Simple and complex micelles in amphiphilic mixtures: a coarse-grained mean-field study
aa r X i v : . [ c ond - m a t . s o f t ] J un Simple and complex micelles in amphiphilic mixtures: acoarse-grained mean-field study
Martin J. Greenall
Institut Charles Sadron, 23, rue du Loess, 67034 Strasbourg, France ∗ Gerhard Gompper
Theoretical Soft Matter and Biophysics, Institute for Complex Systems,Forschungszentrum J¨ulich, 52425 J¨ulich, Germany
Abstract
Binary mixtures of amphiphiles in solution can self-assemble into a wide range of structures whenthe two species individually form aggregates of different curvatures. In this paper, we focus on small,spherically-symmetric aggregates in a solution of sphere-forming amphiphile mixed with a smalleramount of lamella-forming amphiphile. Using a coarse-grained mean-field model (self-consistentfield theory, or SCFT), we scan the parameter space of this system and find a range of morphologiesas the interaction strength, architecture and mixing ratio of the amphiphiles are varied. When thetwo species are quite similar in architecture, or when only a small amount of lamella-former is added,we find simple spherical micelles with cores formed from a mixture of the hydrophobic blocksof the two amphiphiles. For more strongly mismatched amphiphiles and higher lamella-formerconcentrations, we instead find small vesicles and more complex micelles. In these latter structures,the lamella-forming species is encapsulated by the sphere-forming one. For certain interactionstrengths and lamella-former architectures, the amount of lamella-forming copolymer encapsulatedmay be large, and the implications of this for the solubilization of hydrophobic chemicals areconsidered. The mechanisms behind the formation of the above structures are discussed, with aparticular emphasis on the sorting of amphiphiles according to their preferred curvature. ∗ Electronic address: [email protected] . INTRODUCTION Amphiphilic molecules such as block copolymers and lipids can self-assemble into manydifferent structures when dissolved in solution [1, 2]. This phenomenon has recently attracteda great deal of attention [3, 4], driven both by the potential applications of self-assembledamphiphile aggregates in the encapsulation and delivery of active chemicals such as drugsand genetic material [5, 6] and the insights gained into biological systems [7].For solutions of a single type of simple amphiphile, such as a diblock copolymer or simplelipid, it is fairly straightforward to gain a basic understanding of which aggregate will formin a given system [8, 9]. Although a variety of factors, such as the concentration [8, 10]and size [11] of the amphiphilic molecules, may play a role, the shape of the aggregates ismost easily controlled via the architecture of the amphiphile; that is, the relative sizes ofits hydrophilic and hydrophobic blocks. If the hydrophilic component is large (or appearslarge due to its interaction with the solvent), then spherical micelles are seen. However, ifthe hydrophobic block is large, lamellar structures such as vesicles form. For intermediatearchitectures, cylindrical micelles are observed, either as isolated, worm-like structures [12],or branched networks [13].The experimental phenomenology is much richer in binary mixtures of amphiphiles [14],especially those that individually self-assemble into different aggregates [15–17]. Novel struc-tures are observed, such as undulating cylinders and branched, octopus-like aggregates [15].Binary mixtures have been investigated in a wide variety of amphiphile systems. A greatdeal of work has been carried out on lipid-detergent systems [18, 19], and over the pastfew years lipid [7, 20, 21] and block copolymer [12, 15] mixtures have been widely studied.Lipid mixtures are of interest due to their presence in cells and role in biological transportprocesses [22]. In the case of block copolymers, on the other hand, the motivation for theuse of two amphiphiles is that it greatly increases the number of design parameters and givesfiner control over the self-assembly. The architectures and concentrations of both speciesmay now be varied, as may the stage in the self-assembly process at which they are blended[12]. A number of properties of the aggregates may be controlled, such as their shape [12],stability [23], and ability to solubilize hydrophobic compounds [24]. An interesting and re-cent example is the addition of lamella-forming copolymers with a short hydrophilic blockto a solution of longer sphere-formers to increase the solubilization capacity of the resulting2icelles while maintaining their compact and stable spherical shape [23, 24]. In the currentpaper, we study a basic example of such a system: a solution of sphere-forming diblockcopolymers to which an admixture of diblocks with a much shorter hydrophilic block isadded. To study the problem in as simple a form as possible, we consider two copolymerspecies that are formed of the same species A (hydrophilic) and B (hydrophobic), and havethe same length hydrophobic blocks. We focus on the case where the sphere-formers re-main in the majority, and, using coarse-grained mean-field theory, investigate how the smallspherical aggregates formed are modified by the presence of the shorter copolymers. Weperform a broad scan of the system’s parameter space, and study how the core compositionand radius of the micelles are affected by the interactions, concentrations and architecturesof the two polymers.The paper is organized as follows. In the following section, we introduce the coarse-grained mean-field theory (self-consistent field theory) that will be used. We then presentand discuss our theoretical results, and give our conclusions in the final section.
II. SELF-CONSISTENT FIELD THEORY
Self-consistent field theory (SCFT) [25] is a coarse-grained mean-field model that has beenused successfully to model equilibrium [26–28] and metastable [29, 30] structures in polymersblends and melts. SCFT has several features that make it particularly suitable for the studyof the current problem of small binary aggregates. First, its general advantages are that itis less computationally intensive than simulation techniques such as Monte Carlo, yet, forsufficiently long amphiphiles [31], provides comparably accurate predictions of micelle sizeand shape [31–33]. Secondly, as a relatively simple, coarse-grained theory, it will allow us tomodel the broad phenomenology of the system clearly and show how general the phenomenaobserved are likely to be. Furthermore, SCFT has a specific feature that is especially useful inthe current problem: it makes no initial assumption about the segregation of two copolymersof different architecture within the micelle, provided the two amphiphile species are formedfrom the same types of monomer. This will enable us to demonstrate that effects such asthe encapsulation of one polymer species by another within the micelle arise spontaneouslyand do not require further assumptions to be made.To make our discussion more concrete, we now outline the mathematical structure and3ain assumptions of SCFT as applied to the current system. The theory considers anensemble of many polymers. These are modeled as random walks in space, which means thatfine details of their molecular structure are not taken into account [34]. The inter-molecularinteractions are modeled by assuming that the system is incompressible and introducing acontact potential between the molecules [28], the strength of which is fixed by the Flory χ parameter [35].The first step in finding an approximation method (SCFT) for this problem is to thinkof each polymer molecule as being acted on by a field produced by all other molecules in thesystem [28]. Viewing the problem in this way involves no approximations in itself, but hasseveral advantages when computing numerical solutions. First, it transforms the N -bodyproblem of modeling a system of N polymers into N a N , where a is the monomer length and N is the degree of polymerization [28]. Thispolymer contains N B hydrophobic B-monomers and N A = N − N B hydrophilic A-monomers.To consider the effect of different lamella-formers on the self-assembly, we vary the numberof A monomers while keeping N B fixed.We note that our use of the term lamella-former to refer to amphiphiles with shorthydrophilic blocks is not precise, as these molecules might precipitate rather than self-assemble in solution [40] if not mixed with an amphiphile with a larger hydrophilic block4uch as a sphere-former. Furthermore, in aggregates formed from a mixture of sphere-formersand amphiphiles with a short hydrophilic block, the presence of these latter molecules mightlead to the formation of regions of negative curvature rather than the zero-curvature regionsthat would be favored by a lamella-former.All sphere-formers considered contain N B2 ≡ N B hydrophobic monomers, but their overalllength is necessarily greater and is given by αN , where α >
1. As above, the number ofA monomers N A2 is varied with N B fixed in order to investigate the effect of sphere-formerarchitecture on the aggregate properties. For simplicity, the number N S of A monomers ina homopolymer solvent molecule is also fixed at N B . Since we focus on spherical aggregates,we assume spherical symmetry of the calculation box with reflecting boundary conditionsat the origin and outer limit of the system.In this paper, we keep the amounts of copolymer and homopolymer fixed; that is, wework in the canonical ensemble. Applying the procedure described above, we find that theSCFT approximation to the free energy of our system has the form F Nk B T ρ V = F h Nk B T ρ V − ( χN/V ) Z d r (cid:2) ( φ A ( r ) + φ A2 ( r ) + φ S ( r ) − φ A − φ A2 − φ S )( φ B ( r ) + φ B2 ( r ) − φ B − φ B2 ) (cid:3) − ( φ A + φ B ) ln( Q AB /V ) − [( φ A2 + φ B2 ) /α ] ln( Q AB2 /V ) − φ S ln( Q S /V ) (1)where the φ i are the mean volume fractions of the various components. The φ i ( r ) arethe local volume fractions, with i = A or A i = B or B i = S for the A homopolymersolvent. The strength of the repulsive interaction between the A monomers (hydrophiliccomponent and solvent) and B monomers (hydrophobic component) is determined by theFlory parameter χ . V is the total volume, 1 /ρ is the volume of a monomer, and F h is theSCFT free energy of a homogeneous system of the same composition. This last quantitycontains terms arising from the entropy of mixing of the different species [35] and a termproportional to χ describing the interactions of the species in the absence of self-assembly[28]. The architectures of the individual molecules enter through the single-chain partitionfunctions Q i . As an example, that for the homopolymer is given by [28] Q S [ w A ] = Z d r q S ( r , s ) q † S ( r , s ) (2)5here the q and q † terms are single chain propagators [28]. The partition functions of thecopolymer chains are calculated in a similar way. We now recall that the polymer moleculesare modeled as random walks in an external field that describes their interactions with theother molecules in the system. This is reflected in the fact that the propagators satisfydiffusion equations with a field term. Again considering the case of the homopolymer, wehave ∂∂s q S ( r , s ) = (cid:20) a N ∇ − w A ( r ) (cid:21) q S ( r , s ) (3)where s is a curve parameter specifying the position along the polymer and the initialcondition is q S ( r ,
0) = 1. The copolymer propagators are computed similarly, although inthis case the corresponding diffusion equation is solved with the field w i ( r ) and the prefactorof the ∇ q term appropriate to each of the two sections of the copolymer [41]. In the caseof the longer sphere-forming copolymer, the fields must be multiplied by the ratio α to takeinto account the higher degree of polymerization [28].The derivation of the mean-field free energy F also generates a set of simultaneous equa-tions linking the values of the fields and densities. The first of these simply states that allvolume fractions must add to 1 at all points due to the incompressibility of the system;however, we also find the following linear relation w A ( r ) − w B ( r ) = 2 χN [ φ A + φ A2 + φ S − φ A ( r ) − φ A2 ( r ) − φ S ( r )] (4)Furthermore, the homopolymer density is related to the propagators (see Eqn. 3) accordingto [28] φ S ( r ) = V φ S Q S [ w A ] Z d s q S ( r , s ) q † S ( r , s ) (5)The copolymer densities are computed in a similar way, with the integration limits set togive the required amounts of each species.In order to calculate the SCFT density profiles for a given set of polymer concentrations,Eqn. 4 must be solved with the densities calculated as in Eqn. 5 and taking account of theincompressibility of the system. To begin, we make a initial guess for the fields w i ( r ) with theapproximate form of a micelle and solve the diffusion equations to calculate the propagatorsand then the densities corresponding to these fields (see Eqns 3 and 5). New values for thefields are now calculated using the new φ i ( r ), and the w i are updated accordingly [42]. Theprocedure is repeated until convergence is achieved.6he diffusion equations are solved using a finite difference method [43]. To resolve themore complex features of the micelle density profiles, it is necessary to use a relatively finediscretization: a spatial step size of 0 . aN / and a step size for the curve parameter s of0 . III. RESULTS AND DISCUSSION
We divide the results section into four subsections. These focus on the effect on themicelle morphology of, respectively, the lamella-former concentration, the strength of theinteraction between the two species, the lamella-former architecture and the sphere-formerarchitecture.
A. Effect of lamella-former concentration
To begin, we investigate the effect of the gradual addition of lamella-former to a systemof sphere-formers, finding the micelle of lowest free energy using the method of variable7ubsystem size described above. To see the effect of blending two species as clearly aspossible, we consider a pair of strongly mismatched copolymers: a lamella-former with N A = N B / N A2 = 7 N B . The Flory parameter is set to χN B =22 .
5, a high value which causes relatively sharp interfaces to form between the species A andB. We fix the overall volume fraction of copolymer to 10%, to give a reasonable volume ofsolvent around the micelle without making the simulation box so large that the calculationsbecome slow.Figure 1 shows a series of radial cuts through the density profile of the optimum spher-ical aggregate as the volume fraction of lamella-former is increased from 5% to 35% of allcopolymers in steps of 10%. For the lowest of these lamella-former concentrations (Figure1a), the sphere-formers and lamella-formers are homogeneously mixed in the core, and asimple mixed micelle is formed (see the sketch in Figure 2). This structure is little differ-ent from that which would be formed in a system of pure sphere-forming amphiphiles: theconcentration of lamella-formers is not yet sufficiently large to have a strong effect on themicelle morphology. Indeed, for such low lamella-former concentrations, mixed micelles maynot be present [46], with pure aggregates of the two species forming instead.However, as more lamella-forming molecules are added (Figure 1b), their influence onthe core composition of the micelle becomes clear. A polymer with a large hydrophobiccomponent that naturally forms flat bilayers or even structures of negative curvature is in anenergetically highly unfavorable state in a small, positively curved micelle. In consequence,these molecules segregate to the center of the aggregate, where they form a tightly-wrappedbilayer. This structure is sketched in Figure 3. The polymers in the inner leaflet of thisbilayer are in a more favorable negative curvature state, with their hydrophilic componentspointing in towards the center of the micelle. Those in the outer leaflet are also in a morefavorable state than at lower lamella-former concentrations: they are no longer forced intothe core of a compact micelle, but sit in a shell on the outside of the new core region. Here,they are mixed with the sphere-formers, which can no longer form their preferred simplemicelle structure, but strongly prefer the positively-curved surface of the new sphericalaggregate to its core. The new micelle therefore has an inner core of hydrophilic A-blocks,an outer core of hydrophobic B-blocks, and a hydrophilic A corona.As the lamella-former concentration is increased still further, to 25% by volume of allcopolymers (Figure 1c), solvent penetrates into the core of the micelle, as the inner bilayer8f lamella-formers becomes more dominant in fixing the micelle morphology and expandstowards the planar state. This process continues in Figure 1d, where 35% of all copolymersare lamella-forming. Here, a number of the sphere-formers have mixed with the inner leafletof the lamella-former bilayer, meaning that the overall structure has the form of a (very)small bilayer vesicle (see the sketch in Figure 4). Small vesicles with a preferred radius haveindeed been seen in experiments on mixtures of sphere- and lamella-forming amphiphiles[7, 47]. Their existence has also been predicted in recent lattice SCFT calculations by Li et al.[48]. Our current work considers a different region of parameter space to these lattice-basedcalculations, which focus on weakly mismatched amphiphiles and (usually) higher lamella-former concentrations. In consequence, the mechanism by which the preferred vesicle radiusis selected appears to be rather different in the two studies. In our work, strong segregation ofthe two species occurs and the compositions of the inner and outler leaflets of the bilayer arequite different. The small vesicle structure forms as it accommodates both the preferenceof the shorter copolymers for a bilayer structure and that of the longer, sphere-formingamphiphiles for positively-curved surfaces. This is in contrast to the results of Li et al.[48], where the vesicles are larger and the two bilayer leaflets have similar compositions.The individual vesicles therefore have no preferred curvature, and coexistence with a highconcentration of mixed micelles is found to be necessary for vesicle size selection to occur.Highly-curved mixed bilayers may also be seen as the end sections of larger tubular vesicles[49], and have also been investigated using molecular dynamics simulations [50]. Our currentwork gives some broad guidance as to how the system parameters might be varied in orderto encourage or discourage the formation of these structures.We now turn our attention to a system in which the two polymer species have the samearchitectures as before ( N A = N B / N A2 = 7 N B for the sphere-former), but a significantly weaker interaction strength of χN B = 15. The difference betweenthe two systems can be seen even at only 5% lamella-former (Figure 5a). Here, the radialsegregation of the two polymers according to their preferred curvatures is already clearlyunderway. Before, it was prevented at lower lamella-former concentrations by the energeticcost of mixing the hydrophilic A blocks with the hydrophobic B core. As more lamella-former is added to reach 15% (Figure 5b), the behavior of the two systems diverges stillfurther. In the system with stronger repulsive interactions discussed earlier, the A and Bblocks demix in the core, leading to the ABA structure seen in Figure 1b and sketched in9igure 3. In the current system, although some demixing does indeed occur (Figure 5b), theeffect is much weaker, and A- and B-rich regions can no longer be clearly separated. Thisstructure is sketched in Figure 6. For even larger concentrations of lamella-former of 25%and 35% (Figure 5c and d), the small vesicle structure seen before is absent. Instead, sincethe A and B blocks may mix much more freely than before, the core of the micelle is formedof a nearly homogeneous melt of lamella-former.To make our study of the concentration dependence of the binary system more quanti-tative, we calculate the core radius and composition as a function of the ratio φ ′ /φ of thevolume fraction of lamella-formers φ ′ to the total volume fraction of copolymers φ . We definethe core boundary as the radius at which the volume fraction of hydrophobic blocks is equalto 0 .
5, and plot this quantity in Figure 7a for both systems considered above. The core radiiof both species grow steadily and almost identically as the lamella-former concentration isincreased. In the case of the system with χN B = 22 .
5, the growth is associated with theexpansion of the micelle to form a small vesicle, while in the system with weaker interactions( χN B = 15), it arises from the fact that the core is gradually filling with lamella-former. Theonly appreciable difference in radius is seen in the final point, where 40% of all copolymersare lamella-forming. Here, the radius of the χN B = 22 . χN B = 15 system, the amount of A-block inthe core grows steadily as the amount of lamella-former is increased, as in this case thelamella-forming copolymer is simply encapsulated in the center of the micelle. For lowlamella-former concentrations, the χN B = 22 . χN B = 15 system, where some mixing of A and B blocks occurs in the core. In contrast, forlarger amounts of lamella-forming copolymer, the fraction of A-block in the more strongly-interacting system grows more and more rapidly as the preferred aggregate changes from a10losed micelle to an open vesicle.The clearest difference between the two systems is seen in the behavior of the amount ofsolvent in the core as the lamella-former concentration is increased (see the lower two curvesin Figure 7b). In the χN B = 15 system, where the core is largely composed of copolymers,the fraction of solvent remains fairly constant at around 0 . .
03. The fact that this valueis a little higher than might be expected from the density profiles in Figure 5, and also variesslightly, can be attributed to the fact that our simple definition of the core radius means thata thin shell of solvent is always included as part of the core. In the system with strongerrepulsion between the A and B components, the core solvent fraction starts at a similarsmall value, remaining close to this as the lamella-former volume fraction is increased andthe morphology of the system changes from a simple micelle to the ABA structure shownin Figure 3. However, as the fraction of lamella-formers φ ′ /φ is increased towards 40%, thefraction of solvent in the core grows very quickly as the aggregate expands towards a vesicle.We have checked a selection of these calculations for a much more dilute system with anoverall copolymer volume fraction of φ ≈ χN B = 22 . φ ′ /φ is between 15% and 20%,rather than between 10% and 15% in the φ = 10% system. This preference for smallspherical micelles in more dilute systems is in line with the known concentration dependenceof block copolymer solutions [8]. Similar small shifts in the morphology transitions as theoverall copolymer concentration is varied, or no appreciable shifts at all, are observed in allthe systems considered in this paper, where the polymers considered aggregate reasonablystrongly and the free-energy minima associated with the various micelle shapes will berelatively sharp. This is not the case for shorter or more weakly-interacting polymers, wherethe concentration dependence may be quite strong.11 . Effect of interaction strength In the results discussed in the preceding subsection, we found a clear contrast in phe-nomenology between two systems with different levels of repulsion between their hydrophilicand hydrophobic components. To investigate this effect in more detail, we take a systemwith the same copolymer architectures as considered above ( N A = N B / N A2 = 7 N B for the sphere-former), fix the lamella-former fraction φ ′ /φ to 25%,and vary χN B . Figure 8 shows a series of cuts through the density profiles of the smallspherical aggregates formed as χN B is increased from 15 to 30 in steps of 5. For the smallestof these values, shown in Figure 8a, we find a weakly-structured aggregate of the kind shownin Figure 5. Very similar results are also found for the even smaller value of χN B = 12 . χN B is increased to 20 (Figure 8b), the segregation between hydrophilic and hydropho-bic blocks has become much stronger, and the ABA aggregate of Figure 3 is seen, with somepenetration of solvent into the core region. On further increase of χN B , the boundariesbetween the various layers become sharper and sharper as the repulsive interaction betweenthe A and B blocks increases in strength. The radius also increases, and the aggregateopens out to a small vesicle. Finally, in Figure 8d, where χN B = 30, the layers are veryclearly defined, and the calculated density profile resembles very closely the vesicle sketchedin Figure 4. C. Effect of lamella-forming architecture
In the two previous subsections, we considered the effect of blending two strongly mis-matched copolymers, to demonstrate the effects of segregation on the micelle morphologiesas clearly as possible. We now investigate how the micelle shapes change as the mismatchbetween the two polymers is decreased. Specifically, the hydrophilic block size of the lamella-former is gradually increased from the small value N A = N B / N A = N B . The same sphere-formerarchitecture as before is used, with N A2 = 7 N B . Since we wish to focus specifically on theeffects of the lamella-former, we use a slightly higher fraction of these molecules than in thepreceding section, and set φ ′ /φ = 33 . χN B = 15, and first consider the strong lamella-former with N A = N B / N A = 3 N B / χ parameter between the hydrophilic and hydrophobic blocks, but also ashort hydrophilic block of the lamella former. If the length of this block is increased, theeffective strength χN of the interaction between the A and B blocks of the lamella-formerbecomes larger [35] and the two blocks can segregate within the core.As the length of the hydrophilic block of the lamella-former is increased still further, to N A = 2 N B /
3, the mismatch between the two species decreases and the aggregate showsthe first signs of approaching the small micelle favored by the sphere-formers. Specifically,the core radius decreases slightly, and the solvent begins to be expelled from the center ofthe micelle (Figure 9c). This process is complete in Figure 9d, where the lamella former issymmetric and N A = N B . Here, a simple mixed micelle is formed, with no segregation ofthe two species.We now consider a system with the same sequence of polymer architectures as above, butwith a larger repulsive interaction strength χN B = 22 . N A = N B /
4, a small vesicle forms (Figure 10a), in contrastto the weakly-structured aggregate seen for this architecture for the smaller value of χN B (Figure 9 a). As the hydrophilic block length of the lamella-former is increased, and thedegree of mismatch between the two copolymer species lessens, the vesicle contracts (Figure10b and c) until a simple mixed micelle is formed (Figure 10d).As in our discussion of the dependence of the micelle morphology on lamella-formerconcentration, we now plot the aggregate core radii and composition as a function of lamella-former architecture for the two systems studied in this section. In Figure 11a, we showthe decrease of the micelle radius as the length of the lamella-former hydrophilic block isincreased. The lower line shows the behavior of the radius of the system with the smallerFlory parameter χN B = 15. The sharpest change in the radius occurs between the firsttwo points, when the aggregate changes from the weakly segregated structure plotted in13igure 9a to an ABA micelle with the form shown in Figure 9b. This latter structure thengradually contracts as the lamella-former is lengthened until we arrive at the simple mixedmicelle plotted in Figure 9d. This steady contraction with increasing lamella-former lengthis also seen in the χN B = 22 . χN B = 15) but also for all other lamella-formers apart from thelongest with N A = N B .The fraction of the core that is composed of A-blocks displays especially interestingbehavior as the lamella-former hydrophilic block length is varied. In the system with thesmaller Flory parameter χN B = 15, the A-block fraction has a rather high value of around0 .
125 for the short lamella-formers with N A = N B /
4. This is because the system formsa weakly-structured aggregate (Figure 9a) here, with a core composed of lamella-formingcopolymers (Figure 6). As the hydrophilic blocks of the lamella-formers are lengthened,solvent enters into the core and the A-block fraction falls slightly. Further increase of thelamella-former hydrophilic block length causes the fraction of A-blocks in the core to risesteadily. The reason for this is that, as the aggregate shrinks and solvent is slowly expelledfrom the center of the aggregate, the amount of A-blocks changes relatively little. Theseblocks therefore come to constitute a larger fraction of the core. As the lamella-former A-blocks are lengthened still further, we observe a sharp drop in the fraction of hydrophilicmaterial in the core, as the system contracts to form a simple mixed micelle.Some aspects of this behavior are also seen in the χN B = 22 . χN B = 15 system. For the largest A-block lengthsstudied, the fraction of hydrophilic material in the core is much smaller, as the system hasformed a mixed micelle with a predominantly hydrophobic core.To reinforce the above arguments, we also show in Figure 11b the volume fraction of thecore that is composed of solvent as a function of lamella-former hydrophilic block length.For the solution with χN B = 15, the core solvent fraction initially rises as the weaklysegregated structure is replaced by an ABA structure with some solvent in the core. It thenfalls gradually as the aggregate contracts to form a mixed micelle. A similar steady fall14s observed in the more strongly segregated χN B = 22 . N A closes to form a micelle. D. Effect of sphere-forming architecture
To conclude the scan of our system’s parameter space, we now investigate the effect ofthe architecture of the sphere-former on the morphology of the aggregates. In the aboveresults, we focused on strongly mismatched copolymers and so used a highly asymmetricsphere-former with N A2 = 7 N B . We now vary the length of the hydrophilic block of thesphere-forming copolymer over a wide range, starting from a short molecule with N A2 = 3 N B and increasing the number of A monomers until N A2 = 9 N B . The architecture of thelamella-former is fixed, with N A = N B /
4, and, as in all the above cases, the total copolymervolume fraction is kept constant at 10%. Three quarters of these copolymers are sphere-forming, so that φ ′ /φ = 25%. As in our studies of the effect of copolymer concentration andlamella-former architecture, we consider two values of the Flory parameter: χN B = 22 . χN B = 15.In Figure 12a to d, we show cuts through the density profiles of the optimum aggregatesformed when N A2 = 3 N B , 5 N B , 7 N B and 9 N B for χN B = 15. Despite the wide variationin the number of hydrophilic monomers, roughly similar small vesicle structures are formedin all cases, with particularly little change in morphology being observed between N A2 =5 N B and 9 N B . Provided the two copolymer species are sufficiently strongly mismatchedto segregate within the aggregate, there is indeed no reason to suspect that increasing thesphere-former A-block length further should cause major qualitative changes to the form ofthe aggregate, as the sphere-formers have already reached the outer surface and can move nofurther. In fact, the differences between the four panels of Figure 12 can be attributed mainlyto the fact that increasing the length of the sphere-former hydrophilic block at constant φ ′ /φ gradually reduces the amount of sphere-former hydrophobic block, with the result that thehydrophobic core becomes more and more dominated by the lamella-former. In consequence,the core radius of the aggregate increases somewhat, as the lamella-formers push outwardstowards their preferred flat state.The dependence of the aggregate shape on the hydrophilic block length of the sphere-formers is similarly weak for the smaller Flory parameter χN B = 15. Here, aggregates with15he same basic form of an outer layer of sphere-forming copolymers encapsulating a weakly-structured core of lamella-formers are seen for N A = 3 N B , 5 N B , 7 N B and 9 N B (Figure13a-d). As in the χN B = 22 . N B to 9 N B , save for a fallin the density of the outer sphere-former layer of the core and a slow growth in the coreradius. The explanation for these changes is also the same as in the system with a higherFlory parameter. Specifically, the gradual fall in the amount of sphere-former hydrophobicblock means that the core becomes predominantly composed of lamella-forming copolymers,which also causes it to swell.The relative insensitivity to sphere-former architecture observed in both the systemsdiscussed in this section can clearly be seen from plots of the core radius and composition asa function of the sphere-former A-block length (Figure 14). The growth of the core radiusshown in Figure 14a is clearly weaker than that seen in the corresponding plots of Figure 7and Figure 11. Furthermore, the fraction of the core composed of A-blocks (upper lines) orsolvent (lower lines) remains rather close to constant, although a weak growth in the amountof solvent in the open structure of Figure 12 can be seen. This is in line with the relativelyunchanging morphologies plotted in Figure 12 and Figure 13. IV. CONCLUSIONS
Using a coarse-grained mean-field approach (self-consistent field theory) we have mod-eled several aspects of the formation of small, spherically-symmetric aggregates in a solutionof sphere-forming amphiphile mixed with a smaller amount of lamella-forming amphiphile.By varying the interaction strength, architecture and mixing ratio of the amphiphiles, wehave found a range of morphologies. When the two species were similar in architecture, orwhen only a small admixture of lamella-forming amphiphile was added, we found simplespherical micelles with purely hydrophobic cores formed from a mixture of the B-blocks ofthe two amphiphiles. For more strongly mismatched amphiphiles and higher concentrationsof lamella-former, we found complex micelles and small vesicles. Specifically, as the concen-tration of lamella-former was gradually increased in a strongly mismatched system with arelatively high χ parameter, the simple micelle formed at low lamella-former concentrationsgradually expanded, first forming a more complex micelle with both A- and B-blocks in16he core and then a small vesicle. For similar systems with lower Flory parameters, theaddition of lamella-former resulted in the formation of a intriguing micellar structure inwhich a large and relatively unstructured core of lamella-former is surrounded by a layerof sphere-forming copolymers. Were this structure able to be stabilized in experiments, itcould prove to be useful for the solubilization and delivery of hydrophobic compounds, sinceit contains a large amount of hydrophobic blocks while retaining a relatively small size. Theformation of these aggregates was shown to require not only a relatively weak interactionbetween the two copolymers, but also for one of the species to have a very short hydrophilicblock. The other complex micelles and small vesicles were present over a much wider rangeof lamella-formers. The architecture of the sphere-formers was found to have a rather weakeffect on the aggregate morphology.The work presented here provides several examples of the wide range of aggregates thatmay be formed when two amphiphile species that individually self-assemble into aggregatesof different curvatures are mixed, and gives broad guidance as to how the polymer parametersmight be varied in order to form a given structure. Furthermore, several of the structuresshown here show the segregation of amphiphiles according to curvature [7, 20]. Specifically,in many cases, the sphere-forming amphiphiles move to the positively-curved surface of theaggregate. Effectively one-dimensional aggregates such as those considered here are amongthe simplest possible systems in which this phenomenon could take place.Several possible extensions of the current work suggest themselves. First, given the poten-tial for the solubilization of hydrophobic chemicals of the large micelles with lamella-formercores, more realistic interaction parameters and modeling of the polymers (if necessary bymore microscopic simulation methods) could be carried out in order to search for an experi-mental parameter range in which these structures could be formed. Such a study could alsoinvestigate further the formation of small monodisperse vesicles [7] and bilayers of preferredcurvature [49] in binary systems. Second, the study could be extended to mismatched hy-drophobic blocks, to allow comparison with recent experiments [51]. 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5. The volume fractions of lamella-formers as a percentage of all copolymers are (a) 5%, (b) 15%, (c) 25% and (d) 35%. Sphere-formers are shown with thick lines, lamella-formers with thin lines. The hydrophobic componentsare plotted with full lines, the hydrophilic components with dashed lines, and the solvent with adotted line.FIG. 2: Sketch of a simple mixed micelle. Sphere-formers are shown with thick lines, lamella-formers with thin lines. The hydrophobic components are plotted with straight lines, the hy-drophilic components with zig-zag lines, and the boundary of the hydrophobic core is marked witha dashed circle. This structure is seen for weakly mismatched copolymers at all χ parametersconsidered. IG. 3: Sketch of a complex ABA mixed micelle with a hydrophilic A inner core, a hydrophobic Bouter core and a hydrophilic A corona. Sphere-formers are shown with thick lines, lamella-formerswith thin lines. The hydrophobic components are plotted with straight lines, the hydrophiliccomponents with zig-zag lines, and the boundaries of the two cores are marked with dashed circles.This structure is seen for larger χ parameters. IG. 4: Sketch of a small vesicle composed of a solvent center, a layer of hydrophilic A-blocks, alayer of hydrophobic B-blocks and a hydrophilic A corona. Sphere-formers are shown with thicklines, lamella-formers with thin lines. The hydrophobic components are plotted with straight lines,the hydrophilic components with zig-zag lines, and the boundaries of the various regions are markedwith dashed circles. This structure is seen for larger χ parameters. r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a) (b)(c) (d) FIG. 5: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of lamella-former with N A = N B / N A2 = 7 N B . TheFlory parameter is set to the relatively low value of χN B = 15. The volume fractions of lamella-formers as a percentage of all copolymers are (a) 5%, (b) 15%, (c) 25% and (d) 35%. Sphere-formers are shown with thick lines, lamella-formers with thin lines. The hydrophobic componentsare plotted with full lines, the hydrophilic components with dashed lines, and the solvent with adotted line. IG. 6: Sketch of a complex mixed micelle with a weakly structured core formed of lamella-formers and the hydrophobic blocks of sphere-formers. Sphere-formers are shown with thick lines,lamella-formers with thin lines. The hydrophobic components are plotted with straight lines, thehydrophilic components with zig-zag lines, and the boundaries of the hydrophobic core is markedwith a dashed circle. This aggregate is seen for lower χ parameters.
10 20 30 40 φ / / φ r c o r e / a N B / φ / / φ φ c o r e (a)(b) FIG. 7: Core radius and composition of the spherically-symmetric aggregates formed in a solutionof lamella-former with N A = N B / N A2 = 7 N B . (a) Core radius asa function of the ratio φ ′ /φ of the volume fraction of lamella-formers φ ′ to the total volume fractionof copolymers φ . Points corresponding to simple micelles (Figure 2) are marked by closed circles,ABA aggregates (Figure 3) or small vesicles (Figure 4) are marked with open circles, and weakly-structured aggregates (Figure 6) by asterisks. The data for the system with a Flory parameterof χN B = 15 are connected with dotted lines; those for the system with χN B = 22 . r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a)(c) (b)(d) FIG. 8: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of lamella-former with N A = N B / N A2 = 7 N B . Thelamella-former fraction φ ′ /φ is set to 25%. The Flory parameter χN B is varied and takes thefollowing values: (a) 15, (b) 20, (c) 25 and (d) 30. Sphere-formers are shown with thick lines,lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a) (b)(d)(c) FIG. 9: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of sphere-former with N A2 = 7 N B and lamella-formers of varying architecture. The Floryparameter is set to the relatively low value of χN B = 15. The lamella-former fraction φ ′ /φ is setto 33 . N A = N B / N A = 3 N B /
7, (c) N A = 2 N B / N A = N B . Sphere-formers are shown with thicklines, lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a) (b)(d)(c) FIG. 10: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of sphere-former with N A2 = 7 N B and lamella-formers of varying architecture. The Floryparameter is set to the relatively high value of χN B = 22 .
5. The lamella-former fraction φ ′ /φ isset to 33 . N A = N B / N A = 3 N B /
7, (c) N A = 2 N B / N A = N B . Sphere-formers are shown with thicklines, lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. .4 0.6 0.8 1 N A / N B r c o r e / a N B / N A / N B φ c o r e (a)(b) FIG. 11: Core radius and composition of the spherically-symmetric aggregates formed in a solu-tion of sphere-former with N A2 = 7 N B mixed with lamella-formers of various architectures. Thelamella-former fraction φ ′ /φ is set to 33 . χN B = 15are connected with dotted lines; those for the system with χN B = 22 .5 by dashed lines. (b) Frac-tion of the core that is composed of A-blocks for each of the two systems (upper two curves), andfraction of the core that is composed of homopolymer solvent (lower two curves).
7, (c) N A = 2 N B / N A = N B . Sphere-formers are shown with thicklines, lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. .4 0.6 0.8 1 N A / N B r c o r e / a N B / N A / N B φ c o r e (a)(b) FIG. 11: Core radius and composition of the spherically-symmetric aggregates formed in a solu-tion of sphere-former with N A2 = 7 N B mixed with lamella-formers of various architectures. Thelamella-former fraction φ ′ /φ is set to 33 . χN B = 15are connected with dotted lines; those for the system with χN B = 22 .5 by dashed lines. (b) Frac-tion of the core that is composed of A-blocks for each of the two systems (upper two curves), andfraction of the core that is composed of homopolymer solvent (lower two curves). r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a) (b)(d)(c) FIG. 12: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of lamella-former with N A = N B / χN A = 22 .
5. The lamella-former fraction φ ′ /φ is set to 25%. The hydrophilic lengths of the sphere-forming molecules are (a), N A2 = 3 N B (b) N A2 = 5 N B , (c) N A2 = 7 N B and (d) N A2 = 9 N B . Sphere-formers are shown with thick lines,lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) r / aN B1/2 φ ( r ) (a) (b)(d)(c) FIG. 13: Cuts through the density profiles of the spherically-symmetric aggregates formed in asolution of lamella-former with N A = N B / χN B = 15. The lamella-former fraction φ ′ /φ isset to 25%. The hydrophilic lengths of the sphere-forming molecules are (a), N A2 = 3 N B (b) N A2 = 5 N B , (c) N A2 = 7 N B and (d) N A2 = 9 N B . Sphere-formers are shown with thick lines,lamella-formers with thin lines. The hydrophobic components are plotted with full lines, thehydrophilic components with dashed lines, and the solvent with a dotted line. N A2 / N B r c o r e / a N B / N A2 / N B φ c o r e (a)(b) FIG. 14: Core radius and composition of the spherically-symmetric aggregates formed in a solu-tion of lamella-former with N A = N B / φ ′ /φ is set to 25%. (a) Core radius as a function of the hydrophilic blocklength. Points corresponding to simple micelles (Figure 2) are marked by closed circles, ABA ag-gregates (Figure 3) or small vesicles (Figure 4) are marked with open circles, and weakly-structuredaggregates (Figure 6) by asterisks. The data for the system with a Flory parameter of χN B = 15are connected with dotted lines; those for the system with χN B = 22 .5 by dashed lines. (b) Frac-tion of the core that is composed of A-blocks for each of the two systems (upper two curves), andfraction of the core that is composed of homopolymer solvent (lower two curves).