Letter Cite This: ACS Macro Lett. 2018, 7, 42−46
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Theory of Linear−Dendritic Block Copolymer Micelles 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, 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 ∥ St. Petersburg National University of Informational Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia ABSTRACT: A self-consistent field theory is applied to study structural properties of micelles formed upon spontaneous assembly of diblock copolymers comprising soluble dendron block covalently linked to insoluble linear block in selective solvent. The structure of spherical micelles is analyzed as a function of degrees of polymerization of the blocks and number of generations and branching functionality of the dendron block. We demonstrate that for a given molecular mass of blocks both the hydrodynamic dimensions of the micelles and the aggregation number decrease as a function of the degree of branching of the dendron block. However, the number of potentially functionalizable terminal segments of dendrons in the corona exposed to the solution increases compared to the number of terminal groups in the corona of linear diblock copolymer micelle. This result may have important implications for construction of linear−dendritic block copolymer micelles with smart functionalities for targeted drug and gene delivery.
I
equilibrium properties of micelles (size, aggregation number, and total amount of terminal groups) and compare them quantitatively to the features of micelles formed by homologous linear diblock copolymer. While earlier theoretical studies23,24 focused on miktoarm star copolymers with variable numbers of linear A and B branches, here we focus on micelles formed by diblock copolymer composed of an insoluble linear block B with degree of polymerization NB and a soluble dendron block A with degree of polymerization NA. The linear block B as well as spacers and terminal branches of the dendron block A are assumed to be intrinsically flexible with monomer unit size a on the order of the Kuhn segment length. The dendron block A is characterized by the number of generations g = 0, 1, 2,... and functionality of branching points q = 1, 2, ... (with q = 1 or/and g = 0 corresponding to linear chain). The number of monomer units in the dendron block is NA = n(q g + 1 − 1)/(q − 1), where n is the number of monomer units in each spacer. The number of terminal groups per dendron is qg. The solvent is assumed to be moderately good for corona blocks with the second virial coefficient of monomer−monomer interactions va3 ≅ a3. In dilute solutions a spontaneous assembly of such molecules gives rise to micelles with a dense core with radius R formed by blocks B and a solvated corona with thickness D composed of dendron blocks A. In what follows we assume that micelles
n the past decades applications of nanoscale self-assembled micelles of amphiphilic block copolymers as drug and gene delivery systems (mostly for anticancer therapies), has become a popular paradigm.1−4 Such micelles became one of the most promising nanocarriers because their size, stability, and loading capacity can be fine-tuned by proper design of the constituent polymers. A polymer micelle comprises a hydrophobic core which serves as depot for hydrophobic drug and a hydrophilic corona which governs solubility of the aggregate in water, its stealth properties, and overall dimension. Because of the enhanced permeability and retention effect, the overall size of micelles regulates their permeation and retention in the targeted tissues.5 Functionalization of coronal hydrophilic blocks with ligands allows for incorporation of biorecognition mechanisms. Therefore, replacement of linear block copolymers by linear−dendritic macromolecules with soluble dendritic blocks6−9 could increase the number of potentially functionalizable groups while keeping the size of the micelle within prescribed limits. The linear−dendritic block copolymer already demonstrated the stimuli-induced ability to assembly into nanoaggregates of different morphologies.7−9 Nanoscale assemblies of hydrophobically modified cationic dendrons were also proven to be highly efficient in gene therapy.10−12 While systematic relations between properties of micelles and composition of linear block copolymers are established both theoretically and experimentally,1,6,13−20 less is known how architecture of the soluble blocks affects the structure of selfassembled aggregates.21,22 The aim of the present Letter is to predict how branching of the corona blocks affects the © 2017 American Chemical Society
Received: October 4, 2017 Accepted: December 4, 2017 Published: December 15, 2017 42
DOI: 10.1021/acsmacrolett.7b00784 ACS Macro Lett. 2018, 7, 42−46
Letter
ACS Macro Letters
Normalization condition s∫ 0Dc(z)(1 + z/R)2dz = NAa3 provides the equation for corona thickness D
have spherical shape (other micelle morphologies will be considered in our forthcoming publication). The core radius in the micelle with aggregation number p and volume fraction ϕ ≃ 1 of monomer units B is R = a(3pNB/4πϕ)1/3. In the strong segregation limit, the core−corona interface is narrow compared to both D and R, and the corona of the micelle is envisioned as a spherical brush of dendritically branched blocks A tethered to the surface of the core with area s = 3NBa3/Rϕ per dendron. Following the theoretical approach introduced for micelles of linear diblock copolymers,13−17 the equilibrium structural characteristics of linear−dendritic block copolymer micelles are obtained by minimizing the free energy (per molecule), F = Fcorona + Finterface + Fcore. It comprises the corona free energy, Fcorona; excess free energy of the core−corona interface, Fsurface = kBTγs (with dimensionless interfacial free energy per unit area γa2); and the contribution Fcore due to conformational entropy losses of the core-forming blocks B. In the following analysis, the latter term is omitted, which is justified in the range of thermodynamic stability of spherical micelles.19 The free energy of the corona is calculated using the selfconsistent field analytical theory of dendron brushes. Following refs 25−27, we assume a parabolic shape of the self-consistent molecular potential acting in the corona of the micelle, δf{c(z)}/δc(z) = 3kB Tk2(D 2 − z2 )/2a2 . The so-called topological coefficient k is governed by the dendron architecture; c(z) is volume fraction of monomer units A at distance z from the core surface, and f{c(z)} is the density of free energy of excluded volume interactions. The chains are assumed to exhibit the Gaussian elasticity on all length scales, and approximation f{c(z)} ≈ kBTa−3vc2(z) (good solvent conditions) leads to c(z) = 3k2(D2 − z2)/4va2. In the brush of linear chains klinear = π/2NA, and we use the topological ratio η = k/klinear = 2NAk/π as a quantitative measure of the degree of branching of the dendron blocks. The topological ratio η depends on the number of spacers/branches and their connectivity but does not depend on degree of polymerization NA of the dendron block. In what follows we systematically compare associative properties of linear AB diblock copolymers and those of linear−dendritic block copolymers with the same numbers of monomers NB and NA in insoluble and soluble blocks, respectively. The free energy of the corona includes the contributions due to excluded volume interactions and conformational entropy of dendritic blocks27 Fcorona s ⎛ = 2⎜ kBT R ⎝ +
=
∫0
3k 2 2a5
D
−1/3 ⎛ 3D 1 D2 ⎞ D = D0⎜1 + + ⎟ 4R 5 R2 ⎠ ⎝
The introduction of dimensionless ratio x = D/R and eqs 1−3 allows us to express area s and free energy Fcorona as 1/2 ⎛ 3NB ⎞3/2 1 3 1 2⎞ s 3/2⎛ =⎜ ⎟ πη 3/2 1/2 x ⎜1 + x + x ⎟ 2 ⎝ 4 5 ⎠ ⎝ 2ϕ ⎠ a NA v
(4)
( (
F0 ⎛ D ⎞ ⎜ ⎟ kBT ⎝ D0 ⎠
5
(5)
(6)
determines the parameters of equilibrium micelles: interfacial area per molecule s (eq 4), core radius R = 3NBa3/ϕs, and corona thickness D (eq 3). Following the conventional nomenclature we distinguish between starlike (D ≫ R, i.e., x ≫ 1) and crew-cut (D ≪ R, i.e., x ≪ 1) aggregates. The asymptotic scaling dependences for equilibrium parameters of starlike and crew-cut micelles can be obtained by keeping only the dominant term on the right-hand side in eqs 5 and 4 to give ⎧(γa 2)15/11(N /ϕ )10/11(v 2N )−3/11η−18/11, R ≪ D ⎪ B A B p≅⎨ 2 9/5 2 − 6/5 ⎪(γa /N ) (N /ϕ ) v η−6/5 , R≫D ⎩ A B B
(7)
⎧(γa 2)3/11(N /ϕ )2/11N 6/11v1/11η−8/11 , R ≪ D ⎪ B A B D/a ≅ ⎨ ⎪(γa 2)1/5 N 4/5v1/5η−4/5 , R≫D ⎩ A (8)
⎧(γa2)5/11(N /ϕ )7/11(v 2N )−1/11η−6/11, R ≪ D ⎪ B A B R /a ≅ ⎨ ⎪(γa2)3/5 (N /ϕ )N −3/5v−2/5η−2/5 , R≫D ⎩ B A B (1)
(9)
As follows from eqs 7−9, an increase in the degree of branching (increase in η at constant NA) leads to the decrease in aggregation number p, dimensions of micelle corona and core (D and R), and total size of the micelle (D + R). A decrease in the aggregation number p (and in R) is explained by increased repulsions in the corona. A decrease in the corona thickness D results from the decrease in p (weaker repulsions between blocks A) and higher entropic penalty for stretching of dendrons in the corona compared to linear blocks A.
Here 5 ⎛ 2N va 2 ⎞1/3 F 9 k 4s ⎛ D0 ⎞ ⎜ ⎟ D0 = a⎜ A2 ⎟ and 0 = kBT 20 va 2 ⎝ a ⎠ ⎝ ks ⎠
) 2 )
d[Fcorona(x) + kBTγs(x)] dF = =0 dx dx
⎞ c(z)(z + R )2 z 2 dz⎟ ⎠
⎛ 5D 5 D2 ⎞ + ⎟ ⎜1 + 6R 21 R2 ⎠ ⎝
5
Minimization of the total free energy per molecule F = Fcorona(x) + kBTγs(x) (eqs 5 and 4) with respect to x provides the equation for the ratio x = D/R in the equilibrium micelle as a function of topological ratio η and combination of the parameters u = γa2(NB/ϕ)5/2NA−7/2v−3/2. Parameter u is mediated by variations in the system parameters: γ, ϕ, v, NB, and NA, whereas the analytical expressions for η = η(g, q) in brushes of symmetric dendrons with branching activity q and g ≤ 3 are available in the literature.27 In the limit of q ≫ 1, the topological ratio can be approximated as η ≃ qg/2. The numerical solution x(η, u) of the equation
2
D
5
2 2 Fcorona 3 vNA ϕ 1 + 6 x + 21 x = kBT 5 NB x 1 + 3 x + 1 x 2 4 5
f {c(z)}(z + R ) dz
∫0
(3)
(2)
are the thickness and the free energy (per dendron) of planar brush with grafting area s per dendron, respectively. 43
DOI: 10.1021/acsmacrolett.7b00784 ACS Macro Lett. 2018, 7, 42−46
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ACS Macro Letters
x ≪ 1(x ∼ u−2/11) are indicated near the curves. For typical values of γa2 ≃ 1, ϕ ≃ 1, v ≃ 1, and NB = NA, parameter u ≃ 0.01−0.1 if 10 < NB < 100. As is seen in Figure 1, the asymptotic starlike and crew-cut regimes are separated by a wide intermediate region in which apparent exponents could be noticeably different from the values predicted in eqs 7−9. In Figure 2 we present ratio p(η)/p(η = 1) of the equilibrium aggregation numbers for micelles formed by block copolymer
It is instructive to analyze how the overall number of (potentially functionalizable) terminal segments Nt = qgp per micelle changes upon an increase in the degree of branching while keeping NA = const. Assuming high branching activity (η ∼ qg/2) and using eq 7, one finds 2g /11 ⎧ , R≪D ⎪q Nt = pq ∼ ⎨ ⎪ 2g /5 ⎩q , R ≫ D g
(10)
Hence, dendronization of block A (at constant NA) leads to a weaker increase in the number of terminal groups per micelle than in an individual dendron (with qg free branches) due to the decrease in aggregation number p. If, on the contrary, the length of a spacer n is kept constant and dendronization of the corona block is accompanied by an increase in NA, one finds −g /11 ⎧ , R≪D ⎪q Nt = pq g ∼ ⎨ ⎪ −7g /5 , R≫D ⎩q
(11)
That is, the number of terminal groups per micelle decreases as a function of number g of generations in the dendron block. Although the asymptotic analytical expressions in eqs 7−9 provide useful guidelines in block copolymer design, micelles are often found in the intermediate regime between starlike and crew-cut aggregates with comparable core and corona sizes, D ≃ R. To address this case, we supplement the asymptotic power laws with full solution of eq 6 encompassing both limits of starlike (x ≫ 1) and crew-cut (x ≪ 1) micelles. In Figure 1 the ratio x = D/R in the equilibrium micelles with linear (g = 0) and dendritic corona block with number of
Figure 2. Ratios of aggregation numbers p(η)/p(η = 1) (solid line) and total numbers of terminal monomer units Nt(η)/Nt(η = 1) (dashed line) in micelles of block copolymers with dendritic and linear coronal blocks as a function of topological ratio η. Filled circles indicate values of p(η)/p(η = 1) for selected architectures (g, q) of dendritic coronal blocks. Corresponding values of Nt(η)/Nt(η = 1) are shown by empty circles.
with dendritically branched block A and linear block copolymer (solid line), and ratio Nt(η)/Nt(η = 1) of the corresponding numbers of terminal monomers in the corona per micelle (dashed line) as a function of topological ratio η. The values of p(η)/p(η = 1) and Nt(η)/Nt(η = 1) for selected topologies of dendron block (with g and q indicated) are marked by filled and empty circles, respectively. A decrease in u down to u = 0.01 does not affect noticeably the position of the solid line and shifts the dashed line to larger values at η ≳ 2 (not shown in Figure 2). The equilibrium micelle dimensions R(η)/R(η = 1), D(η)/D(η = 1), and [R(η) + D(η)]/[R(η = 1) + D(η = 1)] for the same parameters as in Figure 2 are demonstrated in Figure 3. As follows from Figures 2 and 3, dendronization of the corona block A leads to the noticeable decrease in aggregation number p (and in the total size of the micelle), accompanied by a moderate (subexponential) increase in the number of terminal groups Nt in the corona. For example, aggregation number p reduces almost 2-fold if linear block A is transformed into dendron of the second generation with g = 2, q = 3, or dendron of the third generation with g = 3, q = 2, while the number Nt of terminal monomers in the corona increases by a factor of ≈5. Note that geometrical parameters (core and corona sizes) of these micelles are rather close (see Figures 2 and 3) . According to eq 7, the branching-induced decrease in aggregation number p is rather strong (p ∼ η−β with exponent β = 18/11 and 6/5 for starlike and crew-cut micelles, respectively). For values of q = 2, g = 1−3 (corresponding to the branching ratio η ≃ 1.5−3), dendronization of the corona block leads thereby to 2−3-fold decrease in aggregation number p compared to linear counterparts with the same molecular weight. Therefore, branching of the corona block A
Figure 1. Ratio D/R of corona thickness D to the core radius R in micelles with linear (violet circles) and dendritic coronal block with the same number of monomer units NA. Branching activity q = 3 and g = 1 (η = 1.333, red circles), g = 2 (η = 2.091, blue circles), and g = 3 (η = 3.449, green circles). Dimensionless parameter u = γa2(NB/ϕ)5/2NA−7/2v−3/2. Horizontal line D/R = 1 separates regimes of starlike (D ≥ R) and crew-cut (D ≤ R) spherical micelles. Slopes of corresponding asymptotic dependences are indicated near the curves.
generations g = 1, g = 2, and g = 3 is plotted as a function of parameter u = γa2(NB/ϕ)5/2NA−7/2v−3/2. Branching activity of the dendron blocks q = 3 was kept the same for all values of g. The horizontal line D/R = 1 separates the regimes of starlike (D > R) and crew-cut (D < R) spherical micelles. The slopes of asymptotic power law dependences at x ≫ 1(x ∼ u−2/5) and 44
DOI: 10.1021/acsmacrolett.7b00784 ACS Macro Lett. 2018, 7, 42−46
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ACS Macro Letters
Figure 3. Ratios of the core radii, R/R(η = 1) (black line); corona thickness, D/D(η = 1) (blue line); and total size of the micelle, (D + R)/[D(η = 1) + R(η = 1)] (red line) for micelles with dendritically branched and linear coronal blocks. Filled circles correspond to selected architectures of the dendritic coronal blocks, the same as in Figure 2. Corresponding values of the topological ratio η are indicated by arrows.
corona of micelle formed by homologous linear block copolymer with the same NA and NB. While dendronization of soluble blocks A allows for stable micelles with relatively small overall dimensions, a large number of terminal groups in the corona, and sufficient loading capacity of the core, additional dendronization of insoluble blocks B could increase the number of terminal monomers in the core of the micelle (potentially enhancing the loading capacity for endactive drug species). All this makes dendronized block copolymers attractive candidates for construction of nanocarriers with enhanced loading capacity combined with cellular selectivity and targeting efficiency for drug delivery applications.
could suppress self-assembly, making micelization threshold p ≥ 1 unattainable for block copolymer with relatively short to have p ≳ 1 insoluble block B. The requirement NB ≳ N3/10 A for block copolymer with linear blocks is substituted by a 9/5 for block copolymer with stronger requirement NB ≳ N3/10 A η dendronized (η > 1) corona block A. Dendronization of the soluble block A affects also the critical micelle concentration (CMC). According to ref 31, CMC is specified by the difference in free energies per chain in the equilibrium micelle (F) and unimer state (F 0 ) as ln(cCMC) = (F − F0)/kBT. The dendronization-induced difference in ΔF = F − F0 is governed mostly by the increase in F = F(η). As follows from eqs 7 and 9, free energy per chain in the equilibrium micelle is F(η) ∼ γR2/p ∼ ηα with exponent α = 6/11 and 2/5 for starlike and crew-cut micelles, respectively. Therefore, an increase in CMC due to dendronization of the corona block could be rather noticeable. The general trends in micelle behavior outlined in this study are consistent with the computer simulations.28−30 Extensive Langevin dynamics simulations28 of linear−dendritic block copolymers with number of generations g = 1−5 in the soluble block A indicated the decreased aggregation number p and increased CMC compared to solutions of linear analogues. They also indicated the decreased dimensions, R and D, of the core and corona of spherical micelles upon dendritic branching of the soluble block A. However, a quantitative comparison between our predictions and the results of computer simulations is currently difficult because of relatively short spacers in dendritic blocks in the simulated systems. To conclude: Dendronization of corona blocks A reduces the equilibrium aggregation number and size of micelles compared to micelles formed by linear diblock copolymer with the same numbers of monomers NB and NA in the core and corona blocks. If the equilibrium micelles of linear diblocks are relatively small, branching of the corona block could prevent micellization and keep block copolymer molecules with dendron block A as unimers in the solution. In spite of the decrease in aggregation number p(η), the number of terminal (potentially functionalized) groups in the dendronized corona of micelle increases quite remarkably. The total number of terminal monomers in the dendronized corona could be much larger than that in a single dendron or in a
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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.
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ACKNOWLEDGMENTS This work was supported by Russian Science Foundation, Grant N 14-33-00003.
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