Multicore Unimolecular Structure Formation in Single DendriticâLinear

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Multicore Unimolecular Structure Formation in Single Dendritic− Linear Copolymers under Selective Solvent Conditions 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

‡

S Supporting Information *

ABSTRACT: The conformational and thermodynamic properties of single dendritic−linear copolymers are investigated by analytical models and computer simulations. Applying poor solvent conditions on the dendritic part, these molecules are known to form single unimolecular micelle-like structures. A mean-field model applying the Daoud−Cotton approach and a surface tension argument is presented and suggests the splitting of the unimolecular single-core structure into a multicore structure with increasing dendrimers generation and decreasing solvent selectivity. Monte Carlo simulations utilizing the bond fluctuation model with explicit solvent are performed which show the formation of multicore structures for trifunctional codendrimers of different generations and spacer lengths with linear chains attached to the terminal groups. These findings are aimed to understand the physics of spontaneous self-assembly of codendrimers in various well-defined macro-conformations under change of environmental conditions with potential applications such as drug delivery systems.

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Frechet et al.3,10,11 and Newkome et al.12 in the context of application for the drug delivery system. In contrast to the micellization of many individual amphiphilic molecules, selfassembly in a codendrimer takes place inside a single molecule and is therefore under strong constraints and controlled by the molecular architecture. These systems have been reported to be thermoresponsive undergoing a reversible phase transition.13 At finite concentration small codendrimers can also form multimicellar aggregates in solution14 stemming from insufficient screening of the corona caused by a low number of terminal groups per surface unit of the collapsed core, but this is not the scope of this work. Computer simulations on dendrimers with attached chains have been performed recently by Markelov et al.15,16 using molecular dynamics simulations (MD) and a Scheutjens−Fleer self-consistent field approach. They investigated the effect of grafted end chains onto the dendrimer to act as transport molecule finding both dense core and hollow core conformations. A coarse-grained MD study on PAMAM dendrimers with linear poly(ethylene glycol) (PEG) grafted onto the terminal groups was done by Lee and Larson17 analyzing the effect of chain length and density of grafted chains. Unimolecular micelles were also reported in earlier computational studies of codendrimers.18−20

INTRODUCTION Dendrimers are exactly defined branched molecular structures and therefore attracted experimental and theoretical research in the past with promising applications in medicine, material science, and chemical industry.1−3 Much effort is spent to use dendrimers in drug delivery, especially for cancer treatment.4 Under physiological conditions and neutral pH the water solubility of poly(amidoamine) (PAMAM) dendrimers is, however, limited and can be significantly increased by adding hydrophilic chains such as poly(ethylene glycol) (PEG). This procedure is known as PEGylation and is widely used, not only for dendrimers.5 This treatment also reduces significantly the cytotoxicity of the PAMAM dendrimer.6 A different way of endgroup modification is the attachment of sugar groups to improve the water solubility and reduce toxicity of PAMAM or poly(propyleneimine) (PPI) dendrimers.7 Extensive investigations have been done for these so-called dendritic glycopolymers for therapeutics and diagnostics.8 Adding extensions with different properties to the terminal groups of a dendrimer creates a two-component system. Such codendrimers are particularly interesting because of the large number of terminal groups growing exponentially with the dendrimer’s generation. Usually the dendritic part will be rather hydrophobic and has the tendency to collapse in aqueous environment. As a consequence, hydrophilic chains attached to the terminal groups form a corona screening the hydrophobic part and thus avoid aggregation and decrease cytotoxity in therapeutic applications.9 Such type of self-assembly in codendrimers has been termed unimolecular micelles by © XXXX American Chemical Society

Received: August 5, 2016 Revised: November 10, 2016

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DOI: 10.1021/acs.macromol.6b01712 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Scheme of a dendritic (unimolecular) micelle with a mapping of the simulated systems (left, dendritic monomers colored blue, linear extensions are colored yellow) in poor solvent condition to a spherical (single-core) micelle (middle) with the inner radius R0 of the dendritic core and total radial extension R. The dendritic core is uniquely given by the functionality F, spacer length S, and number of generation G of the dendrimer (right).

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MEAN-FIELD MODELS OF DENDRIMERS AND CODENDRIMERS IN SOLUTION Dendrimers are characterized by the number of branching units originating from the central monomer to the terminal groups called the generation G, the length of the linear chains between the branching points, called spacer length S, and the functionality of the branching units F (see Figure 1). Starting with the central monomer, we consider trifunctional dendrimers with F = 3 are build up iteratively, generation by generation, by adding two spacer chains to every outermost monomer called the branching point. At the final generation the free ends are called the terminal monomers. The overall number of monomers of the dendrimer ND grows exponentially with the generation according to

Unimolecular micelles are usually considered to be composed of a single collapsed core formed by the dendritic part in poor solvent surrounded by a corona of linear chains grafted to the terminal groups (see Figure 1). Increasing the dendrimers generation leads to an exponential increase of the number of grafted linear chains and a high grafting density on the single collapsed core. Thus, the free energy excess due to the densely grafted corona can overcome the free energy gain by the collapse of the dendritic core. As a consequence, the single-core (unimolecular micelle) structure splits into smaller multicore aggregates optimizing the global free energy by reducing the number of linear chains per aggregate and surface interaction area per core. A similar prediction has been made by Borisov and Zhulina21 and Kosovan et al.22 for comb-like copolymers forming pearl-necklace-type conformations of starlike micelles or spherical “crew-cut” micelles. The balance between the free energy of the corona and the surface tension of the collapsed globules will be controlled by the solvent selectivity and the structural parameter of the dendrimer, namely the number of generations and the length of flexible spacers. In this work we present a combined mean-field and scaling model which takes into account the competition between surface tension of the poor solvent-component and the free energy excess of spherical brushes formed by the linear chains in good solvent. To test our analytical predictions, we analyze the structure and properties of a universal model system for codendrimers under a wide range of generations and spacer lengths with attached linear chains in different solvent conditions using the bond fluctuation model (BFM)23,24 with explicit solvent. We report on various forms of self-assembled single-core unimolecular micelles and the transition to multicore structures depending on the generation, spacer length, and solvent selectivity. By analyzing the thermodynamics in terms of free energy and heat capacity, we trace structural transitions by thermodynamic signatures such as peaks of the heat capacity. Mapping the simulation parameters to a PAMAM/PEG codendrimer shows that conventional parameters for both the dendrimer as well as for the attached linear chains could be sufficient to display the predicted behaviors in experimental setups.

ND = 3S(2G − 1) + 1 ≃ 3S 2G

for 2G ≫ 1

(1)

Also, the number of terminal monomers Nends grows exponentially with the generation: Nends = 3·2G − 1

(2)

In the limit of large molecules (2 ≫ 1) we obtain a constant fraction of end groups which can be related with the degree of branching Γ = 2Nends/ND = 1/S. Note that the degree of branching is not a topological measure but depends of the length of spacers and is unity only for S = 1. Since dendrimers have a relatively high self-density, a meanfield model25−28 can be applied which is in very good agreement with computer simulations.25,28,29 Let us define a thread of G·S monomers reaching from the central monomer to an arbitrary end monomer. This results in a number of threads G

N

3·2G

in one dendrimer of GSD ≃ G in the limit of large molecules.25 In the approximation that each thread is equally stretched due to mean-field interactions with all other monomers in the dendrimer the free energy per thread can be written as25 ⎛ N ⎛R 2⎞ -thread N 2⎞ g ⎟ + GS⎜v D + w D ⎟ = ⎜⎜ ⎟ ⎜ R 3 kBT R g 6 ⎟⎠ ⎝ GS ⎠ ⎝ g

(3)

where Rg denotes the radius of gyration of the dendrimer, kB is the Boltzmann constant, T is the temperature, and v and w are the two-body and three-body interaction parameter, respecB


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The surface of the codendrimer is only stable if the total surface tension is larger than zero: Σb + γ > 0 which is equal to γ > c1Nchain 2(11/18)G

where c1 denotes some numerical constant which cannot be obtained from the scaling arguments directly. Thus, increasing the generation requires an exponentially increasing surface tension to remain stable because of the exponentially increasing grafting density of terminal chains. The negative (excess) surface tension induced by the grafted chains can be reduced by splitting the micellar structure into many cores. Denoting the number of cores by k the surface area per core is given by A ∼

∂- b ∼ −A−11/6M11/6Nchain = −Nchainσ 11/6 ∂A

2/3

( Mk )

∼

2G k

2/3

( )

leading to

Σ b ∼ −Nchaink−11/182(11/18)G . Applying the tension balance condition again it follows

γ > c1Nchain 2(11/18)Gk−11/18

(7)

Rearranging the equation and substituting the surface tension by the interaction parameter yields

(4)

k ∼ 2Gϵ−9/11Nchain18/11

where n = ND/S is the number of spacers. In the following we focus on the case where the dendritic monomers are in poor solvent conditions and the end groups stay in good (athermal) solvent conditions. We note that in many applications the linear chains are attached to the dendritic part in order to improve the solubility of the dendrimer in a specific solvent, usually aqueous solutions, and to prevent aggregation of many dendrimers. We expect the formation of a collapsed dendritic part surrounded by the attached chains similar to a polymer micelle. This expectation is supported by simulation snapshots (see Figures 1, 7, and 10). A possible mapping of the dendritic system to a polymer micelle is shown in Figure 1. Surface Tension Balance. A simple argument shows why we can expect multicore structures: The surface energy of a surface dividing two different species with area A is given by -s = γA with γ being the surface tension. The miscibility of the dendrimer and polymer chains can be modeled by the Flory− Huggins theory30,31 with an effective Flory−Huggins parameter χ, which is assumed to be proportional to the effective interaction parameter ε between solvent and monomers. The surface tension at the interface is then given by 32 γ ∼ χ1/2 ∼ ε1/2. For convenience, we will consider γ in units of kBT, i.e., being dimensionless in the following. The free energy of a polymer brush containing M brush chains in the Alexander−de Gennes 33,34 model or self-consistent field approach35 is given by - b ∼ MNchainσ 1/2ν ≃ MNchainσ 5/6 with σ = M/A being the grafting density and Nchain the number of monomers in the grafted chains and using the Flory exponent ν = 3/5 for simplicity. The surface tension induced by the brush is negative and given by Σb =

(6)

(8)

giving a condition of how many cores are formed in a thermodynamically stable system for a given surface tension between the collapsed cores and the surrounding solvent. In these calculation we have assumed a flat-brush approximation for the corona to illustrate the basic argument. In the following we generalize this to spherical brushes which contains the above-discussed case as the limit of large cores. Micellization in Codendrimers in Poor Solvent. Using the theory of Halperin36 for spherical polymeric micelles based on the scaling model of Daoud and Cotton,37 we propose a mean-field model for dendritic−linear copolymer micelles. This model assumes that f chains consisting of Nchain repeating units are grafted to a sphere with radius R0. The ratio between the total radial extension R of a single micelle and the radius of the inner-sphere R0 is given by36 ⎛ ⎞ν R (1/2)(1/ ν − 1) Nchain = ⎜⎜c 2f + 1⎟⎟ R0 ⎝ R 01/ ν ⎠

(9)

where c2 is a numerical constant and ν is the Flory exponent with ν ≃ 3/5 in good solvent conditions for linear chains. The free energy of the chains forming a spherical brush can be expressed by36 - brush R = c3f 3/2 ln kBT R0

(10)

with another constant c3. We consider now the case that k such micelles are formed within a single codendrimer. The overall free energy - can be considered as the sum of the spherical brush part - brush and the surface energy of the collapsed cores -surface - = k - brush + k -surface

(5)

(11)

with

-surface = 4πγR 0 2

Assuming a globular shape of the collapsed dendrimer gives A ∼ M2/3 leading to Σ b ∼ −NchainM11/18. The number of chains M is proportional to the number of end groups in the dendrimer M ∼ 2G (see eq 2), yielding Σ b ∼ −Nchain 2(11/18)G .

(12)

where γ represents the surface tension. Here, we neglect the free energy of “connectors” which must be formed by spacers between the individual cores of the brushes. We also note that C


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multicore state is also favored for weaker selectivity and longer hydrophilic chain length. Since the prefactors cannot be determined by scaling arguments, the questions remains open for which values of these parameters a multicore structure is the equilibrium state. Therefore, we perform coarse-grained computer simulations to test the theoretical predictions and to explore the parameter space to clarify this point.

we implicitly assume all cores to be of the same size which is also a strong simplification as in a three-functional dendrimer the number of monomers in roughly monodisperse cores should be ∼3·2n (n ∈ ). Also, the assumption of fully spherical cores is a simplification as they tend to be stretched toward the “connectors”. The simulation results which are presented below show that these simplifications, nevertheless, reflect the behavior of codendrimers with flexible spacers and the parameters can be discussed even quantitatively. The number of grafted chains in a micelle f is given by f = Nends/k. For strong selectivity, i.e., nonsolvent conditions of the dendritic component, the radius of the k cores is assumed to contain an equally closely packed fraction of the dendritic monomers: R0 =

⎛ 3NDvD ⎞1/3 ⎜ ⎟ ⎝ 4πk ⎠

■

(13)

where vD is the self-volume of a dendritic monomer. Substituting all the micelles’ parameters with the dendrimer’s parametersgeneration G, spacer length S, and the number of micelles kinto eq 11, as well as assuming 2G ≫ 1, yields the free energy = c32(3/2)Gk−1/2 ln[c 2k 2/9S −5/92−(2/9)GNchain + 1] kBT + ε1/2k1/3S2/32(2/3)G

SIMULATION MODEL

We use the bond fluctuation model23,24 (BFM) for simulating flexible polymeric structures under different solvent conditions. In this coarse-grained Monte Carlo method polymers are modeled as connected cubes on a simple cubic lattice. The Monte Carlo step is implemented by moving a randomly chosen monomer in a randomly chosen direction along the principal axis of the lattice. The move is accepted if the new lattice positions are empty and all bond vectors connecting the polymers are in the allowed set; otherwise, the move is rejected. Thermal interactions are taken into account by applying the Metropolis criterion:38 The move is accepted if in addition to the criteria listed above the total change of energy Δ/ is negative or if a randomly drawn number from the interval [0,1) is less than p = exp( −Δ//kBT ). The bond vector set and the excluded volume condition are defined in a way to preserve the local and global topology and ensure cut-avoidance. For the present case a set of 108 bonds with length 2, √5, √6, 3, and √10 is allowed. The BFM has been applied to dendrimers first by Kłos and Sommer.29 Throughout this paper we set the length of a lattice site to unity and define one Monte Carlo step (MCS) as one attempted Monte Carlo move per monomer in average (sweep over all monomers including solvent) to be the time unit. In Figure 2 an example of a generation G = 2 dendrimer with spacer length S = 1 and functionality F = 3 is shown. Simulations have been performed with generations G = 2, ..., 8 and spacer lengths S = 2, 4, 6. Codendrimers are constructed by adding linear chains consisting of Nchain = 24 monomeric units to each terminal group of the original (pure) dendrimer. Note that the

(14)

The equilibrium state is then given by the minimum value of the free energy with respect to the number of micelles k. In general, this equation cannot be exactly solved in the form k(ε,G,S), but only in the implicit form ε(k,G,S). One limiting case which can be considered is the radius of the core being large as compared to the corona. The first-order expansion of the logarithm in eq 14 yields ≈ c 2c32(23/18)Gk−5/18S −5/9Nchain + ε1/2k1/3S2/32(2/3)G kBT (15)

Minimizing the free energy with respect to k leads to k 0 ∼ 2Gε−9/11Nchain18/11S −2

(16)

Here we obtain the same result as from the surface tension argument in eq 8. The equilibrium value of the free energy reads in the same approximation -0 ∼ 2Gε 5/22Nchain 6/11S 0 kBT

(17)

The exponent of the spacer length is explicitly denoted as zero here. This follows already from eq 15 which can be written in the form - = f (kS2) with the minimum solution: kS2 = constant (see eq 16). Thus, -0 cannot depend on S. The origin of the kS2 scaling is the surface-to-volume ratio in both terms of eq 15. The approximation of large cores leads to a flat brush scaling (- brush ∼ σ 5/6Nchain ) which only depends on the area per grafted chain, 1/σ, and thus on the total surface area, A ∼ kS2/3k−2/3 ∼ (kS2)1/3, of the collapsed cores. As soon as curvature becomes important for the free energy of the brush this scaling is broken. The result of eqs 8 and 16 predicts the splitting of the unimolecular single-core structure into a multicore structure with increasing generation and decreasing spacer length. The

Figure 2. Representation of coarse-grained dendrimer of generation G = 2 with spacer length S = 1 and functionality F = 3 in the bond fluctuation model (BFM). The central monomer is colored red, the end monomers yellow, and all other dendritic monomers blue. Solvent is modeled by unconnected monomers colored black. The arrows show possible move directions along the principal lattice axis. D


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significantly but do not change the static and dynamic scaling on scales larger than the Kuhn length.24 The codendrimers are generated with a convergent growth algorithm, and the remaining volume fraction is filled with solvent to ϕ = 0.5. For equilibration, the systems were simulated for 107 MCS increasing the interaction strength ε in steps of 0.05 kBT starting at 0.0 kBT up to 0.8 kBT. This way, the systems with the highest interaction energy are stepwise equilibrated for 1.6 × 108 MCS. Observables and thermodynamic properties were evaluated by simulations running at least 108 MCS at a fixed interaction strength. The equilibration time is about 10 times the Rouse time of the generation G = 7 dendrimer with spacer length S = 4 in good solvent estimated from a simple exponential fit to the autocorrelation function of the threads and from the center-of-mass diffusion. Furthermore, we do not observe any significant change in the extension and the density profiles after these relaxation times. For the case of micelle formation we discuss the equilibrium states later in the Results section in more detail. The BFM algorithm is performed in parallel on graphics processing units (GPU) implemented in the C++ framework LeMonADE (Lattice-based extensible Monte Carlo Algorithm and Development Environment) developed in our group.45 For programming the GPU algorithm the parallel computing CUDA framework by NVIDIA46 was used. GPU simulations are about 100 times faster than sequential simulations on CPU, making a big parameter space accessible for investigations. A detailed explanation of the implementation of the parallelized BFM used in this work can be found in the Appendix.

modifications are long as compared to the spacer length. In the following we refer to dendrimers without end-group modification as “pure” and to molecules consisting of a dendrimer with chains grafted to the terminal units as “extended”. We introduce short-range repulsive interactions between nearest neighbors using an interaction shell proposed by Hoffmann et al.39 for copolymer melts and extended for explicit solvent by Werner et al.40,41 Explicit solvent is modeled by single unconnected BFM units and account for the contact energy to mimic specific solvent conditions. Thereby, even in nonsolvent conditions monomer moves are still possible. The alternative would be using long-range interactions as applied to dendrimers by Kłos 25 to avoid frustrated or frozen conformations. This model was successfully applied to selfassembling of lipid bilayer membranes by Werner et al.,40−42 coil−globule transition of linear chains by Jentzsch et al.,43 and translocation of copolymers with lipid membranes by Rabbel et al.44 Thermal interactions are introduced by repulsive nearest neighbor interactions and can be expressed in terms of a Hamiltonian /nn : /nn =

∑ ∑ εnij i = 0 ⟨ij⟩

(18)

⎧ 0 if lattice point is free or occupied by noninteracting type nij = ⎨ ⎩1 if lattice point is occupied by interacting type ⎪

⎪

where ⟨ij⟩ denotes the pair contacts of the surrounding 24 nextnearest lattice sites, ε corresponds to the interaction penalty for the contact energy, and the first sum runs over all monomeric units in the simulation. The interaction scheme defining interacting and noninteracting types is displayed in Figure 3.

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RESULTS Size of Pure and Extended Dendrimers. The dimension of the dendrimer part is characterized by the radius of gyration Rg defined as ⟨R g 2⟩ =

1 ND

ND

∑ ( ri ⃗ − rcm⃗ )2 (19)

i=1

where the position of the ith monomer is denoted by ri⃗ , ⟨...⟩ is 1 N the ensemble average, and rcm ⃗ = N ∑i =D1 ri ⃗ is the center of D

mass. For the sake of simplicity, we use the shorthand notation Rg := (⟨Rg2⟩)1/2 in the following. Note that Rg does not include the monomers of the linear extensions. The first simulations are aimed to find the solvent regimes in the pure dendrimer systems by using the scaling predictions of eq 4. For finding an approximation for the θ-solvent interaction energy εθ, the results for Rg are rescaled by the predictions for θ-solvent given in the second line of eq 4. The interaction energy at which the lines of Rg(ε) cross each other is defined as θ-temperature. At this point, the rescaled radius of gyration is independent from the generation and the spacer length and thus fulfills θ-scaling as predicted in eq 4. A choice of the data used to determine the θ-interaction energy is shown in Figure 4. This yields the estimate of the θ-interaction energy to be εθ ≃ 0.1 kBT. This corresponds well with the value found in previous work for single chains.43 In Figure 5, the influence of end-group modification on the radius of gyration of the dendritic part in the extended dendrimer is displayed as a function of the number of dendritic monomers in (a) good solvent conditions at εgood = 0.0 kBT and in (b) poor solvent conditions at εpoor = 0.8 kBT.

Figure 3. Interaction scheme used in the simulations: dendritic monomers (blue) interact with solvent (black) and end groups (yellow) via energetic penalty ε, but solvent and end groups have no additional energetic penalty.

Simulations are performed on a cubic simulation box with length 128 lattice units under periodic boundary conditions in all directions. The simulation box was adapted to avoid selfinteractions of the dendrimer even in good solvent. For instance, the radius of gyration (see eq 19) is 2Rg = 80.4 ≪ 128 in the case of the generation G = 7 dendrimer with spacer length S = 4 and attached chains in good solvent at ε = 0.0 kBT. This is a conservative estimation because we do our main investigations in poor solvent where the size of the molecules is certainly smaller; for instance, the radius of gyration of the above-mentioned G7 S4 molecule is only 34.6 lattice units at ε = 0.8 kBT. The total volume fraction on the lattice has been fixed to ϕ = 0.5. This volume fraction is known to be a good model for dense systems still providing enough free volume for moves. Higher volume fractions slow down dynamic processes E


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and thus an increasing number of linear extensions in good solvent, there is a jump in the size of the dendritic part of the codendrimer as compared to the pure dendrimer. This indicates a completely different arrangement of monomers in the codendrimer. As we will show below, the regime G < G* belongs to singlecore (micelle) conformations, while the regime G > G* represents multicore conformations build up by a single codendrimer. The transition at G* appears to be discontinuous, and metastability of the number of cores can be observed starting from G* (see the simulation snapshots in Figure 10). Density Profiles. The monomer distribution is investigated via the radial density distribution around the center of mass Figure 4. Radius of gyration Rg of pure dendrimers rescaled by the scaling predictions for θ-solvent in eq 4 for defining the θ-interaction energy to εθ ≃ 0.1 kBT. Good solvent condition is everything below εθ and poor solvent condition is above εθ.

N

ρ(r ) = ⟨∑ δ(r − | ri ⃗ − rcm ⃗ | )⟩ i=1

(20)

Depending on the species under investigation N denotes either the number of dendritic monomers N = ND, the number of monomers of the chains added to the dendrimer N = Next = NendsNchain, or the number of all monomers in the codendrimer N = Nall = ND + Next. To illustrate the behavior, we have selected two generations: G = 5 and G = 7. In selective solvent this corresponds to the case of a single-core micelle (G = 5 < G*) and that of a multicore conformation (G = 7 > G*), respectively. In Figure 6 we display the results for the pure dendrimer density distribution (blue), the distribution of the dendritic monomers of the extended molecule (black), the distribution of all monomers of the extended molecule (green), and the distribution of the end groups only (red). The density distributions are rescaled according to the spacer scaling used in a previous work.25 The characteristic length scale is defined by the spacer chain with an extension RS ∝ Sα where α is the scaling exponent used in eq 4. The self-density is calculated by S/RS3 ∝ S1−3α and defines the characteristic density scale. We start our discussion with the case G = 5 < G* (a, c, e in Figure 6). The pure dendrimer (blue curves) shows a monotonously decreasing density distribution in good solvent conditions (Figure 6a) and a collapsed soft spherical behavior (globule) in poor solvent conditions (Figure 6c). The spacer scaling matches very good. This behavior has been intensively discussed in previous works.25,29 The density profile of the extended molecule in good solvent condition (Figure 6a, green) is stretched as compared to the

The radius of gyration is rescaled with respect to the meanfield model under good solvent condition given by the first line of eq 4. The data of the pure dendrimer matches with the proposed scaling in accordance to previous work.25 We find a significantly larger size of the dendritic part of the extended dendrimer for all generations in good solvent conditions. The difference between the pure dendrimer and the dendritic part of the codendrimer grows continuously with the generation. Adding linear chains lead to an extension of the core which can be approximated by a line with the slope of 0.253 ≈ 1/4. The increase can be explained by the entropic force of the attached chains pulling on the dendritic core of the molecule to explore a broader space around the molecule. The simplest way of approaching this is to add the additional monomers to the mean-field expression, second and third terms of eq 3, by changing ND to ND + NendsNchain which increases the excluded volume pressure and hence the size of the dendrimer. We note, however, that the slope 0.25 is drawn as a guide to the eyes and not implicating a real scaling behavior predicted from the theory. In the case of poor solvent applied to the dendritic monomers, but not to the extensions, an overlap of the size of the extended dendrimer and the pure dendrimer for small generations up to a transition generation denoted as G* ≃ 6 is observed. In this regime, the data match again with the meanfield prediction for poor solvent in the third line of eq 4, indicating globule-like conformations. For higher generations,

Figure 5. Rescaled radius of gyration of the dendritic part Rg of the extended dendrimer vs rescaled number of dendritic monomers ND in good solvent conditions (a) and in poor solvent conditions (b) for G = 2, ..., 8 dendrimers with varying spacer length S. The rescaling is according to the mean-field predictions of eq 4. We find a two regimes behavior for extended dendrimers in poor solvent conditions. F


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Figure 6. (a−d) Rescaled radial density distribution ρ(r)S3α−1 plotted vs the rescaled radial distance rS−α of the pure dendrimer density distribution (blue), the dendritic monomers of the extended molecule (black), the end groups only (red), and all monomers of the extended molecule (green) of generation G = 5 (left, a, c, e) and G = 7 (right, b, d, f) dendrimers with S = 2, 4, 6 (lines, open circles, full circles). Upper plots (a, b): the whole molecule is in good solvent condition. Middle plots (c, d): the dendritic monomers are in poor solvent conditions. Lower plots (e, f): 2D density plots of the dendritic part (left) and the end groups (right) along an arbitrary axis of G = 5 (e) and G = 7 (f) S = 2 codendrimers with dendritic monomers staying in poor solvent condition.

molecule (green) adopt the one of the pure dendrimer in the core region, whereas the end groups dominate the density completely at the exterior. For the generation G = 7 > G* dendrimers (b, d, f in Figure 6) in the good solvent case (Figure 6b), the density distribution of the whole molecule looks again quite similar to a stretched version of the pure dendrimer. The minimum of the dendritic monomers density (black) is more pronounced than for the generation G = 5 dendrimer. The reason for the stretching is the same as for the G = 5 molecule, but due to the exponentially increasing number of end groups, the effect on the dendrimer is more pronounced at higher generations. Increasing the interaction strength between dendritic monomers and solvent (Figure 6d) leads to a different picture as compared to the smaller molecule. Poor solvent does not yield a collapsed core of the dendritic monomers at the center of the extended molecule, and the plots for the pure dendrimer (blue) and for the dendritic part of the extended dendrimer (black) are no more on top of each other. Instead, a formation

pure dendrimer case resulting in a minimum of the density distribution close to the center of mass followed by a maximum. This is caused by the affinity of the end groups (red) to explore a maximal space to maximize the conformational entropy. Therefore, the density of the dendritic monomers (black) is reduced by the stretching of the whole molecule due to the end-group monomers and change the density profile qualitatively from dense to hollow core. The end groups themselves are distributed over the whole molecule up to the core region but preferentially located at the exterior of the molecule. In poor solvent conditions (Figure 6c) the dendritic monomer density (black) collapses to the pure dendrimers density distribution (blue). The end groups (red) are completely located at the surface of the dendritic part. This separation of monomers can be seen clearly in the 2D density plot in Figure 6e. The rescaling by the characteristic size of the spacers matches well close to the center of mass. Increasing the spacer length the density distribution of the full extended G


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dendritic monomers in a certain cutoff distance.47 We applied rcutoff = √12 and a minimum number of monomers per cluster nmin = 4 in this study. By analyzing the number of clusters in the molecule, the predictions of the mean-field model and the simulation results can be compared. The results of the cluster analysis for ε = 0.8 kBT are shown in Figure 8a together with a fit to eq 16. As expected from the results of the dendrimer sizes there is only one cluster up to generation G = 5. Higher generations show sequentially growing numbers of clusters. The spacer length dependence is in agreement with the mean-field model for S > 2 as indicated by the almost equal fit parameter a for spacer lengths S = 4, 6. Up to generation G = 5 dendrimers develop only one core at an interaction energy εmax = 0.8 kBT. Again, G = 6 = G* separates the single-core state from the multicore state. If we fix the topology and change the interaction energy ε, the prediction for the interaction energy dependence given by the mean-field model can be compared to the simulation results. We have used the analytical large core approximation given in eq 16. The corresponding results for codendrimers with G = 5, 6, 7 and spacer length S = 4 are displayed in Figure 8b. We find a good agreement between the mean-field model and simulation results at interaction energies higher than ε ≈ 0.3 kBT > εθ. The numerical constants are fairly close to the ones in Figure 8a. The formation of micelles starts at interaction energies above εθ. For all interaction energies smaller than ε ≈ 0.3 kBT, especially for interaction energies close to and lower than εθ, no micelles are formed since the surface tension between the collapsed and the dissolved phase becomes zero, and thus the driving force for micelle formation vanishes. The picture we obtain is that for increasing energy complete micelles vanish and the codendrimer rearranges to find a new stable macro-conformation with the fewer cores matching the actual interaction energy. For specific parameters, the number of cores oscillates in a small range of interaction energies, e.g., at ε ≈ 0.55 kBT for a G = 7 dendrimer with S = 4 (see Figure 10). This indicates a coexistence state of macro-conformations for the transition states. In order to investigate such transition states in more detail, thermodynamic properties such as the heat capacity are investigated. Free Energy and Heat Capacity. The free energy difference Δ- between a state with a nonzero interaction energy ε with respect to the zero interaction energy ε = 0.0 kBT (athermal state) is calculated by thermodynamic integration.48 Using the Hamiltonian in eq 18, the free energy difference in a system with interaction energy ε is given by

of an enrichment zone of dendritic monomers (black) in a far distance from the center surrounded almost homogeneously by the end groups (red) is observed. The spacers close to the center are nearly stretched resulting in a monomer density close to zero between the center and the enrichment zone. In the 2D density plot (Figure 6f) one can clearly see the almost ring shaped arrangement of the dendritic part and the broad distribution of the end groups all around the dendritic monomers including nonzero density in the interior of the codendrimer. Again, the difference in the density profiles of the high and the low generation codendrimer can be attributed to the formation of a single-core state (G < G*) as compared with a multicore state (G > G*) in codendrimers exceeding the threshold generation G*. Simulation Snapshots and Cluster Analysis. In Figure 7 we display typical snapshots for generation G = 5 and G = 7

Figure 7. Simulation snapshots of generation G = 5 and G = 7 codendrimers with spacer length S = 2 in good solvent conditions (ε = 0.0 kBT, left column) and poor solvent conditions applied to the dendritic part self-assembled into single-core (single micelle) and multicore structure (ε = 0.8 kBT, right column). Blue monomers depict the dendritic part, yellow monomers end chains, and red monomer the kernel monomer; solvent monomers are not shown.

codendrimers with spacer length S = 2 in good and poor solvent conditions applied to the dendritic part of the molecule. A collapse of the dendritic monomers is observed when increasing the interaction strength ε between the solvent and dendritic part. A notable difference between the dendrimers with G < G* and with G > G* in poor solvent is the number of distinguishable groups of collapsed dendritic monomers. For G > G* multicore structures can be identified. This is in agreement with the observations for the extension of the dendritic part seen in Figure 5 and the density profiles seen in Figure 6. In eq 16, we derived a relationship between the number of cores, the topological properties of the dendrimer, and the interaction energy in poor solvent conditions. The number of micelles k0 can be estimated by evaluating unique clusters of

Δ-(ε) =

∫0

ε

d ε′

∂U ∂ε′

= ε′

∫0

ε

dε′⟨Ncontact|ε′⟩

(21)

where ⟨Ncontact|ε′⟩ is the averaged number of contacts between all interacting monomers at a fixed interaction energy. The results of the integration for codendrimers are plotted in Figure 9. We rescaled the free energy by 2G applying the mean-field power law of eq 17, which is proportional to the total number of monomers in the dendritic part. We find the rescaling with the factor of 2G results in an accurate master curve for systems with equal spacer length as predicted by the mean-field model. This rescaling can be interpreted in the way that the interaction energy per end monomer is almost the same independent of the generation G as consequence of the globule formation assumption at high H


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Macromolecules

Figure 8. (a) Number of cores k0 under fixed interaction strength (ε = 0.8 kBT) plotted versus generation G of the dendrimer. (b) Number of cores k0 under fixed generation G and spacer length S plotted versus interaction energy. Lines corresponds to fits for the mean-field model of established number of cores k0(G,ε,S,Nchain) using eq 16.

formation process is completed, there is a power law region well described by the mean-field theory in large core approximation in eq 17: Δ- ∼ ε 5/22 . The heat capacity per monomer cV is measured by energy fluctuations of a thermodynamic system. In our model the heat capacity is calculated in the following way: cV =

1 ε2 (⟨Ncontact 2|ε⟩ − ⟨Ncontact|ε⟩2 ) ND kBT 2

(22)

In Figure 9b, the heat capacity is displayed as a function of the interaction strength. For small values of ε < εθ a steep ascent is shown below the threshold for the formation of stable micelles. For larger values of ε sharp and well-separated peaks are observed. Comparing the numbers of cores depending on the interaction energy in Figures 8b and 9b, we find the peaks in the heat capacity to be exactly at the transition states with changing number of clusters.

Figure 9. (a) Rescaled free energy difference Δ- of codendrimers with solvent sensitive dendritic core plotted against the interaction energy ε for different generations and spacer lengths. (b) Heat capacity cV shown for the spacer length S = 4 for codendrimers under varying interaction energy ε. Figure 10. Snapshots of multicore coexistence state between five cores (a) and three cores (b) formed by the generation G = 7 and spacer length S = 4 codendrimer at an interaction energy ε = 0.55 kBT showing the most prominent peak in the heat capacity cV (see also Figure 9b).

interaction energies. For a fixed spacer length the generation scaling matches well, but the predicted spacer lengths Δdependence of k T ∼ S 0 for the large core approximation is B

not shown. Rather, we observe an increase of the free energy with an increasing spacer length. This can be attributed to influence of curvature on the brush component which breaks the S2k scaling for the free energy as discussed in the MeanField Models section. Apart from that the mean-field model in large core approximation matches good for high interaction energies (ε > 0.4 kBT) above the θ-point. The free energy grows fast at low interaction energies possibly resulting from the strong rearrangement of the dendritic monomers in the molecule on the way to form micelles. After the cluster

In particular, for the most prominent peak of the G = 7, S = 4 dendrimer we get an averaged number of 4 clusters at an interaction energy of ε = 0.55 kBT. As illustrated in the simulation snapshots in Figure 10, we find some conformations with 5 cores and some with 3 cores. We can conclude that the high peak in the heat capacity is the result of the pairwise merging of four micelles to two single micelles. Other examples are the G = 7, S = 4 codendrimer at ε = 0.41 kBT where we observe a transition from 6 cores to 5 cores and the G = 6, I


Article

Macromolecules S = 4 codendrimer at ε = 0.46 kBT where we find a fluctuation between 2 and 3 cores. It is possible to estimate the free energy barrier between the states in a system with fluctuating number of cores A and B by counting the number of states NA and NB applying the direct counting method.49 This results for the G = 6, S = 4 codendrimer at ε = 0.46 k B T in

( ) ≈ 3 k T.

Δ-2,3 = −kBT log

N2 N3

B

In order to sample states across transitions between discrete stable micelle pattern, large relaxation times in particular for the high generation dendrimers are necessary. The simulations are performed for at least 108 MCS, which is a long simulation time compared to the relaxation time of a thread. We can still not be sure to observe only true equilibrium conformations as there might be a variety of metastable states. We provide movies as Supporting Information showing the effect of increasing and decreasing the interaction parameter during the simulation for a much smaller simulation time illustrating the creation and dissolving of the multicore structures. Apparently, the energy barriers may become rather large between the states of discrete number of cores. In the sense for applications as molecular containers the stability of these states is a promising indication for the stability of the carrier molecule even when the solvent quality is fluctuating.

Figure 11. Sketch of phase diagram for codendrimers with a given length of flexible spacers. The plotted lines G(T,k0) refer to the coexistence curves using eq 16 with values k0 = 1.5, 2.5, .... The constant C contains the dependencies on Nchain and S. The maximum generation corresponds to a dense packing of the dendrimers (see eq 23).

sphere of fully stretched threads according to Using eq 1, this leads to 4π 2 2Gmax Gmax −3 ≃ S 9

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SUMMARY AND CONCLUSIONS We presented a theoretical model which predicts the formation of multicore structures in single codendrimers where the dendritic part is subjected to poor solvent while attached linear chains remain under good solvent conditions. In our model we include the surface tension of spherical micellar cores and the Alexander−de Gennes/Daoud−Cotton-type free energy of the corona formed by the linear extensions. An analytical approach can be achieved using a large-core expansion which ignores curvature effects for the brush formed by the linear chains. The obtained scaling predictions are in rather good agreement with the free energy obtained from thermodynamic integration of the simulated codendrimers. Also, the prediction for the number of cores within a single dendrimer as a function of generation and solvent quality is in reasonable agreement with the simulation data, despite the simplicity of the approach. We have used the bond fluctuation model to prove the predictions of multicore dendritic micelles by simulating dendrimers with linear chains attached to the terminal monomers in explicit solvent. In good solvent conditions for both components the effect of these modifications on the dendrimer conformations are not qualitative and only lead to a gradual stretching of the dendritic part. If the codendrimer is placed in poor solvent but the linear modifications are still under good solvent conditions, unimolecular micellization takes place. Because of the fixed relation between the number of linear chains and the generation of the dendrimer, depending on the solvent quality, a state with more than one core can correspond to the optimal free energy. Our simulations give evidence that states with a well-defined number of cores (multicore structures) are stable. Thus, a single molecule can display various stable macro-conformations which are switchable by environmental conditions. Our essential results are illustrated in a sketch of the phase diagram given in Figure 11. For a given spacer length the upper limit for maximum packing of monomers can be estimated by a

3 ND 4π (GS)3

≃ 1.

(23)

with the solution for short spacer Gmax(S = 1) = 10. Longer spacers shift this limit, however only logarithmically, to higher generations. All calculation done in this work are valid only for G ≪ Gmax. At high temperatures (good solvent conditions) there is no formation of micelles. Codendrimers with a generation smaller than the threshold generation G* form only single core micelles at low temperature. Larger codendrimers are able to form multicore micelles where the number of cores depends on the dendrimers size and the solvent quality/temperature. The lines dividing the regions of k0 = 1, 2, 3, ... are given by eq 16, representing the coexistence curves which we have defined for simplicity by the formal solution for k0 = 1.5, 2.5, 3.5, .... A specific codendrimer with a generation G > G* is supposed to show a change in the numbers of cores when crossing one of these lines by changing temperature/solvent quality. We note that the interpretation of the temperature behavior is according to simple solvents with a usual upper critical solution temperature. In general, the inverse temperature should be interpreted as the effective interaction, ε, per monomer. According to our theoretical model and the simulation results, we expect the two parameters G and ε to be the most important ones to determine the number of cores in a codendrimer. Additional parameters like the corona chain length and spacer length extend the phase space further. The calculation of the heat capacity as a function of the solvent quality displays sharp peaks which can be identified as transition states between different numbers of cores. Since a true coexistence in such a small system cannot be achieved, these states actually corresponds to a dynamic fluctuation between different metastable states and can be identified for instance in snapshots of the simulations. Therefore, codendrimers are also interesting examples to study phase transition in small system unable to form static coexistence. We found a threshold value of the generation G* above multicore micellar states are possible only. Although this depends to some extent on the molecular parameters, our J


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Macromolecules

Figure 12. Sketch of the dual lattice approach as parallelization strategy of the BFM algorithm projected to two dimensions for simplicity. Left: One subset (blue squares) is tried to move in parallel while the other subsets behave inert (red squares). This resolves possible bond conflicts in parallel moves. For every monomer the bond-vector constraints are checked for the attempted move. Next, excluded volume is tested in the destination (gray circles). Rejected moves either by bond-vector (red line) or by excluded volume conflicts (shaded red area) do not participate in further steps. Middle: Possible moves (blue shaded) occupy the targeted position indicated by blue circles on the second lattice. The second excluded volume check on the dual lattice rules out direct collision conflicts (red shaded) which lead to further rejection. Allowed moves (blue shaded) can be performed in parallel without conflicts. Right: Allowed moves of monomers result in updating the lattice and monomer position (green arrow) while rejected moves stay on their initial positions (red arrow).

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APPENDIX. PARALLEL IMPLEMENTATION OF THE BFM In order to efficiently implement the bond fluctuation model (BFM)23,24 on the GPU architecture, as many monomers as possible should be attempted to move independently. Our approach is displayed in Figure 12. In the sequential algorithm the bond-vector constraint and the excluded volume property are respected in each move while all other monomers are fixed. For a massive parallel implementation, two challenges have to be solved: first, the bond conflict arising if two connected monomers moving at the same time and, second, the excluded volume conflict if a move of a monomer violating the simultaneous move of an other monomer. The bond conflict can be avoided by splitting all moveable monomers in subsets (, ), ... with no direct bonds of monomers within the same subset. The excluded volume conflict is resolved by using a dual lattice approach. The first lattice holds the overall information on the system and actual occupation. A second lattice is introduced which serves as instantaneous look-up table for possible moves to rule out excluded volume conflicts during parallel moves. Within the sequential BFM algorithm monomers occupy eight nodes on the sc lattice as efficient look-up table for short-range nonbonded monomer interactions. The parallelized algorithm considers the (virtual) interior node of the monomer as reference point. The latter suffers of the computationally more demanding lattice look-up and of the uncommon addressing of the lattice but provides coherent asynchronous read/write data access which is important for an optimized GPU code.52,53 For implementing the algorithm in C/C++ on GPUs the parallel computing CUDA framework (Compute Unified Device

results indicate that dendrimers with a higher generation and flexible spacers are good candidates for an experimental study of the effect. We can estimate the experimental parameter for a PAMAM/PEG codendrimer. The length of the hydrophilic chains of 24 BFM repeating units can be mapped to a molar weight of PEG between 1600 and 2000 g/mol using experimental findings for the Kuhn length, LK ≃ 7.1 Å, and segment length, bchem = 2.8−3.6 Å, of PEG.50 Using the corresponding mapping for one lattice unit to be 3.4 Å, we obtain radii of gyration between 28−55 Å for pure dendrimers G4S2 and G7S2 in good solvent and 19−36 Å in poor solvent. For (unprotonated) PAMAM dendrimers between G4 and G7 one measured an extension between 20 and 37 Å,51 which fits quite well to the choice of simulation parameters. We note that long spacers are not necessary to observe the multicore structure but flexible spacers are important to realize “connectors” between the cores. The change of macro-conformations can be induced by other experimental parameters such as pH and salt concentration if the linear component displays LCST behavior, such as PEG, or it is charged. This could be interesting to drug delivery type applications if the core can be loaded with a third component which can be stepwise release due to splitting up in more then one core. We also note that the inverse scenario where the attached chains are under poor solvent conditions also leads to interesting states. Preliminary simulation studies indicate macro-conformations where the dendritic part now in good solvent cannot fully cover the extensions forming a compact state giving rise to Janus-type self-assembled structures of the codendrimer. K


Macromolecules

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Architecture) by NVIDIA46 was used incorporated into the LeMonADE project.45 The parallelization of the application is realized by parallel execution units naming threads in an array of blocks in a grid by kernel calls (for details see refs 54−56). One thread will be associated as elementary parallel movement of a monomeric unit checking excluded volume, bond vector constraints, and Metropolis criterion and applying or rejecting the invoking movement. The algorithm works as follows: We randomly chose one subset (, ), ... of the not directly connected monomers and then attempt to move every monomer within the selected subset applying the same conditions as in the sequential algorithm. For each monomer we dice a random displacement along the eight principal lattice axis Δr ⃗ ∈ P±(1,0,0), where P± denotes all permutations and sign combinations and check the new bond vectors to be in the set of 108 allowed bonds in the BFM. If one bond does not match this constraint, the monomer move is rejected; otherwise, the algorithm proceeds. For the excluded volume condition nine positions in the vicinity of the direction of the move, 2Δr,⃗ on the first lattice are checked. If all positions are empty, the monomer is tagged for possible movement and the targeted position is occupied on the second lattice; otherwise, the move is rejected. Tagging for move is no yet sufficient due to possible collisions during asynchronous movement. It is necessary to check again for possible overlap taking into account all movable monomers. This can be done by inspecting excluded space (nine nodes) in move direction 2Δr ⃗ for each monomer again on the second lattice. If at least one node in this region is occupied on the second lattice, the move is rejected. Otherwise, the move can be performed without bond vector or excluded volume conflict. In this step a Metropolis criterion38 can be applied if applicable. In the final step the move is performed to the new position rn⃗ ew = ro⃗ ld + Δr ⃗ , and the occupation of the first lattice is updated. This algorithm is repeated as long as necessary to equilibrate the system and for sampling the observables. As in the sequential algorithm we define one Monte Carlo step (MCS) as one attempted Monte Carlo move per monomer in average. This algorithm can be efficiently implemented on SIMD (single instruction multiple data instruction) machines57 such as GPUs and achieves a performance gain of about 2 orders of magnitude between CPU and GPU under usual code optimization in this work. Throughout this paper we used 5 subsets (2 × dendrimer, 2 × end groups, 1 × solvent) accounting for the movement of noninteracting types in a correct local Metropolis criterion. In the parallelization framework the short-ranged interaction shell uses the local monomer vicinity of 66 nodes. This algorithm ensures cut avoidance, preserves excluded volume effects, and respects the global Metropolis criterion. Similar attempts for parallelization of the BFM and their systematic investigations are published elsewhere recently (see refs 58 and 59).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.W.). ORCID

Martin Wengenmayr: 0000-0002-0627-7129 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft under Contract SO 277/13-1. We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for generous allocations of GPU time and Hauke Rabbel (Leibniz Institute of Polymer Research Dresden) for discussions of the explicit solvent model.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01712. Two videos of a generation G = 5 and G = 7 dendrimer with spacer length S = 4 in heating and cooling procedure (ZIP) L


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