Self-Assembly of Linear-Dendritic and Double Dendritic Block

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

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Self-Assembly of Linear-Dendritic and Double Dendritic Block Copolymers: From Dendromicelles to Dendrimersomes Inna O. Lebedeva,†,‡ 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, 64053 Pau, France ‡ Peter the Great St. Petersburg State Polytechnic University, 195251 St. Petersburg, Russia § Institute of Macromolecular Compounds of the Russian Academy of Sciences, 199004 St. Petersburg, Russia ABSTRACT: Nanostructures formed upon self-assembly of amphiphilic block copolymers with linear and dendronized blocks are promising as nanocarriers for targeted drug and gene delivery. We propose a theoretical framework to predict how dimensions and morphologies of such nanostructures as well as the number of potentially functionalized terminal groups exposed to surrounding media can be controlled by adjusting molecular architecture parameters of dendronized blocks (i.e., number of generations and branching functionality). We demonstrate that dendronization of the soluble block stabilizes spherical micelles with a dendritic corona, whereas block copolymers with dendritic insoluble blocks give rise to cylindrical wormlike micelles or polymersomes even if homologous linear-linear block copolymers assemble into spherical micelles. Spherical micelles with a dendronized corona combine smaller hydrodynamic dimensions with a larger number of terminal groups as compared to micelles formed by homologous linear diblock copolymers. Our findings provide guidelines for design of targetable vector systems in nanomedicine.

1. INTRODUCTION

Furthermore, functionalization of corona hydrophilic blocks with targeting ligands enables exploiting biorecognition mechanisms and thus assuring targeted delivery of drugs to specific tissues or cells. In this perspective, replacement of linear block copolymers by linear-dendritic or double dendritic block copolymers20−23 could substantially increase the number of potentially functionalizable terminal groups that are either exposed to the environment providing multiple sites for molecular recognition or embedded into the core domain and enhance loading capacities by providing more binding sites for the cargo.24 Complexes of siRNA with hydrophobically modified cationic dendrimers were explored in gene therapy, and their high efficiency was proven.25−27 The main features of self-assembly of linear diblock copolymers in selective solvents are well rationalized on the basis of existing theories6−11,14,15 that have also enabled one to explain and predict polymorphism of resulting nanoaggregates as a function of the copolymer composition and strength of intermolecular interactions. These theoretical predictions were convincingly validated experimentally in the past decades. In particular, stimuli-responsive properties of ″smart″ blockcopolymer nanostructures, that is, their ability to change size, shape, and aggregation state as a response to varied

Spontaneous self-assembly of block copolymers in selective solvents represents one of the most elaborated approaches for fabrication of aggregately stable nanoparticles with controlled and predictable multicompartment structures and interactive and stimuli-responsive properties. Accumulated up-to-date experimental knowledge1−5 supported by comprehensive theories6−15 enables obtaining self-assembled polymeric nanoaggregates with almost arbitrary complex shapes and desirable properties by rational design and chemical synthesis of elementary macromolecular building blocks. A brief list of emerging and potential applications of polymeric nanostructures includes smart nanocontainers with programed uptake and release profiles for active substances (cosmetics, agrochemistry), templates of nanoelectronic devices (nanowires, nanodots) or nano- and mesoporous hybrid materials, micellar catalysis, colloidal (bio)nanoreactors, and so on. In the past decades, nanosctructures of amphiphilic block copolymers are actively explored in medicine for anticancer drug and gene delivery purposes.1,16−18 A proper design of the constituent copolymers enables simultaneous control of such properties crucial for optimal performance of the drug nanocarriers as their stability, stealth properties, loading capacity, and overall dimensions. The latter is most important in light of exploitation of enhanced permeability and retention (EPR) of micellar carriers in targeted tumor tissues.19 © XXXX American Chemical Society

Received: January 19, 2019 Revised: April 11, 2019

A

DOI: 10.1021/acs.macromol.9b00140 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

is assumed to be the same for both blocks, is on the order of the Kuhn segment length. The solvent is assumed to be poor for monomer units of block B, whereas for monomer units of block A, it is moderately good, and its thermodynamic quality is characterized by the monomer-excluded volume parameter (second virial coefficient) va3 ≅ a3. Spontaneous assembly of such copolymers in dilute solution is driven by solvophobic interactions between insoluble (associating) blocks B and gives rise to nanostructures with diverse morphologies, for example, spherical or cylindrical (wormlike) micelles or polymersomes (the latter are thermodynamically equivalent to lamellar structures). In such nanostructures, condensed solvophobic blocks B form solvent-poor core domains stabilized by a solvated corona comprising blocks A (Figure 1b). In the strong segregation limit assured by poor solubility of block B, the core−corona interface is narrow compared to the dimensions of both core and corona domains so that a solvated corona of the micelle can be envisioned as a swollen brush formed by blocks A tethered to the surface of the core. In turn, the blocks B form a dry brush tethered to the inner side of the core−corona interface. In the following, we ascribe index i = 1,2,3 to aggregates with (quasi)planar, cylindrical, and spherical morphologies, respectively. The uniform packing condition of blocks B in the core imposes the relation between the core−corona interface area s per block copolymer and the size R of the core domain (that is, the radius of the spherical or cylindrical core or half-thickness of the quasi-planar B domain in lamellae) as

environmental conditions, have been extensively explored both experimentally and theoretically.14 However, much less is known about the effect of the macromolecular architecture, that is, topology (degree and mode of branching or macrocyclization) of one or both soluble and insoluble blocks, on the self-assembling behavior of the block copolymers.28,29 In our recent work,30 we have considered the effect of dendronization of the soluble block on structural properties of spherical micelles formed by lineardendritic block copolymers with soluble dendritic blocks. The aim of the present paper is to develop a comprehensive theory of self-assembly of linear-dendritic and double dendritic non-ionic block copolymers (the latter are also coined as ″Janus-dendrimers″) in selective solvent and to examine theoretically how dendronization of one or both blocks affects structural properties and morphologies of the self-assembled aggregates. The rest of the paper is organized as follows: In Section 2, we present the model of linear-dendritic and double dendritic block copolymers and outline the self-consistent field formalism. In Section 3, we discuss the effect of dendronization of diblock copolymers on the equilibrium properties of spherical micelles (Section 3.1) and on polymorphism of the aggregates formed by linear-dendritic and double dendritic block copolymers (Section 3.2). Our conclusions are summarized in Section 4.

2. MODEL AND FORMALISM 2.1. Model. Consider a diblock copolymer comprising block A with a degree of polymerization NA and block B with a degree of polymerization NB in selective solvent. We distinguish linear-dendritic and double dendritic block copolymers where one linear and one dendritic or two dendritic blocks are linked together through the focal point(s) (Figure 1a). The number of generations in the blocks is gA, gB

s=

iNBa3 , Rϕ

i = 1, 2, 3

(1)

where ϕ ≃ 1 is the volume fraction of monomer units B in the core. The number p of block copolymer chains in one spherical micelle is given by p=

4πR3ϕ 3NB

2.2. Self-Consistent Field Formalism. The equilibrium structural and thermodynamic characteristics of the selfassembled diblock copolymer aggregates of morphology i = 1, 2, 3 can be obtained by minimizing the free energy per molecule (i) (i) (i) F (i) = Fcorona + Finterface + Fcore

(2)

which comprises the corona free energy, Fcorona, the free energy of the core-corona interface, Finterface, and the contribution Fcore arising due to conformational entropy losses of core-forming blocks B. The free energies of the core and the corona domains can be calculated using strong stretching self-consistent field (SSSCF) approximation proposed initially for brushes formed by linear chains34 and generalized later for dendron brushes.31−33 The main prerequisites for the SS-SCF approach are (i) Gaussian entropic elasticity of all linear segments in the brushforming macromolecules on any length scale and (ii) the absence of the dead zone depleted from the end segments of the chains/terminal branches inside the brush. The latter requirement is fulfilled for concave and planar brushes formed by linear or dendritically branched macromolecules but, in a general case, violated for convex brushes. However, as was demonstrated in refs 31−33, for regular dendron brushes

Figure 1. (a) Schematics of linear-dendritic and double dendritic block copolymers. (b) Solution nanostructures formed by copolymers with soluble dendritic blocks in selective solvent.

= 0,1,2,... where g = 0 corresponds to a linear block; the functionalities of the branching points are equal to qA, qB = 1, 2, 3, ... where q = 1 corresponds to a linear block. The linear blocks as well as all linear segments (spacers and branches) of the dendritic blocks are assumed to be intrinsically flexible; that is, the monomer unit size a, which B


Article

Macromolecules grafted onto convex surfaces, an increase in the degree of branching leads to the pronounced decrease in the width proximal to the grafting surface dead zone. Below, we implement SS-SCF approximation to calculate the free energy of core and corona domains in self-assembled aggregates formed by block copolymers with a dendron-like architecture of one or both of the blocks. Our calculation based on SS-SCF approximation remains sufficiently accurate as long as soluble blocks are branched and/or the curvature of the core−corona interface is small or vanishing. 2.2.1. Free Energy of Corona. Within SS-SCF approximation, the self-consistent molecular potential acting on the monomers of corona chains is parabolic, δfint {c(z)} δc(z)

= kBT

3 2 2 κA (D − z 2) 2a5

c(z) =

κA 2NAκA = κlinear π

D

∫0

(D 2 − z 2 )

(6)

c(z)s(z)dz = NAa3

with

i z + R yz zz , s(z) = s ·jjj i = 1, 2, 3 (7) k R { being the z-dependent area per block A in the corona, provides the equation for corona thickness D in spherical and cylindrical geometries, i−1

l −1/3 o 3 D yz ij o o o jj1 + zz , i=2 o o 8 R{ o k o o D = D1·m o −1/3 o o ij 3D 1 D2 yzz o o j o + + 1 , i=3 jj z o o 4R 5 R2 z{ o k n where

(3)

(8)

ij a 2 yz NAv1/3jjj zzz ηA−2/3a (9) k s { is the thickness of a planar brush, i = 1, with grafting area s per molecule. The free energy of the corona includes the contributions accounting for conformational entropy of blocks A, and for excluded volume interactions, i8y D1 = jjj 2 zzz kπ {

1/3

1/3

(i) (i) (i) Fcorona = Felastic + Fint

=

(4)

∫0

D

(i) f elastic (z)s(z)dz +

∫0

D

fint {c(z)}s(z)dz (10)

which is equal to unity for linear chains and is larger than unity (ηA > 1) for any branched architectures of blocks A. In the framework of SS-SCF approximation, the topological ratio η = κ/κlinear of regular dendrons depends on the number of spacers/branches and their connectivity in the macromolecule but does not depend on the spacer length and thus is independent of the overall number of monomer units N in the macromolecule. Moreover, η does not depend on the solvent quality (strength of monomer−monomer interactions) or other system parameters such as interfacial area s or the curvature of core−corona interface. The analytical expressions for η = η(g, q) in brushes of symmetric dendrons with branching activity q (that is, the number of spacers emanating from each branching point) and the number of generations g ≤ 3 can be found in ref 33, whereas for a particular case of q = 2, the values of η for dendrons up to the eighth generation were calculated numerically in ref 31. In the limit of high branching activity, q ≫ 1, the topological ratio for regular dendrons can be approximated as η ≃ qg/2. Under good solvent conditions (i.e., assuming dominance of binary monomer−monomer repulsions in the corona), f int{c(z)} can be approximated as fint {c(z)}/kBT = a−3vc 2(z)

4va 2

Normalization of the polymer volume fraction profile,

where c(z) and f int{c(z)} are the volume fraction of monomer units A and density of the free energy of the excluded volume interactions in the corona at distance z from the surface of the core, κA is the so-called topological coefficient, D is the corona thickness, and kBT is the thermal energy. The topological coefficient κA is defined and can be calculated (analytically or numerically) for various architectures of brush-forming macromolecules by using the conditions of elastic force balance in the branching points. A full description on how to calculate the topological coefficient can be found in refs 31 and 33. For the brush of linear chains, κlinear = π/2NA. As a quantitative measure of the degree of branching of brush-forming corona blocks, we introduce the topological ratio ηA =

3κA 2

f(i) elastic(z)

where f int{c(z)} and are the densities of free energy of excluded volume interactions and of the elastic (conformational) free energy in the corona, respectively. As long as chains in the corona-forming blocks exhibit Gaussian elasticity, the density of elastic free energy can be expressed as (i) (z ) f elastic

kBT

=

3κA 2 T (i)(z) = 2kBT 2s(z)a5

∫z

D

z′c(z′)s(z′)dz′

(11)

where T(i)(z) is the flux of elastic tension per unit area. After performing integration, one gets (i) (z ) Felastic 3κ 2 = A5 kBT 2a

∫0

D

z 2c(z)s(z)dz

(12)

By substituting the polymer volume fraction profile given by eq 6 into eqs 5, 10, and 12, we obtain the free energy of the corona as (i) Fcorona

kBT

(5)

=

(1) Fcorona (s) ijj

kBT

D yz jjj zzzz k D1 {

5

l 5 D o o 1+ , i=2 o o o 12 R o o ·m o o 5D 5 D2 o o o + + 1 , i=3 o o 6R 21 R2 n

(13)

which taking into account eq 3 leads to an explicit expression for the polymer volume fraction profile c(z) in the corona,

where C


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5 (1) i a 2 yz Fcorona 9 k 4s ij D1 yz 9π 2/3 2/3j jj zz j z = = N v j z A j s z kBT 20 va 2 k a { 10 k {

2/3

ηA2/3

(i) Finterface γs(x) = kBT a2 l o sa−2 , i=1 o o o o o o 3/2 1/2 o o o o πηA jijj NB zyzz x 3/2ijj1 + 3 x yzz , o i=2 o 3/2 1/2 j z z j 8 { NA v k ϕ { k = γo m o o o o o 1/2 3/2 o o o πηA jij NB zyz ij 3x yz3/2ijj 3 1 2yzz o o z j z j x x 1 , i=3 + + z j o z j j z j z o o 4 5 zz{ o NA3/2v1/2 k ϕ { k 2 { jjk n

(14)

is the free energy (per molecule) in a planar corona with area s per molecule. 2.2.2. Core Free Energy. The elastic free energy of block B in the core with radius R can be presented as (i) Fcore =

∫0

R

(21)

(core, i) f elastic (z)s(z)dz

where γkBT/a is the surface tension at the core−corona boundary (i.e., surface free energy per unit area of interface) which is with good accuracy equal to that at the interface between the collapsed core are pure solvent. Below, we use parameter γ as a dimensionless surface tension. 2.2.4. Free Energy per Molecule (i = 1,2,3). In a planar-like aggregate (i = 1) with interfacial area s per molecule, the free energy per chain is specified by eqs 14 and 18 as 2

(15)

where the density of the elastic free energy can be written similarly to eq 11, (core, i) (z ) f elastic

kBT

=

3κB 2 2s(z)a

5

∫z

R

z′ϕ(z′)s(z′)dz′,

0≤z≤R

ij a 2 yz F (1) 9π 2/3 = NAv 2/3jjj zzz kBT 10 k s {

(16)

with area per molecule s(z) decreasing as a function of distance z from the core−corona interface as iR − s(z) = s ·jjj k R

z yz zz {

i = 1, 2, 3

l o π 2/8, i=1 o o o o 2 biR 2 NBa 2 o = η = 2 2 ηB ·m π /4, i=2 o o kBT NBa 2 B sϕ o o o 2 o 27π /80, i = 3 n 2

l 5 o o o 9 1 + 12 x o o , i=2 o o o 10 x 1 + 3 x 2 o o (i) 2 o 8 Fcorona(x) vNAϕ o = ·o m o o 5 5 kBT NB o 1 + 6 x + 21 x 2 o o 3 o o , i=3 o o o5 3 1 2 2 o o 1 x x x + + o 4 5 n

( ( ( (

4

(18)

) )

(23) (i) Fcore (x ) N 3vϕ ij η yz = A 2 jjjj B zzzz · kBT NB k ηA { 2

−1 l1 o 3 y i o o x−3jjj1 + x zzz , i=2 o o o 4 k 8 { o o m o −1 o o i 1 3 1 y o o o x−3jjj1 + x + x 2zzz , i = 3 o o 10 k 4 5 { n

(19)

By combining eqs 1, 8, and 9, we express interfacial area s in spherical (i = 3) and cylindrical (i = 2) aggregates as πη i N y s (x ) = 3/2A1/2 jjjj B zzzz · 2 a NA v k ϕ { l 1/2 o 3 y i o o o x 3/2jjj1 + x zzz , i=2 o o 8 { o k o m o 1/2 3/2 o o 3 1 y i 3x y i o o o jjj zzz jjj1 + x + x 2zzz , i = 3 o ok 2 { k 4 5 { n

) )

Combination of eqs 18 and 20 provides an explicit equation for F(i) core(x) as

with morphology-dependent numerical prefactors bi coinciding with that obtained earlier34 for the case of a linear core-forming block (corresponding to ηB = 1). 2.2.3. Interfacial Free Energy. In non-planar geometries of the interface, area s per molecule can be presented as a function of dimensionless ratio x of corona thickness D and core radius R, x = D/R

4 π 2 NBa 2 η + γsa−2 8 s 2ϕ 2 B

and depends on a single independent parameter s. In nonplanar geometries of aggregates with i = 2, 3, the free energy per chain in the corona and core of the aggregate can be formulated as a function of a single independent parameter x = D/R. Using eqs 9,14, and 20, one finds

(17)

and κB = πηB/2NB is the topological coefficient for the coreforming block B. Taking into account uniform density in the condensed core, ϕ(z) = ϕ = const, and using the packing condition, eq 1, one obtains after performing integration in eq 15 (i) Fcore

ηA2/3 +

(22)

i−1

,

2/3

(24)

while contribution of the core−corona interface to the total free energy is given by eq 21. Equilibrium structural and thermodynamic properties (free energy per block copolymer and CMC) for spherical (i = 3) and cylindrical (i = 2) micelles as a function of NB and NA, ηA, ηB, γ, ϕ, and v can be obtained by minimization of the total free energy per chain with respect to x, that is, from the equation

3/2

(i ) (i ) (i ) d[Fcorona (x) + Finterface (x) + Fcore (x)] d F (i ) = = 0, i = 2, 3 dx dx

(25)

(20)

F(i) core(x),

F(i) corona(x),

with and 21, respectively.

to give D

and

F(i) interface(x)

given by eqs 24, 23,


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Macromolecules For lamellar structures (polymersomes, i = 1), direct minimization of the free energy in eq 22 with respect to s dF (1)(s) =0 ds

(26)

provides equilibrium values of s and thus all structural properties and free energy per chain in lamellae (polymersome).

3. RESULTS AND DISCUSSION In what follows, we systematically compare structural properties of self-assemblies of linear AB diblock copolymers and those of linear-dendritic and double dendritic block copolymers with the same number of monomers NB and NA in insoluble and soluble blocks, respectively. 3.1. Spherical Micelles. We start with systematic discussion of properties of spherical micelles formed by linear-dendritic or double dendritic copolymers as a function of molecular masses and topology of both blocks. Part of the results for spherical micelles of linear-dendritic copolymers with associating linear block was reported earlier.30 3.1.1. Equilibrium Parameters of Spherical Micelles: Asymptotic Dependences. In the limiting cases of starlike, D ≫ R, and crew-cut, D ≪ R, spherical micelle asymptotic power law dependences for equilibrium structural and thermodynamic properties of the micelles on NA, NB, and ηA can be derived by keeping only the dominant terms on the lhs in eq 25 (i.e., the corona free energy and excess free energy of the core−corona interface) and neglecting the contribution of conformational entropy of the core-forming blocks. Within this (i) (i) approximation, minimization of [Fcorona (x) + Finterface (x)] provides the following asymptotic power law dependences for the aggregation number

Figure 2. Schematic diagram of states in NA, NB coordinates for diblock copolymers with linear insoluble (core-forming) blocks and linear (ηA = 1, solid lines) or dendronized (ηA ≥ 1, dashed lines) corona blocks. Parameters: γ = ϕ = v = 1. Below the green line, copolymer molecules are in the unimer state. Between the green and brown lines, spherical micelles are starlike. Between the red and brown lines, micelles have a crew-cut shape. The red line separates regimes of spherical micelles from non-spherical aggregates (shaded gray area). Equations for boundaries between different regimes are presented near the lines.

between different regimes are presented near the lines. Dashed lines indicate how the stability regions of spherical micelles shift upon dendronization of the corona-forming block A: an increase in ηA leads to widening of the regimes of crew-cut micelles and unimers at the cost of shrinking regime of starlike micelles. 3.1.2. Critical Micelle Concentration (CMC). In the closed association model of micellization, the critical micelle concentration (CMC) is specified by the difference between the free energies of a block copolymer molecule in the unimer state and in the equilibrium (optimal) micelle. By neglecting the translational entropy of micelles with aggregation number p ≫ 1, the expression for CMC yields

l 15/11 o (NB/ϕB)10/11(v 2NA )−3/11ηA−18/11 , R ≪ D o oγ p≅m o o o (γ /NA )9/5 (NB/ϕB)2 v−6/5ηA−6/5 , R≫D o n

(27)

ln CMC ≃

l 3/11 2/11 6/11 1/11 −8/11 o , R≪D o o γ (NB/ϕB) N A v ηA D/a ≅ m o o o γ 1/5N A4/5v1/5ηA−4/5 , R≫D o n

the corona thickness

+

i R(p = 1) yz −Fsurface(p = 1) + F(p) zz ≅ −γ jjjj z kBT a k {

F (p) kBT

2

(30)

where R(p = 1) ≃ a(NB/ϕ) is the radius of the condensed block B (core of the unimer) and F(p) is the free energy per chain in the equilibrium micelle with aggregation number p. The first (dominant) term in eq 30 does not depend on the architecture of block B and thereby is the same for linear and dendronized core-forming block with NB monomer units. The second term in eq 30 depends on the architecture (and molecular mass) of block A. Because in the equilibrium micelle, Fcorona ≃ Finterface ≃ γa−2kBTR2/p, by using eqs 27 and 29, one estimates CMC in eq 30 as 1/3

(28)

l 5/11 7/11 2 −1/11 −6/11 o ηA , R≪D o o γ (NB/ϕB) (v NA ) R /a ≅ m o o o γ 3/5(NB/ϕB)N A−3/5v−2/5ηA−2/5 , R≫D o n

and the core radius

(29)

in the equilibrium spherical micelles (i = 3). It is instructive to outline the ranges of NA and NB values in which diblock copolymer molecules self-assemble in starlike (D ≫ R) or crew-cut (D ≪ R) micelles or stay as unimers in the solution (the latter condition corresponds to p = 1 in eq 27). In Figure 2, we present the scaling-type diagram in NA, NB coordinates with regimes of thermodynamically stable unimers, spherical starlike, and spherical crew-cut micelles. The diagram also contains the regimes of non-spherical aggregates (discussed later in the paper) separated from the regimes of spherical micelles by a red line. Equations for the boundaries

ln CMC ≃ −(NB/ϕ)2/3 γ l −5/11 o (NB/ϕB)4/11(v 2NA )1/11ηA6/11 , R ≪ D o oγ +m o o o γ −3/5NA3/5v 2/5ηA2/5 , R≫D o (31) n As it follows from eq 31, in the limit of starlike micelles, the free energy per chain in optimal micelles increases upon an increase in NA as F(p) ∼ N1/11 A , and therefore, lnCMC depends E


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Figure 3. Aggregation numbers in spherical micelles formed by linear-dendritic block copolymers (dendritic block A) (a) as a function of NA at NB = 100 and (b) as a function of NB at NA = 100 for varied number of generations in the dendron block A as indicated at the curves. Dashed lines indicate the result obtained not taking into account the contribution Fcore. Stars indicate cross-over points, D = R, between starlike and crew-cut micelles. Other parameters are q = 3, γ = 1, v = 1, and ϕ = 1.

quite weakly on the degree of polymerization NA of the corona block. In the limit of crew-cut aggregates, the dependence of F(p) on NA is rather strong (F(p) ∼ N3/5 A ), but due to comparatively small values of NA (see the diagram in Figure 2), the dependence of CMC is predominantly governed by the degree of polymerization NB of the core-forming block B. In both scenarios, dendronization of the corona leads to the shift in CMC to larger values. Note that eq 31 does not account for the numerical coefficients on the order of unity and merely indicates the trends in the CMC behavior. 3.1.3. Equilibrium Parameters of Spherical Micelles: Full Solution. Equations 27−30 outline the general trends in the equilibrium behavior of starlike (D ≫ R) and crew-cut (D ≪ R) spherical micelles. Below, we present the results of full numerical solution of eq 25, taking into account all three contributions to the free energy, and compare these results to the asymptotic dependences predicted in eqs 27−29. As one can see in Figure 3 and in accordance with eq 27, the aggregation number p is a decreasing function of NA and an increasing function of NB. For given copolymer composition, p decreases upon an increase in the degree of branching of the coronal block A (an increase in the number of generations). Interestingly, taking into account conformational entropy of the core-forming blocks leads to systematically smaller aggregation numbers as compared to the results obtained from the balance of coronal and interfacial free energies only (dashed lines in Figure 3). This effect is explained by the increasing entropic penalty for extension of the core-forming block upon an increase in the aggregation number p and the core size R, and it is more pronounced in crew-cut rather than in starlike micelles. Figure 4 illustrates the effect of molecular mass and dendronization of the corona block A on the dimensions of micelles formed by linear-dendritic block copolymers. An increase in the number of monomers NA in the corona block leads to the increase in the corona thickness D, whereas the core size R, on the contrary, decreases (cf. eqs 28 and 29). As a result, the overall dimensions of micelles moderately increase as a function of NA. Dendronization of the corona blocks (at fixed NA) leads to the decrease in both corona and core sizes because of combined effects of enhanced repulsions in the dendronized corona and its simultaneous compactization. A decrease in the corona thickness D due to branching of block A is more pronounced than simultaneous shrinkage of the core, and as a result, micelles formed by linear-dendritic block

Figure 4. Dimensions of the corona D (solid lines) and the core R (dashed lines) and the overall dimensions R + D (insert) in spherical micelles formed by linear-dendritic block copolymers (dendritic block A) as a function of NA at NB = 100 for varied number of generations in the dendron block A as indicated at the curves. Other parameters are q = 3, γ = 1, v = 1, and ϕ = 1.

copolymers with sufficiently long core-forming B blocks retain their crew-cut shape. The effect of dendronization of the core-forming block on structural properties of spherical micelles formed by double dendritic block copolymers is illustrated in Figure 5. According to eqs 27−29 derived by the neglecting conformational entropy of the core-forming blocks, the properties of spherical micelles should be virtually independent of the topology of block B. However, as one can see from Figure 5a where the results obtained taking into account the conformational entropy of the core-forming blocks are presented, dendronization of block B leads to the noticeable decrease in the micelle aggregation number in the crew-cut domain, whereas for starlike micelles, this effect is indeed negligible. Consequently, micelles formed by double dendritic block copolymers have slightly smaller core and corona dimensions compared to those of their homologues with linear associating blocks as long as D ≪ R, but this difference is negligible for starlike micelles with D ≫ R. As follows from the presented results, dendronization of both corona- and core-forming blocks causes the decrease in the aggregation number p and may completely suppress micellization even though homologous linear-linear (or lineardendritic) block copolymers associate in the form of F


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Figure 5. (a) Aggregation number and (b) dimensions of the corona D (dashed lines) and the core R (solid lines) (the overall dimensions R + D are shown in the insert) in spherical micelles formed by double dendritic block copolymers as a function of NA at NB = 100 for varied number of generations in the core-forming dendron block B as indicated at the curves. Other parameters are q = 3 for both blocks,gA = 1, γ = 1, v = 1, and ϕ = 1.

l 2g /11 o , R≪D o oq Nt = pq ∼ m o o o q2g /5 , R ≫ D n

thermodynamically stable micelles. The effect of dendronization of the corona blocks in this respect is stronger than that of the core-forming block. 3.1.4. Number of Terminal Segments in Corona of Micelle. In addition to sizes of the core and corona, R and D, and aggregation number p, an important parameter is the number of potentially functionalizable terminal groups in the corona. In the case when blocks A are regular dendrons of generation g = 0, 1, 2, ... with branching functionality q = 1, 2, 3, ..., the number of terminal segments per micelle equals Nt = qgp. In Figure 6, we demonstrate how the number of terminal groups per micelle increases upon replacing a linear block A by

g

(32)

Remarkably, dendronization of block A (at a constant value of NA) leads to only a subexponential increase in the number of terminal groups per micelle as a function of the number of generations g because of the concomitant decrease in aggregation number p. It is also instructive to analyze how the aggregation number p and number Nt of terminal groups vary as a function of number of generations g in the dendron block A if the number of monomer units in each spacer, n, is kept constant. In this case, dendronization of the corona block is accompanied by an increase in NA as NA = n

q g+1 − 1 q−1

(33)

As one can see from Figure 7, both p and Nt are decreasing functions of the number of generations g in the corona block A. The decrease in p as a function of g is more pronounced for larger q since it corresponds to larger values of ηA and NA for the same g. However, in terms of Nt, this trend is partially counterbalanced by larger values of qg of terminal groups in dendrons with larger functionality q. As a result, the number of terminal groups relatively weakly increases or even remains the same if linear-dendritic block copolymers with larger functionalities of the dendron blocks are used. These results are again in qualitative agreement with asymptotic estimates obtained from eqs 27 and 33 l −g /11 o , R≪D o oq Nt = pq ∼ m o − 7 g /5 o oq , R≫D n

Figure 6. Number of terminal segments of the coronal blocks per micelles Nt (solid lines) and aggregation number p (dashed lines) as a function of number of generations at different functionalities of the branching points q = 2 and 3 in the dendritic coronal block A and constant composition (NA = 100, NB = 100) of linear-dendritic block copolymers. Other parameters are γ = 1, v = 1, and ϕ = 1.

g

(34)

The effect of spacer length n on the number of terminal groups Nt per micelle as a function of number of generations g is illustrated by Figure 8. Here, the branching functionality is q = 2, and we used values of ηA calculated numerically in ref 31 for g ≤ 9. As what follows from Figure 8, the number of terminal segments per micelle decreases as a function of g at any fixed spacer length n and as a function of n at given number of generations g in the dendron block. This can be again explained by enhanced repulsive interactions in the corona leading to a strong decrease in aggregation number p.

a dendron of the first, second, and third generations with functionalities of branching points q = 2 or 3 keeping NA = const. Evidently, the number of terminal groups is an increasing function of both g and q. This result is in agreement with the following estimates obtained using eq 27 for p and asymptotic dependence η ∼ qg/2, which is valid at high branching activity q ≫ 1, G


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Figure 7. Number of terminal segments of the coronal blocks per micelle Nt (solid lines) and aggregation number p (dashed lines) as a function of number of generations at different functionalities of the branching points q = 2 and 3 in the dendritic coronal block A and constant number of monomers n = 100 in a spacer of dendritic coronal block A. The number of monomer units in the linear B block is equal to (a) NB = 100 and (b) NB = 500. Other parameters are γ = 1, v = 1, and ϕ = 1.

block copolymers with dendritically branched corona block A are plotted. Solid lines indicate binodals for the case of linearlinear block copolymers. Dashed and dotted lines correspond to the binodals for block copolymers with dendritically branched corona block A with varied number of generations and branching functionality. According to Figure 9, an increase in degree of branching of block A results in the shift of the sphere−cylinder−lamella transitions to the range of smaller NA, stabilizing thereby the spherical shape of micelles. This trend is explained by the fact that enhancing the excluded volume repulsion in the dendronized corona favors the formation of aggregates (spherical or cylindrical) with less crowded corona blocks. As a result, micelles formed by copolymers comprising blocks with the same polymerization degrees NA and NB but with a different topology of the corona block A may have completely different morphologies. For example, if wormlike cylindrical micelles are formed by linearlinear block copolymers with some specific composition, a homologous block copolymers with dendritically branched corona block A may form spherical micelles. In order to understand how the degree of branching of the core block B affects the morphology of the aggregates, we constructed the diagram of states for aggregates formed by double dendritic block copolymers whose corona block A was weakly branched (gA = 1 and qA = 3), whereas the architecture of the core block B was varied from linear (solid line) to dendrons of the first and the third generations (dashed lines). It is seen from Figure 10 that in this case, an opposite effect is observed, that is, the binodal lines are shifted to larger values of NA at a constant value of degree of polymerization of the dendritically branched block NB, and this shift becomes more pronounced as the degree of branching of the insoluble block increases. Hence, the range of the copolymer compositions (NA,NB) corresponding to the stability of the non-spherical morphologies (cylinder or polymersomes) widens. Finally, in Figure 11, we present a diagram of morphological states for symmetric double dendritic block copolymers (″Janus-dendrimers″) with the same functionality qA = qB = 3 and numbers of generations gA = gB = 0, 1, 2 in both blocks. As discussed above, dendronization of the insoluble block stabilizes non-spherical morphologies, whereas dendronization of the soluble block causes the opposite effect. As what follows from Figure 11, in symmetric ″Janusdendrimer″, the effect of dendronization of insoluble block B

Figure 8. Number of terminal segments of the coronal blocks per micelle Nt as a function of number of generations at different lengths of spacers n = 10, 20, 30 and fixed functionality of the branching points q = 2 in the dendritic coronal block A. The number of monomer units in the linear B block is equal to NB = 200. Other parameters are γ = 1, v = 1, and ϕ = 1.

3.2. Polymorphism. It is known that diblock copolymers with linear blocks can associate in aggregates of various morphologies in dilute solutions.15 The morphological transitions occur in the crew-cut domain in which the corona thickness is smaller than the core size in the aggregate. The ranges of thermodynamic stability of spherical and cylindrical micelles and polymersomes are separated by binodal lines. The binodals separating ranges of stability of aggregates with morphologies i and i + 1 (i = 1,2) are defined by the condition (35) F (i) = F (i + 1) Equation 35 allows for calculation of binodal lines provided that the free energies F(i) of the aggregates with different morphologies i are known. In the previous section, we have calculated the free energies per molecule in the spherical (i = 3), cylindrical (i = 2), and lamellar (i = 1) aggregates. Below, we apply eq 35 to find binodals for self-assembled aggregates formed by diblock copolymers with dendritic blocks. 3.2.1. Morphological phase diagrams. Figure 9 presents diagrams of morphological states, where binodal lines separating regions of thermodynamic stability of spherical, cylindrical, and lamellar aggregates formed by linear-dendritic H


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Figure 9. Diagrams of morphological states of aggregates formed by block copolymers with linear insoluble block B and linear or dendritically branched with different numbers of generations soluble block A. Solid lines correspond to linear-linear block copolymers, and dashed lines correspond to linear-dendritic block copolymers as indicated. (a) Functionality of the branching point of dendritically branched block A is qA = 3, and the number of generations is gA = 0, 2, 3; (b) functionality of the branching point in the dendritically branched block A is qA = 2, and the number of generations is gA = 0, 2, 6, 8. Blue lines denote binodals for lamella-to-cylinder transitions, and red lines denote binodals for sphere-tocylinder transitions.

Figure 11. Diagram of morphological states of aggregates formed by double dendritic block copolymers (dashed lines) and linear-linear block copolymers (solid lines). Large strokes correspond to the number of generations gA = gB = 2, and small strokes correspond to gA = gB = 3; functionality qA = qB = 3 for both cases. Blue lines correspond to binodals for lamella-to-cylinder transition, and red lines correspond to binodals of sphere-to-cylinder transitions

Figure 10. Diagram of morphological states of aggregates formed by double dendritic diblock copolymers; the soluble block A is a dendron of the first generation, gA = 1 and qA = 3, whereas the architecture of the insoluble block is varied. Solid lines correspond to the case of linear insoluble blocks, and dashed lines correspond to dendritic insoluble blocks with parameters (gB = 1, qB = 2) (large stroke) and (gB = 3, qB = 2) (small stroke). Blue lines denote binodals for lamellato-cylinder transition, and red lines denote binodals for sphere-tocylinder transition.

within ranges of thermodynamic stability of each particular morphology (sphere, cylinder, or lamella), structural properties of the aggregates smoothly vary as functions of the block lengths. For linear-linear block copolymers, the morphological transitions occur in the crew-cut domain; that is, thermodynamically stable aggregates of non-spherical morphologies have a crew-cut shape with R ≫ D. In Figure 12, we presented the core, R, and the corona, D, dimensions (panel a) and overall size, R + D, (panel b) of the aggregates formed by diblock copolymers with linear solvophobic blocks and dendritically branched soluble blocks with different numbers of generations as a function of the degree of polymerization of the soluble block NA. As one can see from Figure 12, an increase in the degree of polymerization

appears to be stronger than the effect of dendronization of the soluble block A. A simultaneous increase in the number of generations in both blocks results in the shift of the binodal lines of sphere−cylinder−lamella transitions to a larger degree of polymerization NA of soluble blocks (or smaller degree of polymerization of insoluble block NB), that is, to extension of the range of stability of non-spherical aggregates. This trend is explained by the dominant effect of conformational entropy losses in the dendritically-branched core-forming blocks. Morphological transitions in block copolymer aggregates are accompanied by abrupt (jump-wise) variations in the dimensions of both the core and the corona domains, whereas I


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Figure 12. (a) Dependences of the core radius, R (dashed lines), and corona thickness, D (solid lines), and (b) overall size of the aggregate, R + D, on the degree of polymerization NA of the soluble block for linear-dendritic block copolymers with degree of polymerization NB = 700 of insoluble linear blocks and different numbers of generations in the dendritic corona-forming blocks; in panel (a), curves correspond to the cases of dendritic corona blocks with parameters qA = 3 and gA = 2 (dots) and qA = 3 and gA = 3 (triangles), respectively. In panel (b), black, red, and blue lines correspond to the aggregate formed by linear-linear diblock copolymers, linear-dendritic block copolymers with the corona block parameters qA = 3 and gA = 2, and linear-dendritic block copolymers with the corona block parameters qA = 3 and gA = 3, respectively.

Figure 13. Dependences of (a) the core, R (dashed line), and the corona, D (solid line) and (b) overall dimensions, R + D, of aggregates formed by double dendritic block copolymers on the degree of polymerization NA of the soluble block, which is the dendron of the first generation gA = 1 and qA = 3. In panel (a), the core-forming block is a dendron with NB = 300, qB = 2, and gB = 3. In panel (b), the architecture of the core-forming block is varied as follows: linear (black line), qB = 2, gB = 2 (red line), and qB = 2, gB = 3 (blue line).

of the soluble block NA results in a monotonic decrease in the dimensions R of the core domain and increase in the size of the corona D within ranges of stability of each particular morphology. Morphological transitions from dendrimersomes to cyindrical and further to spherical dendromicelles are accompanied by abrupt jumps up in the core size and a less pronounced concomitant drop in the corona dimensions. The latter feature is explained by decreasing crowding of the corona blocks upon transformation of the core from a lamella to a cylinder and to a sphere. As long as the aggregates have pronounced a crew-cut shapethat is, their overall size is controlled by the size of the corethe size R + D of the aggregate as a whole follows the same trend as the size of the core R (see panel c in Figure 12). However, upon decreasing the degree of branching of the soluble blocks, the relative contribution of the corona to the overall size of the aggregate increases. As a result, the overall size of the aggregates formed by linear-dendritic block copolymers with weakly branched soluble blocks or by linear-linear block copolymers (black

curve in panel c of Figure 12) may exhibit monotonic growth of the overall size as a function of NA: this growth is continuous within ranges of stability of each particular morphology and is interrupted by jumps up in the points of morphological transitions. Figure 13 illustrates the effect of branching of the core block B on relative dimensions of the core and the corona domains in the aggregates in the range of morphological transitions. As discussed above, increasing branching of the core-forming block leads to the shift of morphological sphere-to-cylinder and cylinder-to-lamella transition to smaller values of the degree of polymerization NB of the core block. Consequently, these transitions occur at a smaller R/D ratio and, as one can see from Figure 13, the morphological transitions from sphere to cylinder and from cylinder to lamella may occur even when the aggregates have a starlike shape, R ≤ D. As discussed before, aggregates of linear-linear block copolymers with D ≥ R always have a spherical shape. J


ÄÅ É ÅÅ i 2/3 y6/5ÑÑÑ−5/16 Å π 3 3 j zz ÑÑÑ z Ñ NAi ↔ (i + 1) = ÅÅÅÅ jjj ÅÅ 8 j 5 zz ÑÑÑ { ÑÑÖ ÅÅÇ k

Macromolecules 3.2.2. Approximate Expressions for Binodals. When the morphological transitions occur in the aggregates of the crewcut shape, the approximate analytical expressions for the binodals can be found using the expansion of thickness D and free energy F(i) corona of the corona up to linear terms in powers of relative curvature D1/R. Expanding the rhs of eq 8, we obtain i (1 − i) D1 yz zz, i = 1, 2, 3 D ≈ D1jjjj1 − 8 R z{ k

{i(i + 1)[(i + 1)2 bi + 1 − i 2bi ]}5/16 ·NB5/8v−9/16 5/8 γ 3/8 ηB

ϕ15/16 ηA1/4

(1) Fcorona (s) ij 5(1 − i) D1 yz jj1 + zz, i = 1, 2, 3 j kBT k 24 R z{

(37)

Then, the free energy (per block copolymer) in the crew-cut aggregates of morphology i can be presented as (1) (i) F (i)(s) ≈ Fcorona (s) + (kBT )γ(s /a 2) + Fcore (s ) (i) + δFcorona (s )

(43)

Analysis of eqs 42 and 43 confirms the trends observed in the morphological phase diagrams in Figures 10 and 11: An increase in the topological ratio ηA (that is, an increase in the degree of branching) of the corona-forming block A leads to the shift of the binodal lines to smaller values of NA. On the contrary, an increase in the topological ratio ηB for the coreforming block causes an opposite effect. If both blocks have the same topology, then an increase in ηA = ηB shifts the equilibrium toward non-spherical aggregates (cylindrical dendromicelles and dednrimersomes). These trends are highlighted in the diagram of morphological states, Figure 14, in which positions of S - C and C - L

(36)

and from eqs 1, 2, and 4 we get (i) Fcorona (s ) ≈

Article

(38)

where (i) δFcorona (s ) =

(1) 5(1 − i) Fcorona(s)D1 , i = 1, 2, 3 24 R

(39)

is the incremental correction to the free energy of the corona arising because of its curvature. The first and the second terms in eq 38 are the dominant ones, and they are independent of curvature. Their balance (1) (s )a 2 ∂Fcorona + kBTγ = 0 ∂s

provides an approximate expression for surface area per molecule in the crew-cut spherical or cylindrical micelles or in polymersomes, ij 3π 2/3 yzij NA yz3/5 2/5 2/5 zzj z v η s1/a ≈ jjj A j 5 zzjj γ zz k {k {

Figure 14. Diagram of morphological states of aggregates of doubledendritic block copolymers in (Y, ηA, ηB) coordinates where v9/16γ−3/8ϕ15/16. The surfaces dimensionless parameter Y = NAN−5/8 B indicate boundaries between regions of thermodynamic stability of spheres (upper half-space), lamellae or dendrimersomes (lower halfspace), and cylinders (region between the surfaces).

2

(40)

which coincides with the approximate solution of eq 26 if the second term (F(1) core(s)) is neglected. After substituting eq 38 into eq 35, we formulate the binodal equations in the form (i) (i) (i + 1) (i + 1) Fcore (s) + δFcorona (s) = Fcore (s) + δFcorona (s )

binodals are presented as a function of two variables ηB and ηA according to eqs 42 and 43. Here, the dimensionless ratio + 1) −5/8 9/16 −3/8 15/16 Yi↔(i + 1) = Ni↔(i NB v γ ϕ is plotted as a function A of ηB and ηA. The slit between the binodal surfaces corresponds to the range of stability of cylindrical (dendro)micelles, whereas upper and the lower domains correspond to spherical micelles and dendrimersomes, respectively.

(41)

where s = s1 is given by eq 40. After simple algebra, one obtains explicit equations for the binodals as ÄÅ É ÅÅ i 2/3 y6/5ÑÑÑ1/2 ÅÅ 3 jj 3π zz ÑÑ i ↔ (i + 1) z ÑÑ = ÅÅÅ jj NB ÅÅ 8 j 5 zz ÑÑÑ k { ÑÖÑ ÅÇÅ {i(i + 1)[(i + 1)2 bi + 1 − i 2bi ]}−1/2 ·N A8/5v9/10 2/5 ϕ3/2 ηA γ 3/5 ηB

4. CONCLUSIONS To summarize, we have developed an analytical self-consistent field theory of self-organization in solutions of linear-dendritic and double dendritic diblock copolymers in selective solvents. This theory enables us to predict how dendronization of one or both (soluble and/or insoluble) blocks affects structural and thermodynamic properties of self-organized solution nanostructures of such copolymers.

(42)

or, equivalently, K


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Macromolecules

insoluble block has a stronger effect on the micelle morphology compared to the dendronization of the soluble block. As a consequence, double dendritic diblock copolymers with an equal number of generations and branching functionalities in both blocks have a stronger tendency to form non-spherical aggregates as compared to homologous linear-linear diblock copolymers. Finally, we anticipate that linear-dendritic and double dendritic block copolymers could be attractive candidates for construction of nanocarriers for targeted drug delivery and our findings could provide insightful guidelines on how to exploit ″topological dimension″ in molecular design of such nanopharmaceuticals.

In particular, we have analyzed in detail the effect of dendronization of the soluble block on the properties of spherical micelles formed by copolymers with a dendritic corona and linear or dendritic core-forming blocks. Such micelles are most interesting for their potential application as drug carriers. We have demonstrated that dendronization of the corona blocks allows obtaining stable micelles of a relatively small size but with a large number of terminal groups that can be functionalized by the targeting ligands. These properties can simultaneously be achieved because dendronization of corona blocks reduces the equilibrium aggregation number and overall dimensions of micelles compared to micelles formed by homologous linear-linear diblock copolymer molecules. In spite of the decrease in the aggregation number, the total number of functionazable terminal groups in the corona increases upon an increase in the degree of branching of the corona-forming block, provided that its molecular mass is kept constant. Although the number of terminal segments in a micelle becomes smaller with increasing number of generations g in a corona block than that in a single dendron, the total number of terminal groups in the dendronized corona could be much larger than in one dendron or in the corona of a micelle formed by homologous linearlinear block copolymers. On the contrary, an increase in the number of generations with a concomitant increase of the degree of polymerization of the coronal block leads to such a strong decrease in the aggregation number that the overall number of terminal groups in the corona also decreases. Moreover, dendronization of the corona block leads to an increase in CMC and, eventually, can prevent micellization at all, if the equilibrium micelles of homologous linear-linear diblock copolymers are relatively small. Dendronization of the insoluble block can significantly increase the number of terminal monomer units in the core. Such terminal groups (properly functionalized) are able to interact with active drugs, thereby increasing the loading capacity of the micelle in spite of a relatively weak decrease in the aggregation number caused by dendronization of the insoluble blocks. Very interestingly, block copolymers with dendronized insoluble blocks may form cylindrical micelles or even polymersomes where the size of the coronal domain, D, exceeds the size of the core, R. This is in pronounced contrast to the solution nanostructures of conventional linear-linear block copolymers where non-spherical morphologies always have a crew-cut shape with D ≤ R. This peculiarity in selfassembly of linear-dendritic or double dendritic block copolymers is explained by a larger conformational entropy penalty for the stretching of dendronized insoluble block in the core domain. Furthermore, we have systematically investigated polymorphism of nanoaggregates formed by linear-dendritic and double dendritic block copolymers and established relations between the degree of branching of one or both blocks and equilibrium morphology of the nanoaggregates. In particular, we have demonstrated that dendronization of the soluble blocks favors formation of spherical micelles, whereas the ranges of thermodynamic stability of cylindrical wormlike micelles and dendrimersomes are shifted to a larger degree of polymerization of the insoluble blocks. On the contrary, dendronization of the insoluble block causes the opposite effect; that is, it leads to the extension of stability ranges of polymersomes and cylindrical micelles as compared to those of spherical micelles. Most interestingly, dendronization of the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Oleg V. Borisov: 0000-0002-9281-9093 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Ministry of Research and Education of the Russian Federation within State Contract N 14.W03.31.0022.

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REFERENCES

(1) Lazzari, M.; Lin, G.; Lecommandoux, S. Block copolymers in nanoscience; Wiley-VCH: Weinheim 2006. (2) Mai, Y.; Eisenberg, A. Self-assembly of block copolymers. Chem. Soc. Rev. 2012, 41, 5969−5985. (3) Schacher, F. H.; Rupar, P. A.; Manners, I. Functional block copolymers: nanostructured materials with emerging applications. Angew. Chem. Int. Ed. 2012, 51, 7898−7921. (4) Gröschel, A.H.; Müller, A. H. E. Self-assembly concepts for multicompartment nanostructures. Nanoscale 2015, 7, 11841−11876. (5) Tritschler, U.; Pearce, S.; Gwyther, J.; Whittell, G. R.; Manners, I. 50th Anniversary Perspective: Functional Nanoparticles from the Solution Self-Assembly of Block Copolymers. Macromolecules 2017, 50, 3439−3463. (6) de Gennes, P.-J. Macromolecules and Liquid Crystals: Reflections on Certain Lines of Research. In Solid State Physics; Academic Press: NewYork, 1978, p. 1−17. (7) Halperin, A. Polymeric micelles: a star model. Macromolecules 1987, 20, 2943−2946. (8) Halperin, A.; Alexander, S. Polymeric micelles: their relaxation kinetics. Macromolecules 1989, 22, 2403−2412. (9) Zhulina, Y. B.; Birshtein, T. M. Conformations of blockcopolymer molecules in selective solvents (micellar structures). Polym. Sci. U.S.S.R. 1985, 27, 570−578. (10) Birshtein, T. M.; Zhulina, E. B. Scaling theory of supermolecular structures in block copolymer-solvent systems: 1. Model of micellar structures. Polymer 1989, 30, 170−177. (11) Zhulina, E. B.; Adam, M.; LaRue, I.; Sheiko, S. S.; Rubinstein, M. Diblock copolymer micelles in a dilute solution. Macromolecules 2005, 38, 5330−5351. (12) Gröschel, A. H.; Schacher, F. H.; Schmalz, H.; Borisov, O. V.; Zhulina, E. B.; Walther, A.; Müller, A. H. E. Precise hierarchical selfassembly of multicompartment micelles. Nat. Comm. 2012, 3, 710− 720. (13) Löbling, T. I.; Borisov, O. V.; Haataja, J. S.; Hanisch, A.; Ikkala, O.; Gröschel, A. H.; Müller, A. H. E. Rational design of ABC triblock terpolymer solution nanostructures with controlled patch morphology. Nat. Comm. 2016, 7, 12097. L


Article

Macromolecules (14) 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. (15) Zhulina, E. B.; Borisov, O. V. Theory of block Polymer micelles: recent advances and current challenges. Macromolecules 2012, 45, 4429−4440. (16) Nishiyama, N.; Kataoka, K. Nanostructured devices based on block copolymer assemblies for drug delivery: designing structures for enhanced drug function. Adv. Polym. Sci. 2006, 193, 67−101. (17) Miyata, K.; Nishiyama, N.; Kataoka, K. Rational design of smart supramolecular assemblies for gene delivery: chemical challenges in the creation of artificial viruses. Chem. Soc. Rev. 2012, 41, 2562−2574. (18) Sakai-Kato, K.; Nishiyama, N.; Kozaki, M.; et al. General considerations regarding the in vitro and in vivo properties of block copolymer micelle products and their evaluation. J. Controlled Release 2015, 210, 76−83. (19) Matsumura, Y.; Maeda, H. A new concept for macromolecular therapeutics in cancer chemotherapy: mechanism of tumoritropic accumulation of proteins and the antitumor agent SMANCS. Cancer Res. 1986, 46, 6387−6392. (20) Wurm, F.; Frey, H. Linear-dendritic block copolymers: the state of the art and exciting perspectives. Prog. Polym. Sci. 2011, 36, 1−52. (21) Blasco, E.; Piñol, M.; Oriol, L. Responsive Linear-Dendritic Block Copolymers. Macromol. Rapid Commun. 2014, 35, 1090−1115. (22) García-Juan, H.; Nogales, A.; Blasco, E.; Martínez, J. C.; Š ics, I.; Ezquerra, T. A.; Pinol, M.; Oriol, L. Self-assembly of thermo and light responsive amphiphilic linear dendritic block copolymers. Eur. Polym. J. 2016, 81, 621−633. (23) Mirsharghi, S.; Knudsen, K.D.; Bagherifam, S.; Niström, B.; Boas, U. Preparation and self-assembly of amphiphilic polylysine dendrons. New J. Chem. 2016, 40, 3597−3611. (24) Fan, X.; Zhao, Y.; Xu, W.; Li, L. Linear-Dendritic Block Copolymer for Drug and Gene Delivery. Mater. Sci. Eng. C 2016, 62, 943−959. (25) Yu, T.; Liu, X.; Bolcato-Bellemin, A.-L.; Wang, Y.; Liu, C.; Erbacher, P.; Qu, F.; Rocchi, P.; Behr, J.-P.; Peng, L. An amphiphilic dendrimer for effective delivery of small interfering RNA and gene silencing in vitro and in vivo. Angew. Chem., Int. Ed. 2012, 51, 8478− 8484. (26) Liu, X.; Zhou, J.; Yu, T.; Chen, C.; Cheng, Q.; Sengupta, K.; Huang, Y.; Li, H.; Liu, C.; Wang, Y.; Pososso, P.; Wang, M.; Cui, Q.; Giorgio, S.; Fermeglia, M.; Qu, F.; Pricl, S.; Shi, Y.; Liang, Z.; Rocchi, P.; Rossi, J. J.; Peng, L. Adaptive Amphiphilic Dendrimer-Based Nanoassemblies as Robust and Versatile siRNA Delivery Systems. Angew. Chem., Int. Ed. 2014, 126, 12016−12021. (27) Liu, X.; Liu, C.; Zhou, J.; Chen, C.; Qu, F.; Rossi, J. J.; Rocchi, P.; Peng, L. Promoting siRNA delivery via enhanced cellular uptake using an arginine-decorated amphiphilic dendrimer. Nanoscale 2015, 7, 3867−3875. (28) Borsali, R.; Minatti, E.; Putaux, J.-L.; Schappacher, M.; Deffieux, A.; Vivelle, P.; Lazzaroni, R.; Narayanan, T. From ″Sunflower-like″ Assemblies toward Giant Wormlike Micelles. Langmuir 2003, 19, 6−9. (29) Steinschulter, A. A.; Gelissen, A. P. H.; Jung, A.; Brugnoni, M.; Caumanns, T.; Lotze, G.; Mayer, J.; Pergushov, D. V.; Plamper, F. A. Facile Screening of Various Micellar Morphologies by Blending Miktoarm Stars and Diblock Copolymers. ACS Macro Lett. 2017, 6, 711−715. (30) Lebedeva, I. O.; Zhulina, E. B.; Borisov, O. V. Theory of Linear-Dendritic Block Copolymer Micelles. ACS Macro Lett. 2018, 7, 42−46. (31) Pickett, G. T. Classical Path Analysis of end-Grafted Dendrimers: Dendrimer Forest. Macromolecules 2001, 34, 8784− 8791. (32) Zook, T. C.; Pickett, G. T. Hollow-Core Dendrimers Revisited. Phys. Rev. Lett. 2003, 90, No. 015502.

(33) Zhulina, E. B.; Leermakers, F. A. M.; Borisov, O. V. Ideal mixing in multicomponent brushes of branched Polymers. Macromolecules 2015, 48, 8025−8035. (34) Semenov, A. N. Contribution to the Theory of Microphase Layering in Block-Copolymer Melts. Zh. Eksp. Teor. Fiz. 1985, 61, 733−742.

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Self-Assembly of Linear-Dendritic and Double Dendritic Block

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