φ, as a function of the reduced radial distance from the center of the cell. The potential reaches its maximum at the origin and decreases monotonously throughout the interior region and beyond. As we discuss in the following, this feature of the electrostatic potential plays a crucial role in the swelling behavior of the molecules. In regard to varying spacer length, we find that the decrease of the potential is sharper for the molecules with shorter spacer chains. Subsequently, at further distances from the interface a very slow drop of the potential is observed. In general, the obtained mean electrostatic potential is typical for penetrable spherical polyelectrolytes examined with the Poisson−Boltzmann theory and a cell model.37,47 Figure 14, in turn, presents the mean electrostatic potential for the dendrimers with various generations at fixed spacer length. Here, the tendency of the potential to flatten in the interior region with increasing generation is observed. Knowing the mean electrostatic potential allows the ∂Φ calculation of the reduced electric field, E = − ∂y . In Figure

interface. In Figure 11 we present the reduced dendrimer effective charge Q eff = fN (1 − fcin )

(4.20)

(4.19)

along with the absolute reduced charge of the condensed counterions, Qcin = f Nfcin, and the reduced dendrimer bare charge, f N, versus dendrimer generation for different spacer lengths. It is seen that in spite of an increase of the absolute reduced charge of the condensed counterions, the reduced dendrimer effective charge increases monotonously as well. Thus, the growth of the reduced dendrimer bare charge with generation number, as given by eq 2.2, is more rapid than that of the absolute charge of the condensed counterions. In the polyelectrolyte regime the reduced dendrimer effective charge is practically independent of spacer length and equal to the reduced dendrimer bare charge due to minor counterion condensation, whereas in the osmotic regime a somewhat smaller reduced dendrimer effective charge is obtained for shorter spacers. Furthermore, Figure 12 indicates that in the osmotic regime the reduced dendrimer effective charge increases more slowly with generation number than the reduced dendrimer bare charge. Actually, the ratio

15 we display the reduced electric field versus the radial distance from the origin for fixed dendrimer generations and varying spacer length. The field peaks at the dendrimer−bulk F


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Figure 14. Reduced mean electrostatic potential, Φ, versus the reduced distance from the center of the cell for the dendrimers with S12 spacer and various generations. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

Figure 16. Reduced electric field, E, versus the reduced distance from the center of the cell for dendrimers with S12 spacer and various generations. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

slip plane one ionic diameter away from the surface of the particle.50,54 Thus, for Y denoting the reduced radius of gyration of the dendrimer and for the ionic diameter on the order of Kuhn length, we approximate the reduced ζ potential as ζ = Φ(Y + l−1) − Φ(D). Although there is some ambiguity in such a choice of the slip surface, we expect that our results reflect qualitatively the generation and spacer dependence of the reduced ζ potential (see Figure 17). The reduced ζ

Figure 17. Reduced ζ potential versus dendrimer generations for various spacers.

potential is found to increase monotonously with dendrimer generation for fixed spacer length, whereas for fixed generation it reveals a slight tendency to decrease for longer spacer chains. The former observation corresponds with the results of Monte Carlo and molecular dynamics simulations of PAMAMs50 as well as with the capillary electrophoresis measurements of the effect of counterions on the mobility of carboxyl-terminated dendrimers.53 C. Swelling of Charged Dendrimers. Based on eq 2.11, the expansion factor of the charged dendrimers in equilibrium can be expressed as R fexp = = (1 + fcs )1/5 R0 (4.21)

Figure 15. Reduced electric field, E, versus the reduced distance from the center of the cell for fixed dendrimer generations and various spacers. The vertical lines indicate the approximate location of the dendrimer−bulk interface.

interface and drops to zero both at the origin and at the boundary of the cell due to the used boundary conditions. In particular, we observe that the height of the peak reveals a spacer dependence such that it decreases for the molecules composed of longer spacer chains. In the interior of the dendrimers existing in the polyelectrolyte regime the electric field is zero at the origin only and, like for uniformly charged solid spheres,51 its radial growth looks linear. For the molecules in the osmotic regime, there is a domain of a finite radius around the origin where the electric field practically vanishes. Such an effect corresponds with the aforementioned flattening of the electrostatic potential in that area and originates from screening of Coulomb interactions caused by the condensed counterions. As a result, in this case the nonlinear behavior of the field inside the molecules becomes apparent. The variations of the electric field are also seen when considering it for the dendrimers with fixed spacer length and increasing generations (see Figure 16). In relation to the phenomenon of electrophoretic mobility,52,53 the Poisson−Boltzmann−Flory approach makes it possible to estimate the ζ potential for dendritic polyelectrolytes. In this respect we consider charged dendrimers as colloids bearing the effective dendrimer charge and with the

where we have introduced the Coulomb steric factor 8πfl 3Δ (4.22) νN defined as the ratio between the electrostatic and the steric term in this equation. Note that fexp > 1 in the whole range of generations and spacers due to Δ > 0, as a consequence of the aforementioned property of the mean electrostatic potential of being a monotonously decreasing function of the radial distance from the center. Thus, the electrostatic term in eq 2.11 represents repulsion, and in accordance with the ideal-gas approach to the condensed counterions,1,3,5,6,20−23,55,56 the theory predicts swelling of the dendritic polyelectrolytes as compared to the size of their neutral counterparts. In particular, in the polyelectrolyte regime where the dendrimer fcs =

G


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factor decrease, whereas the steric elastic one reveals an increase with dendrimer generation. Thus, the dominance of Coulomb repulsion over the steric interaction is suppressed, which leads to a diminishing and nearly spacer-independent expansion factor. The polyelectrolyte and the osmotic regime are also reflected by the rescaled radius of gyration of the charged dendrimers. In Figure 20 we display on the double-logarithmic

charge density is dominant, the electrostatic part takes a 2

3 (fN ) −5 2 Y

, resulting from the analytical solution Coulomb form, of the Poisson−Boltzmann equation (2.9) in the interior region in the limiting-case approximation with m = 0. The generation dependence of the expansion factor for various spacer lengths is displayed in Figure 18. It is seen that

Figure 18. Expansion factor, fexp, and in the inset the Coulomb steric factor, fcs, versus dendrimer generation for various spacers. Figure 20. A log−log plot of R/(bS3/5) versus (N/S)G2. The lines indicate the slopes 0.26 and 0.13.

in the polyelectrolyte regime swelling is enhanced by increasing dendrimer generation up to the crossover generation G7. In this regime, because of unscreened longrange Coulomb repulsion within the dendrimers and the exponential growth of the dendrimer bare charge, the dominance of the electrostatic term over the steric one in the balance of pressures is strengthened (see the inset of Figure

scale R/(bS3/5) versus (N/S)G2 as obtained from our calculations. According to eq 2.6, for neutral dendrimers the corresponding points belong to a straight line with the slope 1/5. For the charged dendrimers our data reveal a tendency to fall on a single curve, which however on the log−log scale can be considered piecewise linear only. In particular, we find that the slopes of the lines are 0.26 and 0.13 in the polyelectrolyte and the osmotic regime, respectively. Thus, our results indicate that for the weakly charged dendrimers some stretching effect occurs in each regime, which influences the scaling behavior of the radius of gyration originally formulated for neutral dendrimers accordingly. In the polyelectrolyte regime the rescaled radius of gyration increases a bit more rapidly with (N/S)G2 as compared with the neutral case, whereas in the osmotic regime the tendency is inverted.

Figure 19. Coulomb elastic factor, fce (open), and the steric elastic factor, fse (filled), versus dendrimer generation for various spacers.

V. SUMMARY Using the Poisson−Boltzmann−Flory approach and a cell model, we have examined the effect of generation and spacer length on the equilibrium properties of the charged dendrimers and counterions. We have found that for the assumed size of the cell there are two regimes of behavior for the molecules. The features of the lower-generation dendrimers are the manifestation of the polyelectrolyte regime characterized by low and almost even spatial distributions of counterions. The attributes of the higher-generation dendrimers reflect the osmotic regime in which pronounced condensation of counterions onto the molecules occurs. In this case the ionic densities are highly inhomogeneous as a consequence. Our calculations indicate that the crossover between the polyelectrolyte and the osmotic regime occurs for G7 dendrimers due to the bare charge of the molecules being sufficiently large to draw the counterions into their volume. As related to the size of the neutral dendrimers, in both regimes the charged molecules exist in swollen conformations because of Coulomb intramolecular repulsion predicted by the theory. Specifically, the expansion factor depends nonmonotonously on dendrimer generation and is maximal at the crossover. The dendrimer effective charge grows monotonously with generation number and reveals a subtle tendency to decrease with decreasing spacer length in the osmotic regime. Our findings about the expansion factor and the dendrimer effective charge corre-

18). The latter effect is also presented in Figure 19, which shows the Coulomb elastic and the steric elastic factor fce = fse =

fcs 1 + fcs

(4.23)

1 1 + fcs

(4.24)

defined as the ratio of the modulus of the electrostatic and the steric term to the elastic one in eq 2.11. The Coulomb elastic factor is found to increase with dendrimer generation and is accomapnied by a drop in the steric elastic one. In particular, in the proximity of the crossover the elastic stress of the dendrimers is mostly balanced by the effective intramolecular electrostatic repulsion. On the other hand, as the number of monomers is enlarged by increasing spacer length and keeping the dendrimer bare charge fixed, the steric part is enhanced relative to the Coulomb one, and a reduction in swelling is obtained. Subsequently, for the dendrimers in the osmotic regime the electrostatic contribution to the balance of pressures is weakened due to pronounced screening of Coulomb repulsion in the interior region by the condensed counterions. Here the Coulomb elastic and the Coulomb steric H


Article

Macromolecules δ Ωele = qk ϕ + kBT ln(Λ3nk ) − μk = 0 δnk

spond to a certain degree with the experimental investigations of PAMAM dendrimers based on small-angle neutron scattering, small-angle X-ray scattering, and a combination of diffusion and electrophoresis NMR.27,49,57 The theory has also made it possible to calculate the electric field and the ζ potential for the dendritic polyelectrolytes. The most striking property of the former is the generation and spacer-dependent peak shape. The ζ potential reveals a monotonous grow with dendrimer generation and is subtly affected by spacer length as well. The theoretically obtained generation dependence of the ζ potential concurs qualitatively with that observed in simulations50 and capillary electrophoresis measurements.53

which leads to the Boltzmann distribution of the ions nk = n0k e−qkϕ / kBT

where n0k = Λ e is the concentration of the kth component at a point where ϕ = 0, and μk = kBT ln(Λ3n0k )

j

V

∇2 ϕ = −ϵ−1(ρext +

Ωele =

(A.1)

j

■

AUTHOR INFORMATION

The author declares no competing financial interest.

■

REFERENCES

(1) Ballauff, M.; Likos, C. N. Dendrimers in Solution: Insight from Theory and Simulation. Angew. Chem., Int. Ed. 2004, 43, 2998−3020. (2) Kłos, J. S.; Sommer, J.-U. Coarse Grained Simulations of Neutral and Charged Dendrimers. Polym. Sci., Ser. C 2013, 55 (1), 125−153. (3) Lin, Y.; Liao, Q.; Jin, X. Molecular dynamics simulations of dendritic polyelectrolytes with flexible spacers in salt free solution. J. Phys. Chem. B 2007, 111, 5819−5828. (4) Iwaoka, N.; Takano, H. Relaxation of a Single Dendrimer. J. Phys. Soc. Jpn. 2013, 82, 064801. (5) Kłos, J. S.; Sommer, J.-U. Simulations of Terminally Charged Dendrimers with Flexible Spacer Chains and Explicit Counterions. Macromolecules 2010, 43, 4418−4427. (6) Kłos, J. S.; Sommer, J.-U. Simulations of Dendrimers with Flexible Spacer chains and Explicit Counterions under Low and Neutral pH conditions. Macromolecules 2010, 43, 10659−10667. (7) Rathgeber, S.; Monkenbusch, M.; Hedrick, J.; Trollsås, M.; Gast, A. Starlike dendrimers in solutions: Structural properties and internal dynamics. J. Chem. Phys. 2006, 125, 204908. (8) Tian, W.; Ma, Y. Theoretical and computational studies of dendrimers as delivery vectors. Chem. Soc. Rev. 2013, 42, 705−727. (9) Thakur, S.; Tekade, R. K.; Kesharwani, P.; Jain, N. K. The effect of polyethylene glycol spacer chain length on the tumor-targeting potential of folate-modified PPI dendrimers. J. Nanopart. Res. 2013, 15, 1625. (10) Deserno, M.; Holm, C. Cell model and Poisson-Boltzmann theory: A brief introduction. In Electrostatic Effects in Soft Matter and Biophysics; Springer: 2001; pp 27−52. (11) Kłos, J. S.; Milewski, J. Dendritic polyelectrolytes as seen by the Poisson-Boltzmann-Flory theory. Phys. Chem. Chem. Phys. 2018, 20, 17818−17828.

(A.4)

is the overall charge density, ρext is the external fixed charge density, and qj are the charges of the ions of the jth type, and the fact that in the identity the surface integral vanishes due to the condition of zero electric field at the surface of the cell, eq A.1 can be transformed into36 ϵ Ωele = − |∇⃗ϕ|2 dV + ρϕ dV V 2 V

∫

∫V nj[ln(Λ3nj) − 1] dV − ∑ μj ∫V nj dV j

(A.5)

The equilibrium state of the system is found by minimizing the functional with respect to the mean electrostatic potential and the ionic concentrations.36,61−64 The first minimization yields the Poisson equation δ Ωele = ϵ∇2 ϕ + ρ = 0 δϕ

(A.11)

Notes

where

j

B

J. S. Kłos: 0000-0002-8518-9556

(A.3)

+ kBT ∑

j

ORCID

∫V |∇⃗ϕ|2 dV = 12 ∫V ρϕ dV + 2ϵ ∮S(V ) ϕ∇⃗ϕ·dS ⃗

∫

∫V n0je−q ϕ/k T dV

*E-mail: [email protected].

consists of the electrostatic energy due to the electric field and to the ideal entropy of mixing of the ions, respectively. By use of the electrostatic identity

j

(A.10)

Corresponding Author

∫

∑ njqj + ρext

B

∫V ρext ϕ dV − 2ϵ ∫V |∇⃗ϕ|2 dV

− kBT ∑

(A.2)

ρ=

j

Substituting eqs A.4, A.8, and A.9 into eq A.5 yields the standard form of the functional19,36,61,62,65

where nj and μj are the concentrations and the chemical potentials of the jth kind of ions present in the system. Denoting the mean electrostatic potential by ϕ, the thermal energy by kBT, and the thermal wavelength by Λ, the free energy ϵ Fele = |∇⃗ϕ|2 dV + kBT ∑ nj[ln(Λ3nj) − 1] dV 2 V V j

ϵ 2

∑ qjn0je−q ϕ/k T ) j

nj dV

∫

(A.9)

Substituting eqs A.4 and A.8 into the Poisson equation (A.6) brings about the Poisson−Boltzmann equation

APPENDIX. POISSON−BOLTZMANN GRAND POTENTIAL For the ions in a cell of volume V with electric constant ϵ enclosed by the surface S(V), it is convenient to use the Poisson−Boltzmann grand potential10,36,58−60

∑ μj ∫

(A.8)

−3 μ k /kB T

■

Ωele = Fele −

(A.7)

(A.6)

which links the mean electrostatic potential with the overall charge density. The second yields I


Article

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K


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