Temperature-Induced Re-Entrant Morphological Transitions in Block

Article Cite This: Langmuir XXXX, XXX, XXX−XXX

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Temperature-Induced Re-Entrant Morphological Transitions in Block-Copolymer Micelles Artem M. Rumyantsev,*,†,‡ Frans A. M. Leermakers,§ Ekaterina B. Zhulina,∥ Igor I. Potemkin,⊥,‡,# and Oleg V. Borisov*,†,∥,¶

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†

Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS UPPA, Pau 64053, France ‡ DWI Leibniz Institute for Interactive Materials, Aachen 52056, Germany § Physical Chemistry and Soft Matter, Wageningen University and Research, Wageningen 6708 WE, The Netherlands ∥ Institute of Macromolecular Compounds, Russian Academy of Sciences, St. Petersburg 199004, Russia ⊥ Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia # National Research South Ural State University, Chelyabinsk 454080, Russian Federation ¶ Peter the Great St. Petersburg State Polytechnic University, 195251 St. Petersburg, Russia S Supporting Information *

ABSTRACT: Using a combination of a mean-field theoretical method and the numerical Scheutjens−Fleer self-consistent field approach, we predict that it is possible to have re-entrant morphological transitions in nanostructures of diblock copolymers upon variation in temperature-mediated solubility of the associating blocks. This peculiar effect is explained by the different rates in variation of the density of the collapsed core domains and the corresponding interfacial energy as a function of the temperature. The theoretical findings are supported by existing experimental observations of reversed sequences of the morphological transitions occurring upon temperature variation in solutions of amphiphilic block copolymers.

1. INTRODUCTION Interest in self-assembly of block copolymers in selective solvents has noticeably increased after the formation of aggregates of different morphologies had been experimentally detected two decades ago.1,2 It was found that amphiphilic block copolymers composed of covalently linked soluble and insoluble chains are capable of composing not only spherical micelles but also wormlike aggregates, flat lamellae, and vesicles (polymersomes).3−5 Further, more careful studies have revealed a number of intermediate structures that emerge in the vicinity of sphere-to-worm and worm-to-vesicle transitions. In the latter case, they are branched worms, octopus-like micelles, and “jellyfish” aggregates consisting of vesicle hemisphere with wormlike threads attached to its edges.6 External stimuli control of aggregates morphology offers opportunities for various practical applications. For instance, switching from worms to either spheres or vesicles results in a dramatic (by up to 5 order of magnitude) decrease in the solution viscosity.7 The degelation is attributed to the disappearance of topological entanglements network formed by long wormlike micelles when they change their morphology and acquire a more compact spherical or vesicle shape.8 Any changes in external conditions (such as temperature, pH, and solvent composition) affecting the solvent quality for even one of the blocks and, hence, the structure of aggregates9 allow one © XXXX American Chemical Society

to efficiently and reversibly govern the solution rheology.7,8,10,11 Recently, an incorporation of photosensitive azobenzene moieties into a diblock copolymer made it possible to trigger worm-to-vesicle transitions by UV/light irradiation:12 solubility of azo groups is known to change under lightinduced cis−trans izomerization.13−15 Vesicular aggregates are promising for drug-delivery needs.16−18 Required species can be compartmentalized in their interior and released on demand by means of transformation of vesicles to wormlike micelles.19 The type of external stimulus that provides the bestperforming control of block copolymer aggregates depends on the presence of ionic groups in the AB molecule. If any of the blocks contains ionic groups, pH and salt are very convenient tools for tuning the micelle properties.20 One can govern selfassembly of (i) aggregates with ionic corona and neutral hydrophobic core via a change of the ionization of the corona affecting its swelling, micelle aggregation number, and morphology and (ii) aggregates with ionic core and neutral soluble corona (so-called complex coacervate core micelles, C3Ms21). In the latter case, core domain is a complex of Received: November 6, 2018 Revised: January 18, 2019

A

DOI: 10.1021/acs.langmuir.8b03747 Langmuir XXXX, XXX, XXX−XXX

Article

Langmuir oppositely charged polyelectrolytes in which density and surface tension are dependent on the salt concentration and degree of ionization of both polyanion and polycation.22,23 The control of the core properties is very efficient because decreasing ionization of the core blocks and/or addition of salt may not only cause morphological transitions in C3Ms but also induce their disintegration.22,24 For neutral AB block copolymers, a temperature change is the easiest and the most frequently used way to control the aggregate morphology. While the solubility of organosoluble polymers usually increases upon an increase in temperature, water-soluble polymers demonstrate both trends: for so-called lower critical solution temperature (LCST) polymers, the quality of water as the solvent decreases as a function of temperature, whereas for so-called upper critical solution temperature (UCST) polymers, the opposite effect is observed.25 Thus, temperature-induced changes of aggregate properties can be achieved both in aqueous medium [where poly(n-isopropylacrylamide),26 poly(N-vinylcaprolactam),27 and poly(ethylene oxide)19 are often used as thermosensitive blocks] and also in organic solvents.9 One can tune solvent quality for either the coronal A-block or the core B-block, while simultaneous temperature-induced control of A and B blocks solubility is possible when they both are thermosensitive,9,27 where it is understood that the A-block should retain solubility to provide colloidal stability of aggregates. Control of the block copolymer aggregates by means of coronal block solubility tuning is demonstrated in ref 19 where thermotropic morphology alternation of poly(2-vinilpyridineb-ethylene oxide) (P2VP-b-PEO) aggregates formed in aqueous medium was detected. Transformation of vesicles to wormlike aggregates under solution cooling from 25 to 4 °C was attributed to enhancing the solubility of coronal PEO blocks. In regards to the core B-block, it should remain insoluble in order to provide aggregation. Disintegration of micelles can be induced by means of the core block solubility improvement. Its solvophobicity defined by the corresponding Flory−Huggins parameter value, χ > 1/2, can also be adjusted when it remains insoluble. Herewith, solvent quality for the core block governs (i) the core degree of swelling measured by, for example, the average polymer volume fraction ϕ̅ in it and (ii) the surface tension at the core−corona interface γthe core properties which are often implicitly considered as independent quantities. For instance, Armes and co-workers reported about wormto-sphere transitions in micelles with poly(2-hydroxypropyl methacrylate) core and poly(glycerol monomethacrylate) corona under solution cooling, which was accompanied by physical hydrogel dissolution.10 Temperature decrease from 21 to 4 °C caused core hydration (moderate swelling, slight decrease in ϕ̅ ), which also led to a reduction of the surface tension γ, and the latter was indicated by the authors as the driver for reorganization of worms into spheres. Lowe’s group observed similar ethanol solution degelation caused by wormto-sphere morphological transition in poly[2-(dimethylamino)ethyl methacrylate]-b-poly(3-phenylpropyl methacrylate) (PDMAEMA-b-PPPMA) micelles, which was induced by heating from 28 to 70 °C. Again, increasing the degree of PPPMA core solvation (ϕ̅ reduction) at changing temperature was found, and the authors proposed the lowering of the interfacial tension γ was primarily responsible for the worm-tosphere transition.8

At the same time, Yu and Eisenberg reported about inverse sphere-to-worm and sphere-to-vesicle transitions in polysterene-b-poly(acrylic acid) (PS-b-PAA) micelles by means of the swelling of the core induced not by temperature changes but by substitution of the organic solvent used for block-copolymer dissolution (before dialysis against water).3 Though the change of the solvent should affect not only the core block solubility but also that of the coronal block, authors explained these changes of morphology by increasing the stretching of the core blocks under their swelling and the fact that core block elongation in nonspherical aggregates is lower. Thus, there exist indications of both vesicle-to-worm-tosphere and inverse transitions upon the aggregates core swelling: the former is caused by surface tension diminution and the latter is due to core block stretching which is known to diminish in nonspherical morphologies. To clarify this issue and resolve apparent contradiction, we here develop a consistent theory of morphological transitions in nonionic block copolymer micelles induced by changes in the solvent quality for the core block. This theory accounts for a simultaneous variation in the core density ϕ̅ and the core surface tension γ as the functions of the Flory−Huggins parameter χ. This single parameter defines the solvent quality which is expected to set by the temperature.28,29 Below, we will show that the simultaneous account for ϕ̅ (χ) and γ(χ) dependencies allows to explain both direct and inverse order of morphological changes induced by the same factor, that is, changing solvent quality for the core chains. It should be mentioned that in the framework of our theory, we consider thermodynamically equilibrium aggregates of block copolymers, so that the results are not applicable to the kinetically frozen systems (e.g., with vitrified micellar cores), where the relaxation processes are too slow to be experimentally detected.30,31 The article is organized as follows. First, the properties of the micelle core are discussed and ϕ̅ (χ) and γ(χ) dependencies that are valid in the wide range of χ, that is, for both moderately and highly poor solvent for the core, are revealed. Second, χ-induced changes in spherical micelles are considered, and it is shown that the geometrical properties of these micelles (the ratio between the corona and the core dimensions) change in a nonmonotonous fashion with increasing χ. Third, we combine our findings on the core properties with the earlier developed theory of morphological transitions in block copolymers micelles20,32 in order to predict feasible re-entrant changes of micelles morphology under decreasing solvent strength for the core block. With that, our findings explain the different order of morphological transitions observed in various experiments. Finally, conclusions are formulated.

2. THEORETICAL MODEL OF THE MICELLAR CORE In the following, we consider nanoaggregates formed by AB diblock copolymers in a selective solvent which is good for block A and a poor solvent for block B. The core domains of the aggregates (micelles) consist of insoluble B-blocks and the Flory−Huggins interaction parameter between the B-monomer and the solvent is χ > 1/2. We recall that χ is the continuous function of the temperature T, increasing for LCST and decreasing for UCST polymers (see refs 28 and 29 for detailed discussion). We follow the Lifshitz theory of polymer globules33 in order to find the core density ϕ̅ and the surface tension γ at the core−corona interface. The latter is calculated B


Langmuir

l o 1 yz2 ij 1y ij o o j zz , jjχ − zzz ≪ 1 χ − o o j γ≃m 2{ k 2{ k o o o o o χ≫1 n χ,

as that at the core−solvent interface owing to the sparse micelle corona swollen in theta or good solvent that contributes small fraction of A-monomers to the interface. 2.1. Scaling Analysis. At low χ values, the internal structure of the globule is akin to the semidilute polymer solution, and the blob (mesh) size is on the order of the thermal blob,34,35 ξ ≃ ξt ≃ |v|−1 ≃ (χ − 1/2)−1; here and below, we express all length scales in a units (a is the statistical segment length), and v is the dimensionless second virial coefficient. Polymer chains have the ideal coil statistics at all length scales, and globule density is equal to the polymer concentration within the blob, ϕ̅ ≃ ξ−1 ≃ (χ − 1/2). The energy of neighboring thermal blobs attraction is on the order of unity (all energies are given in kBT units). Blobs at the globule interface have lower number of neighbors than that in the globule depth, which results in the excess interfacial free energy. The globule surface tension is the ratio between this excess free energy and the surface area ξ2 occupied by one blob at the globule interface, γ ≃ 1/ξ2 ≃ (χ − 1/2)2 (measured in kBT/a2 units). Scaling relationship γ ≃ ϕ̅ 2 is fulfilled only in the regime, when globule density is relatively low. As the solvent becomes very poor, χ ≫ 1, almost all solvent molecules are expelled from the globule and ϕ̅ ≃ 1. In this regime, the connectivity of monomer units into polymer chains does not strongly influence the globule structure, which is controlled by noncovalent (short-range binary attractive) interactions between monomer units. As a result, the thermodynamic properties of the globule are similar to that of a droplet of a low molecular weight liquid in a continuous medium of disparate immiscible liquid (e.g., oil in water emulsion). The surface layer between two of these liquids is known to be very thin,36,37 and the excess surface free energy can be attributed to the unfavorable interactions between the globule (oil) layer and the surrounding solvent (water) layer: γ ≃ χ because the surface area per statistical segment is on the order of unity and χ = χps − (χpp + χss)/2 is the excess free energy. Indeed, when the polymer and solvent beads are placed from their bulks to the interfacial layer, they loose half of the favorable contacts with the beads of the same type, resulting in the ≃−(χpp + χss)/2 term but get extra unfavorable contact providing ≃χps contribution. Here, we do not consider a possible formation of a thin shell consisting of coronal chains adsorbed at the core that could lead to partial screening of unfavorable core−solvent interactions. Because this layer consisting of hydrophilic (and presumably incompatible with hydrophobic B-blocks) A-chains should contain high amount of the solvent (especially in good solvent for corona), it cannot drastically diminish the surface tension at the core−corona interface and change our findings. Substantial renormalization of the surface tension can be expected in multicompartment micelles, where the core is surrounded by a quite dense spherical hydrophobic shell38 or covered with dense hydrophobic patches.39 Results of the above scaling analysis are summarized below l 1y o i o o o jjjχ − zzz, 2{ ϕ̅ ≃ m k o o o o o 1, n

1y ij jjχ − zzz ≪ 1 2{ k χ≫1

Article

(2)

2.2. Mean-Field Consideration. Let us consider a globule of B-chains with the radius far exceeding the width of its interfacial layer, so that it can be treated as an infinitely large one,34 and introduce the x axis being normal to the flat globule−solvent interface. Let x = 0 be the coordinate of the midpoint of the globule surface layer where the globule density is half of that in the interior, ϕ(x = 0) = ϕ̅ /2, while x → −∞ and x → ∞ correspond to globule interior and the outer solution, respectively. We write down the free energy of the globule -{ϕ(x)} as the functional of the globule density ϕ(x) and minimize it in order to find the equilibrium globule properties. -{ϕ(x)} functional -{ϕ(x)}a3 = S

i

∫−∞ jjjjjFvol(ϕ(x)) + ∞

k

a 2(ϕ′(x))2 24ϕ(x)

yz + k(ϕ′(x))2 zzzdx z { ij ∞ pNBa3 yzz zz − μjjj ϕ(x) dx − j −∞ S z{ k

∫

(3)

contains a term Fvol(ϕ(x)) that is responsible for the volume interactions, the gradient terms ∼(ϕ′(x))2 taking into consideration the excess free energy in the globule surface layer caused by inhomogeneities, and a normalization condition (the second integral). Here, ϕ′(x) = ∇ϕ = dϕ/dx is the gradient of the globule density in this effectively onedimensional problem, S is the total surface area of the globule, p is the number of B blocks in the core (p/S being the finite number), and μ is the Lagrange multiplier. The first contribution to -{ϕ(x)} can be written in the Flory−Huggins form40 Fvol(ϕ) = (1 − ϕ) ln(1 − ϕ) + χϕ(1 − ϕ)

(4)

where the term ϕ ln ϕ/NB which accounts for the translational entropy of polymer chains was omitted owing to NB ≫ 1. The two gradient terms in functional 3 have a different physical meaning. The first of them, a2(ϕ′(x))2/24ϕ(x), called the Lifshitz−Edwards entropy,41,42 takes into account the entropy losses of the polymer chains in the vicinity of the globule surface. Roughly, when a chain approaches the globule interface, it should adopt loop conformations in order to return to the globule interior, while conformations retaining orientation (e.g., rodlike) are almost prohibited because a protrusion of the chain into the outer solution results in a high number of unfavorable polymer−solvent interactions. The second gradient term k(ϕ′(x))2 being inhomogeneityinduced correction to Fvol has an enthalpic (energetic) rather than an entropic origin. It was introduced by Cahn and Hilliard43,44 and is widely used for calculation of the surface tension at the interface of two low molecular weight liquids. Its meaning can be clarified as follows. Consider three neighboring planar globule layers parallel to the x = 0 plane with the polymer densities in these layers being ϕ(x − a), ϕ(x), and ϕ(x + a). For simplicity, the cubic lattice with the coordination number z = 6 and site equal to a is used for implementation of the FH lattice model. The energy of

(1) C


Langmuir

ik dϕ 1 zyz = − jjjj 2 + z dx 24 ϕ z{ ka

polymer−polymer interactions per site of the lattice belonging to the middle x-layer ÄÅ ÉÑ Epp Å4 Ñ 1 1 = χpp ϕ(x)ÅÅÅÅ ϕ(x) + ϕ(x + a) + ϕ(x − a)ÑÑÑÑ ÅÇ 6 ÑÖ kBT 6 6

−1

a χpp 6

ϕ(x)ϕ″(x)

(5)

∫0

ϕ(∞) = 0;

max

a 2 (ϕ′)2 + Fvol(ϕ) − μϕ = 0 24 ϕ

dFvol dϕ

(7)

∫0

2

∞

(9)

i k

2

2

zy + 2k(ϕ′(x))2 zzzdx z {

(14)

dϕ

y 1 ij ϕ 3i 1 y2 jj (ϕ ̅ − ϕ)2 zzz = jjjχ − zzz 24ϕ k 6 4k 2{ {

(15)

2.2.2. High-Density Globule, ϕ̅ ≈ 1. When the solvent is very poor for the B-block, χ ≫ 1, the globule contains a small volume fraction of the solvent given by

= ln(1 − ϕ ̅ ) + ϕ ̅ + χϕ ̅ = 0

∫−∞ jjjjj a 12(ϕϕ′((xx)))

ϕ̅

γ≈2

ϕ̅ ≈ 1 − e−χ − 1

and is the function of χ. This result coincides with that of the volume approximation.34 Substitution of eq 8 in the free energy functional 3 allows to find the excess interfacial free energy per surface area unit -{ϕ(x)}a3 = S

(13)

can be found. This result remains reliable until (χ − 1/2) ≪ 1 and can be also written as ϕ̅ = −v/2w with v = 1/2 − χ and w = 1/6 being the second and the third dimensionless virial coefficients in the framework of FH lattice consideration because Fvol(ϕ) ≈ vϕ2 + wϕ3 at low ϕ (linear in ϕ terms are omitted). At a very low globule density ϕ̅ , namely, at rχ ≪ 1/ 24ϕ̅ equivalent to (χ − 1/2) ≪ 1/36r, the Edwards−Lifshitz entropy reliably dominates over the Cahn−Hilliard term, and eq 12 yields the well-known result for the low surface tension of the low-density globule33,34,48

(8)

ϕ̅

}

1y i ϕ ̅ ≈ 3jjjχ − zzz 2{ k

where the boundary conditions at x → ∞ have been taken into account upon the choice of the integration constant. Using another boundary condition corresponding to the globule interior, x → −∞, makes it possible to find the equilibrium globule density ϕ̅ which obeys Fvol(ϕ ̅ ) − ϕ ̅

dϕ(x) dx

The integral in eq 12 for the surface tension cannot be calculated analytically in a general case. Approximate analytical results for globule density and surface tension can be found in the limiting cases of the low density and the high density globule. These cases correspond to the predominance of the first and the second gradient term, respectively, that is, k/a2 = rχ and 1/24ϕ terms in the first brackets in eq 12. 2.2.1. Low-Density Globule, ϕ̅ ≪ 1. In low-density regime, eq 9 can be expanded into a series and globule density

After multiplying eq 7 by ϕ′ and integration once, we get −k(ϕ′)2 −

{

−∞< x χcr, the properties of the macroscopic globular phase coincides with that of a single infinite globule, and the flat interface between two phases is fully analogous to the globule interface. SCF lattice modeling allowed to find (i) the equilibrium composition of both phases ϕconc and ϕdil, (ii) the excess surface energy of this interface γ, and (iii) the interface width Δ found from the sampled density profile ϕ(x) (using definition 13) in a wide χ-range for different values of the polymer chain length N. Modeling results for δϕ = ϕconc − ϕdil, γ, and Δ as functions of the deviation from the critical point δχ = χ − χcr are plotted in Figure 1a−c, respectively. SCF results are compared with theoretical predictions for infinitely long chains, N → ∞, when χcr → 1/2, ϕdil → 0 and hence ϕconc − ϕdil = ϕ̅ (see the case of finite N, when ϕdil ≠ 0, in the Supporting Information). Thus, we fit SCF data with the theoretical values of ϕ̅ , γ, and Δ given by eqs 9, 12, and 13, respectively, and shown with solid black lines in Figure 1. It is clearly seen that SCF data and theoretical results are in good agreement, and regimes of low-density globule and highdensity globule can be distinguished. In the former case, the globule density ϕ̅ and the surface tension γ are linear and squared-law functions of the deviation from the critical point, in accordance with eqs 14 and 15. At high χ values, ϕ̅ saturates while γ grows linearly with χ, as predicted by eqs 16 and 17, with the crossover at (χ − χcr) ≈ 0.5. One can notice a certain deviation of SCF modeling from theoretical predictions in the regions of very high and very low deviation from the critical point. At very high χ, the boundary between the globular phase and the supernatant becomes very narrow and comparable with the lattice size a, see Figure 1c. Therefore, one cannot neglect high order terms in linearization 5 because the condition ϕ′a ≪ 1 is not fulfilled, which makes

Figure 1. (a) Difference between the densities of precipitant and supernatant ϕconc − ϕdil, (b) surface tension γ, and (c) interface width Δ as the functions of deviation form the critical point χ − χcr obtained using numerical SF-SCF results for finite N (symbols) and analytical theory for N → ∞ (solid curve) predicting ϕconc − ϕdil = ϕ̅ . The critical point coordinate χcr for any N was calculated from eq 18.

the second gradient term k(ϕ′(x))2 in the free energy functional approximate. Moreover, the accuracy of lattice SCF modeling also goes down as the interface width Δ tends to the lattice unit size a. Deviation of the modeling data from the mean field theory (MFT) predictions in the vicinity of the critical point, at low δχ = χ − χcr values, should be attributed to the neglected translational entropy of chains in eq 4, defining Fvol which is used in the theoretical consideration (see the Supporting Information, Figure S1). The width of this region δχdev decreases with the increasing chain length N, δχdev ≈ N−1/2. In this region, SCF modeling data show that, for example, (ϕconc − ϕdil) ≈ δχ1/2 at δχ → 0, which is a well-known value of the critical exponent predicted by the mean field-type theories.54 However, at δχ ≤ δχdev, where the SF-SCF data for finite N values are not adequately approximated by the theoretical curve calculated for N → ∞, fluctuations within the system become very strong. This results in the failure of any mean field-based approach so that the SF-SCF data themselves are not fully reliable. The critical region width, where the meanE


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Langmuir field approaches fail, can be estimated with the aid of the Ginzburg criterion.52,53 2.3.1. Ginzburg Criterion. Above the critical point, at χ = χcr + δχ and ϕ = ϕcr, the compositions of the coexisting phases can be found from the equality of the osmotic pressures Π = ϕ· dFvol/dϕ − Fvol and the chemical potentials μ = dFvol/dϕ with Fvol =

ϕ ln ϕ + (1 − ϕ)ln(1 − ϕ) + χϕ(1 − ϕ) N

3. RESULTS AND DISCUSSION 3.1. Spherical Micelles. Considering the cores of the block copolymer aggregates (micelles), we will assume that the local properties of the core (i.e., density and surface tension) are affected by neither stretching of the insoluble B-blocks in it nor by the presence of soluble coronal A-blocks tethered to the core−corona interface. In other words, it is assumed that the core structure is defined purely by short-range binary attractive interactions (so-called volume approximation34) and coincides with that of an infinitely large globule. This type of assumption is generally accepted in the theory of block-copolymer micellization.20 We start with considering spherical micelles formed by AB diblock copolymers with degrees of polymerization of soluble and insoluble blocks NA and NB, respectively. Following the classical approach,55−59 the equilibrium properties of the micelles (i.e., aggregation number p, core radius R, and corona thickness H) can be found via minimization of the free energy per AB block copolymer Ftot. The free energy Ftot expressed in kBT units contains three contributions Ftot = Fcore + Fsurf + Fcorona (24)

(19)

where the translational entropy of chains (the first term) is nonzero, owing to the finite N values. The difference between the coexisting phases in the vicinity of the critical point equals (ϕconc − ϕdil)2 ≈ 6δχ / N .52 Below the critical point, the solution remains homogeneous but fluctuations in it demonstrate unlimited growth. The Fourier transform of the density−density correlation function can be found within the random phase approximation53 S −1(q) =

(qa)2 1 1 + + − 2χ 12ϕ Nϕ 1−ϕ

(20)

In the vicinity of the critical point, substituting χ = χcr − δχ and ϕ = ϕcr, one can find the solution structure factor S(q) ≈

responsible for stretching of core blocks B, the excess free energy at the core−corona interface and the free energy of corona consisting of swollen A-blocks, respectively. As demonstrated in refs,58,59,32 the first contribution is negligible for spherical micelles as long as they are thermodynamically stable with respect to transformation into aggregates of other morphologies (see below). The second (interfacial term) can be presented as Fsurf ≃ γs, it is proportional to the surface tension γ and the core area per AB molecule s ≃ R2/p, where the core radius R is related to the aggregation number p by the space-filing conditions, R ≃ (NBp/ϕ̅ )1/3. The free energy of the corona can be calculated by using the blob ansatz, that is, by treating the corona as a set of densely packed spherical layers of blobs60,61 and using the rule of one kBT per blob

S(q = 0) 1 + q2ξcorr 2

(21)

that has the conventional Ornstein−Zernike form34 with S(q = 0) = (2δχ)−1 and the correlation length ξcorr = N1/4 / 6δχ diverging at δχ → 0. The typical magnitude of fluctuations in the system reads54 ⟨δϕ2⟩ ≃

S(q = 0) ξ3

≃3 6

(δχ)1/2 N3/4

(22)

According to the Ginzburg criterion,53,54 predictions of the MFT remains reliable until (ϕconc − ϕdil)2 ≫ ⟨δϕ2⟩, that is at δχ ≫ δχGinz =

3 2 N

Fcorona ≃

(23)

It is clear that for infinitely long chains, N → ∞, our theoretical predictions are valid at any δχ, while the validity range of the SF-SCF modeling data is limited. For example, for N = 104, fluctuations are negligible at δχ ≫ δχGinz = 0.015, where SF-SCF data are fairly approximated by the theoretical curve, see Figure 1. To conclude, in our theoretical analysis, we have neglected the translational entropy of polymer chains which affects the results for the globule properties (ϕ̅ , γ, and Δ) only in the vicinity of the critical point, at δχdev ≈ N−1/2. This is the reason for the deviation of the SF-SCF modeling data from the analytical theory results (see the Supporting Information). However, this δχ-range approximately coincides with the critical region where no mean field-based approach for finite Nneither the SF-SCF modeling nor analytical theory with the N-dependent term in Fvol responsible for the translational entropy of chainsis applicable. In fact, one can expect Isingtype exponent values of the critical region, but more sophisticated approaches are required to study this issue. For simplicity, below, we entirely neglect the effect of the chain length on the globule properties and use the analytical results derived for N → ∞.

∫R

R+H

dr ≃ ξA(r )

Hy i p lnjjj1 + zzz R{ k

(25)

where the blob size grows linearly with the distance from the center, ξA(r ) ≃ r s /R . We consider only limiting cases of the theta solvent (vA = 0 and wA ≃ 1) and athermal solvent (vA ≃ 1) for corona chains, in order to avoid intermediate regimes when the solvent is effectively theta for the inner, dense corona layers and good for the outer layers.32 Then, the blob size is related to local polymer concentration as φ A (r) ≃ vA(1−2ν)/νξA(1−3ν)/ν(r) and the condition of conservation of the number of monomer units in the coronal A-blocks leads to a closed equation for the corona thickness vA (1 − 2ν)/ νp−(1 − ν)/2ν [(R + H )1/ ν − R1/ ν ] ≃ NA

(26)

with ν = 1/2 and ν = 3/5 for the theta solvent and athermal solvent, respectively. Equilibrium parameters of the micelles corresponding to the free energy minimum as a function of the aggregation number can be found analytically in the limiting cases of starlike (“st” index, R ≪ H) and crew-cut (“cc” index, R ≫ H) micelles. 3.1.1. Starlike Micelles, H ≫ R. In this regime, eq 26 yields a corona thickness H ≃ vA2ν−1NAνp(1−ν)/2 and the free energy of corona scales as Fcorona ≃ p ln(H /R ) according to F


Article

Langmuir

Table 1. Properties of Starlike and Crew-Cut Spherical Micelles in the Low Core Density and High Core Density Limits low density core, (χ − 1/2) ≪ 1

high density core, χ ≫ 1

starlike micelles, H ≫ R

pst ≃ NB4/5(χ − 1/2)8/5 Rst ≃ NB3/5(χ − 1/2)1/5 Hst ≃ vA2ν−1NAνNB2(1−ν)/5(χ − 1/2)4(1−ν)/5

crew-cut micelles, H ≪ R

pcc ≃ R cc ≃

pst ≃ NB4/5χ6/5 Rst ≃ NB3/5χ2/5 Hst ≃ vA2ν−1NAνN2(1−ν)/5 χ3(1−ν)/5 B

NB2(χ − 1/2)2(4ν − 1)/(2ν + 1) (vA

2ν − 1

pcc ≃

ν 6/(2ν + 1)

NA )

NB(χ − 1/2)(2ν − 1)/(2ν + 1)

R cc ≃

(vA 2ν − 1NA ν)2/(2ν + 1)

Hcc ≃ (vA

2ν−1

ν 3/(2ν+1)

NA )

(χ − 1/2)

NB2χ 6ν /(2ν + 1) (vA 2ν − 1NA ν)6/(2ν + 1)

NBχ 2ν /(2ν + 1) (vA 2ν − 1NA ν)2/(2ν + 1)

Hcc ≃ (vA2ν−1NAν)3/(2ν+1)χ(1−ν)/(2ν+1)

2(1−ν)/(2ν+1)

Figure 2. (a,b) Graphical representation of equation γ3ν−1/ϕ̅ 2ν+1 = f for the cases of the theta solvent, ν = 1/2, and athermal solvent, ν = 3/5 for the coronal block A. The horizontal dotted red line represents the particular case f = 1.5, st ↔ cc crossover points between starlike and crew-cut regimes are denoted by red dots. (c,d) Corresponding H/R ratios.

eq 25. The equilibrium aggregation number within the accuracy of a logarithmic factor is given by iN y pst ≃ γ 6/5jjjj B zzzz k ϕ̅ {

f≃

iN y R st ≃ γ 2/5jjjj B zzzz k ϕ̅ { Hst ≃ vA

NB 2ν + 1

(31)

4/5

which is independent of the solvent quality χ for the core block. Micelles remain starlike as long as f ≫ γ3ν−1/ϕ̅ 2ν+1, and the st → cc crossover occurs at γ3ν−1/ϕ̅ 2ν+1 ≃ f. Remarkably, in the low core density regime, the ratio ϕ̅ 2ν+1/γ3ν−1 ≃ (χ − 1/ 2)3−4ν increases with increasing χ, that is, a decrease in the solvent quality makes micelles “more starlike”. In contrast, in strongly poor solvents, the ratio ϕ̅ 2ν+1/γ3ν−1 ≃ χ−(3ν−1) decreases with increasing χ that leads to a decrease in the relative corona thickness H/R. Thus, in the starlike regime, as the solvent monotonously becomes poorer for the coreforming block (χ growth), the ratio H/R changes in a nonmonotonous fashion, while the aggregation number monotonously increases, see Table 1. 3.1.2. Crew-Cut Micelles, H ≪ R. Linearization of eqs 26 and 25 results in H ≃ v A ( 2ν− 1 ) / ν N A s − ( 1− ν) /2 ν and Fcorona ≃ H / s . Minimization of Ftot with respect to p results in

(27)

The core and corona dimensions of starlike micelles equal

2ν − 1

(vA 2ν − 1NA ν)5

3/5

i NB zy ν 3(1 − ν)/5j j z

NA γ

jj zz k ϕ̅ {

(28) 2(1 − ν)/5

(29)

Scaling laws for pst, Rst, and Hst in the low- and the highdensity core cases are presented in Table 1. To know whether or not a result is applicable for the starlike micelles limit, it is of interest to write the H/R ratio as ij ϕ ̅ 2ν + 1 yz Hst ≃ jjjjf 3ν − 1 zzzz R st k γ {

1/5

iN y γ 6ν /(2ν + 1) pcc ≃ jjjj B zzzz 2ν − 1 ν 6/(2ν + 1) NA ) k ϕ ̅ { (vA 2

(30)

where we have introduced the function G

(32) DOI: 10.1021/acs.langmuir.8b03747 Langmuir XXXX, XXX, XXX−XXX

Article

Langmuir N γ 2ν /(2ν + 1) R cc ≃ B ϕ ̅ (vA 2ν − 1NA ν)2/(2ν + 1)

(33)

Hcc ≃ (vA 2ν − 1NA ν)3/(2ν + 1)γ (1 − ν)/(2ν + 1)

(34)

condition NB ≫ (χ − 1/2)−2 corresponding to the collapse of individual B-blocks should be satisfied. At χ → 1/2 + 0, micelles dissociate because of weakening of hydrophobic attraction of B-blocks, and the first st → cc transition takes place only if the χ-coordinate of the first red point is larger than the disintegration threshold χdis ≃ 1/2 + 1/ NB . For this reason, the re-entrant transitions cc → st → cc are more feasible under θ solvent conditions for the coronal blocks A, when the function γ3ν−1/ϕ̅ 2ν+1 exhibits a minimum at χΘmin ≈ 1.03, whereas in a good solvent, this function has a minimum at χ+min ≈ 0.67, and it is quite challenging to detect the first crossover because micelles are already crew-cut even near the micelle formation threshold, χdis. It should be challenging to experimentally detect these crossovers because changes in micelle dimensions are quite weak, see eqs 30 and 35. However, these nonmonotonous changes in the micelle geometry are an indication of possible re-entrant morphological transitions in micelles which are much easily observable in real experimental systems. 3.2. Polymorphism of Aggregates. As the aggregation number of crew-cut spherical micelles grows, the core blocks become more stretched. Change of aggregate morphology from spherical (S, i = 3) to wormlike (W, i = 2) and further to vesicular (V, i = 1) diminishes conformational entropy losses, Fcore, in the core but causes Fcorona growth because lower volume is available for each coronal chain in a vesicle (polymersome) than in a wormlike micelle and in a wormlike micelle than in a spherical crew-cut micelle. Following refs 32 and 20, we consider a wormlike micelle as an infinite cylinder neglecting the edge effects. Similarly, vesicles are treated as infinite flat lamellae. In the vicinity of morphological transitions, the Fcore term and possible corrections to the corona free energy remain small as compared to Fsurf and Fcorona.20,32 For this reason, the value s of the core surface area per chain and the main, morphologyindependent contribution in the corona thickness H0 are the same for all types of aggregates, and results derived for the crew-cut spherical micelles can be used

Power law predictions for the micelle properties in moderately and strongly poor solvents for the core block can be found in Table 1. The ratio between the corona thickness and the core radius reads

ij ϕ ̅ 2ν + 1 yz Hcc ≃ jjjjf 3ν − 1 zzzz R cc (35) k γ { and resembles that derived in the starlike limit (see eq 30), except that the bracket power exponent 1/5 is substituted by 1/(2ν + 1). However, this difference does not qualitatively change the behavior of the H/R ratio, which increases as a function of χ at low χ and diminishes at high χ. These changes are accompanied by a steady growth in the aggregation number pcc. 3.1.3. Crossover and Re-Entrant cc → st → cc Transitions. As follows from eqs 30 and 35, we can find the crossover points between starlike and crew-cut micelles, R ≃ H, from the condition γ3ν−1/ϕ̅ 2ν+1 ≃ f. Both γ and ϕ̅ are monotonous functions of χ, but the ratio γ3ν−1/ϕ̅ 2ν+1 changes nonmonotonuosly and exhibits a minimum as a function of χ. Below, we use exact results for the core density ϕ̅ and surface tension γ defined by eqs 9 and 12. In Figure 2a,b, we plot the dependencies of γ3ν−1/ϕ̅ 2ν+1 as a function of χ for theta and good solvents, and the intersection of these curves with a horizontal line corresponding to a fixed value of f are the crossover points. Remember that f depends only on the lengths of A and B blocks and solvent quality for coronal chains A. In the particular case of f = 1.5 depicted in Figure 2, with the red-dashed curve, micelles are crew-cut in the vicinity of χ = 1/ 2 and at high χ, whereas in the intermediate range of χ, they acquire a starlike shape (see Figure 3). Thus, an increase in χ 1/(2ν + 1)

iN y s ≃ vA 2(2ν − 1)/(2ν + 1)jjjj A zzzz k γ {

2ν /(2ν + 1)

(36)

H0 ≃ γ (1 − ν)/(1 + 2ν)vA 3(2ν − 1)/(2ν + 1)NA 3ν /(2ν + 1)

(37)

One can also estimate the main, i-independent term in the free energy F0tot ≃ Fsurf ≃ Fcorona 0 Ftot ≃ vA (2ν − 1)/ νNAs−1/2ν

Figure 3. Schematic representation of transitions between crew-cut and starlike regimes of spherical micelles upon decreasing solvent quality for the core block (increasing χ − 1/2 value).

(38)

In order to find the binodals of S ↔ W and W ↔ V (i) transitions, the morphology-induced correction δF(i) tot = Fcore + (i) (1) δFcorona to the Ftot should be calculated. The F(i) core term can be written as

results in cc → st → cc transitions: the nonmonotonous change of the H/R ratio calculated from eqs 30 and 35 for starlike and crew-cut regimes is shown in Figure 2c,d, H/R = 1 in the crossover points. At much lower f values corresponding to, for example, a larger length of hydrophobic block NB, the red-dashed curve would lie lower than the black one in Figure 2a,b and micelles would remain crew-cut in the whole range of χ. It is worth noting that our consideration is applicable if the core block length is large enough to assure formation of micelles. In the low core density limit, this implies that the

R i2 N = bii 2 2 B2 NB s ϕ̅ l o π 2/8, i = 1 (V) o o o o NB o 2 = 2 2 ×m π /4, i = 2 (W) o o o s ϕ̅ o o 2 o 27π /80, i = 3 (S) n

(i) Fcore = bi

(39)

where the aggregate core radius equal to Ri = iNB/sϕ̅ , according to space-filling conditions, depends on the H


Article

Langmuir

Figure 4. Graphical solution of eqs 43 and 44 defining the binodals for the V ↔ W and W ↔ S transitions at f ̃ = 1; (a) case of the theta solvent, ν = 1/2, and (b) case of the good solvent, ν = 3/5, for the corona chains.

Numerical solution of eqs 43 and 44 at f ̃ = 1 is shown in Figure 4. One can specify the regions of stability of spherical and wormlike micelles and vesicles. The narrow range of wormlike micelles stability with γ(7ν−2)/(2ν+1)/ϕ̅ 3 values differing only 21/10 = 2.1 times at the boundaries should be highlighted.20 The possibility of re-entrant morphological transitions V → W → S → W → V with decreasing solvent strength (increasing χ) schematically shown in Figure 5 is similar to cc → st → cc crossovers found in the case of spherical micelles.

morphology and the numerical coefficients bi take into account the radial distribution of the B-block ends within the core.62 Generalization of eqs 25 and 26 to the case of arbitrary morphology yields the corona thickness and the corresponding free energy ÄÅ ÉÑ ÅÅÅ (1 − ν)(i − 1) Hi ÑÑÑ ÑÑ Hi ≃ H0ÅÅÅ1 − ÅÅÇ 4ν R i ÑÑÑÖ (40) ÄÅ ÉÑ Hi ÅÅÅ (i − 1) Hi ÑÑÑ (i) Å ÑÑ Fcorona ≃ Å1 − s ÅÅÅÇ 4 R i ÑÑÑÖ (41) Because Hi/Ri ≪ 1, one can identify the second terms in square brackets of these equations with the small morphologyinduced corrections and check that at a zeroth approximation, these results coincide with eqs 37 and 38. However, H0 = H1 > 1 2 3 H2 > H3 and Fcorona > Fcorona > Fcorona , owing to the morphology-induced correction: repulsion of the coronal chains is the strongest in vesicles and the weakest in spheres, as expected. After transformations, the following δF(i) corona term can be distinguished (i) δFcorona ≃−

Figure 5. Schematic representation of morphological transitions in micelles upon decreasing solvent quality for the core block (increasing χ − 1/2 value).

(1) (i − 1) FcoronaH1sϕ ̅ 2i 2νNB

Indeed, in the low core density regime, the l.h.s. of binodals expressed by eqs 43 and 44 scales as ≃(χ − 1/2)−(7−8ν)/(2ν+1), that is, they decrease as a function of growing χ. This indicates the possibility of V → W → S transitions upon an increase in χ. In this regime, close to χ = 1/2, when the γ ≃ ϕ̅ 2 relationship is fulfilled, increasing the core density accompanied by the surface tension growth results in this sequence of morphological changes irrespective of the driving force, provided that ϕ̅ and γ growth occur simultaneously. For instance, the same transitions were theoretically predicted for C3Ms with not very dense cores, ϕ̅ ≪ 1, where the core tightening (accompanied by surface tension growth, γ ≃ ϕ̅ 2) is achieved via increasing ionization of core polyelectrolyte blocks and/or decreasing concentration of salt.22 The experimentally found sequence of C3Ms morphology changes supports these findings.23 Moreover, this branch of transitions when the core swelling favors nonspherical morphologies was probably observed by Yu and Eisenberg in ref 3. In contrast, in the range of large χ (i.e., in the high core density approximation), the ratio γ (7ν−2)/(2ν+1) /ϕ̅ 3 ≃ χ(7ν−2)/(2ν+1) increases as a function of χ, thus revealing the inverse order of the morphology changes, S → W → V. Most likely, these transitions were found in refs 10 and 8, where enhancing of the core block solvation (corresponding to a decrease in χ) promoted vesicle-to-worm and worm-to-sphere

(i − 1) (2 − 3ν)/(1 + 2ν) (vA 2ν − 1NA ν)7/(2ν + 1) ≃− γ ϕ̅ 2i NB (42)

δF(i) tot

δF(i+1) tot

Equating = for i = 1 and i = 2, one can get the following binodals for the V ↔ W and W ↔ S transitions, respectively γ (7ν − 2)/(2ν + 1) ϕ̅

3

γ (7ν − 2)/(2ν + 1) ϕ̅

3

= 2f ̃ (43)

=

20 ̃ f 21

(44)

with f ̃ independent of χ f̃ =

C0 (vA 2ν − 1NA ν)11/(2ν + 1) π2

NB 2

(45)

and controlled by the copolymer composition and solvent quality for the coronal block A. Here, C0 is the numerical constant on the order of unity, unknown because of the scaling approximation used for calculation of the corona free energy but the same for both the binodals. I


Article

Langmuir transformations of aggregates. Because these authors reported about the core block solvation rather than swelling, it is natural to expect that the core remained quite dense. In this regime, ϕ̅ ≈ 1 and the changes in γ govern the morphological transition. At the appropriate lengths of A and B blocks and solvent quality for A-chains, providing f ̃ = 1, the whole cascade of transitions, V → W → S → W → V, may occur upon a decrease in the solvent quality for the core-forming block in a wide range. However, to the best of our knowledge, there were no reports about experimentally detected re-entrant polymorphism because it is challenging to vary the solvent quality for the core block from almost theta to very poor and to perform a thorough investigation of low density core regime (close to the onset of the micelle formation).

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.M.R.). *E-mail: [email protected] (O.V.B.). ORCID

Artem M. Rumyantsev: 0000-0002-0339-2375 Frans A. M. Leermakers: 0000-0001-5895-2539 Igor I. Potemkin: 0000-0002-6687-7732 Oleg V. Borisov: 0000-0002-9281-9093

4. CONCLUSIONS We have developed a theory of neutral block copolymer aggregates polymorphism induced by changes in the solvent strength for the core block at a fixed solubility of the corona forming block. It is based on a combination of the mean-field theory and the numerical Scheutjens−Fleer SCF modeling approach for unraveling of dependencies of the density ϕ̅ and interfacial tension γ of the collapsed domains formed by insoluble blocks on solvent quality expressed as a function of the Flory−Huggins parameter χ. The latter is assumed to be a monotonous function of temperature or solvent composition. Both analytical theory and modeling show that two different regimes of hydrophobic core can be distinguished. In the moderately poor solvent, at (χ − 1/2) ≪ 1, the so-called scaling regime is implemented, and core density and surface tension reveal linear and quadratic dependence on the deviation from the θ-point, respectively: ϕ̅ ≈ (χ − 1/2) and γ ≈ (χ − 1/2)2. When core blocks are highly insoluble, χ ≫ 1, the core contains a low amount of solvent, ϕ̅ ≈ 1, while the surface tension grows linearly with the Flory−Huggins parameter, γ ≈ χ. Thus, our approach takes into consideration that improving the solvent quality for the core block simultaneously leads to (i) a decreasing core density and (ii) a lowering core−corona interfacial tension. An interplay of these factors at appropriate lengths of the soluble and insoluble blocks may lead to reentrant morphological transitions for block-copolymer aggregates. More specifically, when the solvent is very poor for the core block, the improving of the solvent quality results in fairly negligible core solvation (ϕ̅ ≈ 1), whereas surface tension γ diminution triggers vesicle-to-worm-to-sphere transitions, V → W → S. As soon as the solvent becomes moderately poor, combined effects of the surface tension decrease and considerable core swelling (γ ≈ ϕ̅ 2 in the scaling regime, at ϕ̅ ≪ 1) induces reverse sphere-to-worm-to-vesicle transitions, S → W → V, that is, the impact of a drop of ϕ̅ overcomes a decrease in γ. Hence, increasing the solvent strength for the core block (which remains, however, insoluble) may result in re-entrant V → W → S → W → V transitions. Some steps of these morphological transitions were already detected, while direct observation of re-entrant morphology changes remains a challenging experimental task.

■

Theoretical calculations of globule density ϕ̅ , surface tension γ, and interface width Δ with account for the translational entropy of chains (at finite length N) and their comparison with SF-SCF data (Figure S1) (PDF)

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS A.M.R. and O.V.B. thank the ANR for funding within the MESOPIC Project ANR-15-CE07-0005. E.B.Z. acknowledges support by the Ministry of Education of the Russian Federation within State contract no. 14.W03.31.0022. I.I.P. thanks the Russian Science Foundation project no. 15-1300124, and the Government of the Russian Federation within Act 211 contract no. 02.A03.21.0011 for the financial support.

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b03747. J


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