Shear-induced self-assembly of linear ABC triblock copolymers in solution: creation of 1D cylindrical micellar structures

Abstract

Although the creation of nanostructures from solution self-assembly block copolymers is one of the most promising approaches, the formation of 1D cylindrical structures remains a challenge. In this work, shear flow is introduced to create 1D cylindrical micellar structures based on solution self-assembly of linear ABC terpolymers. The dissipative particle dynamics method is used to explore the whole morphological space. Firstly, 7 spherical (0D), 10 cylindrical and 1 oblate spherical (1D), 5 lamella and 3 oblate cylindrical (2D) micellar structures are summarized from the total 315 morphologies. Secondly, several 1D cylindrical micelles provided by this simulation have the potential to become interesting single- or double-walled nanotubes and cylindrical co-micelles. Finally, the shear rate, the concentration and the solvophilic block length as 3 key factors of controlling the creation of multi-dimensional structures are given, which are important to the formation of 1D, 2D and 0D structures, respectively. In fact, this work uses a simple shear means to promote the reorientation and rearrangement of self-assembly morphologies, and selectively builds 0D (dot), 1D (tube) or 2D (sheet) nanostructures. These results are helpful to understand the formation of complex micelles by shear-induced self-assembly of linear ABC triblock copolymers and for tailoring new 1D cylindrical morphologies.

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1. Introduction

Self-assembly of block copolymers in order to prepare well-defined nanostructures has gained significant scientific interest during the last few decades.¹ These exquisitely ordered nanostructures have potential applications in drug release,² microelectronic materials,³ and so on. Recently, ABC triblock copolymers, which have two representative topological structures of linear (l-ABC) and star ABC (μ-ABC), have been given more attention⁴ because of their expansive spectrum of accessible morphologies. Originally, concentric morphologies are generally regarded as the default structure adopted by l-ABC,⁵ however, effectively suppressed by μ-ABC.⁶ It leads to a vast array of fascinating morphologies.^4,7 Thus, most works have been focused on the self-assembly of μ-ABC, including several crucial simulation and theoretical studies.^8–13 Laschewsky and co-workers later revealed an inspiring result that l-ABC also form nonconcentric micelles.^14–18 In addition, Müller et al. also found several novel micelles (vesicle, toroid, undulated ribbon) from l-ABC.¹⁹ Obviously, the potential of the simple l-ABC was underestimated. Dissipative particle dynamics (DPD) simulations were used to explore the phase diagram of l-ABC in solution and rich micelles (such as raspberry-onion, helix-on-sphere, etc.) were understood further.^20–22 Furthermore, combining with other simulations and theories,^9,10,23–27 knowledge on the self-assembly behaviours of linear ABC triblock copolymers has been enhanced.

Among all exquisite morphologies, 1D nanostructures have attracted considerable attention because of their wide potential applications as flow-intensive drug or dye delivery systems,²⁸ liner scaffolds for mineralization,^29,30 templates for creating linear arrays of nanoparticles,³¹ and so on. Moreover, the preparation of 1D nanostructures by self-assembly of block copolymers has also become a hot topic in this field. For example, Wang and co-workers utilized the diblock copolymer of polyferrocenyldimethylsilane (PFS) to build cylindrical micelles in solution and to form co-micelles by adding a different block copolymer.³² Through a combination of crystallization-driven self-assembly of block copolymers based on PFS and selective micelle corona cross-linking, Rupar et al. created monodisperse cylindrical seed micelles (about 130 nm) that grow unidirectionally.³³ He’s group successfully prepared the nanotubular (hollow cylindrical) micelles via self-assembly of a poly(styrene-b-4-vinyl pyridine-b-ethylene oxide) triblock terpolymer in binary organic solvents with the assistance of solution thermal annealing.³⁴ In fact, most diblock copolymers prefer to form spherical micelles than cylindrical micelles.³² The investigation of triblock copolymers creating 1D nanostructures is still rare.³⁴ Therefore, the fabrication of 1D nanostructures via block copolymer self-assembly remains a challenge. In addition, the nonequilibrium conditions like shear flow are commonly encountered during processing. The understanding of shear-induced self-assembly of block copolymers has attracted a lot of attention.^35–37 Considering the ability of shear flow, i.e., the new assembled morphologies of block copolymers and the different orientation, we believe that the shear should be an effective approach to create 1D nanostructures of linear ABC triblock copolymers, which has not been performed, to the best of our knowledge.

In this paper, we provide the first simulation of shear-induced morphologies of linear ABC triblock copolymers with different block sequences and block lengths by the DPD technique, and demonstrate that the shear is a good means of tailoring 1D cylindrical micellar structures based on our simulated whole morphological space.

2. Method and model details

DPD is a coarse-grained particle-based simulation technique, which allows larger length and a longer time scale. DPD particles obey Newton’s equation of motion, and the forces between pair non-bonded DPD particles include a conservative force F^C, a dissipative force F^D, and a random force F^R, respectively. A string of DPD particles bonded by a harmonic spring force F^S are always used to describe the polymer. Therefore, the total force is expressed by

The four forces are respectively given by

where r_ij = r_i − r_j, r_ij = |r_ij|, e_ij = r_ij/|r_ij| and v_ij = v_i − v_j. ζ_ij is a Gaussian random number with zero mean and unit variance. α_ij is the repulsion parameter between bead i and j, which reflects the chemical characteristics of interacting beads. γ and σ are the friction constant and noise strength, respectively. To ensure that the system satisfies the fluctuation-dissipation theorem and corresponds to the Gibbs Canonical ensemble, only one of the two weight functions w^D and w^R can be chosen arbitrarily and this choice fixes the other one. There is also a relation between the amplitudes (σ and γ) and k_BT. It is w^D = (w^R)² and σ² = 2γk_BT, where k_B is the Boltzmann constant and T is the temperature.³⁸ Simple forms for w^C = (1 − r_ij)² = w^D = (w^R)² and σ = 3 (i.e., γ = 4.5) are chosen, and Newton equations for all beads are integrated by a modified version of the velocity-Verlet algorithm with λ = 0.65.³⁹ In addition, l_(i,i+1) is the bond length between connected beads i and i + 1. Here, the spring coefficient k_S = 4 and the balance bond length l₀ = 0 are chosen. For easy numerical handling, the cutoff radius (r_c), the bead mass (m), and the temperature (k_BT) are chosen as the unit of the simulated system.

Fig. 1 gives our coarse-grained models for l-ABC (x-y-z), l-BAC (y-x-z) and l-ACB (x-z-y) which consist of the solvophilic block (A), the weakly solvophobic block (B) and the strongly solvophobic block (C), x, y and z are the number of beads A, B and C, respectively. The solvent is represented by an individual bead S. As for the interaction parameters (Table 1), we still follow those used by our previous works,^20–22 which are firstly defined to describe the self-assembly of the experimental miktoarm terpolymer.^6,8 However, Sheng et al. think that the interactions between the poly(perfluoropropylene oxide) (F) block and other two blocks are very strong and provide a suite of gentle parameters.¹⁰ Here, our main aim is to obtain the universal rule of governing shear-induced morphologies for l-ABC. Therefore, we still adopt these present parameters for better comparison with our previous work.

Fig. 1 The coarse-grained model of linear ABC triblock copolymers with different block sequences (left) and the simulation box with the initial constitution (right).

Table 1 Repulsion parameters (DPD unit) in this work

	A	B	C	S
A	25	45	90	27
B	45	25	75	50
C	90	75	25	120
S	27	50	120	25

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To avoid the finite size effect, our simulations are performed in a larger cubic box of size (30r_c) (ref. 3) containing 81 000 DPD beads with random distribution at ρ = 3. The Lees–Edwards periodic boundary condition is used to model shear flows and maintain a steady linear velocity profile with gradient τ = dv_x/dr_y.^40,41 It also indicates that the shear direction is along the x axis. Different shear rates, τ = 0.01, 0.05, 0.1, 0.15 and 0.2, are chosen to check the effect of shear rate on the morphologies. The concentrations (φ, the volume fraction) of l-ABC in solution are 0.1, 0.2 and 0.3, respectively. The time step Δt is 0.03 and a total of 1–2 × 10⁶ DPD time steps are carried out to guarantee the equilibration for each system.

3. Results and discussion

The self-assembly morphologies of ABC triblock copolymers in the solvent is controlled in large by parameters such as concentration, block length and ratio, block sequences, interactions, and external factors (pH, temperature, pressure, shear, confined conditions, etc.). Obviously, the expansive parameter space can result in a boundless array of possible structures and chemical functionalities. It is difficult to find one special structure among them. Therefore, we fixed all the interaction parameters and focused on the effect of shear rate (τ) and block length (x, y and z) on the morphology. For one of the 3 sequences (l-ABC, l-BAC and l-ACB), there are 7 kinds of linear terpolymers with different block lengths (2-2-2, 2-2-8, 2-8-2, 2-8-8, 8-2-2, 8-2-8 and 8-8-2), 5 kinds of shear rates (τ = 0.01, 0.05, 0.1, 0.15 and 0.2) and 3 kinds of concentration (φ = 0.1, 0.2, 0.3). Considering all 3 sequences, a total of 315 equilibrium micellar morphologies (see ESI†) are provided by our simulations. It is interesting that several shear-induced 1D cylindrical micelles are created.

Fig. 2 gives 5 basic morphological schematics (sphere, oblate sphere, cylinder, oblate cylinder and lamella) summarizing from the full morphological phase diagrams (see ESI†), which belong to three dimensionalities, respectively. It is a logical choice to ascribe the familiar spherical (dot), cylindrical (tube) and lamellar (sheet) micelles to zero-dimensional (0D), one-dimensional (1D) and two-dimensional (2D) structures, respectively, the same as the definition used by Bockstaller et al.⁴² The oblate sphere and cylinder compared with the spherical (0D) and cylindrical (1D) micelles especially show the obvious increase in the radical dimension and, therefore, are defined as 1D and 2D, respectively, which are more than the pristine structure. Next, we introduce these morphologies with the different dimensionality according to the above classes in detail.

Fig. 2 Schematics for zero- (0D), one- (1D) and two-dimensional (2D) micellar structures by abstracting from all the phase diagrams.

For the 0D structure, about 8 types of spherical micelles are summarized from the whole phase diagram with 315 morphologies and the results are given in Fig. 3. All these 0D spherical micelles have been found in our previous simulations^20–22 and other experiments.^14–18 For example, S⁰_i, the typically concentric onion-like arrangement, does not have independent access to the exterior of the micelle. Judging from the appearance, S⁰_ii, S⁰_iii, S⁰_iv and S⁰_v are the familiar raspberry micelles. However, they have different cores as shown in Fig. 3. For example, S⁰_v is a big CSC micelle containing a small reverse CSC inside. However, S⁰_iii has only a core of B block.

Fig. 3 Morphologies represented by the symbols for 0D micellar structures. Blue, red and yellow represents A-, B- and C-rich domains, respectively. For clarity, several morphologies use the isodensity surface and omit the A block. In symbols, S denotes the sphere, a superscript 0 represents 0D and the subscripts (i–vii) represent the different morphologies. (i) Spherical core–shell–corona (CSC) micelle; (ii) raspberry micelle; (iii) raspberry with a core of A block; (iv) raspberry with a core of C block; (v) a big CSC micelle containing a small CSC inside; (vi and vii) spheres with different sizes in gel.

As for the significant 1D structure, there are about 11 types of micellar morphologies created by the shear flow as shown in Fig. 4. Among them, all 3 micelles of oblate sphere (O¹_S), capsule (C¹_a) and worm (mC¹) in the first row are often provided by the common self-assembly of triblock copolymers despite shear flows. The following 8 micelles with notable cylinder characteristics are introduced in detail: C¹_i and C¹_ii are the normal cylindrical core–shell–corona morphologies, the only difference is that the shell of the former constitutes of B block and the latter of A block. C¹_iii and C¹_iv are the cylindrical CSC micelles with a cylindrical core. The core of C¹_iii is C block and the shell of C¹_iii is embedded by the dispersed C-rich domain, which is also found by concentration-induced self-assembly of linear ABC terpolymers.²² The difference between C¹_iii and C¹_iv is that the core of C¹_iv is B block. C¹_v and C¹_vi are also regarded as cylindrical CSC micelles. The difference is that C¹_v and C¹_vi have two strips of shell, not like the continuous ones in typical cylindrical CSC micelles. C¹_vii is a single raspberry cylinder and C¹_viii a gel of A block containing the different cylinder of B and C block, which is very similar to S⁰_vi and S⁰_vii in 0D structures. For the above 1D cylindrical micelles, if B and (or) C blocks can be selectively melted away, C¹_i and C¹_ii would change into single-walled nanotubes, and C¹_iii and C¹_iv would create a double-walled nanotube. The most interesting phenomenon is that C¹_viii forms a parallel co-micelle, which has been found from the self-assembly of PFS block copolymers.³²

Fig. 4 Morphologies represented by the symbols for 1D micellar structures. Blue, red and yellow represents A-, B- and C-rich domains, respectively. For clarity, several morphologies use the isodensity surface and omit the A block. Os, Ca, mC and C denote the oblate sphere, capsule-like, multi-core and cylinder morphologies, respectively. Superscript 1 represents 1D and the subscripts (i–viii) represent the different cylinder structures. (i and ii) Cylindrical CSC with different shells; (iii) cylindrical raspberry micelle with a cylindrical core of A and C block; (iv) cylindrical micelle with a cylindrical core of A and B block; (v) two strips of B block are linked to a cylindrical core of C block by A block; (vi) two strips of C block are pasted on a cylindrical core of B block; (vii) single raspberry cylindrical micelle; (viii) different cylinders in the gel. For portions, the side view is given on the right.

Among all 8 kinds of 2D structures in Fig. 5, L²_i and L²_ii are the typical 5 layers morphology, which have the different order of ACBBCA (L²_i) and ABCCBA (L²_ii), respectively. L²_iii is a lamella like L²_i, the difference is that the left 3 layers of L²_iii show a bend due to the variation of interfacial tension. mL²_i is a one-piece lamella consisting of alternate solvophobic blocks B and C (layers) linked by the solvophilic block A, and the thickness of single layer (B or C) is about a third of the lamella. As for mL²_ii, it has not the average thickness and shows a spindle-like structure from the side view. Oc²_i, as a normal oblate cylindrical micelle, can be regarded as a typical cylinder flattened by external force. Oc²_ii has a little variation, i.e., two cylindrical cores of B block. Obviously, Oc²_iii is a curving oblate cylinder. Obviously, for the outer shapes of 2D structures, the interfacial tension plays a crucial role. Among them, L²_iii, Oc²_ii and Oc²_iii are the main products depending on the shear flow.

Fig. 5 Morphologies represented by the symbols for 2D structures. Blue, red and yellow represents A-, B- and C-rich domains, respectively. L, Oc and mL denote the lamella, oblate cylinder and multi-layer morphologies, respectively. Superscript 2 represents 2D and subscripts (i–iii) are as follows: for lamella, (i and ii) lamella with a middle of block B and C, respectively; (iii) lamella with a curving shell. For oblate cylinder, (i) flat cylindrical CSC micelle with a core of block C; (ii) flat cylindrical CSC with two parallel cores of block B; (iii) curving flat cylindrical CSC with a core of block C. For multi-layers, (i) alternate blocks A, B and C form a lamella; (ii) alternate blocks A, B and C form a plump lamella. The side view is given on the right.

Based on the above simple and direct class in Fig. 2, we provide the whole morphological phase diagram for linear ABC triblock copolymers under the different concentrations and shear flows, and the results are shown in Fig. 6 and 7, respectively. We also select the given region in the phase diagrams of φ = 0.1 (see the rectangles in Fig. 6) and enlarge them to gain more detailed morphological information labelled by the symbols of Fig. 3–5 (see the right of Fig. 6). Although solution self-assembly of block copolymers represents one of the most promising approaches to prepare well-defined nanostructures with different shapes,³³ the formation of non-centrosymmetric 1D or 2D nanostructures remains a major challenge. At present, there is a small quantity of experiments^33,34,43 and simulations^22,44,45 on the aggregation of triblock copolymers that luckily provides access to approximate 1D structures. However, it is very rare for triblock copolymers to form volume-produced 1D nanostructures. On the other hand, it is known that shear flow can change the orientation of the microphase to create globally aligned structures.^35,36 Therefore, we first utilize the shear flow to control the self-assembly of linear ABC terpolymers in solution and introduce the fabrication of 1D structures. From the foregoing simulation morphologies, we have found several novel 1D cylindrical micelles.

Fig. 6 Morphological phase diagrams of linear ABC terpolymers under the different shear rate at concentration φ = 0.1. The 3 numbers listed in y-axis represent the length of block A, B and C, respectively. The values of x-axis are the different shear rate. The detailed phase diagrams on the right are the enlarged area of the rectangle on the left.

Fig. 7 Morphological phase diagrams of linear ABC terpolymers under the different shear rate at concentration φ = 0.2 and 0.3. The 3 numbers listed in y-axis represent the length of block A, B and C, respectively. The values of x-axis are the different shear rate.

Then, by analyzing the phase diagrams of Fig. 6 and 7, the elementary judgements in mass are summarized as follows. Firstly, compared with our previous simulation without shear,²² the area of phase diagrams held by the blue oblate spheres and cylinders (the so-called 1D structures) show a prominent increase, especially for the larger shear rate (τ ≥ 0.1). For example, in Fig. 6c, all l-BAC triblock copolymers (including all 7 block compositions) form the 1D cylindrical morphologies after the shear rate is more than 0.1 (τ ≥ 0.1). It is also worth noting that, among these 1D structures, the number of cylindrical micelles is far more than those of oblate spheres. The widely blue domains dominating the phase diagrams testify that introducing the shear is a successful alternative to external stimuli to create a batch of cylindrical 1D structures in virtue of the advantage of solution self-assembly of linear ABC triblock copolymers. Secondly, as shown in Fig. 6 and 7, it is clear that the dark sphere defined as 0D structure occupies the limited domain in phase diagrams, which mainly converge in the small shear rate (τ ≤ 0.05). The other interesting phenomenon is that the linear ABC terpolymers with the longer solvophilic A block have great ability to keep the spherical morphology when increasing the shear rate. In fact, those terpolymers with longer solvophilic A block prefer to form S⁰_vi and S⁰_vii structures that B and C block assembles the unattached spheres distributing in the gels of A block and the different ratio of B and C block would result in the different spherical size. Obviously, the longer solvophilic A block can build the powerful shield against the effect of solvent and shear on the morphology, and the protected B and C block can easily go through the formation pathway of nucleation and growth.²² However, the longer solvophilic A block can bring positive factors to play only for the smaller shear rate. Once the shear rate exceeds a given value, the system would self-assemble into a 1D structure like C¹_viii, whose B and C block turns into the different cylinders distributing in the gels of A block. For example, l-BAC (2-8-2) at φ = 0.1 forms the 0D morphology of S⁰_vii when the shear rate τ ≤ 0.05. Whereas when we further increase the shear rate (τ = 0.1), the 1D morphology of C¹_viii appears.

The sphere-embedded gel configuration of ABC terpolymers⁴⁶ and their application in drug release² have been given increasing attention. As for the cylinder-embedded gel configuration, the corresponding report is still rare, to the best of our knowledge. Thirdly, no matter what the block sequences, the so-called 2D structures including the oblate cylinder and the lamella also occupy the relative marked domain in phase diagrams shown in Fig. 7, mainly for the larger concentration of φ = 0.2 and 0.3. For example, the phase diagrams of l-ACB drawn in Fig. 6a and 7a, respectively, show a considerable amount of differences. In detail, all the l-ACB (x-z-y is equal to 2-2-2, 2-2-8 and 2-8-2) terpolymers at φ = 0.2 (Fig. 7a) assemble to the oblate cylindrical micelles under all the shear rates, however, at φ = 0.1 (Fig. 6a) they form the same oblate cylindrical micelle only under four conditions, i.e., x-z-y is 2-8-2 and τ = 0.15, x-z-y is 2-2-2 and τ = 0.1, 0.15 and 0.2, respectively. In addition, compared with the oblate cylinder, the lamellar morphologies are more frequently found at the concentration of φ = 0.3.

Taking l-ACB (2-2-2, 2-2-8 and 2-8-2) under τ = 0.01 as examples, for l-ACB (2-2-2 and 2-2-8) terpolymers they provide the oblate spherical morphology at φ = 0.1 (Fig. 6a), the oblate cylindrical ones at φ = 0.2 (Fig. 7a) and the lamellar ones at φ = 0.3 (Fig. 7d), respectively. The l-ACB (2-8-2) experiences the morphological variation from the cylinder to the oblate cylinder and the lamella with the increase of concentration from 0.1 to 0.3. In fact, the oblate cylindrical micelle is not the specific product of shear-induced self-assembly of ABC triblock copolymers, and in the concentration-induced morphologies it is also expressed frequently.

Finally, the long-coexistence of different morphological micelles, such as sphere/cylinder and cylinder/oblate cylinder, often appears in the phase diagrams in Fig. 6 and 7. It has to be emphasised that each schematic (sphere, oblate sphere, cylinder, oblate cylinder and lamella) in Fig. 2 corresponds to a series of different morphologies. For example, the dark sphere represents an aggregate of seven 0D morphologies in Fig. 3, and the blue cylinder is a deputation containing ten 1D cylinder-like morphologies (except for the oblate sphere O¹_S) in Fig. 4. Therefore, we can zoom in on the phase diagrams, like in the right portion of Fig. 6, to search for more coexisting micelles. It is interesting that the mixtures of two similar morphologies, such as C¹_vii/C¹_a, Oc²_i/C¹_a and C¹_ii/C¹_a, can be searched. The phase diagrams in Fig. 7 can also find many coexisting morphologies like the above mentioned, if we display them based on the detailed morphological symbols (in Fig. 3–5), not the schematic shapes (in Fig. 2). Of course, our 315 equilibrium micellar morphologies in the ESI† can approximately support the above results.

We can summarize several factors that influence the creation of multi-dimensional micellar morphologies through the self-assembly of linear ABC terpolymers in solution. The important one is the shear rate (τ). The reorientation and rearrangement of the morphologies, which can easily result in the mass formation of 1D structures, significantly depend on the larger shear rate. However, the small shear rate (τ = 0.01) hardly alters any morphologies, that is, very similar to those from solution self-assembly without shear. The next one is the concentration (φ). The larger concentration is a top priority to build the 2D structures (oblate cylinder and lamella). The lamellar morphologies are almost from the concentration of φ = 0.2 and 0.3. At the small φ = 0.1, there is a lack of the lamella for all linear ABC triblock copolymers and shear rates, and the limited numbers of the oblate cylinder only for l-ACB (2-2-2) and l-ABC (2-2-2). The last one is the length of the solvophilic A block. The longer solvophilic A block has the obvious shield ability and protects the self-assembly from the effect of solvents and shear rates (only in the given range), which is important for the formation of 0D spherical micelles like S⁰_vi and S⁰_vii.

4. Conclusions

In summary, combined with our previous work,^20–22 we further utilize DPD simulation method to give insight into the shear-induced self-assembly morphologies of linear ABC triblock copolymers in solution, where A is a solvophilic block, B and C are weak and strong solvophobic blocks, respectively. The shear flow (introduced by the Lees–Edwards periodic boundary condition) has a remarkable influence on the morphologies, especially for the creation of 1D cylindrical micelles. The detailed phase diagrams show that the larger shear rate is a key factor to fabricate the 1D structure and the smaller one hardly produces any influence. However, the shear rate is not the one and only factor of controlling the morphology, the concentration and the block length also play a significant role in forming the special structure. For example, the larger concentration is a preferential choice to obtain the 2D structure, especially the lamella. The longer solvophilic block prefers the formation of “spheres in gel” (0D). In addition, several 1D cylindrical micelles (such as C¹_i–viii) can be further processed (melting one or two blocks away) and should become the single-, double-walled nanotubes, and the cylindrical co-micelles, respectively. In fact, this work provides a simple and effective approach, which is that introducing the shear flow promotes the reorientation and rearrangement of self-assembly morphologies, to selectively build 0D (dot), 1D (tube) or 2D (sheet) nanostructures. These results are helpful to understand the formation of complex micelles by the shear-induced self-assembly of linear ABC triblock copolymers, and can assist experiments to create new 1D cylindrical morphologies.

Acknowledgements

All the authors appreciate the financial support from the Foundation of CAEP (No. 2014B0302040, 2014-1-075) and the National Nature Sciences Foundation of China (No. 11402241).

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Footnotes

† Electronic supplementary information (ESI) available: The equilibrium micelle morphologies. See DOI: 10.1039/c5ra23474c

‡ These authors contributed equally to this work.

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