Article Cite This: Macromolecules XXXX, XXX, XXX−XXX
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Peculiarities of Polyrotaxanes Collapse: Polymorphism of Globular Structure and Stability of Unimolecular Micelles Alexey A. Gavrilov† and Igor I. Potemkin*,†,‡,§ †
Physics Department, Lomonosov Moscow State University, Moscow 119991, Russian Federation DWI - Leibniz Institute for Interactive Materials, Aachen 52056, Germany § National Research South Ural State University, Chelyabinsk 454080, Russian Federation
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‡
ABSTRACT: In the present paper we investigate the behavior of polyrotaxanes under bad solvent conditions by means of computer simulations. Polyrotaxanes can be viewed as amphiphilic macromolecules with annealed sequences because the rings can freely move along the backbone. We showed that, depending on the ring inclusion ratio and the rings radius, one can observe various structures. If the rings radius is small (i.e., enough to fit only one monomer unit), in bad solvent we observed the formation of a collapsed core surrounded by long loose tails and loops with an increased ring density. Upon increasing of the backbone−solvent incompatibility, the core grows as well as the number of rings located on its surface; the latter leads to a partial screening of the backbone−solvent interactions. When the rings are large enough to fit several monomer units, the backbone packs inside the rings, which allows for complete isolation from the solvent. If the number of rings (i.e., the inclusion ratio) is large enough to fit all the monomer units, unimolecular cylindrical micelles are formed in bad solvent. Such micelles are formed by the backbone packed inside the rings that are stacked on “top” of each other. We showed that the transition from the coil to the unimolecular micelle is sensitive to the solubility of the rings: if the rings tend to distribute along the backbone, higher incompatibilities are required for the unimolecular micelles formation compared to the case when the rings tend to form aggregates. In the state of unimolecular micelles the backbone is completely isolated from the solvent, which makes the micelles soluble even at rather high concentrations.
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chains with amphiphilic monomer units.11 Amphiphilic dendrimers,2,12 arborescent,13 or comblike14 macromolecules are other examples of systems forming unimolecular micelles via proper topology design. In the present paper we study the properties of a new type of amphiphilic macromolecules that reveal a variety of globular structures including unimolecular micelles. Structurally, the macromolecules are neither linear nor branched, having movable (annealed) rather than covalently linked (quenched) “sequence” of soluble units. Namely, we investigate the behavior of polyrotaxanes under bad solvent conditions by means of computer simulations. Polyrotaxanes are polymers with threaded rings capable of sliding along the chain and bulky end groups keeping the rings threaded;15−18 the rings are usually cyclodextrins, which are cyclic molecules with a hydrophilic outside and hydrophobic inside. The latter feature allows for threading such molecules onto a hydrophobic backbone and also using them as promising candidates for being nanoreactors for catalysis.19 The presence of hydrophilic and hydrophobic parts makes polyrotaxanes intrinsically amphiphilic. As was mentioned
INTRODUCTION The polymeric micelles have attracted a great deal of attention because of their promising applications in various areas, including targeted drug delivery systems, nanoreactors, emulsion stabilizers, and others.1−3 The “simplest” and the most studied systems are solutions of amphiphilic diblock copolymers which aggregate into micelles above the critical micelle concentration. Consequently, their size and size polydispersity due to the random nature of the micellization are sensitive to the concentration variation.4,5 This limits application potential of the micelles, when a precise size control and integrity are needed in a wide range of concentrations. The natural way to overcome such a disadvantage is to prevent aggregation of the amphiphilic macromolecules at any concentrations via formation of solvophobic core and solvophilic shell within single macromolecules, that is, the so-called unimolecular micelles.2 Because the formation of such micelles does not rely on the self-assembly of a large number of macromolecules, their usage could help to overcome many challenges of using micelles as delivery carriers. There are few ways to stabilize the unimolecular micelles. For example, protein-like sequence design6−9 or “umbrella” concept10 allow prevention of aggregation of linear chains. Single-chain micelles have also been observed for polymer © XXXX American Chemical Society
Received: October 31, 2018 Revised: December 6, 2018
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DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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parameter between alike beads aαα = 200 was obtained:30 χαβ = (0.273 ± 0.007)Δaαβ. Similar relation holds for aαα = 25, so for simplicity we assume that χαβ ≈ 0.3Δaαβ. There are three χαβ parameters in our system: χsolvent−backbone, χsolvent−ring, and χbackbone−ring. The latter one was always zero (i.e., abackbone−ring = 130), while the other two were changed (see below). The χsolvent−backbone value was varied from 0 to 9 with a step of 0.3 to mimic different solvent quality; the system was relaxed for 7 × 106 steps at each value of χsolvent−backbone. For each system, we performed three independent runs (except for the systems studied in Figure 4, where seven runs were performed for better averaging) and compared the results. The structures were always of the same type, neglecting the randomness of the point along the chain where the collapsed core is formed and the presence of loose loops instead of tails and vice versa. The polyrotaxanes were simulated as polymer chains of the length 400 with threaded rings. A single polyrotaxane chain was placed into a simulation box of the size 403 (the number density was equal to 3 and the total number of particles was equal to 192 000), thus mimicking dilute solution conditions. The radius of the rings and the inclusion ratio were varied. The inclusion ratio is the ratio between the studied number of rings to the maximum number of rings the chain backbone can carry. The chain model is in general similar to that discussed in our previous work,31 but in the present work the beads composing the rings are of the same size as the backbone monomer units, and to vary the radius of the rings, we changed the number of beads in each ring. Because the rings have the same thickness as a backbone monomer unit, the inclusion ratio is then simply the number of rings divided by the number of monomer units in the backbone because the maximum number of rings the backbone can carry is equal to its length in monomer units. We tested three types of rings consisting of N = 10, 14, and 19 beads. The smallest ring size was chosen so that the ring could slide freely along the backbone, but only one monomer unit can fit inside such rings due to the interaction with the ring beads; the other two were large enough to fit multiple monomer units. We can consider the rings with N = 10 as a model of α-cyclodextrins (consisting of 6 glycosyl units) which have been used in a number of works dealing with polyrotaxanes;15,17,32−35 the rings with N = 14 and 19 then correspond to γ- or δ-cyclodextrin (8 or 9 glycosyl units) and to ζ- or η-cyclodextrin (11 or 12 glycosyl units) based on the ring contour length. Three values of inclusion ratio were studied: 0.025, 0.05, and 0.1, which corresponds to 10, 20, and 40 rings. To prevent unthreading, both ends of the backbone were cross-linked to an additional ring with the same size as the threaded rings. The amphiphilic nature of the rings is accounted for by the interaction parameters: we assumed that both the solvent beads and backbone beads have affinity for the rings. We tested two regimes of the rings solubility, which we call regularly soluble rings and highly soluble rings. In the former the repulsion parameter asolvent−ring is equal to 130, while in the latter it was equal to 120. We found that in the case of regularly soluble rings (asolvent−ring = 130, χsolvent−ring = 0) the rings tended to form aggregates along the chain even when χsolvent−backbone was also 0. We suppose that the reason for the observed behavior is the size of the solvent beads being equal to the ring beads (which is the method feature), and the rings are attracted because of the depletion attraction (effective entropic force) similar to the effect observed in colloidal
earlier, because the rings can freely move along the backbone, the chain basically has an annealed sequence in terms of the position of the hydrophilic nodes (it changes over time as the rings move due to the thermal fluctuations) that can be adjusted depending on the conditions.
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SIMULATION METHOD AND MODEL The simulations were carried out using dissipative particle dynamics (DPD) method. DPD is a version of the coarsegrained molecular dynamics;20−23 it is a well-known method which has been used to simulate properties of a wide range of polymeric systems, such as single chains in solutions,24 polymer melts,25 and networks.26−28 In the DPD method, macromolecules are represented in terms of the bead-andspring model (each coarse-grained bead usually represents a group of atoms), with beads interacting by a bond-stretching force (only for connected beads), a conservative force (repulsion), a dissipative force (friction), and a random force (heat generator). The soft core repulsion between i- and jth beads is equal to Fijc
l o o aαβ (1 − rij/R c)rij/rij , =o m o o o 0, rij > R c n
rij ≤ R c
where rij is the vector between the ith and jth bead, aαβ is the repulsion parameter if the particle i has the type α and the particle j has the type β (α, β = monomer unit, ring bead, or solvent bead; all combinations are possible), and Rc is the cutoff distance, which represents the size of each bead in real units. Rc is basically a free parameter depending on the volume of real atoms each bead represents;23 Rc is usually taken as the length scale, that is, Rc = 1. All possible pairs of beads at distances closer than Rc to each other interact through the soft core repulsion force. One should also note that the solvent in DPD is taken into account explicitly; the size of all beads (including solvent) in the system is equal. The dissipative and random force in the DPD method act as the thermostat; we used the widely used form for these forces described in ref 23. If two beads (i and j) are connected by a bond, there is also a simple spring force acting on them: rij Fijb = −Kb(rij − l) rij where Kb is the bond stiffness and l is the equilibrium bond length. To simulate nonphantom bonds (which is necessary to keep the rings threaded), we used the approach described in the works:27,29 if the bonds are stiff enough and the volume repulsion parameter is high, the probability of bonds passing through each other is extremely low, and the topology violations do not happen during the simulation. The parameters we used include the following: aαα = 130 (the repulsion parameter between any alike beads, i.e., solvent− solvent, backbone−backbone, and ring−ring, is equal to 130), Kb = 130, and l = 0.4. Harmonic angle potential Ua = Ka(φ − φ0)2 with Ka = 30 and φ0 = 180° was additionally used to keep the rings circular. By changing the repulsion parameter aαβ between beads of different types α and β (α ≠ β), we can set incompatibility between these bead types. It was shown23 that the Flory− Huggins interaction parameter χαβ is linearly related to the difference of the DPD repulsion parameters Δaαβ = aαβ − aαα (α ≠ β and aαα = aββ); the relation for the case of the repulsion B
DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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one is a dense core and the other is a short (its length is ∼50 monomer units) tail with a higher concentration of the rings. Because the ring distribution is random when the chain starts to collapse, there can be noncollapsed loops with an increased ring concentration instead of tails. At higher values of χ (3.6 or higher) a spherical globule is formed in which the rings are located on its surface (Figure 1b); the length of the loose loops and tails is significantly decreased, which is followed by their disappearance at even higher incompatibilities (χ = 9, Figure 1c). At higher inclusion ratios (0.05 and 0.1) the behavior does not change significantly, but the effect of the presence of the rings becomes more obvious: since the amount of monomer units exposed to the solvent decreases with the increase in the inclusion ratio, the transition to the globular state becomes very smooth: the chain exhibits partially collapsed states even at rather high χ-values of ∼3 (Figure 1d). These states are characterized by the presence of a small collapsed core surrounded by long loose tails and loops. The position of such a core along the chain is random and depends on the rings positions during the onset of the collapse. One should note that in the system of regularly soluble rings the fraction of collapsed part is larger compared to the system in which the rings are highly soluble due to the fact that the rings in the former case readily form compact clusters. The difference can clearly be seen in Figure 1e,f: even at the highest studied χvalue of 9 there are loose rings in the case of highly soluble rings, but all the rings are located on the surface of the globule with regularly soluble rings or form long clusters. If the ring radius increases (N = 14), at low inclusion ratio (0.025) the chain behavior is similar to that for the case of N = 10: upon increasing the χ-value, we observe the formation of a spherical globule with loops and tails similar to those depicted in Figure 1a,b. However, we observed two differences, which can be mainly observed in the case of regularly soluble rings: first, the rings have increased tendency to aggregation (which is due to the increased depletion attraction between the rings because of their increased size), and second, starting from certain values of χ, the chain is packed inside the rings. Figure 2a depicts the chain conformation at χ = 2.1; we can clearly see that there is a cluster consisting of several rings inside which the chain forms a helix. We assume that the helical structure is formed in the case when the ring radius was not large enough to fit two monomer units in the ring plane, but is still larger than one monomer. In that case, to fit inside the rings, the backbone always has to move in one “direction” along the ring cluster even in the packed state. The helix shape is obviously the shortest one and allows for the largest portion of the backbone to be isolated from the solvent. Such structure allows complete screening of the polymer−solvent interactions but is unfavorable from the entropic point of view; therefore, it can be observed only at large enough values of incompatibility. We should emphasize that explicit solvent is necessary to obtain such structures in simulation. No significant changes were observed upon further increase in the incompatibility. Increasing the inclusion ratio (to 0.05) reveals that the packing of polymer chain inside the rings can be observed at lower values of χ around 0.9−1.2 in the case of regularly soluble rings (see Figure 2b). Moreover, at χ-values larger than ∼5.1 the packing was observed even for highly soluble rings (Figure 2c), while at lower incompatibilities (χ ∼ 4.5−4.8, Figure 2d) we obtain structures with a collapsed core surrounded by loose tails and loops similar to those obtained for N = 10. One
suspension; the role of the depletant is played by the solvent itself. However, the exact nature of such an attraction is unclear and is out of the scope of the present work; what is important is that in real systems there may be other factors influencing the distribution of the rings along the chain (like the amphiphilic nature of the rings or hydrogen bonding), but ultimately the main result of all the factors is how easily the rings aggregate, and to cover different situations, we studied two different regimes of rings solubility. From the entropic point of view, it is more favorable for the rings to be randomly distributed along the backbone; however, the presence of the depletion attraction results in the formation of rings clusters, especially if the rings are large. The interaction with the solvent allows us to balance the depletion attraction; the ring−solvent repulsion parameter of 120 for the case of highly soluble rings was chosen so that no aggregation was observed even for the largest rings for athermal solvent conditions (χsolvent−backbone = 0). In what follows, χsolvent−backbone for simplicity will be referred to as χ. We should note here that the obtained results concerning the position of the transition points are qualitative rather than quantitative. The used model is deliberately rather generic and does not take into account specific interactions of real systems; our intention was to study the general physical behavior.
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RESULTS AND DISCUSSION When the ring radius is small (N = 10) as well as the inclusion ratio (0.025), the presence of rings has only a minor influence on the chain collapse. The main effect is related to the fact that the rings can partially screen the polymer−solvent interactions, which is especially evident in the case of highly soluble rings. Figure 1a shows a typical chain conformation at χ = 2.1; we can clearly see that the chain can be divided into two regions:
Figure 1. Chain conformations for the case of highly soluble rings for N = 10 and the inclusion ratio of 0.025 at (a) χ = 2.1, (b) χ = 5.4, and (c) χ = 9. For the inclusion ratio of 0.1 for (d) highly soluble rings, χ = 3.3, (e) highly soluble rings, χ = 9, and (f) regularly soluble rings, χ = 9. C
DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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Upon increasing the χ-value to 9, we again observed the formation of a small exposed globule while the major part of the backbone was packed inside the rings (see Figure 2g for an example of such conformation). Increasing the rings radius to N = 19 has two general effects on the system behavior: first, the enhanced depletion attraction promotes the formation of long ring clusters, and second, the volume available for the backbone inside such clusters is much larger. These two factors facilitate the formation of densely packed structures inside the rings. For example, for regularly soluble rings at the inclusion ratio of 0.05 we can observe long ring clusters with backbone folded inside them at χ ∼ 0.6−0.9 (Figure 3a). If we further increase the χ to ∼1.8, the chain
Figure 2. Chain conformations for N = 14 and (a) inclusion ratio 0.025, regularly soluble rings, χ = 2.1; (b) inclusion ratio 0.05, regularly soluble rings, χ = 1.2; (c) inclusion ratio 0.05, highly soluble rings, χ = 5.7; (d) inclusion ratio 0.05, highly soluble rings, χ = 4.5; (e) inclusion ratio 0.05, regularly soluble rings, χ = 4.5; (f) inclusion ratio 0.1, regularly soluble rings, χ = 1.5; (g) inclusion ratio 0.1, regularly soluble rings, χ = 9.
Figure 3. Chain conformations for the case of regularly soluble rings for N = 19 and the inclusion ratio of 0.05 at (a) χ = 0.9 and (b) χ = 1.8 and the inclusion ratio of 0.1 at (c) χ = 0 and (d) χ = 0.6. The rings are rendered semitransparent in the latter picture for better visibility.
should note that for the inclusion ratio of 0.025 the packed structures for highly soluble rings were observed only in one system. Obviously, the reason for that is that the increased inclusion ratio promotes the formation of ring clusters, which facilitates the backbone packing. At the highest studied χvalues of 9 and the inclusion ratio of 0.05 for the case of highly soluble rings, the system had the structures similar to that depicted in Figure 2c: a globular core surrounded by single rings and a cluster of rings containing a folded chain part attached to the core. For the regularly soluble rings at high χvalues we observed an increased probability of the formation of several ring clusters instead of one (Figure 2e) or one larger cluster. At the highest studied inclusion ratio of 0.1 almost the entire backbone can fit inside the rings. For the case of regularly soluble rings, at high enough χ-values (∼1.5 or more), the chain forms small collapsed core attached to long ring clusters filled with densely packed backbone (Figure 2f). Upon increasing of the incompatibility, the ring clusters merge, and finally, at very high χ-values the chain is mainly packed inside the rings, while some portion of it forms a small globule (Figure 2g). The behavior of the system with highly soluble rings is similar to that observed at the lower inclusion ratio of 0.05: the formation of the densely packed chain regions inside the ring clusters happens only at rather high χ-values (the transition is similar to that depicted in Figure 2c,d), namely, at χ ∼ 4.8−5.1.
adopts its final conformation with only a small globular core not being inside the rings (Figure 3b), which do not change significantly even at χ = 9. It is easy to understand from the results for the inclusion ratio of 0.05 that doubling this value would lead to the situation when the chain can be completely fitted inside the rings. When the inclusion ratio is equal to 0.1 and the rings are regularly soluble, there are large ring clusters even at χ = 0 (Figure 3c). It is evident that the chain conformation inside the rings is rather stretched; upon increasing of the χ-value, the chain backbone starts to retract inside the rings, and at χ ∼ 0.6−0.9, the backbone is completely hidden inside the rings, forming a cylindrical micelle (Figure 3d). We can also see that there is a region inside the cylinder in which the backbone is not densely packed because the volume inside the rings is larger than the backbone volume. This implies that other molecules that have affinity for the backbone can be fitted inside, and also even lower inclusion ratios can be used to obtain conformation with completely isolated backbone. In the case of highly soluble rings, there are no ring clusters at χ = 0; instead, the rings are randomly distributed along the backbone. Starting from χ ∼ 2.7, small collapsed backbone core starts to emerge. It grows to χ ∼ 3.6, and finally at χ ∼ 3.9 the backbone starts to pack inside the rings starting from the chain parts located close to the collapsed core. A further D
DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules increase in χ value increases the fraction of monomer units packed inside the rings; at high χ values all the monomer units are hidden inside the rings, and the structure starts to look like that depicted in Figure 3d, with the only difference being the rings that are cross-linked to the backbone ends do not participate in the formation of the main ring cluster but freely rotate in the solvent (like in Figure 2c). To characterize the described transitions in a more quantitative way, we calculated the dependencies of the solvent−backbone interaction energy on the χ value. This quantity basically reflects how well the backbone is isolated from the solvent. Three systems in which the backbone packing inside the chain occurs were studied: (1) N = 14, inclusion ratio is equal to 0.175; (2) N = 19, inclusion ratio is equal to 0.0825; (3) N = 26, inclusion ratio is equal to 0.035. The latter one was added to demonstrate the influence of the increasing ring size; all these systems were chosen so that the number of rings was just enough to fit the entire backbone, and the rings were highly soluble. Seven independent runs were carried out to ensure good averaging. The behavior of the system with N = 14 was found to be the simplest (see Figure 4,
For the system with N = 19 we observed much more interesting behavior. The solvent−backbone interaction energy (Figure 4, top) clearly shows the two transitions described above: from a swollen chain to a chain with collapsed core and from a chain with collapsed core to a cylindrical micelle with completely isolated backbone. We can also see that the averaging error is significantly increased around the second transition, which is due to the fact that in each individual case the transition occurs at slightly different values of χ because of the random position of the collapsed core and rings. Therefore, the averaging somewhat “smears out” the curve, while the transition is actually more abrupt for each individual system. The dependence of the radius of gyration on χ for this system (Figure 4, bottom) also reveals an interesting feature: there is a minimum observed at χ ∼ 3.5. Obviously, the increase in the gyration radius at higher values of χ is related to the onset of the packing of the backbone inside the rings because such conformation is more elongated compared to the spherical globular core observed at lower values of χ. We should mention that a special relaxation procedure was utilized to obtain more consistent results: the χ value in the region of the second transition (from 3 to 6) was slowly changed (with a step of Δχ = 0.003) during 100 × 106 integration steps. Such procedure mimics a realistic system cooling which happens at a finite speed. Finally, the behavior of the solvent−backbone interaction energy for the system with N = 26 and the inclusion ratio of 0.035 is in general similar to that observed in the system with N = 19 with two detectable transitions. The first transition for N = 26, however, is shifted toward smaller values of χ and is somewhat more abrupt compared to N = 19, which can be attributed to the less effective screening of the solvent− backbone interactions because of the lower inclusion ratio. Surprisingly, the position of the second transition is almost the same as for the case of N = 19, while we expected it to be shifted toward smaller χ as well. This is probably due to the fact that the globular core is rather large at such low inclusion ratios, so a large portion of the backbone is already isolated from the solvent even without packing inside the rings (compare the blue and red curves in Figure 4, top). Another possible reason is the influence of the ends of the micelle: the backbone there is not isolated from the solvent, and in shorter micelles the impact of this factor is larger, thus making the packing less energetically favorable. A special relaxation procedure was utilized for the case of N = 26 as well: the χ value was slowly changed from 1.5 to 6 (with a step of Δχ = 0.003) during 600 × 106 integration steps. We found that the system with N = 26 is more prone to the formation of kinetically trapped states compared to the system with N = 19; in these states one or two rings were located on a small loop outside of the unimolecular micelle. We attribute that to the topological restrictions during the packing into larger rings. Such structures were considered metastable and were not used for averaging. Relaxation with a slower change (namely, 4 times slower) of the χ value (which is more realistic) compared to the case of N = 19 seems to reduce the number of defect structures. We should also mention that in some rare cases we observed the unthreading of the rings since the special rings cross-linked to the chain ends (which served the role of bulky groups) became overall less rigid because of the increase in the contour length. These systems were restarted with a different random seed.
Figure 4. Dependence of the solvent−backbone interaction energy for the three studied systems (top) and the dependence of the backbone radius of gyration (bottom) on the χ value for the system with N = 19 and the inclusion ratio of 0.0825.
top); only one transition is present: from a swollen coil to the unimolecular micelle state. No intermediate state with a collapsed core was observed; this can be attributed to a rather good screening of the backbone by the rings because of the high inclusion ratio, which allows the chain to be in a swollen state even at rather high χ values. E
DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules The formation of the cylinder structure observed at large N and inclusion ratios (Figure 3d) means that the polyrotaxanes with large rings have a unique property: they essentially remain solvophilic even at large polymer−solvent incompatibilities by forming unimolecular micelles. Let us test the aggregation of systems with different inclusion ratios. To do that, we placed 27 micelles prepared from polyrotaxane chains with N = 19 and the inclusion ratio of 0.025, 0.05, and 0.0825 at χ = 6 with highly soluble rings in a box with the size 603. The system was relaxed for 14 × 106 steps; such time is enough for the micelles to move the distance equal to the half-size of the simulation box. The resulting system states are depicted in Figure 5 and Figure 6 (top).
Figure 6. Final state of the system prepared from polyrotaxane chains with N = 19 and the inclusion ratio of 0.05 at χ = 6. The system contains 27 unimolecular micelles (top) or 64 unimolecular micelles (bottom, relaxed for 30 × 106 steps). The polyrotaxanes are rendered semitransparent for better visibility.
The system with the lowest inclusion ratio behaves somewhat similar to just bare polymers in bad solvent: large spherical aggregates are formed. They are, however, stabilized by the rings on the chain surface, so there is more than one aggregate in the box. At the intermediate inclusion ratio of 0.05, the chains have only small exposed cores (see Figure 3b), and these cores merge to reduce the surface energy. The long ring clusters stabilize the surface of such enlarged cores, so again several aggregates are present in the box. It is also interesting that the free ends of the ring clusters attach to the collapsed cores of other chains to reduce the area of contact with the solvent, so there are some “trees” of polyrotaxanes in the system (can be observed in Figure 6, top). One should note that the formation of such structures is mostly due to the
Figure 5. Final state of the system with 27 unimolecular micelles prepared from polyrotaxane chains with N = 19 and the inclusion ratio of 0.025 (top) and 0.0825 (bottom) at χ = 6. The polyrotaxanes are rendered semitransparent for better visibility. F
DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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usage of rings as the end-capping bulk groups, which do not cover the free ends of the ring clusters effectively. We tested an additional system with an increased concentration (64 micelles in a box of the same size, Figure 6, bottom); the size of the trees as well as aggregates formed by the collapsed cores is increased. Finally, when the number of rings is enough to fit the entire backbone (Figure 5, bottom), we observe no aggregation (except for some alignment of the cylindrical unimolecular micelles, which might as well be the method feature as for the case of the formation of ring clusters) so such structures remain soluble even at rather large concentrations.
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CONCLUSIONS Summarizing, in this work we investigated the behavior of polyrotaxanes in solvent of different quality by means of coarse-grained simulations. We showed that if the rings are large enough to fit several monomer units and the inclusion ratio is sufficient, unimolecular cylindrical micelles are formed in bad solvent. Such micelles are formed by the backbone packed inside the rings that are stacked on top of each other. This configuration completely isolates the backbone from the solvent which makes the micelles soluble even at rather high concentrations. We showed that the transition from the coil to the cylindrical micelle is very sensitive to the solubility of the rings: if the rings tend to distribute along the backbone, significantly higher incompatibilities are required for the unimolecular micelles formation compared to the case when the rings tend to form aggregates because of the depletion attraction. In the former case we observed the formation of a dense collapsed core with loose tails at intermediate values of incompatibility which was followed by packing this core into the rings at higher values of χ resulting in the unimolecular micelles. The observed structures have great potential to be used as stimuli-responsive carriers for drug delivery: the cavity formed by the rings can hold various molecules that have affinity for the backbone. We hope that our work will further facilitate experimental investigations of polyrotaxanes and their applications.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Alexey A. Gavrilov: 0000-0002-0916-8219 Igor I. Potemkin: 0000-0002-6687-7732 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The financial support of the Russian Foundation for Basic Research and the Government of the Russian Federation within Act 211, Contract No. 02.A03.21.0011, is gratefully acknowledged. The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.
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DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.macromol.8b02326 Macromolecules XXXX, XXX, XXX−XXX
