Dendrimers in Solution of Linear Polymers: Crowding Effects

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

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Dendrimers in Solution of Linear Polymers: Crowding Effects Martin Wengenmayr,*,† Ron Dockhorn,*,† and Jens-Uwe Sommer*,†,‡ †

Leibniz Institute of Polymer Research Dresden, Hohe Strasse 6, 01069 Dresden, Germany Institute for Theoretical Physics, Technische Universität Dresden, 01069 Dresden, Germany

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‡

ABSTRACT: We study dendrimers embedded in a solution of linear chains with various degrees of polymerization and concentrations. We distinguish the term “crowding”, addressing the impact of the polymer environment on the dendrimer, from the term “semidilute”, which refers to the state of the polymer environment only. Depending on the length of linear chains, two regimes are observed. Dendrimers in a solution of chains much shorter then their own size display good solvent characteristics at all concentrations. In a dense solution of long chains the dendrimer conformation statistics changes qualitatively. It displays Gaussian scaling with respect to the spacer length instead of the cube root behavior expected for a compact globule. We have found a very good description of the simulation data by using a geometrical ansatz where for crowding given by the matching of the characteristic length scales of the two subsystems; i.e., the ratio of the dendrimer size and the correlation length of the semidilute solution provides the relevant scaling variable. Our study reveals that dendrimer conformations in the strongly crowed state neither are in a compact globular state nor do they correspond to a mean-field θstate. Instead, a novel scaling variable combining the properties of both species reflects the conformation statistics.

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INTRODUCTION Mixing polymers of the same chemical composition has been found to induce secondary interactions between the species depending on branching or grafting. Maybe the most prominent example is the autophobic dewetting of densely grafted polymer chains in contact with a melt or solution of the same polymer.1−5 Also, mixtures of hyperbranched polymers and linear polymer solutions show effects originating from the difference in polymer architecture: Experiments on dendrimer and linear chain mixtures with many different chemical realizations of poly(amidoamine) dendrimers6 have shown large differences in the vapor−liquid equilibrium between chemically compatible dendrimers and linear polymers in solution. The effective interactions between different architectures can be strong enough to induce phase separation,7,8 which is also relevant for the application of dendritic polymers as processing aid in extrusion processes.9 In the latter case it was shown that a small volume fraction of dendritic polyethylene (PE) obtained from chain walking catalysis significantly reduces the apparent shear stress in linear PE and postpones the onset of surface instabilities after extrusion. This is explained by segregation of droplets of the strongly branched polymer to the surface of the molten polymer phase forming a lubricating layer. The question remains open whether hyperbranched polymers, in particular dendrimers, display a similar autophobic behavior as it is well-known from polymer brushes exposed to melts of the same species or if the origin of architectural-driven phase separation is rather different. Autophobicity leads to depletion attraction between two brush-coated surfaces. Taking this idea to dendrimers, © XXXX American Chemical Society

depletion attraction would give rise to architectural-driven phase separation, and as dendrimers are small molecular objects with a finite mixing entropy, this would lead to a nontrivial phase diagram. However, in contrast to depletion attraction between hard nanosized spheres mixed with linear chains,10−13 it is not only the volume exclusion causing depletion. Rather, it is the free energy effort of the linear chains to penetrate into a dendrimer phase which is under high constraint. Roby and Joanny14 worked out a mean-field theory and a blob model for mixtures of branched and linear polymers based on Zimm−Stockmayer hyperbranched polymers15 and Gaussian chain statistics. They suggested a phase diagram of an isolated hyperbranched polymer dissolved in polymer chains predicting a good solvent, θ-solvent, and poor solvent regime as well. Scaling and mean-field theory for star polymers in solvent of high molecular weight16 predict two regimes: a good solvent regime with swollen stars for low solvent concentration and a collapsed regime for high solvent concentration. Using the random phase approximation, Fredrickson, Liu, and Bates17 calculated entropic corrections to the Flory− Huggins free energy in particular for blends of chemically identical branched and linear polymers. On the basis of the Gaussian approximation for the conformations of the molecules, demixing was predicted for short spacer between branching points, which most likely goes beyond the Gaussian approximation of the authors. These works, however, do not Received: January 3, 2019 Revised: February 26, 2019

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DOI: 10.1021/acs.macromol.9b00010 Macromolecules XXXX, XXX, XXX−XXX

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the state of the macromolecular target is influenced by its macromolecular surrounding. This term is frequently used in the context of biological polymer systems with a rather nonspecific meaning but not unlike to our definition. In this study we analyze the conformational properties of dendrimers immersed in a solution of linear polymers for various dendrimer generation, spacer length, length of linear polymers, and polymer concentrations up to the melt state. We use scaling arguments to rationalize the conformational changes of dendrimers under crowding conditions in a solution of long polymer chains. Our analysis is consistent with a geometrical picture of crowding, which is given by the matching of the characteristic length scales of the two subsystems. In the melt state the dendrimer is not simply collapsed but displays an ideal chain scaling with respect to the spacer length similar to a θ-point state as predicted by meanfield theory. However, the appropriate scaling variable forces a different dependence from the dendrimer’s generation, indicating that a simple screening of the second virial coefficient cannot explain this particular state. Our analysis of the density profiles concurs with the picture of strong penetration of linear chains inside the dendrimer’s volume. The analysis of the pairwise effective interaction potentials between dendrimer and linear chain points to an entropic loss of the linear chains during crowding, however not sufficient to collapse the flexible dendrimer even for high generations.

address the case of dendrimers. Moreover, the Gaussian statistics as the reference state for all calculated corrections cannot be realized in dendrimers of higher generation as we will point out in our work. Lue and Prausnitz18 investigated pairwise molecular interactions of a single dendrimer and a single linear chain by Monte Carlo simulations. They concluded that a dendrimer in dilute solution displays much weaker interpenetration as compared with two interacting linear chains. In the same work a lattice cluster theory has been applied for the high concentration regime including effective interactions between linear polymer and dendrimer. It was found that the dendrimer shows a lower critical temperature for the liquid−liquid phase transition occurring for repulsive interactions as compared to the linear polymer in particular for high-generation dendrimers. Another method to investigate pairwise effective interactions is the Gaussian soft sphere model (GSSM). Dendrimers and their behavior in various environments have been investigated with this method intensively.19−22 In mixtures of Gaussian soft spheres and linear polymers solutions depletion attractions have been found.23,24 In atomistic simulations on pairs of poly(amidoamine) dendrimers25 depletion attractions were reported as well, but the shape of the interaction potential was claimed not to be simply Gaussian. To consider high concentrations of highly branched polymers and linear polymers, the GSSM is not applicable as it does not account for multibody interactions and density dependence of the interaction potentials being only valid up to the overlap concentration.26 Atomistic simulations are limited in the systems size and the accessible parameters due to the enormous computational effort. On the other hand, for polymer brushes the origin of depletion effects should be related to the conformational entropy of the polymers and are thus universal. Therefore, coarse-grained models and universal theoretical approaches like mean-field and scaling concepts can be applied. Hence, we use in this work a coarse-grained polymer model making a large parameter space accessible. On the other hand, the polymer model reflects the chain conformations instead of an effective potential accounting for multibody interactions and density dependence of the interaction potential. The impact of the density in pure dendrimer solutions27,28 already showed distinct changes in the molecular conformations with increasing density. There is still not much known about the conformational properties of dendrimers immersed in linear polymer solutions. For the study of solutions of different species of macromolecules it is important to characterize the state where the target, like a dendrimer or a multimolecular acervate, is influenced by the surrounding matrix of polymers. The term “semidilute” has been coined to express the impact of other polymers on a target polymer of the same species and length in a homogeneous solution. For a heterogeneous solution this is not fully adequate. For instance, and as we will show in our work, the solution of linear polymers can be semidiluted and therefore controlled by strong excluded volume interactions, but the dendrimer is not affected if its dimension is smaller than the correlation length of the semidilute solution. On the other hand, the state of homogeneous concentration, i.e., matching of the target self-concentration and matrix concentration, can already be in a state of strong deviation from the target’s conformations. Therefore, we will use the term “crowding” in this work which defines the range where

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MODEL AND METHOD In this study we consider a single trifunctional dendrimer in a solution of linear polymers with various degrees of polymerization N and volume fractions φ. We use coarse-grained lattice Monte Carlo simulations utilizing the bond fluctuation model (BFM).29,30 This method has been already intensively used to investigate dendrimers with flexible spacers,31 dendrimers in solvents of varying quality,32 dendrimers with end-group modifications,33 and solutions of dendrimers.28 While all essential properties of the flexible polymers are conserved, the simulation method is efficient enough to simulate large systems over very long time scales as compared to atomistic or molecular dynamics methods. The trifunctional dendrimer is characterized by the total number of monomers ND = 3S(2G − 1) + 1

(1)

given by the generation G, and spacer length S, between the branching units as depicted in Figure 1a. We consider dendrimers with generation G = 3−7 and spacer lengths S = 1, 2, 4, 8 in solutions with nC linear chains with degree of polymerization N and total number densities vN vn N φ = φD + φC = 0V D + 0 VC . Here, v0 is the volume of a monomer which is eight lattice points for a single BFM unit (see Figure 1b). Note that only one dendrimer is placed in the simulation box; thus, the total density is varied by changing the number density of the linear chains (ND ≪ nCN). We apply an implicit solvent model here. Implicit solvent simulations are adequate for athermal solvent34 and for the analysis of equilibrium properties. For a comparison of methods see also ref 35. The athermal solvent model used in our work also implies that the thermal blob size36 is of the order of a single monomer. The simulations have been performed in a cubic simulation box with a volume V = 1283 elementary cells under B


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conditions: all new bond vectors have to be part of the BFM bond vector set,30 and all new lattice points have to be empty to ensure both excluded volume conservation and cut avoidance preserving the topology of the system. If all conditions are met, the move is valid and accepted; otherwise, it is rejected. Then, the next monomer is randomly chosen, and the algorithm is repeated. One Monte Carlo step (mcs) is defined as one attempted monomer move of every monomer on average. Conformations were saved every 104 mcs, and at least 108 mcs were performed for sampling the observables. Equilibration was done for at least 2 × 107 mcs. A parallel version of this procedure is used and explained in detail in our previous work.33

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RESULTS AND DISCUSSION Dendrimer Size. First, we consider the radius of gyration 1 2 RD2 = of the dendrimer in units of the 2 ∑ij (ri − rj) (2ND )

average bond length RD/b in different solution conditions. Because the bond length b decreases slightly with concentration, these units take into account only the size change by variation of the conformations. A thread is defined as the minimum linear path along the dendrimer’s contour starting from the central (core) monomer and ending at a terminal monomer containing GS monomers (see Figure 1a). The number of threads in the dendrimer is thus given by z = z(G ) =

ND ∼ 2G GS

(2)

We assume that a thread is the primitive structure providing the elasticity of the dendrimer and all threads contribute equally, and the elastic free energy of a thread is given by Fel ∼ RD2/RD,id2 ∼ RD2/GS with RD,id2 ≃ GS.32,38,39 Then the meanfield free energy per thread reads F=

RD2 (GS)2 (GS)3 + vz + wz 2 3 GS RD R D6

(3)

Here, v is the two-body interaction parameter and w is the three-body interaction parameter. We use units of kBT for the energy, where kB is the Boltzmann constant and T is the absolute temperature. In terms of the mean-field approximation constant prefactors are systematically suppressed since we are interested in the scaling relation for the resulting equilibrium solutions. We note that the free energy per thread is form invariant with the free energy of a single chain of length GS within the Flory model. The dendrimer property enters formally in renormalized virial coefficients only. Three regimes with respect to v have to be distinguished: good solvent v > 0, θ-solvent v = 0, and poor solvent v < 0. Minimizing the free energy with respect to RD in the different solvent regimes results in the following scaling predictions for radius of gyration of the dendrimer:32,39

Figure 1. (a) Sketch of dendrimer and linear chain with the system parameters: generation G, spacer length S, and chain length N. (b) Exemplary view on a simulated system in the bond fluctuation model of a dendrimer (blue) G = 2 and S = 1 and linear chain (yellow) N = 4. (c) Simulation snapshot of a G = 5, S = 4, and N = 16 system with a total density φ = 0.1. The core monomer is colored red.

l z1/5(GS)3/5 , if v > 0 o o o o o RD ∼ o m z1/4(GS)1/2 , if v = 0 o o o o o 1/3 1/3 o o n z (GS) , if v < 0

full periodic boundary conditions. All simulations and evaluations were performed using the LeMonADE framework developed in our group.37 The simulations are performed using the following Monte Carlo procedure: Starting form an initial configuration created with a convergent growth algorithm, a monomer is chosen randomly to perform a move. First, a random direction along the principal lattice axis is drawn (see arrows in Figure 1b). Second, the new position is checked for the following

(4a,b,c)

We note that for the first relation (good solvent) the exponent 3/5 corresponds to the Flory exponent ν in the mean-field approximation. These equations display in particular spacer scaling; i.e., when all length scales are normalized by the C


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fraction φ the dendrimer obeys good solvent behavior (see Figure 2a). On the other hand, θ-solvent scaling is found for mixtures for long linear chains, typically N > GS, and high volume fractions close to the melt state as shown in Figure 2b. These findings are supported by comparison with simulations of S = 4 dendrimers in athermal and in θ-solvent33 (green triangles in Figure 2). To estimate the agreement with the scaling prediction, we used the mean-squared deviation from the fit to these reference simulations. We note that short chains at higher volume fractions lead to shrinking of the dendrimer without the signature of θ-scaling. This is also shown in Figure 3b for various chains lengths.

extension of the spacers in the corresponding solvent condition, invariant results are obtained for extensions and density distributions as shown earlier.31,32 It is worth noting that for dendrimers in the θ-state this result is not identical to the state of an ideal Gaussian structure. The ideal conformations are simply given by the ideal conformation of a thread, i.e. RD,id 2 ∼ b2GS ∼ b2 ln ND

(5)

Thus, even the compact globule is more extended as compared to the ideal state. The essential difference being the missing zfactor which is of the order 2G/2 here. One should therefore consider the fact that the ideal “super-compact” state of a flexible dendrimer hardly exists. This point will become important in our discussion of the crowding effect below. In Figure 2, we display the rescaled radius of gyration as a function of the number of monomers in the dendrimer under different linear chain densities. Good solvent scaling according to eq 4a is displayed in Figure 2a, while θ-scaling according to eq 4b is shown in Figure 2b. We found that for solutions with short linear chains up to moderately high values of the volume

Figure 3. Crossover scaling plot of radius of gyration of dendrimer RD for (a) different dendrimer sizes and (b) different chain lengths over reduced volume fraction φ/φ*D normalized by the bond length b from dilute to dense solution emphasizing the impact of the surrounding solution of linear chains with different lengths N on the excluded volume screening for long chains. Figure 2. Rescaled radius of gyration, RD, as a function of the scaled degree of polymerization of the dendrimer in a solution of linear chains of various length N and concentrations φ: (a) Short chains and low to moderate concentration preserve the good solvent scaling of the dendrimer. (b) Long chains compared to GS in the melt-like state lead to θ-scaling for the dendrimer size. The length units are normalized by the weakly concentration-dependent size of the averaged bond length b.

These results indicate two clearly defined regimes for dendrimers immersed in solutions of linear polymers: a swollen state, which corresponds to good solvent scaling, and a screened state, typical for θ-solvent conditions. The latter fulfills in particular the relation RD ∼ S1/2 for the overall size of the dendrimer; see the mastering of data for different values of S in Figure 2b. We did not observe a collapsed state, i.e., D


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Macromolecules RD ∼ ND1/3 ∼ S1/3, according to eq 4c, as claimed in earlier theoretical40,41 and experimental42 works. Our results are in agreement with previous findings for dendrimer melts.28 We note that similar regimes are found in planar polymer brushes termed swollen wet brush and screened wet brush.2,3 This is also a proof that the picture of a compact globule which is composed of relatively short but Gaussian chains is not valid. In the latter case the spacers would display Gaussian behavior, RS ∼ S1/2, but not the globule itself; i.e., the relation RD ∼ S1/3 would be valid instead. The effective θ-solvent-like behavior is consistent with screening of the second virial coefficient as predicted by the random phase approximation for linear polymers.43,44 We discuss the consequences of the screening argument further below. We just note that the mathematical formalism (see Doi and Edwards43) can be easily extended to any Gaussian structure, such as dendrimers, and leads to the renormalized excluded volume of v → v/N. However, the random phase approximation implies that the Gaussian state is the state of reference in the absence of the second virial coefficient which is not the case for strongly branched polymer such as dendrimers as discussed above. By applying the Flory− Huggins model, Raphael et al.16 arrived at a complete screening of the excluded-volume interactions including the higher order virial coefficients for the case of star polymers. This leads to a collapsed state of stars in a melt of long chains. We note that this approach assumes mean-field type mixing of the macromolecule in the chain-like solvent which results in a screening factor of 1/N in all higher virial coefficients. The origin is the reduction of the translational degree of freedom of the linear chains as compared to a simple solvent. Because we do not observe a collapsed state of the dendrimers in our simulation, we cannot support this argument. In the Conclusions section we briefly present an alternative derivation for star polymers based on the findings in this work. We note further that densely grafted bottle brushes show only partial screening of excluded volume of the side chains, increasing the effective backbone stiffness.45 Therefore, the nature of the screening by linear chain crowding is not clear. A straightforward approach to consider the crowding by linear chains is to consider the self-concentration (overlap concentration) of the dendrimer as characteristic scale for the crossover to the crowded state. It is given by φD* = ND/RD0 3 ∼ b−3ND2/5(GS)−6/5 ∼ z 2/5(GS)−4/5

φ≫φ *

D RD0 2(φ /φD*)x ∼ RDθ 2 ⎯⎯⎯⎯⎯⎯→ RD0 2φ xz −2/5x(GS)4/5x ∼ z1/2GS

(8)

From this we obtain x = −1/4. We note that this exponent is the same as for the well-known scaling of semidiluted linear chains,44,46 which is not surprising since spacer-scaling determined the exponent. The solution is fully consistent with eq 4b simultaneously for the variables ND and G. Therefore, in the θ-state we expect the following power-law behavior of the normalized squared radius of gyration of the dendrimer: RD2 RD0 2

ij φ yz zz ∼ jjjj z j φD* zz k {

−1/4

if φ ≫ φD*

(9)

The results for the crossover scaling of the radius of gyration RD are shown in Figure 3. In Figure 3a, the length of the linear chains is fixed to N = 128, and the dendrimer generation and spacer length are varied under different linear chain concentrations. Equal generations and different spacer length show a good scaling, but different generations break the scaling. The slope of RD with respect to the reduced density φ/φD* is in fair agreement with the prediction of −1/4. In Figure 3b, we display the results for the variation of the length of the linear chains fixing G = 5 and S = 4. Very short chains reduce the overall extension of the dendrimer as compared to the athermal case, but they are not sufficient to screen excluded volume interactions even at high volume fraction of φ = 0.5. We note that φ = 0.5 is known to correspond to the melt state in the BFM.30 For chains longer than the thread size N ≫ GS and for φ ≫ φD* the dendrimers extension follows the predicted asymptotics, as given by eq 9. The breakdown of the dendrimer’s scaling with different generations raises the question if the choice of the scaling variable is appropriate. The definition of the crossover density where crowding sets in was based on the assumption of a homogeneous concentration of the binary solution. Alternatively, a geometrical picture can be applied as illustrated in Figure 4.

(6)

where the size of an isolated dendrimer in good solvent is denoted by RD0 (see eq 4a). We introduce a scaling function for the radius of gyration RD2 = RD0 2f (φ /φD*)

(7)

Figure 4. Geometric picture of dendrimer in a semidilute solution of linear chains. On the left-hand side, the free dendrimer’s extension is smaller than the correlation length (mesh size) ξ of the linear chains in the semidilute state corresponding to φ < φ+. The right-hand side illustrates crowding with φ > φ+.

In the limiting case of a solution much more diluted as compared to the dendrimer self-concentration (φ ≪ φ*D) the size of the dendrimer, RD, is equal to the size of an isolated dendrimer in good solvent, and hence f(φ ≪ φD*) = 1. We note that the state of the linear chains is not directly related to φ*D. Very long chains can be already in the semidilute state while short chains can be diluted. For concentrations exceeding the self-concentration of the dendrimer (φ ≫ φD*), we assume excluded volume screening and in particular the relation RD ∼ S1/2. This can only be obeyed if the crossover function f(φ/φ*D) attains a power law in the asymptotic region, i.e., for φ ≫ φ* in the form (φ/φD*)x:

This would suggest to consider the ratio of the mesh size (correlation length) ξ of the semidilute polymer solution and the size of the dendrimer as the relevant scaling variable y′ = ξ/RD0. Here, the correlation length44 is given by ξ ∼ φν/(1−3ν) ∼ φ−3/4, where in the last relation the approximate Flory value ν = 3/5 has been used. Note that for RD0 ∼ ξ the solution is not homogeneous since the dendrimer’s selfE


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of gyration is for all systems ξ/Rg0 ≈ 0.6 being a reasonable value. The crowded regime substitutes the mean-field θ-solution given by eq 4b. The corresponding scaling plot is displayed in Figure 6 together with the expected slope according to eq 13.

concentration, given by φ*D in eq 6, is higher as the linear polymer solution. We denote this point of crowding by φ+, which is given by φ+ ∼ RD0−4/3 ∼ z −4/15(GS)−4/5

(10)

The last relation results again from substituting the prediction of the dendrimer size in good solvent according to eq 4a. In analogy to the argumentation above, we propose the scaling for the radius of gyration under crowding following the scaling function RD2 = RD0 2f+ (φ /φ+)

(11)

which leads to the following asymptotics in the crowded state φ ≫ φ+: RD2 ∼ (GS)φ−1/4z1/3

(12)

Because we know that the Gaussian scaling for spacers should be fulfilled in the strongly crowded state, we claim again RD ∼ S1/2 for φ/φ+ ≫1. Thus, the asymptotic exponent is again x = −1/4, but the dependence from z is different now. In Figure 5 we display our data using the geometric scaling Figure 6. Scaling plot for the dendrimer size in melts of long linear chains using the crowding scaling of eq 10.

Density Distributions. In the following we consider density distribution functions of both species in heterogeneous solution. The radial density distribution function with respect to the center of mass of the dendrimer r⃗COM, ρ(r) = ⟨∑ni=0δ(r − | r⃗COM − r⃗i|)⟩, for the dendrimer, ρD, and for the chains, ρC, is shown in Figure 7a for selected parameters. The case of zero concentration of linear chains is given by the results for a dendrimer in athermal solvent (black line). For comparison, we also display the results for a dendrimer in the collapsed state as obtained from direct simulations in poor solvent (black dashed). Poor solvent was simulated with explicit monomeric solvent and repulsive nearest-neighbor interactions between dendritic monomers and solvent, where simulation details and data can be found in our previous work.33 In Figure 7a, we display the results for the dendrimer with generation G = 5 and spacer length S = 4 in solution of short and long linear chains of low and high volume fraction. The density profiles of dendrimers mixed with linear chains are slightly more compact than the athermal density profiles but differ strongly from the results for collapsed dendrimers in poor solvent. Dendrimer density profiles are similar to those of polymer brushes at curved surfaces47 decreasing very smoothly as compared to star polymers16 with a power law profile following ρS(r) ∼ r−4/3. The interpenetration of linear chains into the dendrimers is akin to the interpenetration of linear chains in a polymer brush.3 Thus, spacer length and branching point functionality control the interpenetration of linear chains similar to the chain length and the grafting density in a polymer brush. Short linear chains fully interpenetrate the dendrimer even for low linear chain volume fractions. Long linear chains interpenetrate only partially for low volume fractions. The interpenetration behavior is shown more pronounced by the normalized density overlap presented in Figure 7b. Here, we display the product of the densities of the chains and the dendrimer normalized by the overall volume fraction ρDρC/φ. Short chains (full lines) show high overlap which takes its

Figure 5. Scaling plot for crowding with the assumption of the geometrical crossover concentration φ+ as given by eq 10. All data correspond to Figure 3.

variable. The agreement with the simulation data is much improved as compared to the φD* scaling (see Figure 3). This indicates that the matching of the two characteristic length scales of both subsystems, namely, the correlation length of the semidilute solution and the dendrimer size, controls the onset of crowding effects on the dendrimer. In the limit of melt concentration of linear chains, φ → 1, this argument predicts the following relation: R+2 ∼ z1/3 GS

(13)

The correlation length ξ for a specific set of parameters can be estimated using the data of Figure 5. Fitting the factor a in Rg(φ) = aφ−1/8 to the unscaled data clearly showing the power law behavior and calculating the intersection point with the constant Rg0 yields the crossover number density φ+. We obtain the absolute value of ξ using ξ ≃ φ−3/4, and the dimensionless ratio between the correlation length and radius F


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Figure 7. (a) Density distribution functions ρD,C for a generation G = 5 dendrimer with spacer length S = 4 dissolved in short chains N = 4 and long chains N = 128 under low (φ = 0.1) and high (φ = 0.5) volume fractions. The diluted state is an isolated dendrimer in athermal solvent, and the collapsed case is an isolated dendrimer in poor solvent. (b) Density overlap ρDρC for the same dendrimer with varying volume fraction of linear chains.

Figure 8. (a) End monomer distribution ρE versus the reduced center of mass distance r/RD for generation G = 5 dendrimer with spacer length S = 4 in athermal solution and in solutions of short and long chains with low and high volume fraction, respectively. (b) Position of maximum of end monomer distribution rmax versus linear chain volume fraction for generation G = 5 dendrimers of various spacer lengths.

maximum value in the center of the dendrimer. For mixtures with long linear chains interpenetration dominates in intermediate distance to the dendrimer center at r ≈ RD. In dendrimers about half of the monomers belong to the outermost spacers as the number of monomers increases exponentially ND ∼ 2G. Hence, the distribution of end monomers displays the positions of a large amount of monomers being the least restricted monomers with only one connection to another monomer. Examples of end monomer distributions are shown in Figure 8a. First, we note that the end monomers are distributed over the whole dendrimer with a distinct maximum in the dendrimer interior rmax < RD. The distribution becomes more compact both for longer chains and for higher volume fractions of the same chain length. The maximum position of the distribution is displayed in Figure 8b as a function of the concentration for various spacer lengths. We conclude that the end monomers fold back stronger with increasing spacer length. Furthermore, with increasing crowding by linear chains the maximum position shifts toward the center of the dendrimer. This shift is

most distinct for dendrimers with short spacers and indicates a stronger back-folding due to crowding. According to the concept of spacer scaling developed in previous work,32,39 we expect a common distribution if the radial density of the dendrimer is rescaled by the self-density of a spacer S/RS3 ≃ S1−3ν and if the radial distance is rescaled by the characteristic extension of the spacer chain Sν. The corresponding scaling plots are shown in Figure 9. The good solvent case (short chains and low concentrations) is displayed in Figure 9a, while the expectations for the crowded regime (long chains and higher concentration) are displayed in Figure 9b. The rescaling is in fair agreement for the exterior part of the molecules. It was seen already from Figure 7 that the densities for different spacer lengths differ significantly within the interior of the dendrimer. Particularly, the densities in the core region r/(bSν) < 1 and for small spacer length, S < 4, do not follow this scaling behavior. We should note that for S = 1, where the deviations are strongest, this region corresponds to 1−2 bond lengths, being a distance smaller than the Kuhn G


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Figure 10. Potential of mean force of a generation G = 5 dendrimer with spacer length (a) S = 1 and spacer length (b) S = 4 for various concentrations of linear chains with degree of polymerization of N = 32 > GS with respect to the center-of-mass to center-of-mass distance of dendrimer and linear chain. The results for the dilute solution are obtained by considering only the dendrimer and a single test chain. The distance is rescaled by the sum of the radius of gyration of the dendrimer RD and the chain RC for the given concentration.

Figure 9. Rescaled dimensionless density distributions of generation G = 5 dendrimer in the (a) good solvent regime with short chains N = 4, 8 and low volume fractions and (b) crowded regime with long chains N = 64, 128 ≫ GS and high volume fractions.

length. We also note the rescaling in Figure 9b corresponds to the scaling under crowding conditions in eq 12. Interaction Free Energy. The depletion of linear chains inside the dendrimer in the crowded state corresponds to an entropic penalty of chains for intruding the dendrimer’s volume. To quantify this effect, the interpenetration free energy is calculated. We note that our heterogeneous polymer solution is athermal and thus the free energy change is equal to the negative entropy change in temperature units. Essentially, the potential of mean force W(rc2c) as a function of the distance between the center of mass of the dendrimer and the center of mass of a linear chain, rc2c, is calculated by using umbrella sampling and the weighted histogram algorithm method.48 In Figure 10, the potential of mean force between dendrimers with generation G = 5 and spacer length S = 1 (see Figure 10a) and S = 4 (see Figure 10b) with linear chains of length N = 32 > GS is shown for various concentrations of the solution. The center-to-center distance rc2c is rescaled by the sum of the radius of gyration of the dendrimer RD and the linear chain RC. The effort a chain has to spend for interpenetrating the dendrimers volume is reduced by increasing the volume fraction of surrounding linear chains

converging for high densities. For the case of dilute solutions, the energy required to push a single chain into the center of a dendrimer is (WS1,dilute(rc2c) ≈ 10 kBT, WS4, dilute(rc2c) ≈ 7 kBT). Only in this limiting case can the dendrimer be considered as nearly impenetrable.18 Increasing the volume fraction of the surrounding chains reduces the insertion barrier for complete overlap to a few kBT, and linear chains can interpenetrate the dendrimer up to its center (see also Figure 7). The energy required to interpenetrate large parts of the dendrimer’s volume is even much lower: For example, in the range 0.5 < rc2c/(RD + RC) < 1 the barrier is lower than 1 kBT. The high interpenetration can explain the screening of the excluded volume for the threads and hence leads to the crowded regime as demonstrated above. The entropic force acting between the two molecules can be defined as f=−

∂W (rc2c) ∂rc2c

(14)

When reducing the center-to-center distance rc2c between dendrimer and the linear test chain, the potential of mean force shows an inflection point. This corresponds to nonmonotoH


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Macromolecules

Interestingly, for S = 1, the free energy profile in Figure 10a displays even a weak minimum for higher concentration. This minimum is located at the overlap distance between the chain and the dendrimer and corresponds to a weak attractive force. The attractive interaction between dendrimer and linear chain coincides with a significant shift of the maximum of the end monomer distribution toward the center of the dendrimer of the S = 1 dendrimers which was shown in Figure 8b. A possible explanation is the reduction of elastic free energy of the dendrimer due to an effective back-folding of the end monomers by the osmotic pressure of the surrounding chains. This effect is most pronounced for the short spacer dendrimers because their spacers are most stretched also clearly seen in Figure 8b. However, the attractive force is less than 0.1 kBT per lattice spacing being not sufficient to form stable aggregates.

nous behavior of the force with a maximum value outside of the dendrimer’s center. The center-to-center force is displayed in Figure 11. From this we conclude that a linear chain that

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CONCLUSIONS If a polymer solution contains two different species, the classical limits of dilute and of semidilute solution lose their absolute meaning. In the present case long linear chains can be strongly overlapping before they influence the conformational state of the embedded dendrimer. The central question of this work was to find an appropriate scaling variable which describes the impact of linear chains on a single dendrimer and thus to predict the state of dendrimers in this case. Generally, we found that dendrimers immersed in a solution of chemically identical linear chains displays crowding effects with a signature similar to a θ-state if the linear chains are long as compared to the length of a thread N ≫ GS and at high concentration. We have shown that Gaussian scaling of the dendrimer size with respect to the change in the spacer length, RD ∼ S1/2, is valid in the limit of melt concentration of the linear chains. For none of the conditions studied in this work up to the melt state of the linear chains was a collapsed state of the dendrimer with a signature of RD ∼ S1/3 found. Using the picture of a polymer brush, this implies that for the parameters of functionality, generation, and spacer length we investigated there is no “dry-dendrimer” regime, and autophobic attractions between dendrimers should be weaker as compared to planar brushes. First studies of two dendrimers in a melt of linear chains provide indications for soft depletion attraction. To quantify the nature of the crowding effect and to find an appropriate scaling function for the size of the dendrimer embedded in a solution of long chains, we have tested two assumptions. In the first case it is assumed that the dendrimer conformation is influenced by the interaction with the matrix chains if the concentration of linear chains matches the overlap concentration, i.e., the self-concentration of the dendrimer. The corresponding scaling variable leads to the theoretical θstate of the dendrimer in the limit of strong crowding according to the mean-field theory. This, however, is not consistent with the simulation results with respect to the dendrimer’s generation. As an alternative approach we considered a geometrical argument given by the matching of characteristic length scales as the signature for crowding. The corresponding scaling variable is given by RD/ξ, i.e., the ratio of the dendrimer size and the mesh size (correlation length) of the semidilute solution. Here, it is implicitly assumed that the linear chains are larger than the size of the dendrimer (or a thread of the dendrimer) to form a semidilute state before the dendrimer is impacted. This model leads to very good scaling with respect to the dendrimer generation for our simulation results.

Figure 11. Effective force according to eq 14 between the dendrimer and linear chains for a G = 5 dendrimer with spacer length S = 1 (a) and spacer length S = 4 (b) for various concentrations of linear chains (N = 32) with respect to the center-of-mass to center-of-mass distance of dendrimer and linear chain.

interpenetrates the dendrimer can be forced by a sufficiently high osmotic pressure of the surrounding polymer solution to favor the center of the dendrimer rather than the shell of the dendrimer. This tendency is stronger for dendrimers with long spacers having lower self-densities and a lower force peak to overcome. Such a behavior is not untypical for effective potentials in polymer systems. As an example, we mention the insertion of nanoparticles in planar polymer brushes.49 The reason is that small objects can be absorbed in soft environments with a finite free energy effort related to the change in conformational entropy of the total system. Once inserted, the homogeneity of the system leads to zero force. This is perfectly true for extended 3D systems such as networks and gels. For brushes and dendrimers the density profile is not homogeneous but displays a plateau-like behavior which leads to a finite force behind the insertion threshold. Similar force−distance curves were also observed for the interaction of charged dendrimers among each other by Huissmann, Likos, and Blaak.50 Comparing the spacer length dependence of the force our results display the same trend as those of ref 50. I


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Macromolecules ORCID

According to this analysis, the dendrimer in the crowded state follows ideal scaling with respect to the spacer length, but a different scaling with respect to the generation: R+ ∼ z1/6(GS)1/2. Thus, the crowded state of dendrimers in solution of long flexible chains seems not to correspond to the θ-state as known for linear chains and weakly branched polymers. Our analysis of the radial density distribution functions around the dendrimer’s center of mass concurs with the picture of strong interpenetration into the dendrimer volume, in particular for the case of longer spacers. Following the arguments by Roby and Joanny,14 a compact state of the dendrimer is expected for long enough linear chains due to complete screening of all viral coefficients. Thus, the absence of the collapsed state in our simulations indicates that the ideal mixing entropy given by the Flory−Huggins model which is used by Roby and Joanny14 is apparently not valid for linear chains inside the dendrimer’s volume. On the other hand, the screening of the second virial coefficient only, which corresponds to the random phase approximation, seems not to be sufficient. This would lead to the mean-field θ-state which is not displayed in our simulations either. To understand the possible failure of this concept, it is important to note that for dendrimers the ideal chain statistics, the Gaussian state, does not correspond to the θ-state but rather to a nonphysical state with RD,id ∼ ln ND, more compact than the collapsed state. Thus, the Gaussian state cannot be taken as the reference when considering the corrections due to screened excluded volume effects. In fact, the ideal state where space filling is not relevant does not exist for real dendrimers. For comparison with the existing theoretical models for star polymers it is interesting to apply the concept of geometric screening here as well. The radius of the star in good solvent51 is given by the scaling relation Rs ∼ Ns3/5f−2/5, where Ns denotes the total number of monomers in the star. Using the same geometric argument which led to eq 10, we arrive at φ+s ∼ Ns−4/5f 8/15. The corresponding scaling variable for the star embedded in a semidilute solution of linear chains is then given by ys = φ/φ+s . We obtain for the size of the star in the limit of strong crowding Rs(φ) ∼ Ns1/2φ−1/8f−1/3, again for the assumption that the arms of the star display screened statistics. For the melt case, i.e. φ = 1, a compact state of the star, i.e. Rs ∼ Ns1/3, is reached for fc ∼ Ns1/2. We note that the last condition is well below the packing limit of a star. An experimental test of these predictions as well as those obtained for the dendrimers in our work could be obtained by scattering techniques using labeling of the diluted star polymers or dendrimers respectively in a solution of long linear chains. Our findings in this work show that crowding of the dendrimer by linear chains cannot be mapped to simple excluded volume screening effects but can be nevertheless described very well by a geometric scaling argument. As a result, the dendrimer displays Gaussian scaling with respect to the length of spacer threads but poor solvent scaling with respect to the number of spacers. The crowded state thus has a novel scaling behavior different from the classical states (collapsed or θ) predicted by mean-field arguments.

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Martin Wengenmayr: 0000-0002-0627-7129 Ron Dockhorn: 0000-0002-5268-5430 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft under Contract SO 277/17. We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for generous allocations of CPU and GPU time.

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Dendrimers in Solution of Linear Polymers: Crowding Effects

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