Micelles of Gradient vs Diblock Copolymers: Difference in the Internal

Nov 7, 2016 - We performed computer simulations to reveal a difference in internal structures of micelles formed by AB gradient copolymers and equival...
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Micelles of Gradient vs Diblock Copolymers: Difference in the Internal Structure and Properties Vitaly S. Kravchenko† and Igor I. Potemkin*,†,‡ †

Physics Department, Lomonosov Moscow State University, Moscow 119991, Russian Federation DWI − Leibniz Institute for Interactive Materials, Aachen 52056, Germany

‡

ABSTRACT: We performed computer simulations to reveal a difference in internal structures of micelles formed by AB gradient copolymers and equivalent diblock copolymers in a selective solvent. In contrast to distinct core−shell structure of the diblock copolymer micelles (DCM), the soluble and insoluble monomer units are less segregated in the gradient copolymer micelles (GCM). Furthermore, the concentration of the soluble units in the GCM has a maximum at the core−corona interface. The maximum is a consequence of loop formation near the interface due to the broad distribution of the insoluble units along the chain and their assembly into the core of the micelle. As a result, the interfacial area per one gradient copolymer chain is larger than the area of the diblock copolymer, and the aggregation number of the GCM is smaller. Worsening of the solvent quality (increase of attraction between the insoluble groups) enlarges the aggregation number of the DCM. On the contrary, the aggregation number of the GCM practically does not change. Furthermore, the corona of the GCM becomes less swollen because more and more insoluble units join to the core and aggregate in the corona upon solvent worsening. In other words, the GCM become smaller. Such behavior is known as a “reel in” effect detected for gradient copolymer micelles at temperature elevation.39

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INTRODUCTION Micelles of amphiphilic (block)copolymers represent a versatile tool for many applications including carriers for drug delivery,1−3 efficient catalysts,4,5 templates for synthesis and deposition of metallic nanoclusters,6 etc. Structurally “simplest” polymer micelles are formed by conventional diblock copolymers in selective solvents: insoluble blocks constitute a dense core of the micelle, and the soluble blocks form swollen corona which suppresses aggregation of the micelles.7−9 Micellar morphology is primarily controlled by chemical composition of the copolymer. The spherical micelles are stable in very broad range of the compositions: from strongly asymmetric copolymers with short solvophobic block (so-called hairy or star-like micelles) up to the inverse case of short soluble block (crew-cut micelles).7−11 Only in the case of very short soluble block, wormlike (or cylindrical) micelles and bilayers or vesicles are formed.12−15 In strongly selective solvents (an analogue of the strong segregation regime in the melts), the size of the micelles and their aggregation number (the number of chains per micelle) are controlled by a balance © XXXX American Chemical Society

between (i) the stretching of the blocks in the core and corona and (ii) interfacial energy at core−corona interface.7−9 As a result, the aggregation number of the star-like micelles increases with the increase of the length of the insoluble block and of the surface tension coefficient.8 However, once the micelles are formed, it is difficult to manipulate their structure (size and shape) with external stimuli (pH and temperature), if we do not deal with the pretransition composition of the copolymer when small variation of the temperature (or pH) triggers morphological transformation. In other words, the conventional (di)block copolymer micelles can be classified as “strong” ones in which effective attraction between the insoluble blocks is pretty strong sometimes even leading to a glassy state like in the case of polystyrene core in aqueous environment.12 To make the micellar systems more susceptible to the external stimuli, one needs to weaken the association energy. There are Received: October 6, 2016 Revised: November 7, 2016 Published: November 7, 2016 A

DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

NA/N·100%. Such choice of the primary structure allows simulation of long enough polymer chains which aggregate into micelles whose aggregation number is sufficiently small. On the other hand, the aggregation number of diblock copolymers in selective solvents depends not only on the fraction of monomer units but it is also proportional to the contour length of the chains.41 Therefore, to be able to compare two sorts of the micelles, an “intermediate” number of beads per chain was considered, N = 166. The number of macromolecules in the simulation box of the linear sizes Lx = Ly = Lz = 100σ were n = 80. Here, σ is the diameter of the bead. The second type of the primary structure corresponded to the so-called linear gradient copolymers (LGC): the fractions of A and B beads linearly decrease and increase along the chain, respectively.44 The average fractions of A and B units in such copolymer are equal, f = 50%, and the aggregation number of the LGC micelles is larger than in the case of the asymmetric EGC if they are formed by the chains of the same length, N = 166. Therefore, to deal with smaller micelles, we have considered shorter polymer chains, N = 56, 72, 90, and 110 to obtain few micelles of the equilibrium aggregation number in the simulation box. The number of the chains in the simulation box of the linear sizes Lx = Ly = Lz = 180σ were n = 150. Aggregation ability of both EGC and LGC were compared with equivalent diblock copolymers (DBC). Equivalency means that the numbers of soluble and insoluble beads in the diblock copolymers is the same like in EGC and LGC.45 Initial conformation of each chain in the simulation box corresponded to a two-dimensional helical one. The chains were arranged into a parallelepiped stack. The chemical bonding between monomer units is simulated using the sum of FENE and Lennard-Jones potentials:45

few ways to design such systems. For example, incorporation of charged soluble groups into the hydrophobic core-forming block weakens the association energy due to electrostatic repulsion of similarly charged groups.16 Another way deals with the sequence design of monomer units along the chain. Amphiphilic macromolecules with sophisticated primary structures (including natural polymers) are able to form “weak” micelles,17,18 whose association−dissociation free energy (its absolute value) is much smaller and can be controlled by external stimuli. Gradient copolymers are one of the examples of the macromolecules which can form the “weak” micelles. Conventional AB gradient copolymers exhibit a gradual change in monomer composition from predominantly one species (A) at one end of the chain to predominantly the other species (B) at the opposite end, unlike with block copolymers, which have an abrupt change in composition. Therefore, we can anticipate that the micelles composed by the gradient copolymers do not have a distinct core−corona structure due to the presence of the soluble and insoluble units in the core and corona, respectively. Thus, one of the main aims of the current manuscript is to reveal a detailed internal structure of the gradient copolymer micelles. Intensive studies of association behavior of the gradient copolymers in selective solvents have started quite recently.19−32 In particular, it was revealed that the gradient copolymers have better solubility in solutions than the diblock copolymers leading to high critical micelle concentration.24,33,34 Similarly to the diblock copolymers,15 morphological transitions from the spherical to the cylindrical micelles and vesicles were found for gradient copolymers of different chemical structures.35−38 At high polymer concentration, the gradient copolymer micelles can interconnect into network-like structure because of association of insoluble groups in overlapping coronae.29 Very interesting phenomenon was detected for the spherical micelles of gradient copolymers in dilute solution comprising gradual shrinkage of the micelles upon the temperature change (the so-called “reel in” effect).39−41 It has to be mentioned that such effect is absent in the diblock copolymer micelles and is a feature of the primary structure of the gradient copolymers. Similar shrinkage can also be induced by pH.42,43 Another aim of the current article is a demonstration of the reel-in effect in computer simulations and its explanation. The article is organized as follows. In the next section we propose two models of the gradient copolymers and briefly describe details of computer simulations. Then we compare the computer simulation results on aggregation behavior and internal structure of the gradient copolymer micelles with those of the diblock copolymer micelles.

⎛ ⎛⎛ σ ⎞12 ⎛ σ ⎞6 ⎞ ⎛ r ⎞2 ⎞ 1 2 ⎜ E = − KR 0 ln⎜1 − ⎜ ⎟ ⎟⎟ + 4ε⎜⎜ ⎟ − ⎜ ⎟ ⎟ + ε ⎝r⎠ ⎠ 2 ⎝ R0 ⎠ ⎠ ⎝⎝ r ⎠ ⎝

where K = 10, R0 = 1.3, ε = 1, and σ = 1. Pairwise attraction between insoluble beads is modeled by Lennard-Jones potential with interaction parameter εAA and cutoff radius rcut = 2.5. Interactions of the soluble beads with each other (B−B) and with the insoluble beads (B-A) are described by “soft spheres” potential Uij, ij = BB, AB ⎧ ⎛⎛ ⎞12 ⎛ ⎞6 ⎞ ⎪ 4εij⎜⎜ σ ⎟ − ⎜ σ ⎟ ⎟ + εij , r ≤ 2.5 ⎝r⎠ ⎠ Uij = ⎨ ⎝⎝ r ⎠ ⎪ ⎪ 0, r > 2.5 ⎩

The values of the parameters εAB and εBB were fixed and corresponded to the exclude volume repulsion, εAB = εBB = 0.01, and interactions between insoluble beads were variable. They quantify selectivity of the solvent. We used Brownian dynamics (BD) algorithms implemented in LAMMPS46 package with damping parameter set to 10 time steps. The time step is equal to Δt = 0.0001. To study aggregation properties of the systems, we use the clustering algorithm given in ref 47. This algorithm allows calculating the average size and the size distribution function of the micelles.

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COMPUTER SIMULATIONS Two types of primary structures of AB gradient copolymer were considered. The first one corresponded to an exponential decay of the sort of monomer units along the chain (the socalled exponential gradient copolymer, EGC). Here, the probability g(n) that the nth bead along the chain is of the sort A (insoluble) is done by the exponential function g(n) = exp(−n/NA), where NA is the average number of A beads per chain. In our simulations, the average fraction of insoluble monomer units was always smaller than the fraction of soluble ones, f = 23, 30, 35%. This parameter is defined as the ratio of NA to the total number of monomer units in the chain N, f =

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RESULTS AND DISCUSSION EGC versus DBC. Annealing of the initial structure of each sort of the copolymers folded into parallelepiped stacks lasted during 108 simulation steps. Snapshots of the systems are

B

DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 1. Snapshots of DBC (left) and EGC (right) after 108 simulation steps at different values of the interaction parameter εAA = 0.6, 0.7, and 0.8 (top-down). The average fraction of insoluble (red) beads is 30%. Q is the average aggregation number of the micelles.

number of the DBC and EGC micelles is plotted as a function of εAA. We can observe very fast grows of Q with εAA. For example, 4-fold increase of the aggregation number Q (from 9 to 40) of DBC micelles is achieved upon variation of εAA from 0.6 to 0.8. On the other hand, the aggregation number of the EGC micelles practically does not change (4 to 6) within this range despite of the same number of insoluble beads. There are at least two physical reasons for the difference in the aggregation numbers. To clarify them, we visualized typical conformations of the single chains in DBC and EGC micelles, Figure 3. Insoluble (red) and soluble (blue) units are well segregated in the case of DBC micelles. In contrast, the EGC chain has some loops near the core−corona interface and some of the soluble units present in the core because of the particular primary structure of the gradient copolymer. Thus, if we compare the area of the core−corona interface for these two

shown in Figure 1 for different values of the interaction parameter εAA = 0.6, 0.7, and 0.8 measured in units of the thermal energy kBT. This parameter quantifies the strength of attraction of insoluble monomer units resulted in micelle formation: the larger the parameter, the stronger the attraction (the poorer the solvent for the A beads). Micelles of DBC and EGC are shown in the left and the right columns of Figure 1, respectively. We can observe that the average aggregation number of the micelles Q depends on the primary structure of the copolymer and selectivity of the solvent (εAA). The aggregation number grows with εAA for both sorts of the copolymers. However, the EGC micelles are always smaller than the DBC ones (at fixed value of εAA) and this difference enhances with the increase of εAA, Figure 1. More quantitative information on the difference in the aggregation numbers can be extracted from Figure 2. Here the average aggregation C

DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The second physical reason why the aggregation number of the EGC micelles is smaller than the DBC ones comprises the difference in the lengths of core-forming blocks. DBC has always longer core-forming block than that of the EGC (some of the insoluble beads are located in the corona). As a result, the aggregation number, which is proportional to the length of the insoluble block,8 is smaller. Effect of the average fraction of monomer units on the aggregation number is demonstrated in Figure 4. In accordance with the theoretical predictions,8 Q of the DBC micelles is increasing function of f for all values of εAA, Figure 4 (left). On the other hand, the aggregation number of the EGC micelles weakly depends on f (23, 30, and 35%) at least in the range εAA = 0.6−0.9, where the DBC micelles reveal considerable growth with f, Figure 4. Such behavior of the EGC micelles can be explained by formation of loops at the core−corona interface. Internal structures of the DBC and EGC micelles can be quantified by the concentration of soluble and insoluble beads as a function of the radial coordinate r, Figure 5. The curves were obtained via averaging over the polar and the azimuth angles of the spherical coordinates so that the density profiles are spherically symmetric. The dashed lines in Figure 5 correspond to gradual decay of concentration of core-forming insoluble groups (red) and weakly changing concentration of corona-forming blocks (blue) in diblock copolymer micelles. In contrast, the EGC micelles reveal faster concentration decay of the insoluble groups (the core of the micelle is smaller than in the case of the diblock copolymer). Concentration of the soluble groups is nonzero in the core of the EGC micelle and has a maximum. This is the maximum results from the formation of loops at the core−corona interface: the loops enrich the interface by the soluble units. The height of the peak is sensitive to the attraction between the insoluble units: the stronger the attraction, the higher the peak, Figure 5 (left). It means that stronger attraction brings more and more A-beads into the core sometimes forming few loops per one chain. Similar peak was predicted recently for planar morphologies using mean-field calculations.44 Thus, the peak position can serve as a quantitative measure of the size of the core in the case of gradient copolymer micelles. The intake of the chains into the core, which can be accompanied by weak increase of the aggregation number, explains the so-called reel in effect:39 decrease of the size of the gradient copolymer micelles at elevating temperature. Figure 6 depicts both the decrease of the size of the EGC micelles with the attraction energy (worsening of the solvent quality) and their characteristic snapshots which demonstrate difference in the structure: higher attraction between insoluble monomer units is responsible for more compact corona. LGC versus DBC. The behavior of the linear gradient copolymer (LGC) in selective solvents is qualitatively similar to that of the EGC. In particular, the difference in the average aggregation number of the DBC and LGC micelles grows with the increase of the attraction energy, Figure 7. The LGC molecules do not aggregate up to εAA = 0.5 because they are shorter than the EGC ones (56 vs 166 beads) and the number of solvophobic units in the LGC chain is smaller. Further increase of εAA from 0.6 to 0.8 leads to drastic changes of Q: 15 to 20 (LGC) vs 20 to 78 (DBC). The typical snapshots of the LGC micelles are shown in Figure 8. To establish a difference in the micellar structures visually, we compare LGC and DBC micelles of similar sizes. For this, the number of beads in the chain and attraction energy

Figure 2. Average aggregation number of the DBC (black circles) and EGC (red squares) micelles as a function of the interaction parameter εAA. The fraction of the insoluble beads is 30%.

Figure 3. Typical conformations of diblock copolymer (left) and exponential gradient copolymer (right) in the corresponding micelles. Green circle depicts a loop formed by soluble block near the core− shell interface.

chains, we can observe that the EGC area is larger than the DBC area because of the loop(s), Figure 3. As a result, the average aggregation number will be smaller. Increasing attraction between A beads (parameter εAA) leads to further decrease of the core−corona interface in the case of diblock copolymer via elongation of the core- and corona-forming blocks and to the increase of Q, Figure 2. However, in the case of the gradient copolymer, there are two competing effects which slow down increase of Q with εAA. On one hand, increased attraction between monomer units tends to the decrease of the area of core−corona interface (like in the DBC case). On the other hand, more and more insoluble beads located in the corona are “sucked” into the core with formation of additional loops at the interface and increase of the interfacial area. These two effects nearly compensate each other and the aggregation number weakly changes at least in the range εAA = 0.6−1.1, Figure 2. D

DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 4. Average aggregation number of the DBC (left) and EGC (right) micelles as a function of the interaction parameter εAA for different content of insoluble beads.

Figure 5. (left) The volume fractions of insoluble (red) and soluble (blue) beads as functions of the distance r from the center of mass of the micelles: EGC (solid) and DBC (dashed) micelles. f = 30%, εAA = 0.8. The snapshots of the single micelles demonstrate difference in the size. (right) The volume fraction of the soluble beads in EGC micelle as a function of r at different values of the attraction energy εAA.

between insoluble units in diblock copolymer micelles were taken to be smaller, Figure 8. We can see a distinct segregation of core and corona in the DBC micelles and presence of soluble units in the core and insoluble units in the corona of the LGC micelles. Also, increased concentration of the soluble groups is seen at the core−corona interface. The physical reason for the decreased aggregation number of the LGC micelles in comparison with the DBC ones is demonstrated in Figure 9 via visualization of typical conformations of the chains. Some of them reveal surface activity of the middle segments (left image) and some of them form loops (right image); both result in increase of the interfacial area per chain (decrease of the aggregation number).45

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CONCLUSIONS The effect of monomer distribution (gradient vs block) on the structure of micelles and aggregation behavior of the macromolecules in selective solvents have been investigated using molecular dynamics simulations. We have demonstrated that the size and the aggregation number of the gradient copolymer

Figure 6. Size of the EGC micelle as a function of the attraction energy εAA at f = 30%.

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DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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per chain is relatively small, Figure 3. On the contrary, the gradient copolymers form loops in the interfacial region due to the broad distribution of the solvophobic groups along the chain and their tendency to form the core, Figure 3. As a result, the interfacial area per chain in the gradient copolymer micelles is larger, so that the number of the constituent chains in the micelle is smaller. The presence of the loops in the interfacial region is responsible for the maximum of concentration of the soluble groups whereas the concentration of insoluble groups gradually decays from the center of the micelle. In contrast to the diblock copolymer micelles, segregation of the different groups in the gradient copolymer micelles is less pronounced even in strongly selective solvent. Susceptibility of the size of the gradient copolymer micelles to the change of the solvent quality is provided by the presence of insoluble units in the corona. The solvent worsening brings some of the solvophobic groups into the core of the micelle and induces aggregation of other insoluble groups in the corona. As a result, the corona becomes less swollen and the total size of the micelle decreases. This “reel in” effect has been detected experimentally upon temperature increase.39 The specific internal structure of the gradient copolymer micelles, lower association−dissociation energy (as compared to diblocks) together with susceptibility to the external stimuli attributes them to very promising systems for drug delivery. In particular, we can anticipate that the presence of insoluble groups in the corona can be useful for more efficient uptake of solvophobic guest (macro)molecules because they do not need to overcome the entropic barrier in the course of diffusion through the corona to bind to the insoluble core like in the case of diblock copolymer micelles.

Figure 7. Average aggregation number of the DBC (black squares) and LGC (red circles) micelles as a function of the interaction parameter εAA. The average fraction of the insoluble beads per chain is 50%; the total number of the beads is N = 56.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Igor I. Potemkin: 0000-0002-6687-7732

Figure 8. Typical snapshots of the LGC (upper row) vs DBC (bottom row) micelles. Side view, distribution of insoluble beads, and crosssection through the center of mass are shown on the left, middle, and right images, respectively. LGC: N = 90, εAA = 0.7; DBC: N = 80, εAA = 0.5;.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support from the Russian Science Foundation, project # 15-13-00124. The simulations were performed on multiteraflop supercomputer Lomonosov at Moscow State University.

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DOI: 10.1021/acs.jpcb.6b10120 J. Phys. Chem. B XXXX, XXX, XXX−XXX