Letter Cite This: ACS Macro Lett. 2018, 7, 811−816
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Scaling Theory of Complex Coacervate Core Micelles Artem M. Rumyantsev,† Ekaterina B. Zhulina,‡,§ 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, France ‡ Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004 St. Petersburg, Russia § National Research University of Information Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia ∥ Peter the Great St. Petersburg State Polytechnic University, 195251 St. Petersburg, Russia ABSTRACT: We propose scaling theory of complex coacervate core micelles (C3Ms). Such micelles arise upon electrostatically driven coassembly of bis-hydrophilic ionic/nonionic diblock copolymers with oppositely charged ionic blocks or bishydrophilic diblock copolymers with oppositely charged macroions. Structural properties of the C3Ms are studied as a function of the copolymer composition, degree of ionization of the ionic blocks, and ionic strength of the solution. It is demonstrated that at sufficiently large length of the polyelectrolyte blocks the C3Ms may exhibit polymorphism; that is, morphological transitions from spherical to cylindrical micelles and further to lamellar structure or polymersomes may be triggered by increasing salt concentration. A diagram of states of micellar aggregates in the salt concentration/ionization degree coordinates is constructed, and scaling laws for experimentally measurable properties, e.g., micelle aggregation number and core and corona sizes, are found.
M
properties, which could be readily experimentally checked, is absent. In this Letter, we develop scaling theory of C3Ms with the aim (i) to predict equilibrium properties of spherical micelles such as aggregation number, core radius, and corona thickness and (ii) to provide a theoretical description of polymorphism of nanoaggregates with CCC domains. C3Ms are formed via coassembly of AB block copolymers containing soluble neutral A block and polyelectrolyte B block with oppositely charged polyions C.5,6 Comicellization of AB and AC block copolymers offers another opportunity to obtain micelles with hydrophilic corona consisting of swollen A chains and the core being a coacervate of B and C polyion blocks.7 In both cases, we assume all chains to be flexible, with statistical segment lengths on the order of the monomer unit size a, and restrict consideration to stoichiometric CCCs. B and C blocks/ chains are weakly charged quenched polyelectrolytes with equal degrees of ionization f+ = f− = f ≪ 1 and equal lengths NB = NC. The solution contains also a monovalent salt with concentration cs. The micelle core is the complex coacervate, globally neutral, and stabilized by the fluctuation-induced attraction between oppositely charged polyelectrolytes which is counterbalanced by short-range repulsions of the monomer units. We assume
icelles of amphiphilic block copolymers are considered as promising nanocarriers for drug and gene delivery.1−3 However, hydrophobic cores of these micelles are able to efficiently incorporate only hydrophobic species,4 while biomacromolecules, such as DNA and proteins, are ionic and hydrophilic. The use of complex coacervate core micelles or C3Ms5 (also called block−ionomer complexes6 or polyion complex micelles7 or micellar (inter)polyelectrolyte complexes8) instead of conventional nonionic counterparts responds to this challenge: ionic bioactive compounds can be easily solubilized by the complex coacervate core (CCC) which has a high affinity for them.9,10 Extra benefits of C3Ms over hydrophobic core micelles are (i) capability of their formation directly in aqueous medium facilitating drug/gene uptake and enabling to avoid toxic impurities of organic solvents in the core and (ii) high sensitivity to external stimuli, i.e., pH and salt concentration, which allows triggering nanoaggregate disintegration accompanied by drug/gene release without loss of their bioactivity owing to mild release conditions.11 Other C3M applications are synthesis of metal nanoparticles, nanowires and quantum dots, mesoporous inorganic materials, catalysis, design of biosensors, etc.11 In view of steadily increasing experimental interest in C3Ms, an understanding of principles governing their assembly, structure, and response to external stimuli, in particular medium salinity, is highly important for their applicationoriented design. However, theoretical works devoted to C3Ms remain scarce,12 and consistent scaling theory providing insight into their behavior and quantitative predictions of their © XXXX American Chemical Society
Received: April 26, 2018 Accepted: June 18, 2018
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DOI: 10.1021/acsmacrolett.8b00316 ACS Macro Lett. 2018, 7, 811−816
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ACS Macro Letters
where the corona thickness H can be found from the normalization condition
that water is close to the theta solvent for uncharged monomer units of the polyelectrolyte blocks. The internal structure of the coacervate is similar to semidilute solution of neutral polymers:13−15 the CCC can be envisioned as an array of densely packed oppositely charged electrostatic blobs, and correlation length within it is equal to the blob size, ξ ≃ ξel−st ≃ (uf 2)−1/3. Here u = e2/aϵkBT is the Bjerrum length (all length scales are expressed in a units), in aqueous medium u ≈ 1. The addition of salt does not affect the coacervate structure as long as the Debye screening length rD ≃ (ucs)−1/2 is larger than the size of the electrostatic blob ξel−st. Equality rD ≃ ξel−st allows specifying threshold salt concentration c*s ≃ u−1/3f4/3 (in a−3 units) above which salt-induced screening of Coulomb interactions results in their weakening and CCC swelling, i.e., growth of the correlation length ξ beyond rD. At cs ≫ cs*, the attraction between oppositely charged correlation blobs can be described as an outcome of the short-range attraction between chain segments of size rD. Since the energy of two-body screened Coulomb attraction between oppositely charged blobs W ≃ ξuf 2r2D (all energies are given in kBT units) is on the order of the thermal energy, we get ξ ≃ cs/f 2.16 The average volume fraction (concentration) of monomer units in the CCC ϕ ≃ ξ−1 is equal to the concentration within the correlation blob. Therefore l 1/3 2/3 * o o o u f , cs ≪ cs ϕ≃m o o o f 2 /cs , cs ≫ cs* n
∫R
R+H
dr = ξA(r )
pst ≃ γ
6/5
R st ≃ γ
4/5
(NB/ϕ)
2/5
3/5
(NB/ϕ)
≃
≃
l 2 8/15 o o (uf ) , cs ≪ cs* o o(f 2 /cs)8/5 , cs ≫ cs* n
o NB4/5m o
l 2 1/15 o o (uf ) , cs ≪ o(f 2 /c )1/5 , c ≫ o s s n
o NB3/5m o
(4)
cs* cs*
(5)
l 2 4(1 − ν)/15 o , cs ≪ cs* o o (uf ) Hst ≃ vA2ν − 1NAν NB2(1 − ν)/5m o o o(f 2 /cs)4(1 − ν)/5 , cs ≫ cs* n
(1)
i R + H yz zz p lnjjj k R {
(3)
with polymer density within the corona φ A (r) ≃ v(1−2ν)/ν ξA(r)(1−3ν)/ν being the function of the blob size ξA(r) A and under good (athermal) solvent conditions of the second virial coefficient vA ∼ 1. Equation dF/dp = 0 has an analytical solution only in two limiting cases, R ≪ H and R ≫ H, corresponding to so-called starlike (“st”) and crew-cut (“cc”) micelles. In both cases, the Fcore term is small compared to Fsurf and Fcorona and can be omitted. For starlike C3Ms, R ≪ H, we get with the account of eq 1 asymptotic scaling laws for the low and high salt regimes:
The surface tension of the coacervate complex scales as γ ≃ ξ ≃ ϕ2 in kBT/a2 units.16 These scaling theory results for the CCC properties16 account explicitly for connectivity of polymer charges,17−19 in contrast to popular Voorn−Overbeek coacervation theory20 which disregards the polymer nature of macroions and thus leads to erroneous power-law dependences of the coacervate structural properties on the key control parameters. Upon deriving eq 1 the salt concentration mismatch between the CCC and the surrounding solution has been neglected, which is justified at f ≪ 1.16 For highly charged chains, salt ions are expelled from dense coacervate because of the excluded volume interactions,17,21 and therefore, ϕ(cs) and γ(cs) dependencies are different.22,23 Equilibrium properties of spherical C3Ms can be found from minimization of the free energy F per AB block copolymer with respect to aggregation number p.24 The free energy F = Fcore + Fsurf + Fcorona contains three terms responsible for deformation of the core-forming blocks, an excess free energy of the core− corona interface, and corona free energy, respectively. The radius of the spherical micelle core is R ≃ (NBp/ϕ)1/3, and stretching of polyions within the core costs free energy Fcore ≃ R2/NB. The core area per one chain is s ≃ R2/p and Fsurf ≃ γs. Finally, Fcorona accounts for the short-range repulsions of monomers and stretching of chains in the corona. We denote the Flory exponent for the A block by ν and consider here the cases of theta and good solvent with νΘ = 1/2 and ν+ = 3/5, respectively. The corona of the spherical micelle consists of spherical layers of densely packed blobs with the size ξA(r) ≃ r s /R increasing from the center to the edge.25,26 Corona free energy is equal to the number of blobs per one A block
∫R
φA (r )ξA2(r )dr
≃ vA(1 − 2ν)/ νp(1 − ν)/2ν [(R + H )1/ ν − R1/ ν ] ≃ NA
−2
Fcorona ≃
R+H
(6)
As follows from eqs 4−6, an increase in the degree of ionization of the ionic blocks leads to stronger coacervation and, hence, to an increase in both core density ϕ and surface tension γ. For these reasons, both the micelle aggregation number and the dimensions increase as f grows. Remarkably, a weak increase in the core size as a function of f is a result of counterbalance between growing aggregation number and increasing core density. Addition of salt, on the contrary, weakens attraction between polyions within the core and in the high salt regime, cs ≫ c*s , leads to a simultaneous decrease in p, R, and H. For given block lengths NA, NB and fixed u and vA values, the micelle core should be dense enough to ensure starlike micelle geometry, R ≪ H. This is the case at f ≫ f st↔cc in salt-free 0 solution (subscript “0”) or at f ≫ fst↔cc at high salt concentration where f 0st ↔ cc ≃ u−1/2(NB1+ 2ν /vA5(2ν − 1)NA5ν)3/2(3 − 4ν) , cs ≪ cs* f st ↔ cc ≃ cs1/2(NB1+ 2ν /vA5(2ν − 1)NA5ν)1/2(3 − 4ν) , cs ≫ cs* (7)
and the latter threshold can be also presented as f ≃ f st↔cc (cs/c*s )3/2. Equivalently, the salt concentration cs should 0 not exceed the threshold value cst↔cc ≃ c*s (f/f st↔cc )2/3 to keep s 0 st↔cc grows as ∼f 2 upon an micelles in the starlike shape. Here cs increase in f. Beyond this boundary, at cs ≫ cst↔cc , the micelles acquire s crew-cut shape with a bulky core and thin corona, R ≫ H. Linearization of eqs 2 and 3 results in Fcorona ≃ H/ s and H ≃ v(2ν−1)/ν NAs−(1−ν)/2ν. Thus, equilibrium aggregation number p A reads st↔cc
(2) 812
DOI: 10.1021/acsmacrolett.8b00316 ACS Macro Lett. 2018, 7, 811−816
ACS Macro Letters pcc ≃
Letter
l 2 2(4ν − 1)/3(2ν + 1) o , cs ≪ cs* o (uf )
o NB2(vA2ν − 1NAν )−6/(2ν + 1)m o
o o(f 2 /cs)2(4ν − 1)/(2ν + 1) , cs ≫ cs* n (8)
and the corresponding core and corona dimensions obey the following scaling laws R cc ≃
Hcc ≃
l 2 (2ν − 1)/3(2ν + 1) o , cs ≪ cs* o (uf )
o NB(vA2ν − 1NAν )−2/(2ν + 1)m o
o o(f 2 /cs)(2ν − 1)/(2ν + 1) , cs ≫ cs* n
l 2 2(1 − ν)/3(2ν + 1) o , cs ≪ cs* o (uf )
(9)
o (vA2ν − 1NAν )3/(2ν + 1)m o
o o(f 2 /cs)2(1 − ν)/(2ν + 1) , cs ≫ cs* n (10)
Hence, the effect of f and cs on equilibrium properties of starlike and crew-cut micelles is qualitatively similar (though the scaling exponent values are different): an increase in f leads to increasing aggregation number and micelle dimensions, whereas an increase in cs leads to a decrease in p, H, and R, with the remarkable exception of Rcc = const{cs, f} at ν = 1/2. In the latter case, a decrease in the aggregation number is exactly countervailed by the core swelling. Figure 1 shows the dependencies of the aggregation number (a) and core and corona dimensions (b) of spherical micelles on salt concentration cs at f ≫ f st↔cc , i.e., when micelles in salt0 free solution are starlike. At cs < cs* neither the structure of the coacervate core nor the micelle physical properties are considerably affected by salt. Above the c*s threshold the polymer concentration in the core ϕ decreases, and the aggregation number p and the corona thickness H decrease as well due to weakening of the coacervation-induced aggregation. Finally, at cs ≫ cst↔cc ≃ c*s (f/f st↔cc )2/3 the micelles s 0 acquire crew-cut shape with R ≫ H. This st → cc transition occurs only if the ionic block is long enough to ensure p(cst↔cc ) s 2ν−1 ν st↔cc ≫ 1, i.e., at N1/2 micelles B ≫ vA NA. Otherwise, at cs ≥ cs disassemble into unimers. Such salt-induced disintegration of C3Ms has been repeatedly experimentally observed.6,27 A decrease in ionization of the core-forming blocks f results in a narrowing of the stability range of starlike micelles which entirely disappears (c*s < cs < cst↔cc ) at f ≃ f st↔cc . At f ≪ f st↔cc s 0 0 the micelles are crew-cut at cs = 0, and their physical properties follow laws 8−10 at any cs as long as pcc ≫ 1. As follows from eqs 9 and 10, the ratio Rcc/Hcc ≫ 1 increases with increasing cs and/or decreasing f, which is indicative of possible micelle polymorphism at high cs/low f. The addition of salt can trigger the transformation of spherical crew-cut micelles into cylindrical ones and further to lamellae (or polymersomes). The driving force for these morphological transitions is a decrease in the stretching of the core-forming block B.28 However, the formation of nonspherical micelles leads not only to a diminution of Fcore but also to an increase in Fcorona since a volume available for coronal chains decreases and their short-range repulsion grows. The interplay between these trends determines positions of the morphological transition lines (binodals). Introducing index i denoting the micelle morphology with i = 1 for lamellae (“L”), i = 2 for cylinders (“C”), and i = 3 for spheres (“S”), one can express the aggregate core radius as Ri = iNB/sϕ. Below we show that the equilibrium area per chain s is
Figure 1. Equilibrium properties of spherical C3Ms under theta (Θ) and good (+) solvent conditions for coronal chains as functions of salt concentration cs: (a) aggregation number p and (b) core radius R and and N1/2 corona thickness H. Schematic log−log plot at f ≫ f st↔cc 0 B ≫ 2ν−1 ν vA NA.
virtually independent of i. The free energy penalty for the core block stretching
l o π 2/8, i = 1 (L) o o o N o o 2 Fcore = = 2 B2 m π /4, i = 2 (C) o NB sϕ o o o o o 27π 2/80, i = 3 (S) (11) n decreases in the sequence S → C → L, and bi are numerical coefficients found in ref 29. The surface term, Fsurf = γs, is independent of i. The corona free energy and its thickness can be calculated as performed in eqs 2 and 3 with an account for the radial dependence of the blob size ξA(r) = s1/2(r/R)(i−1)/2 for arbitrary core geometry. At small relative curvature, with the accuracy of linear in H(i)/R ≪ 1 terms, one obtains ÅÄ ÑÉ (i − 1) H (i) ÑÑÑÑ H (i) ÅÅÅÅ (i) Fcorona ≃ 1 − Å Ñ 4 s ÅÅÅÅÇ R ÑÑÑÑÖ (12) ÄÅ É Ñ ÅÅ (1 − ν)(i − 1) H (i) ÑÑÑÑ H (i) ≃ vA(2ν − 1)/ νNAs−(1 − ν)/2νÅÅÅÅ1 − Ñ ÅÅ 4ν R ÑÑÑÑÖ ÅÇ biR i2
(13)
These equations reveal increasing corona free energy upon S → C → L transitions. In the zero-order approximation both H(i) ≃ H(1) and Fcorona ≃ H(1)/ s do not depend on i, which allows us to find s ≃ v2(2ν−1)/(2ν+1) (NA/ϕ2)2ν/(2ν+1) for the A 813
DOI: 10.1021/acsmacrolett.8b00316 ACS Macro Lett. 2018, 7, 811−816
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ACS Macro Letters aggregate of arbitrary morphology. One can specify i-dependent contribution to the total free energy δF(i) = F(i) − γs − H(1)/ s being the small correction δF (i) = bii 2
NB 2 2
sϕ
− C0
(i − 1) ϕ(5 − 4ν)/(1 + 2ν)(vA2ν − 1NAν )7/(2ν + 1) 2i NB (14)
with unknown numerical coefficient C0 independent of the aggregate morphology, i, and emerging as a consequence of scaling approximation for the corona free energy.24,28 Since we focus on morphological transitions triggered by salt concentration and/or core block ionization, it is convenient to write down the binodal equation δF(i+1) = δF(i) as ϕ(i + 1) ↔ i = αi(1 + 2ν)/(7 − 8ν)NB2(1 + 2ν)/(7 − 8ν) (vA2ν − 1NAν )−11/(7 − 8ν)
Figure 2. Diagram of C3M morphologies (schematic log−log plot) at 2ν−1 ν N1/2 B ≫ vA NA; st ↔ cc crossover, S ↔ C and C ↔ L binodals, and L ↔ uni boundary are given by eqs 7, 15, and 17. The crossover between the low and the high salt regimes, c*s ≃ u−1/3f4/3, is shown in green dashed line.
(15)
with the numerical coefficients αi given by αi =
=
l −1/2 −3/4 o NB , cs ≪ cs* o ou f uni ≃ m o o o NB−1/4cs1/2 , cs ≫ cs* n
2i(i + 1) [bi + 1(i + 1)2 − bii 2] C0
l1, i = 1 (C ↔ L) π2 o o m o 2C0 o 21/10, i = 2 (S ↔ C) n
(17)
At f ≤ f uni attraction between oppositely charged B and C blocks becomes weak; the free energy gain upon their complexation is on the order of the thermal energy; and micellization is suppressed (see refs 19 and 31 for more details). Remarkably the S → C → L morphological transitions in C3Ms provoked by an increase in cs or decrease in f and followed by disintegration of micelles are caused by simultaneous diminution of the core density ϕ and surface tension γ ≃ ϕ2. In a narrow range of the block copolymer ν st↔cc composition, N1/2 > v2ν−1 (NB), starlike B A NA and f > f 0 micelles formed at low salt concentration can be successively transformed into crew-cut spherical and cylindrical micelles, further to polymersomes, and, eventually, disassembled upon increasing salt concentration. Our theoretical predictions are in line with experimental findings27 on C3Ms formed from poly(N-methyl-2-vinylpyridinium)-b-poly(ethylene oxide) (PM2VP-b-PEO) and poly(acrylic acid) (PAA). In the case of nearly equal lengths of ionic blocks, i.e., PM2VP41-b-PEO204 and PAA47, the saltinduced morphological transition from spherical to long wormlike micelles was detected. Another interesting point was an observation of large clusters in the vicinity of the saltinduced disintegration of C3M.32 In the framework of our theory this can be interpreted as formation of nonspherical aggregates in the range between spherical micelles and unimers corresponding to the regions of C and L in Figure 2. Our theory also allows obtaining semiquantitative numerical estimations of boundaries shown in Figure 2 for any particular case, e.g., NA = 100, vA = 1 (athermal solvent), and NB = 500, assuming all unknown numerical coefficients in scaling laws equal to unity. In the low salt regime, f uni ∼ 0.01, f C↔L ∼ 0.125, and f S↔C ∼ 0.38, while at high salt concentrations f uni ∼ C↔L S↔C 0.22c1/2 ∼ 0.50c1/2 ∼ 0.72c1/2 s ,f s , and f s with cs expressed in M. Although the above theory was developed for the case of symmetric coacervate with equal (and low) degree of the polyion ionization, f+ = f− = f, it can be generalized to the case
(16)
The peculiarity of the binodal equations for S → C and C → L transitions, ϕ3↔2 = ϕS↔C and ϕ2↔1 = ϕC↔L, is their difference only in numerical coefficients. This implies that the binodal lines are parallel to each other in cs−f coordinates in both high and low salt concentration regimes. The range of stability of cylindrical micelles in the low salt regime, c s ≪ c s* , is specified as f S ↔ C /f C ↔ L = (α2/α1)3(1+2ν)/2(7−8ν) that is approximately 2.1 and 3.04 in theta and good solvent for coronal chains, respectively. At cs ≫ cs* the region of equilibrium wormlike micelles becomes even more narrow, f S↔C/f C↔L = (α2/α1)(1+2ν)/2(7−8ν), that is, 1.28 in theta solvent and 1.45 in a good solvent. Alternatively, at a given f value, the salt concentrations corresponding to these binodals, cC↔L /cS↔C = (α2/α1)(1+2ν)/(7−8ν), differ only by the s s factors of 1.64 and 2.1 under good and theta solvent conditions, respectively. Hence, increasing degree of ionization of the core blocks provokes L → C → S transitions, whereas addition of salt induces an opposite sequence of morphology changes, S → C → L. One should also remember that instead of lamellae that usually precipitate from the solution due to van der Waals attractive forces28 the polymersomes can be observed in experiment.30 Our findings on the micelle polymorphism are summarized in the diagram of states (Figure 2) where the horizontal part of binodals corresponds to the low salt regime, whereas their asymptotic behavior f(i+1)↔i ∼ c1/2 at cs ≫ cs* s follows from eqs 1 and 15. The aggregation number in the crew-cut micelles is much larger than unity in the vicinity of the S → C binodal provided 2ν−1 ν that N1/2 B ≫ νA NA. This inequality ensures the formation of cylinders and lamellae under the addition of salt and/or decreasing ionization prior to the disintegration of the aggregates into unimers. Aggregation is favorable as long as the number NB/g of correlation blobs per one core block is large (with g ≃ u−2/3f−4/3 at cs ≪ cs* and g ≃ c2s f−4 at cs ≫ cs*), which yields f ≫ f uni with 814
DOI: 10.1021/acsmacrolett.8b00316 ACS Macro Lett. 2018, 7, 811−816
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ACS Macro Letters
(9) Nolles, A.; Westphal, A. H.; de Hoop, J. A.; Fokkink, R. G.; Kleijn, J. M.; van Berkel, W. J. H.; Borst, J. W.Encapsulation of GFP in Complex Coacervate Core Micelles. Biomacromolecules 2015, 16, 1542−1549. (10) Blocher, W. C.; Perry, S. L. Complex Coacervate-Based Materials for Biomedicine. WIREs Nanomedicine and Nanobiotechnology 2017, 9, 835. (11) Voets, I. K.; de Keizer, A.; Cohen Stuart, M. A. Complex Coacervate Core Micelles. Adv. Colloid Interface Sci. 2009, 147−148, 300−318. (12) Kramarenko, E. Yu.; Khokhlov, A. R.; Reineker, P. Stoichiometric Polyelectrolyte Complexes of Ionic Block Copolymers and Oppositely Charged Polyions. J. Chem. Phys. 2006, 125, 194902. (13) Spruijt, E.; Leermakers, F. A. M.; Fokkink, R.; Schweins, R.; van Well, A. A.; Cohen Stuart, M. A.; van der Gucht, J. Structure and Dynamics of Polyelectrolyte Complex Coacervates Studied by Scattering of Neutrons, X-rays, and Light. Macromolecules 2013, 46, 4596−4605. (14) Shusharina, N. P.; Zhulina, E. B.; Dobrynin, A. V.; Rubinstein, M. Scaling Theory of Diblock Polyampholyte Solutions. Macromolecules 2005, 38, 8870−8081. (15) Wang, Z.; Rubinstein, M. Regimes of Conformational Transitions of a Diblock Polyampholyte. Macromolecules 2006, 39, 5897−5912. (16) Rumyantsev, A. M.; Zhulina, E. B.; Borisov, O. V. Complex Coacervate of Weakly Charged Chains: Diagram of States. Macromolecules 2018, 51, 3788−3801. (17) Perry, S. L.; Sing, C. E. PRISM-Based Theory of Complex Coacervation: Excluded Volume versus Chain Correlation. Macromolecules 2015, 48, 5040−5053. (18) Qin, J.; de Pablo, J. J. Criticality and Connectivity in Macromolecular Charge Complexation. Macromolecules 2016, 49, 8789−8800. (19) Delaney, K. T.; Fredrickson, G. H. Theory of Polyelectrolyte Complexation − Complex Coacervates Are Self-Coacervates. J. Chem. Phys. 2017, 146, 224902. (20) Radhakrishna, M.; Basu, K.; Liu, Y.; Shamsi, R.; Perry, S. L.; Sing, C. E. Molecular Connectivity and Correlation Effects on Polymer Coacervation. Macromolecules 2017, 50, 3030−3037. (21) Li, L.; Srivastava, S.; Andreev, M.; Macriel, A. B.; de Pablo, J. J.; Tirrell, M. V. Phase Behavior and Salt Partitioning in Polyelectrolyte Complex Coacervates. Macromolecules 2018, 51, 2988−2995. (22) Spruijt, E.; Sprakel, J.; Cohen Stuart, M. A.; van der Gucht, J. Interfacial Tension between a Complex Coacervate Phase and its Coexisting Aqueous Phase. Soft Matter 2010, 6, 172−178. (23) Qin, J.; Priftis, D.; Farina, R.; Perry, S. L.; Leon, L.; Whitmer, J.; Hoffman, K.; Tirrell, M.; de Pablo, J. J. Interfacial Tension of Polyelectrolyte Complex Coacervate Phases. ACS Macro Lett. 2014, 3, 565−568. (24) Borisov, O. V.; Zhulina, E. B.; Leermakers, F. A. M.; Müller, A. H. E. Self-Assembled Structures of Amphiphilic Ionic Block Copolymers: Theory, Self-Consistent Field Modeling and Experiment. Adv. Polym. Sci. 2011, 241, 57−129. (25) Daoud, M.; Cotton, J. P. Star Shaped Polymers: A Model for the Conformation and Its Concentration Dependence. J. Phys. (Paris) 1982, 43, 531−538. (26) Birshtein, T. M.; Zhulina, E. B. Conformations of Chains Grafted to the Spherical Surface. Polym. Sci. U.S.S.R. 1983, 25, 834− 840. (27) van der Kooij, H. M.; Spruijt, E.; Voets, I. K.; Fokkink, R. G.; Cohen Stuart, M. A.; van der Gucht, J. On the Stability and Morphology of Complex Coacervate Core Micelles: From Spherical to Wormlike Micelles. Langmuir 2012, 28, 14180−14191. (28) Zhulina, E. B.; Adam, M.; LaRue, I.; Sheiko, S. S.; Rubinstein, M. Diblock Copolymer Micelles in a Dilute Solution. Macromolecules 2005, 38, 5330−5351. (29) Semenov, A. N. Contribution to the Theory of Microphase Layering in Block-Copolymer Melts. Sov. Phys. JETP 1985, 61, 733− 742.
of weak asymmetry of the polyions charges via substitution f = f+ f− .33 This approximation remains applicable as long as the ratio f+/f− or the inverse one is lower than 2÷3. Otherwise, coacervate ceases to be homogeneous because the sizes ξ±el−st ≃ (uf2±)−1/3 of polyanion and polycation electrostatic blobs as well as polymer densities within them become unequal, and the blob picture of the symmetric coacervate14−16 is violated. In conclusion, the scaling theory of complex coacervate core micelles was developed. It was shown that addition of salt weakens correlation attraction within the core, which leads to a simultaneous decrease of the core density and in surface tension and resultant changes in C3M structure, namely, decrease in the size and aggregation number of spherical C3Ms and their subsequent disintegration. At high degree of ionization of the core blocks f and/or their short length NB, micelles are starlike in the absence of salt, whereas upon an increase in cs the micelles acquire crew-cut shape. At lower f and/or larger NB values the micelles are crew-cut at cs = 0, and increasing solution salinity induced consequent sphere → cylinder → lamellae transitions succeeded by dissolution of aggregates at higher cs.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Artem M. Rumyantsev: 0000-0002-0339-2375 Oleg V. Borisov: 0000-0002-9281-9093 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A.M.R and O.V.B. acknowledge support from the ANR MESOPIC Project ANR-15-CE07-0005 and Russian Science Foundation, Grant No. 14-33-00003.
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REFERENCES
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