Dissipative Particle Dynamics Study of Electrostatic Self-Assembly in

Article pubs.acs.org/Macromolecules

Dissipative Particle Dynamics Study of Electrostatic Self-Assembly in Aqueous Mixtures of Copolymers Containing One Neutral WaterSoluble Block and One Either Positively or Negatively Charged Polyelectrolyte Block Karel Šindelka,† Zuzana Limpouchová,† Martin Lísal,‡,§ and Karel Procházka*,† †

Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University in Prague, Hlavova 2030, 128 40 Prague 2, Czech Republic ‡ Laboratory of Chemistry and Physics of Aerosols, Institute of Chemical Process Fundamentals of the ASCR, v. v. i., Rozvojová 135/1, 165 02 Prague 6-Suchdol, Czech Republic § Department of Physics, Faculty of Science, J. E. Purkinje University, Č eské Mládeže 8, 400 96 Ú stí nad Labem, Czech Republic S Supporting Information *

ABSTRACT: The paper describes the general features and trends of the electrostatic assembly (EA) of block polyelectrolytes. We performed computer simulations of the associative behavior of aqueous mixtures of diblock copolymers containing one neutral water-soluble block and one either positively or negatively charged polyelectolyte (PE) block. While the neutral block is readily soluble in water, the hydrophilic vs hydrophobic nature of the neutral backbone of the PE block and the compatibility of the blocks vary in a broad range. We investigated the role of (i) electrostatics, (ii) solvophobicity of the PE block, (iii) compatibility of the polymer blocks, and also (iv) compatibility of small ions with the polymer blocks. We employed the dissipative particle dynamics (DPD) method and used the generally recognized formula (J. Chem. Phys. 1997, 107, 4423) for recalculating the Flory−Huggins interaction parameters in DPD parameters of soft coarse-grained repulsion forces. The Coulomb interactions are described by the rigorous expression derived for the exponentially smeared charge with a fairly low charge decay length λ = 0.2. A low λ value has been used to reproduce the behavior of small counterions as realistically as possible at the DPD level. We compared the self-assembling behavior of charged and neutral copolymers for all the systems. The conclusions of the study can be briefly outlined as follows: Even though long-range electrostatic interactions are a prerequisite for electrostatic self-assembly and the increase in entropy due to liberation of mobile counterions represents an important driving force in all cases, the solvent quality for the PE backbone and incompatibility of blocks play an important role and substantially modify the association process. Only dimers containing one positively and one negatively charged chain are formed in systems with readily soluble PE blocks. The formation of large core−shell associates assumes (in addition to the effect of electrostatics) significantly unfavorable interactions of PE segments with water and with segments of the water-soluble block. The presence of opposite charges on different chains promotes the formation of associates, i.e., both the fraction of associates and association number increase, but the latter increase is fairly small (taking into account the value that is attained in corresponding neutral system).

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INTRODUCTION

developed model block PE−surfactant nanoparticles for use in medicine at approximately the same time,2 the interaction of PEs with oppositely charged PEs, surfactants, and other organic (usually multiply charged) ions has been broadly studied and a number of systems (either fully biocompatible formulations for medical use and nanoparticles for nanotechnologies or welldefined synthetic model systems) have been developed and investigated experimentally.3 Recently Lutz et al.4 published a

The coassembly of double hydrophilic copolymers containing polyelectrolyte (PE) and neutral water-soluble blocks with oppositely charged surfactants or PEs has been attracting the interest of a number research groups for almost two decades. Electrostatic association (EA) usually leads to the formation of core−shell nanoparticles that can be exploited in various technical and biomedical fields. Following the pioneering work of Kataoka and his co-workers, which was the first group that prepared electrostatically stabilized nanoparticles based on biocompatible polypeptides for drug-delivery purposes in the early 19 nineties1 and Kabanov and his co-workers, who © XXXX American Chemical Society

Received: May 15, 2014 Revised: August 5, 2014

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dx.doi.org/10.1021/ma501018x | Macromolecules XXXX, XXX, XXX−XXX

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soluble blocks, whose importance has been recognized, but whose effect has not been systematically studied, because the research has been aimed mostly at development of suitable and readily applicable nanosized carriers for drug delivery systems. The amphiphilicity of PEs is a result of their chemical structure. Many synthetic and natural PEs (e.g., poly(methacrylic acid), sulfonated polystyrene) contain a fairly hydrophobic backbone−very often a strongly hydrophobic hydrocarbon chain and their solubility and properties in aqueous solutions depend strongly on the content of pendant hydrophilic groups that are either permanently charged or can be ionized. The intricate interplay of antagonistic effects: (i) the hydrophobicity of the backbone on the one hand and (ii) the presence of electric charges on the chain on the other hand give rise to the very specific properties of aqueous PE solutions. The rich conformational behavior of “quenched” (i.e., strong) PEs can be exemplified by the formation of pearl-necklace structures in a certain range of solvent properties and charge densities,9 or by analogous (although more complex and less well understood) pH-dependent conformational behavior of “annealed” (i.e., weak) PEs, e.g., poly(methacrylic acid), PMAA, in aqueous buffers.10 During the past decade, it has been shown that the dissipative particle dynamics (DPD) simulation method is a versatile and fairly successful tool for studying the behavior of polymer systems11 and interesting examples of successful studies of specific self-assembling copolymer and PE systems have also been published.12 In addition to papers on PEs that include electrostatics, there exist studies that use an alternative approach and do not treat electrostatics explicitly.13 In a recent paper, we studied the pH-dependent selfassembly of a diblock copolymer formed by a PE block with a strongly hydrophobic backbone and a readily water-soluble block, namely the poly(2-vinylpyridine)-block-(ethylene oxide) sample using both soft coarse-grained repulsion and the Coulomb interaction potentials.14 We obtained results that compare well with the experimental data,15 suggesting that DPD simulations (when properly applied) yield a good prediction of the global features of general trends in the behavior of complex polyelectrolyte systems and provide an explanation for these features. In this paper, we study the EA of double hydrophilic copolymers by mesoscopic coarse-grained DPD simulations, namely the behavior of double-hydrophilic symmetric diblock copolymers containing a highly soluble neutral block and a charged PE block. The studied mixtures contain 50% positively charged A(+) 5 B 5 and 50% negatively charged A(−) 5 B 5 copolymer species, together with the corresponding numbers of both types of counterions. Symbols A and B stand for the coarse-grained units of the PE and neutral block, respectively. While we maintained the hydrophilic properties of B-block constant, we varied the hydrophobicity of the PE backbone and incompatibility of the two blocks. The study is intended to improve understanding of the following aspects of the association process: (i) the effect of hydrophobicity of the Ablock on the association number, structure and compactness of associates, (ii) the effect of incompatibility of the two copolymer blocks, (iii) the net role of electrostatics, and (iv) the compatibility of ions with other components (which to some extent emulates specific ion effects). (v) Last, but not least, we would like to elucidate the intricate interplay among the above effects.

paper on EA-based onion micelles, which offer similar applications as neutral multicompartment micelles.5 Here we focused on small self-assembled polymeric nanoparticles and did not consider a number of relatively large polymeric structures (e.g., layer-by-layer films and capsules) that are currently fabricated on a large scale. The self-assembly of polymers, including EA, has been studied by fine coarse-grained simulations, but because the size and complexity of the system usually exceed the capabilities of the most powerful modern supercomputers, a relatively low number of problems have been addressed so far. In addition to lattice Monte Carlo simulations of the self-assembly of neutral chains,6 several specific problems of EA have been studied at a relatively detailed level by fine coarse-grained simulations (usually using a bead−spring model with the Lennard-Jones interaction potential), e.g., the interaction of surfactants with a single chain in infinitely dilute solutions,7a,b the association of two oppositely charged chains,7c the interaction of one or more chains with oppositely charged surfaces and various charged objects, e.g., with a cylinder or a hexagonally packed array of infinitely long rods.7d−f The biologically relevant interaction and assembly of charged globular macro-ions and proteins with flexible PE of opposite charge have been successfully studied by Linse et al.7g,h and a similar (more detailed) system containing patchy globular proteins (with charged patches) has been investigated by Huang et al.7i One of the most complex problems studied by fine coarse-grained molecular dynamics simulations is described in the paper on the self-assembling behavior of a mixture of PEs, polyampholytes and counterions published recently by Dobrynin et al.7j Theoretical description of the association of surfactants with oppositely charged PEs and the self-assembly of two oppositely charged PEs has been proposed by Hansson.8a,b The interaction of PEs with charged globular objects has been studied theoretically by several research groups.8c−e Voets et al.8f recently reviewed the use of SCF calculations to describe EA. Let us briefly outline the current concept of the EA of oppositely charged PEs. The presence of opposite charges on different chains is a necessary prerequisite for electrostatic association in polymer systems; however, the net contribution of electrostatics to the Gibbs function is fairly small and is not the main driving force, because the Coulomb forces are well balanced: If two charges interact over a given distance, the interaction energy is the same irrespectively of whether it involves (i) two small ions, (ii) a charged group on the chain and a small ion, or (iii) two charged groups fixed on the polymer chain. Because the charges on the PE chains are mutually interconnected, the cooperative character of electrostatics promotes self-assembly, but the main driving force has entropic origin: even though the association of several bulky polymer chains is accompanied by a small decrease in the entropy, large numbers of small mobile counterions, which were localized close to the multiply charged chains and compensated the non-negligible electric charge of the macroions at distances that were as small as possible, are liberated upon mixing the solutions and formation of associates in bulk and their translation entropy increases considerably. However, neither the electrostatics nor the entropy of the liberated ions controls the formation of associates (association number, size, shape, and inner structure). They affect the association process, but there exist other factors, such as the amphiphilicity of the PE chains and the incompatibility of chemically different, waterB


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We controlled the solvent quality for A- and B-blocks of A5B5 by parameters aAS and aBS, respectively, and the incompatibility between A and B blocks by parameter aAB. Other details regarding the DPD method are given in the Supporting Information of our previous paper.14 Simulation Details. We mimicked the studied aqueous copolymer solutions using A5B5 immersed in a mixture of solvent particles, counter- and co-ions. One half of chains have positively charged A blocks (A(+)5B5) and the second half negatively charged A blocks (A(−)5B5). The volume fraction of A5B5 in the system is Ft = 0.0512. For comparison, we also performed simulations for neutral diblock copolymers. In the DPD method, the following reduced units were used: rc is the unit of length, the unit of mass is the mass of a DPD particle and the unit of energy is kT; these terms are used throughout this work. All the DPD simulations were carried out at a total particle density of ρ = 3 in a cubic box of 253 with noise amplitude σij = 3 and time step Δt = 0.05. Using the reduced units, we set the repulsion parameter between like particles at aii ≡ ajj = 25. We assumed that the neutral B block is soluble (aBS = 25), while the interactions of the backbone of the A block with the solvent vary in the range aAS = 25−37.5 and also the incompatibility between the A and B blocks varies in the same range, i.e., aAB = 25−37.5. The above parameters correspond to the bead− bead chi-interaction parameters χAS = 0−3.8, χAB = 0−3.8 and χSB = 0, respectively. These values are comparable with those used by other authors of computer studies of copolymer self- and coassembly.6,12a For the harmonic spring potential, we used the spring constant K = 4 and equilibrium distance r0 = 0. We further assumed that the values of aij for the counter- and co-ions are the same as those for the solvent particles. In addition, we used the decay length of the charge λ = 0.2 and Bjerrum length λB ≅ 1.10, which corresponds to an aqueous environment.17a The long-range electrostatic interactions were treated using the Ewald sum with cutoff relc = 3, real-space convergence parameter αES = 0.975 and reciprocal vector range nmax = (5,5,5).17b Simulations typically started from random configurations but, in a few cases, simulations were also initiated from the associated state. After an equilibration period of 2 × 106 time steps, we typically ran (20−50) × 106 time steps for aggregated systems and 5 × 106 time steps otherwise. DPD trajectories were generated using the GNU program DL_MESO,17c followed by postprocessing to evaluate the quantities of interest.

MODEL AND METHODS

Dissipative Particle Dynamics (DPD). In simulations, a DPD particle represents a fluid element which contains a number of solvent molecules, solvated ions or several polymer segments. We thus treated the solvent (S), counterions (CI) and co-ions (I+ and I−) as different types of DPD particles and represented the diblock copolymer by two flexible (linearly connected) chains of DPD particles with lengths n = 5 and m = 5, connected end-to-end and composed of A and B types of DPD particles. We modeled the DPD particles as being purely repulsive and the ij-pairs of particles interacting via a soft repulsive potential16 uijsr

2 aij ⎛ rij ⎞ = rc⎜1 − ⎟ (rij < rc) 2 ⎝ rc ⎠

(rij ≥ rc)

=0

(1)

where aij is the maximum repulsion between particles i and j, rij is the separation distance, and rc is the cutoff radius. Diblock copolymers A(+)5B5 and A(−)5B5 consist of DPD particles connected in a chain using harmonic spring potentials: uihs, i + 1 =

K (ri , i + 1 − r0)2 2

(2)

which act between adjacent particles i and i + 1 in addition to the soft repulsive interaction. In eq 2, K is the spring constant and r0 is the equilibrium distance. A value of K/(kT) between 2 and 4 (k is the Boltzmann constant and T is the temperature) and r0 = 0 are typically utilized.16,17 In the system of interest, the hydrophobic A+- and A−-segments of A(+)5B5 and A(−)5B5 chains, respectively, the counterions and the coions are charged. Since the DPD particles are modeled as soft particles, the charge is spread over a finite volume using the Slater smearing charge distribution17

f (r ) =

⎛ 2r ⎞ exp⎜ − ⎟ ⎝ λ⎠ πλ qe

3

(3)

The charge smearing prevents the collapse of beads on top of each other in the case of unlike point charges.17 In eq 3, q is the relative charge, e is the electron charge, and λ is the decay length of the charge. The electrostatic interaction between charged particles i and j is then defined by qiqj uijel = λBkT[1 − (1 + βrij) exp(− 2βrij)] rij (4)

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RESULTS AND DISCUSSION Theoretical Considerations. In this study, we pursue the following working hypothesis: (i) If the nonionized PE backbone is highly soluble, the entropy increase due to liberation of small ions upon mixing the positively and negatively charged components drives EA of the chains. However, only dimers and small aggregates are formed because, after neutralization (compensation) of the electric charges, the small associates formed as a result of cooperative Coulomb interactions between electric charges on the A+ and A− blocks are soluble. The enthalpy is insignificant because electrostatic forces are balanced and the EA process does not require the formation of compact nanosized Adomains to minimize the number of interactions of the A-segments with water molecules. The entropy term, T(ΔS1+ ΔS2), which comprises two main contributions: (a) “entropy gain” (ΔS1) due to liberated ions and (b) “entropy penalty” (ΔS2) due to association of copolymer chains (which is mild for small aggregates, but severe for large ones), becomes important. The process is both entropy-driven and entropy-controlled, but the two decisive entropy contributions that drive the process

where λB = e2/(4πε0εrkT) is the Bjerrum length (ε0 is the dielectric constant of a vacuum and εr is the relative permittivity of the reference medium), qi and qj are their partial charges, and β = 5/(8λ). Groot and Warren16a mapped the DPD model onto the Flory− Huggins (FH) model and established a link between aij and the chiparameter χij:18

aii + ajj ⎞ rc ⎛ ⎟ χij = 2Cρrc 3⎜aij − ⎝ 2 ⎠ kT

(5)

where ρ is the total particle density and C is a constant depending on ρ. Assuming aii = ajj and using the equation of state for the soft repulsive DPD fluid together with the compressibility value for ambient water, Groot and Warren derived an expression for likerepulsive parameters: ajjrc aiirc 75 ≡ = kT kT ρrc 3 (6a) They further obtained linear relationships between aij and χij for ρrc3 = 3: aijrc ar = ii c + 3.27χij (6b) kT kT C


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Figure 1. Weight distribution functions, Fw(AS), of association numbers AS for double-hydrophilic systems, with aAS = 25, aBS = 25, aA−CI = 25, and aB−CI = 25; (a) neutral system, (b) charged system with corresponding amounts of counterions, and (c) charged system with added salt (csalt = 5 vol %). The red curves correspond to the system with compatible A and B blocks (aAB = 25) and green curves to that with incompatible blocks (aAB = 37.5).

different types of averaging (number- and weight-average distributions) are described in detail in the Supporting Information. As expected, the double-hydrophilic neutral copolymer dissolves in a common solvent in the form of individual chains (unimer) and does not form associates. Nonnegligible populations of species (“pseudo-associates”) with association numbers 2 (and 3with a considerably lower population than that of the unimer) reflect the instantaneous arrangements of chains in the simulation box that emulate random collisions of chains (which are quite frequent at a relatively high concentration of 5 vol %). Similar “collision pairs” have been reported by other authors in simulations of polymers in good solvents.6e In addition to ref 6e, detailed explanation of the formation and identification of “close random pairs” and low associates can be found in the Supporting Information. In agreement with our working hypothesis, the equimolar mixture of double-hydrophilic block copolymers with one neutral and one either positively or negatively charged PE block contains an important fraction of dimers (40%) and significant fractions of small associates with association numbers of 3 to 6 (in total approximately 35%) and still has a high content of single chains (20%). The weight fraction of associates with As = 3 is approximately 15% and decreases with increasing As. Note that we give the weight distribution function and the weight of the chain or associate is proportional to its molar mass, which means that number fractions of associates with As > 3 are very low. The simulations also confirm the generally recognized fact that the addition of salt screens the electrostatic interactions and weakens the tendency toward self-assembly, i.e., it hinders the formation of dimers and promotes the dissolution of individual chains (see 40% of unimers vs 30% of dimers). Because it is very common that two incompatible homopolymers are soluble in the same solvent, but their solutions phase-separate at higher concentrations, we also performed simulations for a double-hydrophilic copolymer with highly incompatible blocks, i.e., for the following set of parameters: aAS = 25, aBS = 25, aAB = 37.5, aA−CI = 25, and aB−CI = 25, where the other parameters the same as in the previous case. The results are also shown in Figure 1 (green curves). It is obvious that the changes are negligible. This finding is understandable, because all the segments are soluble and their density is low both in unimers and in low associates. Therefore, different blocks do not form distinct segregated domains and incompatibility of blocks plays negligible role. Comment on terminology: In the studied systems, various associates differing in association number and in size form. The terms “low” and “high” associates refer to the association number, while the terms “small” and “large” refer to the size

(ΔS1) and control the equilibration of the system (ΔS2) are different. (ii) If the PE backbone is insoluble and the A and B blocks are moreover highly incompatible, enthalpy plays important role and large associates are formed because the system endeavors to minimize the number of unfavorable interactions between A-segments and both water molecules and B-segments. The self-assembly is driven by a combination of entropy and enthalpy and controlled predominantly by entropy, analogous to the association of neutral amphiphilic systems.19 However, the presence of opposite electric charges on different chains strengthens and modifies the EA, favoring associates with approximately equal numbers of positively and negatively charged chains. Note that the increase in the entropy (ΔS1) due to liberation of mobile ions, which promotes the association, is proportional to the total number of released ions only, but it is not related to (and hence it does not affect) the association number of the formed nanoparticles. The entropy decrease (ΔS2) depends on the association number, but it plays less important role than the former one. Therefore, we expect that electrostatic interactions affect mainly the associate-to-unimer mass ratio, but have only a second-rate impact on the association number as compared to the effects of the hydrophobicity of the backbone and incompatibility of the blocks. Computer Simulation Study. In the first series of simulation runs, we studied the EA of several positively and negatively charged diblock copolymers, each composed of two readily soluble blocks A (PE, either polycation or polyanion) and B (neutral). To obtain information on the net effect of electrostatic interactions, we also performed simulations for the corresponding neutral double hydrophilic copolymer and compared the results. We carried out the same comparison for all the studied amphiphilic copolymers in selective solvents for B (i.e., in solvents with deteriorating quality for A). Because both blocks interact favorably with the solvent (water), first we assumed that A and B are mutually compatible and set the values of all the interaction parameters equal, aij = 25. As explained in the methodology part, in all the PE systems we set the electric charge on each segment A equal to the elementary charge e, either (+e) for a polycation, or (−e) for a polyanion and delocalized it using the charge decay length λ = 0.2. The weight distribution functions of the association numbers, Fw(As), for the neutral and the charged systems, the latter both without and with the added salt, are depicted in Figure 1, parts a, b, and c (red curves). The evaluation of the distribution function during DPD simulations and physical meaning of D


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which is affected not only by As, but also by swelling of associates. In the next part of our systematic study of the effects of (i) hydrophobicity of the PE block A and (ii) compatibility of blocks A and B, we performed an extensive series of simulations for aAS ranging from 25 to 37.5 and we varied aAB in the same range for each of them. In addition to the simulations for λ = 0.2, we also performed a few simulations for λ = 0.67, i.e., for the value used by some other authors (see the Supporting Information). As explained in our recent paper,14 we prefer λ = 0.2, because we believe that the DPD treatment of charged subunits of the system, both of charged polymer segments and particularly of small ions, is more realistic. Small ions play an important role in the studied self-assembly and we do not want to excessively delocalize the charge and thus suppress the shortrange ion−ion correlation effects. In our simulations, we use the Bjerrum length λB ≅ 1.10. For λ = 0.2, the entire charge of the ion is localized inside the bead representing the solvated ion (i.e., inside the sphere with cutoff radius rc < λB). The decay length λ = 0.2 is actually the largest value that satisfies this condition and simultaneously the second condition that the electrostatic potential of two charged beads at λB equals to the Boltzmann factor kT (as shown in the Supporting Information). As the coarse-grained polymer beads represent in reality the multiply charged parts of the polymer chain, the smearing does not seem to be so important. However, the interaction between opposite permanent charges on different chains occurs at fairly short distances within compact A-domains. Because for λ = 0.67, the charge is smeared within a sphere of radius ca. 3, the electrostatic potentials for the two considered λ values differ (according to the Gauss law) at distances of r ∈ (0, 3), i.e., in a crucial range of distances with respect to the studied selfassembly. Therefore, we investigated the effect of smearing (see the Supporting Information). It is instructive (as suggested by the reviewer) to evaluate the Debye screening length, λD, but it is fair to say that the calculation of the ionic strength and λD for polymers is not straightforward from the theoretical point of view. If we assume that charges of individual coarse-grained polymer units are sufficiently far from each other and approximate the charged block by a sequence of interconnected single-charged beads (which is a limiting, albeit commonly used, approximation used in PE theories for ionic strength calculation), then we can estimate the Debye screening length λD ca. 0.686rc. However, as the charges in associates with even association numbers are well compensated at short distances, we may also assume that, e.g., in a system with aAS = aBS = aAB = aA−CI = aB−CI = 25 approximately 60% of associates with even association numbers (see later) do not contribute to the ionic strength. In this case, we get λD ca. 0.82rc and the addition of salt reduces it to ca. 0.6rC. Even though we cannot guarantee that the above outlined assumptions on the partial compensation of charges are 100% fulfilled, it is evident that the estimated values are reasonable and that addition of salt screens electrostatic interactions in all studied systems. All the simulated systems together with the parameters used are comprehensively listed in Table 1. In addition to the already discussed system in a good common solvent, we studied: (a) two systems in mild selective solvents (for aAS = 32.5 and 35, respectively) and (b) one system in a strong selective solvent, aAS = 37.5.

Table 1. Studied Systems with a Given Combination of Parameters aAS and aAB Marked by X in the Appropriate Cellsa aAB 25.0 aAS

25.0 32.5 35.0 37.5

Xa X Xa X

30.0 X Xa X

32.5

35.0

37.5

Xa,b X

Xa X X Xa

X X

a

Interaction parameter aBS = 25 was used in all cases. In most cases (if not indicated by superscript b), small ions are incompatible with A and compatible with the solvent and with B (aA−CI = 37.5, aS−CI = aB−CI = 25). In systems Xb, the specific effect of small ions was modelled by their compatibility with insoluble segments A (aA−CI = 25). For exponential smearing of electric charges, in most cases we used (without superscript a) the charge decay constant λ = 0.2; λ = 0.67 was used in Xa systems (the data are given in the Supporting Information).

The results of simulations for marginal selective solvent (aAS = 32.5) are depicted in Figures 2 and 3. Figure 2 shows the weight distributions of the association numbers, Fw(AS), and Figure 3 depicts the radial density profiles (RDP) of the polymer segments A, A+, A−, B, solvent S, and counterions CI+ and CI− as functions of the distance from the center of gravity of neutral and charged associates with association numbers, AS = 5 and 10, respectively. To determine the effect of compatibility of the blocks, we performed simulations for systems with both compatible (aAB = 25), slightly incompatible (aAB = 32.5) and incompatible blocks (aAB = 37.5). As the curves for aAB = 32.5 and aAB = 37.5 are essentially the same, those for aAB = 37.5 are not shown. The distributions of the association numbers for systems with neutral block A are depicted in parts a and d of Figure 2 for compatible and incompatible blocks, respectively. The monotonously decreasing curves indicate that unassociated unimer chains predominate in aqueous solution in both cases. However, the fractions of low associates (with low AS) in the equilibrated systems are statistically relevant. This finding indicates that the interaction parameters used correspond to the onset of the self-assembling process on the aAS scale (a slightly better solvent for the neutral system than the critical selective solvent, c.s.s.). It is interesting to compare Fw(AS) and the structure (i.e., RDP) of low associates formed by copolymers differing in the compatibility of blocks A and B. For the copolymer with fully compatible blocks, less convenient A−S contacts are replaced, not only by A−A, but also by A−B contacts, which results in the formation of solvent-depleted domains with fairly interpenetrating blocks A and B. The association process does not lead to particles with a strongly segregated core−shell structure. On the basis of RDP in Figure 3, we can conclude that associates with relatively low AS are reminiscent of ill-defined “crew-cut” micelles with cores formed by a mixture of blocks A and B (rich in A with admixtures of B) and thin shells formed by B blocks−see the significantly lower density of B segments than that of A segments at normalized distances 0 to 2 and a slightly higher density of B segments at distances 2 to 3 in Figure 3a. Because the fractions of associates with high AS are low (particularly in the neutral system) and analogous associates are also formed in a more selective solvent with aAS = 35 (and their fractions are higher), we did not analyze the structure of high associates for this solvent. We performed this for a system in a slightly more selective solvent with aAS = 35 and the relevant data, their E


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Figure 2. Weight distribution functions of association numbers, Fw(AS), for a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains in a marginal selective solvent with aAS = 32.5, aBS = 25, aA−CI = 32.5 and aB−CI = 25; the upper row of parts a to c is for systems with compatible A and B blocks (aAB = 25) and the bottom row, parts d to f, for slightly incompatible A and B blocks (aAB = 32.5); the pairs of parts a and d depict systems with neutral A blocks; parts b and e, a mixture of chains with charged A blocks without added salt; and parts c and f, analogous systems as in the previous figures with added salt (csalt = 5 vol %).

Figure 3. RDPs of associates with AS = 5 (for the neutral systems) and 10 (for the charged systems) formed in a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains in a marginal selective solvent with aAS = 32.5, aBS = 25, aA−CI = 32.5 and aB−CI = 25; the upper row of parts a to c corresponds to systems with compatible A and B blocks (aAB = 25) and the bottom row, parts d to f, to slightly incompatible A and B blocks (aAB = 32.5); the pairs of parts a and d depict systems with neutral A blocks; parts b and e, a mixture of chains with A charged blocks without added salt; parts c and f, analogous systems as in the previous figures with added salt (csalt = 5 vol %). Neutral systems: A-blocks, red; B-blocks, green; solvent, black. Charged systems: A+, red; A−, blue; B, green; small ions CI+, magenta; CI−, dark green; S, black.

blocks, but the simulated distribution curve is very broad and association numbers of non-negligibly populated associates range from units to almost a hundred chains. The oscillatory character preferring even to odd association numbers reflects the fact that associates with zero net charge are more stable that those with nonmatching positive and negative charges. The ensemble-averaged density profiles (Figure 3b) for associates with As = 10 indicate a fairly well-developed core−shell structure (better than that in the neutral system). The effect of incompatibility of the polymer blocks and the effect of the salt shown in Figures 2 and 3 can be summarized as follows: The worsening of interactions between A and B blocks restricts the formation of associates with high AS with partially intermixed insoluble domains and, as expected, promotes the core−shell structure. Consequently, the range of relevant association numbers shrinks. The largest association number of non-negligibly populated associates in the incompatible system does not exceed 35 and the fractions of all the small associates increase compared with the system with

appropriate analysis and conclusions are presented in the following parts of the paper. As aAS = 32.5 corresponds to a very mild (marginal) nonsolvent for A, the formed polymeric particles are swollen by the solvent−particularly the neutral ones; electrostatic interactions attract the poorly soluble A blocks closer together and the compactness of the insoluble cores increases. Figure 2d shows that increasing incompatibility of the copolymer blocks (aAS = 32.5) hinders the formation of associates with high AS. Figure 3d shows that it also promotes the segregation of blocks and the formation of better defined core−shell associates (results for highly incompatible blocks A and B with aAB = 37.5 indicate better segregation, but they do not differ much and are not shown). The inspection and comparison of the distribution functions Fw(AS) in mixtures of oppositely charged A(+)5B5 and A(−)5B5 species with compatible and slightly incompatible blocks, respectively, in a mildly selective solvent for B is noteworthy (Figure 2, parts b and e). Figure 2b shows that dimers are the most populated species in systems with compatible A and B F


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Figure 4. Weight distribution functions of association numbers, Fw(AS), for a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains in a mild selective solvent with aAS = 35, aBS = 25, aA−CI = 35, and aB−CI = 25; the upper and bottom rows correspond to aAB = 25 and 35, respectively. Concentration of added salt in parts c and f is 5 vol %. The sequence a numbering of figures are the same as in Figure 2. While the data in Figures parts a, d, e, and f are based on simulations for a system with 240 chains in a box of the size 253 (similarly to most data presented in this communication), the data in parts b and c are based on simulations for a larger system with 490 chains in an appropriately larger box of the size 323 (to get the same volume fraction of polymer beads).

promotes the association of oppositely charged A blocks from different chains (Figure 4e). The function Fw(AS) contains a well-pronounced peak with maximum for AS approximately 35, which indicates that the self-assembling process obeys the “closed association scheme”.19b (iii) The addition of salt (Figure 4f) screens electrostatic interactions and weakens the association tendency (both the association number and the total fraction of associates decrease slightly). Because simulations for compatible systems in “marginal and mild selective solvents” yield a surprisingly broad distribution of association numbers, we studied the dynamic behavior of systems close to c.s.s. in detail. The danger of freezing in a local energetic minimum for a long time is low because the solvent is still fairly good for both blocks. However, it cannot be a priori precluded that the system could phase-separate on a macroscopic length scale, which could be manifested by slow oscillations between two states. Therefore, we studied the dynamic behavior and structural characteristics of high and low associates in detail using time-dependent fluctuation of the weight-averaged association numbers, ⟨AS(t)⟩w. The equilibrium parts of simulation trajectories for the systems, whose distributions Fw(AS) are shown in Figure 4, parts a and b, are depicted in Figure 5a and reveal that the ⟨AS(t)⟩w value of associates with compatible blocks in solvents close to c.s.s. fluctuates quite randomly and rapidly in equilibrated systems and that the fluctuations are large, indicating that the neutral system does not freeze in local minima and the simulation yields well-equilibrated data. Figure 5b shows the fluctuations in the charged system (the AS-distribution of which is depicted in Figure 4b). The fluctuations are large. Association number fluctuates in most cases in the range (100, 250), but large positive fluctuation up to 400 appear several times. They are consistent with asymmetric tail of AS-distribution. Large fluctuations in both cases (neutral and charged system) prove dynamic character of the equilibrium in spite of the fact that the average number of charged associates in the simulation box is permanently equal to unity. RDPs of high associates with AS = 100 in a system with compatible blocks A and B, i.e., with aAS = 35, aBS = 25, and

compatible blocks. The oscillations between even and odd association numbers are important (see 20% dimers vs 1% unimers and 3% trimers) indicating that, in low associates, a relatively large mismatch of electrostatic charges due to one excessive chain with noncompensated charges strongly destabilizes the system. The addition of salt lowers the association tendency analogously to the system with compatible blocks. As expected, the systems in a mildly selective solvent with aAS = 35 (Figure 4) show a more pronounced tendency toward association than those with aAS = 32.5, but the solvent selectivity (i.e., that for the neutral system) is still in the region close to the onset of the association process (slightly worse than c.s.s.). Because the behavior, especially that of the system with compatible blocks (Figure 4a), is very interesting, we will analyze and discuss the data in more detail. The solution contains the unimer and a broad distribution of associates with almost all AS up to ca. 200, which suggests that the association obeys the “open association scheme”.19b The presence of charges (Figure 4b) promotes the formation of associates with high association numbers, but the effect is not dramatic. Although, the simulations with 240 chains indicated that associates with As > 240 would probably form under given conditions and therefore we performed two very timeconsuming simulations for systems with 490 chains in an appropriately increased box size to ensure the same volume fraction 0.05 of polymer beadsboth for systems without and with added salt. The results obtained confirmed our suspicion concerning the formation of large associates. The curves in parts b and c of Figure 4 are based (in contrast to most data presented in this communication) on simulations for a large system with 490 chains and it is why the statistics is slightly poorer that in other cases (the simulation took almost two months). The addition of salt (Figure 4c) slightly suppresses EA, analogously to the other studied systems. The simulation data in Figure 4d show that (i) increased incompatibility of the blocks (aAB = 35) restricts the formation of high associates in the neutral system (similarly to the solvent with aAB = 32.5); nevertheless, fractions of associates up to AS approximately 30 are statistically relevant. (ii) The electrostatics G


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negatively charged A segments, respectively. The B segments are represented by green color. Note that the size of the beads was reduced in comparison with size of the elementary DPD unit and reflects the 3D perspective to obtain a comprehensive picture. In all cases, not only pictures of complete associates (bottom row), but also separate clouds of A beads are depicted (upper row). Parts a and b of Figure 7 show the most frequently observed ensembles of several more or less uniform associates of roughly spherical shapes coexisting with a low fraction of single chains. Figure 7c depicts the coexistence of a small spherical associate with an elongated and strongly curved large one and with a dimer. Figure 7d depicts the presence of only one large object. Their shapes are aspherical: elongated, quite irregular, and considerably curved. The A segments in the insoluble domains have quite high density. The structures are stabilized by a thin layer of B segments (they are reminiscent of “crew-cut” nanoobjects), but some B segments partially intermix with A segments. The structures of several (frequently appearing) large associates formed in the neutral system with compatible blocks are shown in Figure 8 and those for the analogous charged system are depicted in Figure 9. It is obvious that large irregular objects with different shapes (roughly spherical, elongated ellipsoidal, curved and even vesicle-like structures) coexist with small ones in the solution. The cores of the large objects are formed preferentially by A blocks. Soluble B blocks (even though compatible with A) only partially intermix with A (mainly in the neutral system, which is reflected in the slightly lower density of the green beads on the surface of the neutral associates compared with the charged onesan effect observable by the naked eye), but concentrate at the periphery of the associates and form a thin stabilizing shell. Having inspected the shapes of randomly chosen large associates, we evaluated (as also suggested by the reviewer) the average principal components of the gyration tensor as functions of AS and some shape descriptors of large associates (asphericity) in a system with compatible blocks A and B, i.e., for aAS = 35 and aBS = 25 and aAB = 25 (Figure 10a) and compared them with analogous characteristics for typical core− shell systems formed by chains with incompatible blocks for systems in selective solvents (aAS = 35, aBS = 25, aAB = 35; Figure 10c) and (aAS = 37.5, aBS = 25, aAB = 37.5; Figure 10d). It is obvious that the associates formed in systems with compatible polymer blocks (Figure 10a) are on average prolong objects and the deviations from spherical symmetry are significant and increase with AS. The number-average distribution function Fasp(b) of asphericity parameter, b = λz2 − 1/2(λx2 + λy2) of associates with AS ∈ (90, 100) is shown in Figure 10b. The values were averaged from 90 to 100 to yield better statistics. It is obvious

Figure 5. Equilibrium parts of the simulation trajectories showing the frequent and random fluctuations of the weight-averaged association number ⟨AS⟩w in the (a) neutral and (b) charged system with aAS = 35, aBS = 25, aAB = 25, aA−CI = 35 and aB−CI = 25. Comment on graphical representation of fluctuations: The zoomed parts of fluctuations are based on all independent configurations of the simulation box (used for calculation of average characteristics) obtained in a pertinent part of the simulation run with increments of 104 steps, i.e., the frequency of fluctuations is related to the real dynamic behavior of the system. The fluctuations depicted in bottom curves are based on a lower number of less densely selected configurations (obtained with increments of 106 steps) and therefore the fluctuations seem slower because their frequency does not correspond to that in the zoomed part of the curves. The curves shown in the bottom part of the figure demonstrate a randomly fluctuating behavior of the system on a longer time scale only. Data in part a are based on the simulation for a system with 240 chains in a box 253, while data in part b are based on the simulation for a larger system with 490 chains in a larger box 323 (to get the same volume fraction of polymer beads).

aAB = 25 are plotted as functions of the distance from the gravity center in Figure 6. The plots show that the high associates are quite large because the region of non-negligible density extends to normalized distances r > 6. This means that the size characteristics of the associates are comparable with the contour lengths of the individual chains, which implies that high associates are either (i) large roughly spherical microgellike particles with significantly interpenetrated A and B segments or (ii) significantly elongated linear or curved objects (in an extreme case, they may resemble vesicles). In the latter case, the angular averaging could appreciably obscure information gained by RDPs. Therefore, we inspected a large collection of simulation snapshots before discussing RDP. We focused on pictures that appear very frequently as repeating self-similar replicas of the simulation box and can be classified as “typical” pictures. The examination of high numbers of instantaneous configurations is time-consuming and does not replace proper statistical analysis of the simulation data; however, it can provide useful additional information on the system of interest. Figure 7 depicts several snapshots of the simulation box for the charged compatible system with aAS = 35, aBS = 25, and aAB = 25, which show the “typical” configuration of the system, i.e., structural arrangements that appear frequently during the simulation run. Red and blue beads represent the positively and

Figure 6. RDPs of high associates with AS = 100 for systems with compatible blocks in a mild selective solvent (aAB = 25 and aAS = 35). The sequence of figures for (a) neutral, (b) charged, and (c) charged salted systems and the colors of the individual curves are the same as in Figure 3. H


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Figure 7. “Typical” simulation snapshots showing various associates frequently formed during simulation runs for the system with aAB = 25, aAS = 35, and aBS = 25. Upper row: the clouds of core-forming A segments, bottom row: whole associates.

35) are plotted in Figure 11 for AS = 20 (for neutral system) and 30 for the mixture of charged species. They reveal that these associates have a distinct core−shell structure and that electrostatics promotes the segregation of different blocks. Two typical snapshots of the charged system with incompatible blocks (aAB = 35), and aAS = 35 and aBS = 25 in Figure 12 confirm the conclusions drawn from Fw(AS): the aggregates are smaller (their association number is lower, but their numbers are consequently higher) and less polydisperse. They coexist with unimers and dimers and acquire distinct core−shell structures. Inspection of a large ensemble of snapshots indicates that large associates are not present (in general, i.e., not only in the simulation pictures shown here, which is also consistent with conclusions from Fw(AS)). Comparison of the structures of the regular associates in Figures 7 and 12 shows that the shells of the associates formed by copolymers with incompatible blocks are significantly expanded, mainly because A and B blocks do not mix and entire B blocks create the soluble shell thicker as compared with the case when parts of B blocks (compatible with A) intermix in the core. The associates are dynamicthere exists a fast exchange of unimer chain between them. The associates partially dissociate and reform, which can be documented by fast fluctuations of the average number of associates in the simulation box, ⟨NA⟩, which is shown in the Supporting Information. Using information on the shapes of the large associates, we can return to RDP depicted in Figure 6, and start to analyze and discuss their shapes. We begin with the neutral compatible system (Figure 6a). The density profiles of the A and B blocks acquire similar shapes, except that the density of the A blocks is approximately twice as high at short distances from the center of gravity of the associate and decays faster with increasing distance. Both curves cross at approximately r = 4.5 and the density of the B blocks is higher than that of the A blocks at large distances. The density profiles in Figure 6a suggest that (i) both blocks interpenetrate quite a bit at both short and large distances and that (ii) the large, poorly soluble inner domains are swollen and contain a non-negligible content of water and thin (B-rich) outer layers stabilize the nanoparticles in the solution. However, it is necessary to modify and refine the conclusions as follows: As the associates are elongated and

Figure 8. Large neutral associates for aAS = 35, aBS = 25, and aAB = 25. Upper row: the clouds of core-forming A segments. Bottom row: whole associates.

Figure 9. Large charged associates for aAS = 35, aBS = 25, and aAB = 25. Upper row: the clouds of core-forming A segments. Bottom row: whole associates.

that the shape (and therefore the asphericity) of objects fluctuates during the simulation run, but a high fraction of large associates are appreciably aspheric objects. The principal components of associates in systems with incompatible blocks in selective solvents are depicted in Figure 10, parts c and d. It is evident that the core−shell associates are not absolutely spherically symmetrical, but the deviations are quite small, they do not almost depend on AS and slightly decrease with increasing solvent selectivity. The angularly averaged RDPs for regular associates (with medium AS) formed by chains with incompatible blocks (aAB = I


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Figure 10. Number-average principal components of the gyration tensor, λx2, λy2 and λz2 for a system with compatible blocks A and B with aAS = 35 and aBS = 25 and aAB = 25 (a) and for salt-free systems formed by chains with incompatible blocks with aAS = 35, aBS = 25, aAB = 35 (c) and for aAS = 37.5, aBS = 25, aAB = 37.5 (d); the number-average distribution function Fasp(b) of the asphericity parameter, b = λz2 − 1/2(λx2 + λy2) of associates with AS ∈ (90, 100) for a system with aAS = 35 and aBS = 25 and aAB = 25 (b). Straight lines in portions a, c, and d are just guidelines for eyes.

Figure 11. RDPs of regular associates with medium AS = 20 and 30 for neutral and charged systems, respectively, with incompatible blocks (aAB = 35) in a mild selective solvent with aAS = 35 and aBS = 25. The sequence of figures (a) for neutral, (b) charged, and (c) charged salted systems and colors of individual curves are the same as in Figure 3.

curved objects, the center of gravity often occurs at the periphery and in some cases even outside the associates. The conclusion that the central domain of the associate is swollen by water and that the A and B segments are strongly intermixed within the whole particle is not correct, because angular averaging of functions for objects that strongly differ from spherical symmetry exaggerates the effect of intermixing and obscures information gained from RDP. The presence of charges modifies the structures of associates (see Figure 6b). We observed an appreciable increase in the density of the poorly soluble domains. The intermixing of the A and B blocks in the charged systems substantially decreases, because the “mixing” and close approach of positively and negatively charged beads A from different chains is reinforced by electrostatics. The profiles indicate the formation of betterpronounced core−shell structures. The addition of the salt (Figure 6c) screens the electrostatic interactions. The density of the insoluble domains decreases slightly in the salted systems, but the basic structures of the associates do not change much. The most important part of the study from the point of view of applications concerns the copolymer composed of incompatible blocks in a highly selective solvent for neutral block B. Since the A−S interactions are inconvenient according to the working hypothesis, the neutral system itself should selfassemble and the electrostatics should promote the formation of associates, but only to a limited degree. Therefore, comparison of the results for neutral and electrically charged

Figure 12. Two “typical” snapshots for charged system with incompatible blocks (aAB = 35), and aAS = 35, and aBS = 25. Upper row: the clouds of core-forming A segments. Bottom row: whole associates.

J


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Figure 13. Weight distribution functions of association numbers, Fw(AS), for a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains with incompatible A and B blocks (aAB = 37.5) in a strongly selective solvent with aAS = 37.5, aBS = 25, aA−CI = 37.5, and aB−CI = 25: (a) the system with neutral blocks A; (b) a mixture of chains with charged blocks A without added salt; (c) an analogous system with added salt (csalt = 5 vol %).

Figure 14. RDPs of associates with AS = 40 for neutral and AS = 55 for charged systems, respectively, with incompatible blocks (aAB = 37.5) in a strongly selective solvent aAS = 37.5 and aBS = 25. The sequence of figures for (a) neutral, (b) charged, and (c) charged salted systems and the colors of the individual curves are the same as in Figure 3.

that the nanoparticles have core−shell structure in all cases and that the blocks are well separated from each other. The profiles of oppositely charged blocks (Figure 14b) in the core coincide well, which illustrates that the electric charges are compensated at short distances inside the core. Small ions (which are incompatible with A) do not enter the core, but they interact favorably with B and hence their concentration in the shell is comparable with that in the bulk solvent. Note that the number of particles A in a neutral associate with given AS is equal to the sum of A+ and A− and hence the scale in (Figure 14a) (for the neutral system) is different from that used in Figures 14b and c (for charged systems). In the last part of the paper, we study the specific effect of ions on the self-assembly. The ions interact unfavorably with nonpolar polymers, however their interaction with polar atoms or polar groups in various water-soluble polymers is often favorableit concerns both nonpolar polymers (such as polyoxyethylene) and of course PEs. The nonpolar PE backbone bears ionizable groups which are polar and hydrophilic even in their nonionized state. The interaction of different ions with polymers depends on chemical nature, properties and structure of both components (tetramethylamonium ions and other ions with organic ligands interact usually more favorably with organic polymers than small nonpolarizable inorganic ions). Although there exist a number of specific studies of different ion−polymer systems (mostly focused on flexible polymer electrolytes and some of them aimed at comparison the effects for a series of ions20), at present it is almost impossible to ascribe reliable interaction parameters to individual systems. Our ambition is to show that specific effects of ions, which can be modeled via their compatibility with A segments, play important role. We study two limiting cases only: ions which are fully compatible or incompatible with A segments. The results are shown in Figure 15, which compares the distribution functions of the association numbers, Fw(AS) for a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains in a selective solvent with aAS = 35 for counterions that are compatible only with B blocks (aA−CI = 37.5, aB−CI = 25, red curves) and counterions that are

copolymers is of great importance. Before studying the electrostatically affected self-assembly, it is necessary to investigate whether the ionization of the A blocks (i.e., appearance of charges of the same sign) and the consequent electrostatic repulsion between otherwise insoluble (neutral) A blocks leads to the dissociation of the originally neutral associates and to dissolution of the copolymer chains, i.e., to the process observed experimentally.15 We studied this topic in detail in our recent study14 and we found that, in a system with comparably inconvenient interactions of the neutral backbone of the PE block with water, a degree of ionization exceeding 50% (in the present case a 100% charged strong PE is involved) led to quantitative dissociation of the associates and full dissolution of all the copolymer chains. Hence the question has already been answered once for all and we do not need to study the behavior of separate PE solutions, either polycations or polyanions. Figure 13a depicts Fw(AS) for a system of neutral A5B5 chains with strongly incompatible blocks (aAB = 37.5) in a strong selective solvent for B with aAS = 37.5. A pronounced selfassembling tendency is obvious from the shape of Fw(AS). The curve is bimodal: the first maximum corresponds to the unimer and the second well-separated and well pronounced peak corresponds to the associates with AS ranging from 10 to 70. The maximum weight fraction of associates is attained for AS approximately 40 and half-width approximately 20. In accordance with our working hypothesis, the effect of electrostatics is rather small. The presence of opposite charges on different chains promotes the association, but the association number of the most populated associate shifts only slightly toward higher values (AS increases by approximately 15) and the half-width increases only marginally (Figure 13b). However, the fractions of associated chains almost double and the unimer disappears from the mixture (the small peak of the dimers is still visible). The salt effect is slight, except that the total fraction of associated chains decreases slightly (Figure 13c). The density profiles of associates formed by copolymers with incompatible blocks are shown in Figure 14. They demonstrate K


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The parameters of the model are as follows: interactions of the neutral block with the aqueous solvent are always good, but those of the neutral PE backbone with water vary in a broad range (from favorable to strongly unfavorable). The compatibility of the blocks also varies. The Coulomb interactions between the DPD units were described by the electrostatic potentials between the exponentially smeared elementary charges delocalized exclusively inside the DPD units. We compared the results for neutral systems with the mixtures of charged systems for all the sets of parameters to obtain information on the roles of all the relevant effects (electrostatics, amphiphilicity of the neutral analogous system and incompatibility of the blocks) and on their intricate interplay. The performed simulation study confirms our working hypothesis: Copolymers with a highly soluble neutral block B and a PE block A which contains a highly soluble neutral backbone (equimolar mixtures of positively and negatively charged chains) electrostatically coassemble, but do not form large associates. In this case, only dimers are formed because the segments of the A+/A− domains interact favorably with water, irrespectively of the overall electric charge. The process is both entropy-driven and entropy-controlled, but the entropy contributions, which drive the process and control it, are different. The main driving force reflects the appreciable entropy increase due to the liberation of small counterions in the bulk solvent upon the formation of dimers and mutual compensation of opposite charges on the different chains forming the dimer. The controlling term consists in a relatively small entropy decrease due to the association of copolymer chains, which is smaller for dimers than for higher associates. The mixtures of copolymers with hydrophobic backbone of the PE A block (both with A+ and A− blocks) and a highly soluble neutral B block (which is incompatible with A) exhibit different assembling behavior. The simulation study shows that these systems form multimolecular core−shell associates, which is in agreement with results of a number of experimental studies.3 The driving force consists of two contributions which reflect the decrease in enthalpy and increase in entropy. The enthalpy contribution reflects minimization of the number of unfavorable interactions of segments A with water during the formation of compact spherical cores A with minimum interface-to-volume ratio. The entropy contribution is caused by liberation of counterions analogously to the previously discussed case. The association number and size of the associates are controlled by a complex interplay of several contributions mostly of entropic origin, similarly to neutral amphiphilic copolymers in selective solvents.19 Simulations for systems with neutral A blocks (with the same set of interaction parameters as for the mixture of charged copolymers) and comparison of the results of corresponding charged and neutral systems show that the hydrophobicity of the PE backbone and incompatibility of the blocks play an important role and appreciably affect the electrostatic coassembly. The neutral amphiphilic copolymer with incompatible A and B blocks forms a core−shell associate with fairly high association number in a strongly selective solvent for B blocks (aAS = 37.5, aBS = 25, and aAB = 37.5). The presence of charges of the same sign on the A segment causes their dissociation in single copolymer chains, as was shown in our recent computer-based and experimental studies.14,15 The presence of opposite charges on different chains promotes the assembling process: (i) electrostatic attraction pulls the A+ and A− blocks closer together, increasing

Figure 15. Weight distribution functions of association numbers, Fw(AS), for a 50%-to-50% mixture of A(+)5B5 and A(−)5B5 copolymer chains in a selective solvent with aAS = 35, aBS = 25, and aAB = 35 for CI which are incompatible with A and compatible with B (aA−CI = 37.5 and aB−CI = 25−red curves) and CI compatible with both A and B (aA−CI = 25 and aB−CI = 25−green curves): (a) mixture of chains with charged blocks A without added salt and (b) with added salt (csalt = 5 vol %).

compatible both with A and B blocks (aA−CI = 25, aB−CI = 25, green curves). Because small compatible ions penetrate in the A-domains, they partially compensate the charges of the A blocks and suppress the association of the polymer chains (Figure 15a). The addition of the salt, which is incompatible with block A, has only a small effect. However, the addition of the salt with compatible ions results in a considerable decrease of the association number. The efficient screening of electrostatic forces reduces the oscillatory character of Fw(AS) in both cases. Comments on Systems with Compatible Blocks. In this paper, almost equal effort was devoted to studying systems with compatible blocks and to much more practically important and considerably more frequent systems with incompatible blocks. However, the association of copolymers with compatible blocks is a largely neglected topic. We have recently experimentally studied the complexation of metallacarborane COSAN (which has potentially very important medical applications)21a,b with poly(ethylene oxide), PEO,21c,d and poly(2-ethyloxazoline), PEOX, and with their block copolymers20e in aqueous buffers. The interaction of PEO with COSAN leads to the formation of insoluble complexes, while the complex with PEOX is soluble. As the two homopolymers are compatible, the complex of COSAN with the PEO−PEOX copolymer in aqueous media represents a neutral system with compatible blocks in a selective solvent for the PEOX− COSAN complex. In our experimental study we were slightly surprised when we found that the prepared nanoparticles were relatively large swollen nanogel particles with interpenetrating blocks of both types.21e We explained the observed structure by the fact that both PEO and PEOX units can interact with a relatively large COSAN molecule, giving rise to an additional compatibilization effect. Even though the role of COSAN as an efficient compatibiliser was important, the present study shows that the intermixing of chains is a general feature of systems with compatible blocks in selective solvents.

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SUMMARY AND CONCLUSIONS The computer study of the electrostatic coassembly of equimolar mixtures of oppositely charged symmetric block PEs with one PE block (either strong polycation or strong polyanion) and one readily water-soluble neutral block in aqueous media was performed using the DPD simulation method. L


Macromolecules the density of the insoluble domains and (ii) the cooperative effect of multiple charges on flexible macro-ions amplifies the effect of electrostatics. Both the average association number and the fraction of associates increase, but the increase in the former is relatively small compared with the average association number of the neutral system. The effect of compatibility of A and B blocks is also important and interesting. The compatibility of the blocks together with bad solvent quality for the A block promotes the intermixing of the two copolymer blocks and formation of large microgel-like particles (especially in case of the neutral A block). Opposite electric charges on the A segments suppress the intermixing of the A and B segments because they strengthen the attractive interaction between two A segments relative to A−B interaction. As shown in the Supporting Information, a narrow smearing of the electric charge on beads A slightly favors coassembly, as concerns both the average association number and the fraction of associates, because the electric charges compensate one another in compact A cores at the short distances at which the electrostatic potentials for charge decay lengths λ = 0.2 and 0.67 differ (are stronger for λ = 0.2). The differences between the simulation results for λ = 0.2 and 0.67 are not negligible, but they do not affect the general coassembling scheme at either the qualitative or semiquantitative level. In the Supporting Information (and also in our previous paper14) we have shown that narrow smearing (with λ = 0.2) provides more realistic behavior of small ions at the DPD level than λ = 0.67 and, what is essential, the charge is smeared within a sphere with radius slightly smaller than the Bjerrum length λB and therefore the dependence of the electrostatic potential on reduced distance satisfies the classic definition of λB, i.e., uelij (rij = λB) = kT, while the value 0.67 violates this requirement. In summary, the DPD study shows that the hydrophobicity of the PE backbone and incompatibility of the blocks significantly affect the electrostatic coassembly. The DPD results for related incompatible systems in bad solvents for A blocks generally obey the experimentally observed trends of complex behavior caused by the interplay of different factors: (i) the neutral amphiphilic system with poorly soluble A block self-assembles and forms spherical core−shell particles which coexist in equilibrium with the unimers, (ii) the presence of charges of the same sign on A block promotes dissolution with the formation of single chains, and (iii) chains with oppositely charged A blocks (when mixed) coassemble electrostatically and form core−shell particles with slightly higher association numbers than those of the neutral associates; in this case the fraction of unimer decreases appreciably and only a small fraction of dimers coexists in equilibrium with large associates. We believe that the performed work demonstrates that DPD (when correctly applied and appropriately parametrized) is a suitable tool for studying the pronounced changes accompanying self- and coassembly in copolymer and particularly in polyelectrolyte systems. From the qualitative point of view, the most obvious changes in the self-assembling behavior of copolymer chains with incompatible blocks induced by electrostatics occur in the region of interactions close to c.s.s., where the electrostatic interactions can provoke multimolecular electrostatic association (obeying the closed association scheme) in systems whose the neutral analogues are still soluble and almost do not associate (see Figures 4d,e).

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

Article

S Supporting Information *

Data equilibration, ensemble average and fluctuation of the number of associates in the simulation box, effect of charge smearing, specific effect of ions, and recognition criterion and classification of associates. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author

*(K.P.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Czech Science Foundation (Grants P106-12-0143 and P106-13-02938P) and by Grant Programme of the Ministry of Education, Youth and Sports (Project No. LH12020 and LK21302). Access to the computing and storage facilities of MetaCentrum, (LM2010005), and CERIT-SC computing (Operational Program Research and Development for Innovations, Reg. No. CZ. 1.05/3.2.00/08.0144) is greatly appreciated.

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