Spontaneous formation of complex micelles from homogeneous solution

# Spontaneous formation of complex micelles from homogeneous solution

## Abstract

We present an extensive computer simulation study of structure formation in amphiphilic block copolymer solutions after a quench from a homogeneous state. By using a mesoscopic field-based simulation method, we are able to access time scales in the range of a second. A “phase diagram” of final structures is mapped out as a function of the concentration and solvent-philicity of the copolymers. A rich spectrum of structures is observed, ranging from spherical and rodlike micelles and vesicles to toroidal and net-cage micelles. The dynamical pathways leading to these structures are analyzed in detail, and possible ways to control the structures are discussed briefly.

###### pacs:
47.57.Ng, 61.25.H-, 64.75.Yz, 83.10.Rs, 73.30.+y,83.80.Qr

Amphiphilic molecules in solution such as lipids or amphiphilic block copolymers self-assemble into a variety of structures, e.g., spherical or cylindrical micelles, lamellae, and vesicles(1). In the case of single-component amphiphiles, vesicular and toroidal structures are energetically less favorable than lamellar and cylindrical structures due to the energy penalty for bending. Nevertheless, they may be stabilized by entropic or kinetic factors and still form spontaneously. These structures provide new opportunities for designing soft materials with enhanced functionalities for various applications, such as complex micro release systems or templates for nanodevice fabrication(2); (3); (4); (5); (6); (7). A detailed understanding of the aggregation process is crucial to understand and eventually control their formation.

In the past decades, a number of experimental studies have revealed the rich diversity of micellar morphologies displayed by amphiphilic systems. Besides spherical, rodlike or wormlike micelles, micelles with various special topologies have been observed, such as unilamellar and multilamellar vesicles, onion vesicles, genus vesicles(2); (3); (4); (5); (6); (7); (8); (9); (10), toroidal micelles with one or several rings, and net or cage micelles(11); (12); (13); (14). The experiments indicated that the micellar structures and size distributions not only depend on molecular parameters, i.e., the chain length of the amphiphilic molecules, the hydrophilic-to-hydrophobic ratio, the molecular stiffness and the intermolecular interactions, but also on system parameters such as the concentration, and on kinetic factors such as the diffusion ability of the amphiphilic molecules and the details of the manufacturing process. Unfortunately, detailed dynamical information on the process of spontaneous micelle formation is scarce. Only a few groups have captured the process of spontaneous vesicle formation in solutions of amphiphile mixtures and proposed a possible pathway of vesicle formation (15); (16); (17); (18); (19).

According to this “standard” pathway (PC), the amphiphilic molecules first self-assemble into small spherical micelles, these then coalesce to rods, the rods transform themselves to bilayers, and finally, the bilayers bend around and close up to vesicles. The last two steps are driven by the rim energy of the bilayers. The mechanism has been confirmed by computer simulations of different coarse-grained models(20); (21); (22); (23); (24); (25). It clearly contributes to the formation of vesicles in amphiphilic systems. However, it cannot explain the existence of complex toroidal structures, since no force pushes rodlike micelles with two detached end caps to form rings, let alone cage structures. In a recent paper, we have reported the existence of an alternative pathway of vesicle formation in copolymer solutions(26) (PG). In this pathway, the micelles do not coalesce, but simply grow by attracting copolymers from the solution. Once a critical micelle size is exceeded, copolymers start to flipflop such that the micelle core becomes solvent-philic (“semivesicle” state). Finally, solvent diffuses inside the core, and the semivesicle swells into a vesicle. The two pathways are illustrated in Fig. 1 (see below for simulation details). The pathway PG may provide a possible route to toroidal structures.

In this letter, we report on an extensive systematic study of structure formation in a single-component amphiphilic diblock copolymer system. The final self-assembled structures depend strongly on the copolymer concentration and the interaction parameters. By varying the latter over a wide range, we can map out the final topologies in a unifying “phase diagram”. The simulations allow to investigate the formation process in detail. Both pathways PC and PG described above can be observed, depending on the copolymer concentration. We find that vesicles and rodlike micelles may form irrespective of the pathway, but toroidal structures only form via the pathway PG. Our results thus demonstrate that complex structure formation is not only controlled by the molecular packing parameters, but also, crucially, by the details of the segregation kinetics.

Complex vesicle formation has also been studied recently by Sevink and Zvelindovsky(24); (27). They considered copolymers made of two incompatible blocks that were both basically solvent-phobic. As a result, the copolymers aggregated to compact, internally structured droplets (onion vesicles) with a relatively low solvent content (a few percent(27)). In contrast, in this work, we focus on copolymers with strongly solvent-philic components, and on open and hollow structures.

We consider a system of amphiphilic diblock copolymers (copolymer volume fraction ) with solvent-phobic blocks (chain fraction ) and solvent-philic blocks (chain fraction ), immersed in a solvent (28). The monomer interactions are characterized in terms of Flory-Huggins parameters , , and . A compressibility modulus ensures that the local density (polymer plus solvent) is roughly constant. The time evolution of the system is modeled with External Potential Dynamics(29), a dynamic density functional theory which locally conserves densities and is approximately valid for Rouse-type chain dynamics, but neglects hydrodynamics and reptation (see Ref.(26) for a compilation of the dynamical equations). The relevant dynamical model parameters are the mobility coefficients and of the solvent and the copolymer.

The parameter and were variable. The other model parameters were set to , , , , and . The high values of and ensured that the -blocks segregate well from the -blocks and the solvent. Copolymers with short -blocks were used to stabilize bilayer structures in the regime where is swollen with solvent. The remaining parameters, and (the unperturbed radius of gyration of the chains), set the length scale () and the time scale () of the simulation. Mapping these to real copolymer solutions such as, e.g., those studied in Ref. (30) (cms and nm), we find that our time unit corresponds to roughly .

On the technical side, the parameters of the simulation were as follows: The time step for integration of the dynamical equation was chosen , on a spatial grid with grid size . The contours of the copolymer chains were discretized with steps. A small Gaussian noise was added to mimick the effect of thermal fluctuations(26). The longest total simulation time was , corresponding to 0.72 second.

Fig. 2 shows a diagram of final structures, obtained after quenching the system suddenly from an initially perfectly homogeneous state. The solvent-philicity ranges from positive to negative, in order to represent a wide class of amphiphilic block copolymers from nonionic to ionic. In addition, the copolymer volume fraction was varied in the region where interesting structures were observed.

In order for structure formation to take place, the copolymer concentration must exceed a certain critical value, which can be identified with the CMC (critical micelle concentration). At lower copolymer concentrations, the translational entropy of the copolymers prevents them from aggregating. The shape of the final structures depends on the solvent-philicity of the -block, (). At moderate (), the system favors bilayer structures and the copolymers aggregate to vesicles. As the solvent quality for the -block increases (), the -block swells. Its radius roughly scales with(31) ( is the chain length). This in turn decreases the critical packing parameter(1), the bilayers become unstable and give way to cylindrical and spherical structures. Consequently, the copolymers aggregate to rods and/or spheres in most of the parameter region. Close to the CMC, however, more complex structures are formed: Ring micelles, toroidal micelles, and even cage-like micelles.

To understand why these complex micelles appear, one must inspect the pathways of structure formation in more detail. Both pathways to vesicle formation, PC and PG, are observed in our system (Fig. 1). Likewise, rod formation also proceeds via the two distinct pathways micelle coalescence (PC), or (anisotropic) micelle growth (PG), depending on the copolymer volume fraction (Fig. 2): Coalescence takes place at . Rings and toroidal micelles emerge at much lower copolymer volume fraction, and their formation is clearly driven by a growth mechanism. Fig. 3a shows the pathway to ring formation, where spherical micelles first grow into small disks, a hole then nucleates at the center of the disks, and finally, the perforated disks evolve into rings. Even closer to the CMC, the same mechanism leads to the formation of toroidal micelles: The micelles first grow into semivesicles or small vesicles, then several holes appear in the vesicle shells, until finally, the perforated vesicles grow into toroidal micelles (Fig 3b,c). The number of holes depends on the size of the embryo vesicle at the time of breakup. Close to the CMC, the dynamical stability of large vesicles increases: The initial number of micellar nuclei is small and they are far apart, hence the vesicles are not perturbed by the environment and break up late. As a result, large cage micelles can be obtained in the vicinity of the CMC (Fig. 3c).

Next we examine the early stages of micelle aggregation. To this end, we define the segregation parameter , where and are local monomer densities, and is the volume. Looking at as a function of time, we find that the copolymer segregation proceeds in an almost step-like fashion after a well-defined “incubation time” (Fig. 4, inset). Fig. 4 shows the incubation times as a function of for different copolymer volume fractions . Remarkably, most curves collapse onto a single power law function with the exponent after a simple shift of by . Only for the lowest copolymer volume fractions, and , do the data fail to collapse; however, the slopes of these curves, shifted by and plotted in a double logarithmic way, are still comparable to .

A similar power law behavior has been observed previously in a simulation study of vesicle formation in two dimensions(26). It was explained in terms of the Cahn-Hilliard theory for spinodal decomposition, and was identified with the spinodal for macrophase separation between polymer and solvent. In the present case, however, is significantly lower than that spinodal(32), it seems rather related to the CMC (cf. Fig. 2). It is worth noting that data collapse and power law behavior is observed in a range of regardless of the final structure and the dynamical pathway of structure formation. This suggests that the characteristics of the initial stage of segregation are universal and related to a spinodal-type instability.

Once created, the micelles grow by attracting copolymers from the solution. In the pathway PC, the growth is supplemented by micelle coalescence. The question arises under which conditions this happens. In fact, most fusion events take place at early times (see Fig. 1 a1,a2). At later stages, they are impeded by two factors: The formation of a well-segregated, swollen -corona at the surface of the micelles, and the emerging copolymer depletion zone around the micelles. Hence the density of micelle nuclei at the early stage is a candidate quantity that might select between pathways. The crossover between pathways is observed at the copolymer volume fraction . Taking into account that a large fraction of copolymers assembles into droplets almost simultaneously (at the time ), and that these droplet nuclei roughly have the diameter , we can estimate the average distance between droplets at given copolymer content(33) via . At , is of the order . Hence the copolymers in solution are in contact with several droplets, they are attracted by all of them, and can serve as bridges that mediate fusion.

In the last stage, the structures ripen. As long as they are still small (semivesicle state), classical Oswald coarsening is observed (Fig. 1a3-a4, b3-b4): Small structures dissolve, large structures grow, driven by the competition of bulk and surface free energy. Once the structures have locally assumed their favored toroidal or bilayer structure, the Oswald process stops and the ripening is governed by much weaker thermodynamic forces, such as, e.g., those associated with the bending energy. The time scales of these processes are very slow and out of reach for our simulation method. Therefore, we have carried out a set of simulations using external potential dynamics with locally non-conserved (but globally conserved) densities. This dynamical model is less realistic, but much faster, such that we could also assess later stages of the aging process. Specifically, we studied the evolution of a system containing two vesicles with different initial sizes in two and in three dimensions. In the case of ring micelles or two dimensional vesicles, the contribution of the bending energy favors a uniform size distribution: The energy of a single ring of radius is proportional to ( being the bending rigidity), and the total energy for a fixed number of rings is minimal if all rings have the same diameter. Indeed, our two dimensional simulations showed that the sizes of the two rings converged in the course of the simulation: The small ring grew at the expense of the large ring. For three dimensional vesicles, the situation is different: The bending energy of a vesicle is , independent of its size. The total bending energy only depends on the number of vesicles, not on their size distribution. Consequently, we did not observe any sign of size uniformization, nor size disproportionation, in the three dimensional simulations. The same behavior has been observed in experiments(34).

To summarize, we have investigated the formation of toroidal micelles in copolymer solutions, and shown that such micelles may form in the vicinity of the CMC by a pathway that proceeds via the nucleation, growth, and subsequent breakup of vesicles. The competition of copolymer aggregation and self-assembly generates a kinetic trap that opens a route to manufacturing highly complex metastable structures. Several methods to control these kinetic traps are conceivable: Controlling the number of initial nuclei by planting seeds (prenucleation)(35); (36), quenching from different initial states(e.g., quenching from a vesicle state), doping with additional components to tune the bilayer properties, or working with amphiphilic mixtures.

X.H. thanks the Alexander von Humboldt foundation for a research fellowship. The simulations were carried out at the Paderborn center for parallel computing.

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