Grafted Polyrotaxanes - American Chemical Society

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Grafted Polyrotaxanes: Scaling Theory and Molecular Dynamics Simulations Holger Merlitz,*,†,‡ Wei Cui,† Chan-Fei Su,† Chen-Xu Wu,*,† and Jens-Uwe Sommer‡ †

Department of Physics and ITPA, Xiamen University, Xiamen 361005, P. R. China Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany

‡

ABSTRACT: We analyze the scaling properties of grafted polyrotaxanes and compare the resulting scaling laws with molecular dynamics simulations. These molecules display a resemblance to bottle brushes, but the mobility of the rotors yields a backbone tension that is uniform along its contour. Properties like brush extension, backbone tension, and rotor mobility as functions of rotor density and degree of polymerization of grafted chains are well covered by a simple scaling model. The dynamics of rotor release after a breakup of the polyrotaxanea process recently observed in laboratory experimentsis studied in detail. It consists of three stages: a rapid backbone retraction, a driven rotor expulsion, and a diffusive transport. The complexity of this process, which may be exploited for drug delivery systems, is asking for further in-depth simulation studies. fragments,12−14 a grafted polyrotaxane releases a large number of its rotors into the environment. Such a complex could therefore qualify as a drug-delivery system.7,8 In the present work, we apply the scaling theory of polymer brushes, initially developed by Alexander15 and deGennes,16 to grafted polyrotaxane complexes. In many aspects, grafted rotaxanes behave similar to bottle brushes (which have their side chains grafted onto fixed anchors). The scaling theory of bottle brushes, with a particular attention to their persistence lengths, has been studied theoretically and via Monte Carlo simulations in earlier works.17−20 These investigations have been extended to copolymer bottle brushes21−23 as well as dendritic bottle brushes.24 While conventional bottle brushes display a backbone tension that is highest in the middle of the molecule and dropping toward the ends,25 grafted rotaxanes have mobile anchor points which rearrange in such a way that the backbone tension is uniform along the backbone axis. These differences are a consequence of the “spillover” of the polymer brush at the two backbone ends, and they diminish asymptotically toward the limit of very long backbones or comparably short chain lengths of the grafted polymers.11,13,14 Yet, the additional degree of freedom of sliding components appears to alter the viscoelastic properties of grafted polyrotaxanes as opposed to bottle brushes with static anchor points.9 The backbone stiffness is as well affected by the local anchor-point density, and a temporary rearrangement allows for a locally enhanced flexibility, even hairpin conformations.7

I. INTRODUCTION Polyrotaxanes are molecular complexes, consisting of linear polymeric backbones and ringlike molecules, named rotors. As rotors may serve doughnut-shaped cyclodextrins, which, under favorable solvent conditions, self-thread onto the backbone with which their inner walls interact in terms of hydrophobic attraction. After this process of self-organization is completed, the ends of the backbone are capped with stopper groups, which sterically prevent the rotors from sliding off the backbone.1−3 Polyrotaxanes are interesting because the threaded rotors are shielding the backbone from its solvent. Further more, a nanoengineering of the polyrotaxane’s interactions with its environment is possible through functionalization of the rotors.4 If the density of threaded rotors is sparse, and these rotors are subsequently grafted onto a surface, then a sliding grafted polymer layer is formed.5 The stiffening of these molecules has recently been studied by Pinson et al. using an exactly solvable theoretical model.6 On the other hand, experiments have shown that hairpin conformations are possible as well in free polyrotaxane molecules, if their rotor density remains significantly below the saturation density.7 Another degree of complexity is added to the polyrotaxane when polymers are densely grafted onto the rotors to create cylindrical polymer brushes with sliding anchors.8,9 The osmotic pressure generated by the brush then leads to an exceptionally high stiffness of the complex, which allows the assembly of highly ordered films under flow.10 Moreover, the mechanical stress generated by the brush has been shown to destabilize the complex to a degree that may even cause a rupture of the backbone.11 While in the case of ordinary bottle brushes such a breakup would lead to the creation of only two © XXXX American Chemical Society

Received: January 5, 2014 Revised: May 6, 2014

A

dx.doi.org/10.1021/ma500032a | Macromolecules XXXX, XXX, XXX−XXX

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(green), and grafted chains (N = 64, white). The backbone is capped at its ends by oversized stopper monomers (yellow). The MD simulations were carried out using the open source LAMMPS molecular dynamics package.27 The equation of motion of each monomer is defined by a Langevin equation:

In this work, we derive scaling relations for the brush layer extension in good solvent, the backbone tension, and the mobility of the rotors as a function of rotor density and degree of polymerization of the grafted chains. The validity and limitations of the scaling approach are studied via a comparison with molecular dynamics (MD) simulations. We further analyze the dynamics of rotor release after a breakup of the backbone in considerable detail. The paper is organized as follows: In section II we first present a simple coarse-grained model of a grafted polyrotaxane, in which the coordinate system is defined and the MD simulation setup as well as details of the simulation procedure are presented. The scaling theory of grafted polyrotaxanes is derived in section III. A comparison between scaling theory and MD simulations is carried out in section IV.1, followed by simulations of the rotor release in section IV.2. A summary of our findings is offered in section V.

m

(1)

with the bead size b and the potential depth ϵ. In the present work, the LJ potential is truncated at its minimum, rc = 21/6b (and shifted up by ϵ), so that only repulsive interactions remain and the polymers are approximately athermal. The monomer diameter b defines our system’s length unit, and the parameter ϵ defines the energy unit. The monomer’s unit mass (m = 1) and the unity Boltzmann constant complete the system of LJ units, which is applied throughout the paper. The beads are connected through a nonlinear spring potential ⎡ ⎛ r ⎞2 ⎤ KR 0 2 ⎢ UFENE(r ) = − ln 1 − ⎜ ⎟ ⎥ ⎢⎣ 2 ⎝ R 0 ⎠ ⎥⎦

dt 2

+ζ

dri ∂U =− + Fi ∂ri dt

(3)

where ri is the (absolute) coordinate of the ith monomer, U is the total conservative potential, and Fi is a random external force without drift and a second moment proportional to the temperature and the friction constant ζ. In the LJ system of units, the time is defined in units of τ = (ϵ/(mb2)1/2, which is proportional to the period of free micro-oscillations of two beads within their mutual LJ potential. For our simulations, the temperature T = 1, a time step Δt = 0.0015τ, and the friction coefficient ζ = τ−1 are implemented. The friction leads to an overdamped motion on length scales of the bead size and hence to a Brownian motion under the approximation of an immobile solvent, i.e., approximately Rouse dynamics.28 The simulation box is fully periodic to eliminate the possibility of spurious boundary effects. Initially, the backbone is given a linear conformation; rotors are threaded uniformly onto that backbone, the ends of which are capped by oversized stopper monomers. The grafted chains are mounted onto each of the eight rotor monomers in radial direction. What follows are 10 million simulation steps (i.e., 1.5 × 104 LJ times) to relax and equilibrate the polyrotaxane. Later on, trajectories of 50 million simulation steps (7.5 × 104 LJ times) are carried out, during which 5000 molecular conformations are stored for following analysis. Since the conformations of the entire molecule are varying dynamically, it is necessary to define local coordinate systems. This is achieved as shown in Figure 1: Each rotor has a center

II. POLYMER MODEL AND MD SIMULATIONS The MD simulations employ a coarse-grained bead−spring polymer model inside an implicit, approximately athermal solvent. Pair interactions are implemented using a LennardJones (LJ) potential of the type ⎡⎛ b ⎞12 ⎛ b ⎞6 ⎤ ULJ(r ) = 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠

d2ri

(2)

also known as the finite extensible nonlinear elastic (FENE) potential,26 with a spring constant K = 30 and a maximum extension of R0 = 1.5. In our simulations, this parameter choice yields average bond lengths between a = 0.97 (relaxed polymer) and a = 1.05 (polymer under high tension). No further bond potentials (e.g., bond or dihedral angle potentials) are implemented; i.e., the chains are almost fully flexible and freely jointed, apart from the steric exclusions enforced by the beads. The core of the polyrotaxane is a backbone chain of Nb monomers. Rotors are made of 8 monomers of ringlike architecture and threaded onto the backbone chain. In reality, these rotors are rigid, but for our simulation model it is unnecessary to fix the bonds because the chains, grafted onto each of the 8 rotor monomers, generate a significant radial pull which stabilizes the ringlike shape of the molecule. Both ends of the backbone are capped with single, oversized monomers of diameter 3 to prevent the rotors from slipping off the thread. In our simulations, we vary the number Nr of rotors that are threaded onto the backbone as well as the degree of polymerization N of the grafted chains. The table of contents graphic displays a snapshot of such a grafted polyrotaxane model, with a backbone chain (red, Nb = 180), Nr = 60 rotors

Figure 1. Local coordinates: each rotor has a center of mass (Rcm), which is projected onto the backbone contour to define the origin of a local coordinate system. The tangent to this contour at the origin defines the axial (z) orientation of the coordinate system, pointing toward the direction of increasing backbone bead indices. The radial distance of the anchor monomer of the grafted chain to the z-axis is denoted rg with ⟨rg⟩ ≈ 2 (monomer diameter).

of mass, which does not in general fall precisely onto the contour of the backbone. A projection of that center of mass onto the backbone then defines the origin of the coordinate system, and the tangent to the backbone defines the orientation of the z-axis, which points toward the direction of increasing backbone bead indices. Radial distances are measured with respect to that axis. Since the backbone is under a significant B


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rs 4/3 − rg 4/3 ∼ N (rgσ )1/3 = N (Ng /2π )1/3 σr1/3

tension, once the rotors are grafted, its wormlike chain contour bends insignificantly over the domains typically occupied by the rotor and its brush. The backbone longitudinal coordinate of the sliding rotor is derived from the origin of the rotor’s local coordinate system, through its tape measure distance from the backbone’s first bead, integrated along the chain’s contour.

If the brush is extended, so that rs ≫rg, this relation simplifies to the power law rs ∼ N3/4Ng1/4σr1/4

III. SCALING THEORY OF GRAFTED POLYROTAXANES The scaling theory of nonplanar polymer brushes has been discussed before by Halperin17 and applied to bottle brushes by Rouault and others.18,19 In the present work, we use the blob picture in the context of threaded and grafted rotors, effectively yielding a bottle brush with sliding anchors. We first of all make a couple of simplifying assumptions, and at the end of this section we briefly discuss the limits of their validity. Locally, we may regard the polyrotaxane as a straight cylinder (neglecting any bending of its contour), with threaded rotors of average distance Δz = L/Nr, L denoting the contour length of the backbone and Nr being the total number of rotors. Be Ng the number of chains grafted symmetrically onto each rotor (Ng = 8 is a fixed constant during our simulations), at an average radial grafting distance rg to the cylinder’s central axis. In this way we may define the average grafting density as σ = NrNg/ (2πrgL), which can be separated into the product of linear grafting densities σ = σrσg. Here, σr = Nr/L = (Δz)−1 is the (variable) longitudinal density of threaded rotors, and σg = Ng/ (2πrg) is the (constant) axial density of chain anchor monomers grafted on each rotor at the radius rg. Scaling theory of correlation (or concentration) blobs refers to semidilute polymer solutions and assumes the coexistence of two length scales, yielding a short-range regime, in which chain conformations behave as in dilute solvent, and a long-range regime, in which these conformations are significantly altered by interchain interactions. The blob size defines the distance at which both regimes are crossing over. In the context of planar polymer brushes, this picture leads to the Alexander−deGennes scaling model with blobs that are of uniform size and perpendicularly stretching away from the substrate onto which the chains are grafted.15,16 With a cylindrical polymer brush, the average area available to each single chain depends linearly on the radial distance r to the central axis of the cylinder, and so does the cross-sectional area of each blob,17 S(r) ∼ r/(rgσ), where r > rg. The linear dimension of the blob should then scale with the square root of that area, i.e. ⎛ r ⎞1/2 ξ(r ) ≈ ⎜⎜ ⎟⎟ ⎝ rgσ ⎠

(6)

(7)

The radial thickness of the brush layer therefore increases slowly with the density σr of threaded rotors and the number Ng of grafted chains per rotor and sublinearly with the degree of polymerization N of the grafted chains. Next, we evaluate the free energy of the grafted layerto which each blob contributes ∼kBT. The total number of blobs along a chain is Nblob ∼

∫r

g

rs

⎡ 2σrNg ⎤1/2 1/2 ξ −1 dr = ⎢ ⎥ (rs − rg1/2) ⎣ π ⎦

(8)

which in case of extended brushes simplifies to the power law Nblob ∼

σrNgrs ∼ N3/8(Ngσr)5/8

(9)

The total free energy of the brush is derived after multiplication with the total number NrNg of chains N3/8(NgNr)13/8 Fbrush ∼ KBT L5/8

(10)

This free energy of the brush is also generating a tension of the backbone. In ordinary bottle brushes, this backbone tension is highest in the middle of the backbone and dropping toward the both ends25a result of the spillover of the grafted chains into the void areas beyond the backbone ends. This is different in the case of sliding rotors, which always rearrange in such a way that the tension along the backbone remains uniform. This necessarily leads to a nonuniform rotor density along the backbone (Figure 2). The (uniform) bond force of the backbone scales as fb kBT

∼

N3/8(NgNr)13/8 L13/8

= N3/8(Ngσr)13/8

(11)

(4)

To determine the layer thickness, we have to integrate over the radial array of blobs, summing up their individual diameters. Here we make use of the fact that, in good solvent, the number g of monomers in each blob has to scale according to Flory’s theory as ξ ∼ g3/5. This implies that ξ5/3ξ−1 is the number of monomers of the chain per blob size, and we simply integrate along the (radially stretched) chain until we hit its degree of polymerization N (mass balance), yielding29 N∼

∫r

g

rs

ξ 5/3ξ −1 dr Figure 2. Average distance between neighbored rotors: Contrary to ordinary bottle brushes, this distance varies along the backbone. Inset: bond forces are constant along the backbone. Here, the length of grafted chains was N = 128, the number of backbone monomers Nb = 180, and the number of rotors Nr = 100.

(5)

Given the grafting distance rg and the chain length N, this integral delivers the radius of the brush surface, rs, as well as the radial layer thickness H = rs − rg, yielding C


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This backbone tension increases quite rapidly with the linear density σr of threaded rotors and the number Ng of chains grafted on each rotor and rather slowly with the degree of polymerization N of the grafted chains. Since the rotors are threaded onto the backbone, but not bonded to it, they are capable of sliding along the backbone axis. The mobility of these sliders is not a function of the length N of chains grafted onto them, as long as the system is in its brush regime: A motion of two neighbored rotors against each other affects the conformations of involved chains only locally and dissipates over length scales of the correlation length, that is, the blob size just above the grafting point. The latter, however, is independent of N. The fluctuation of a rotor about its average position is then only a function of the free distance it can relocate before hitting into a neighbored rotor and scaling as σz ∼ σr −1

(12)

III.1. Limitations of the Scaling Picture. A necessary condition for the brush to enter its universal scaling regime is a sufficient overlap between the grafted chains. Since in our simulations we vary the linear density of threaded rotors, we estimate the threshold density at which the brush enters its scaling regime. Consider the longitudinal extension of the coil that is created by the chains grafted on a single rotor. It approximates the radius of gyration of a single chain in that direction, which is ≈18−1/2N3/5 (the three-dimensional radius of gyration of the isotropic coil of a free chain would amount to ≈6−1/2N3/5). Two neighbored rotors have to approach each other to distances that do not exceed twice that longitudinal coil extension, so that Δz = σr −1 ≲

2 3/5 N 3

Figure 3. Scaling properties as a function of threaded rotor density σr. (a) The radial layer thickness according to eq 6. (b) The backbone bond forces according to eq 11, scaled with the free-backbone forces (see text). (c) The longitudinal mobility of rotors according to eq 12. Symbols are MD simulations; dashed lines are scaling predictions.

(13)

Another limitation is posed by the boundary effects at the backbone ends: As a result of polymer spillover, the rotor density increases near these ends, so that the usage of the average rotor density for scaling predictions is accurate only in the case of long backbones, whenever these molecules approach one-dimensional systems of infinite length, so that boundary effects remain small.

well for all systems, so that the scaling relation eq 7 would have delivered a less accurate agreement of the data with the scaling law. In Figure 3b, we plot the average bond forces of the backbone as a function of the rotor density σr. The bond force is easily computed from the bond length and the derivative of the bond potential eq 2. The absolute bond force may to some extent depend on simulation details like the time step, so that we have rescaled every force with the average bond force taken from a bare backbone, simulated under the same conditions (and yielding average forces of 2.83kBT per LJ length). Our results yielded average bond forces that were 1.5 times as strong as the bare backbone’s forces for the system (N, Nr) = (16, 20) and exceeding 17.5 times the bare backbone’s forces for the dense system (128, 100); see also Figure 2 (inset). For the plot, the extended-brush approximation eq 11 was applied and f b divided by N3/8, which should yield a slope of 13/8 (dashed line), as long as the extended brush limit is sufficiently approximated by the system. At low rotor density, the data deviate from the scaling prediction, which can be accounted to the fact that the grafted chains are not yet inside the brush regime as approximated by eq 13. Additionally, at N = 16 and N = 32, the extended brush approximation is not fully justified, leading to a certain vertical shift of the data points, which depends on N. Yet, the basic predictions of the scaling

IV. RESULTS IV.1. Scaling Properties of Grafted Polyrotaxanes. The MD simulations covered polyrotaxanes with grafted chains of length N ∈ {16, 32, 64, 128} and Nr ∈ {20, 40, 60, 80, 100} threaded rotors (additionally Nr = 30 for the N = 128 system). In this section, the number of backbone monomers was held constant at Nb = 180, and therefore the backbone’s contour length (which varied somewhat with its tension) about an average value of L ≈ 180. Figure 3 summarizes the scaling properties of the various systems. In Figure 3a, the radial thickness of the grafted layer is plotted as a function of the density σr of threaded rotors. Following eq 6, we plot the difference rs4/3 − rg4/3 (rs being the radius of the brush surface, rg the radial grafting distance), scaled with N. This expression should scale ∼σr1/3, as represented by the dashed line. In this case, we have approximated rs by the averaged radial coordinate of the chain ends, which varied between rs ≈ 6.36 in the system (N, Nr) = (16, 20) and rs ≈ 35.4 in the system (128, 100). Since rg ≈ 2, the extended-brush approximation rg ≪ rs was not satisfied D


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ordinary bottle brushes, may be compromised to a degree that they tend to break apart.8,12 Considering the significant increase of backbone tension shown in Figure 3b and eq 11, it is quite natural to assume that covalent bonds of the backbone of densely threaded polyrotaxanes may break. Our MD simulations employ bond potentials which prohibit such a failure, but it might be an interesting exercise to break up selected bonds and analyze the dynamics of unthreading of the rotors. For our simulations, we have selected polyrotaxanes with Nr = 80 threaded rotors of different grafted chain lengths, relaxed the conformations entirely, and then removed the stopper monomers on the both ends of the backbone. What followed was an unthreading of the rotors, which may be divided into three stages: Initially, the backbone, being a loaded entropic spring, is retracting rapidly and thereby releasing some of the rotors near its both ends. Figure 5 displays two snapshots: at the time of

theory are well reproduced: The backbone tension depends weakly on the degree of polymerization N, but strongly on the density σr of threaded rotors. Figure 3c summarizes the mobilities of the rotors when sliding along the backbone axis. These data were computed by analyzing the histograms of coordinates of each rotor, taking their widths (defined as twice the standard deviation from the average position) and taking the average over all rotors. As discussed during the derivation of eq 12, there exists no dependence of this mobility on the degree of polymerization N of grafted chains, as long as these are in the brush regime. Deviations from the prediction (dashed line) therefore exist predominantly in the cases of short chains (N = 16 and N = 32), in combination with low rotor densities. Otherwise, the slope of eq 12 is well reproduced. The simulation results indicate that the scaling theory of grafted polyrotaxanes delivers accurate predictions for their fundamental properties, as long as the conditions for the brush regime, eq 13 and the extended brush approximation rg ≪ rs are satisfied. The increase of the persistence length of the backbone as a result of its increasing tension has been analyzed before in various studies of bottle brushes,17−20 and here we only add a couple of example computations. In Figure 4, the radii of

Figure 5. Initial stage of rotor release: after removal of the stopper monomers, the backbone (red) retracts rapidly to unthread rotors (green) that are located near its ends. Upper frame: before release; lower frame: 150 LJ times after release. Here, Nr = 80 and N = 128. Monomers of grafted chains (white) are reduced in size for clarity. Figure 4. Root-mean-square radius of gyration of the backbone as a function of the backbone length. Here, σr = 0.33 and N = 32. In red: fit of eq 14. Inset: at different rotor densities. Here, Nb = 180 and N = 32. Red lines indicate the persistence length after eq 14.

release and shortly after (monomers of the grafted chains are only shown as tiny white dots for clarity). This initial backbone retraction, during which the rotors propagate only marginally, occurs within time scales of about 100 LJ times. Its dynamics does also depend on the initial tension of the backbone, which, in turn, is a function of the degree of polymerization of the grafted chains, eq 11. Figure 6 summarizes the progress of unthreading of grafted rotors, with various degrees N of polymerization of the grafted chains. Each of the curves is averaged over four independent simulation runs. The initial number of 80 threaded rotors has decreased below 60 after just a few 100 LJ timesthis is a result of the aforementioned backbone retraction, and the curves indicate that the speed of this process is increasing with N due to the increasing backbone tension. What then follows is the second stage of unthreading: A forcefully driven expulsion of rotors off the backbone due to the osmotic pressure of the brush. The resulting driving force is proportional to the number of blobs and therefore ∼N3/8 according to eq 9. However, the friction acting on each rotor is, according to Rouse, proportional to its mass and hence ∼8(N + 1), so that the resulting terminal velocity of the rotor would under constant dragscale approximately as ∼N−5/8, i.e.,

gyration of the backbones of polyrotaxanes at a given rotor density σr = 0.33 and degree of polymerization N = 32 are plotted. This plot may be used to estimate the persistence length lp of the backbone via a standard formula, derived for the wormlike chain model30 Rg2 =

Nblp 3

− lp 2 +

2lp3 Nb

−

⎛ ⎞⎞ 2lp 4 ⎛ ⎜1 − exp⎜ − Nb ⎟⎟ ⎜ l ⎟⎟ Nb 2 ⎜⎝ ⎝ p ⎠⎠

(14)

The fit yields a value of lp ≈ 150 LJ units, similar to the backbone length of the polyrotaxanes analyzed in Figure 3. This persistence length is a nontrivial function of the density σr of grafted rotors, as shown in the inset: Both radius of gyration and persistence length are increasing rapidly with σr, which is a consequence of the steep increase of the backbone tension as shown in eq 11. IV.2. Dynamics of Unthreading. Laboratory experiments have shown that the stability of grafted polyrotaxanes, just like E


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t0)−1 is observable (dashed lines). In this diffusive regime, the release times should now scale proportional to L2, but since it is only the final stage which is diffusion driven, a single master curve is not achievable after a scaling of the time coordinate with L2. Figure 7 shows three snapshots, taken during advanced stages of unthreading. The released rotors form a cloud of

Figure 6. Number of threaded rotors (initially 80) as a function of simulation time after removal of the stopper monomers. During the final stage of the process, the curves are gradually crossing over into the (solitary) diffusion mode with a power exponent according to eq 16. Inset: with different backbone lengths. Here, the initial rotor density was kept constant at σr = 0.44, with grafted chains of length N = 32. Figure 7. Advanced stages of rotor release: After 1000 LJ times, the tangle of free rotors extends along the longitudinal axis of the polyrotaxane (left frame). After 6500 LJ times (center frame), the backbone (red) is in the process of coiling up, while the tangle of released rotators begins to diffuse into all directions. After 50 000 LJ times (right frame), the cloud forms a dilute solution. Here, Nb = 270, Nr = 120, and N = 32.

decreasing upon an increase of N. The constant drag condition is poorly satisfied, since the pressure generated by the brush is rapidly decreasing during the process of unthreading, and this particular scaling regime is not observable at all. Yet, Figure 6 reveals that there exists a crossover of the dynamics, taking place after a couple of hundred LJ times, at which rotors of higher masses begin to unthread slower than those of lower masses. The final stage of the unthreading process sets in when the osmotic drift of the brush ceases, and the system enters into its mushroom regime as indicated in eq 13. From now onward, the motion of the remaining threaded rotors turns diffusive. Note that a single rotor, threaded on a backbone, would display a Brownian motion and satisfy the one-dimensional diffusion law ⟨[z(t ) − z(t 0)]2 ⟩ = 2D(t − t0)

molecules which initially expands somewhat into the longitudinal direction of the backbone (left frame)a consequence of rotors that are continuously being pushed off the backbone. At the same time, the backbone is contracting and gradually coiling up. Later on, the cloud incrementally diffuses into all directions while the backbone, being now released of the osmotic pressure of the brush, turns fully flexible. If the chains grafted onto the rotors are short, then the system enters its diffusive regime rather early with the backbone coiling up despite of accommodating still a significant number of rotors. Such a coiling is likely to influence diffusive motion of threaded rotors and hence their release. On the other hand, those systems with extended brushes are inhibited by a thick tangle of molecules that pile up near the backbone ends. The assumption of a free, one-dimensional diffusion, made to derive eq 16, is then hardly justified, and therefore computer simulations are indispensable to obtain accurate numbers for the multiple rotor release under varying external conditions. Figure 8 summarizes the average washing times of the systems. Here we define τ10 as the time after rotor release, at which 90% of the rotors have left the backbone. When considering the complexity of the involved stages of unthreading, simple scaling laws are not to be expected here. The plot shows that, as a function of backbone length, the washing times appear to scale approximately as ∼NB3/2. This is certainly not the result of any universal scaling behavior, but rather an effective power law which averages over the various different unthreading stages. As a function of the degree of polymerization of the grafted chains (inset), no approximate power law is observable. We point out once more that the numerical simulation procedure employed here is based on the Rouse approximation

(15)

where z(t0) indicates the walker’s position at the initial time and D ∼ 1/m stands for the Rouse diffusion coefficient of the rotor of mass m = 8(N + 1). Starting from a random position on the backbone, the average time31 the walker would remain on a backbone of length L would scale as Δt ∼ L2/(12D) ∼ L2m, and the fraction of threaded walkers of an ensemble of equally prepared systems would drop inversely proportional with that time, which implies Nthreaded ∼

mL2 t − t0

(16)

This scaling relation would strictly hold only under the condition of solitary walkers, that is in the asymptotic limit of low rotor density in which interactions between sliding rotors are rare. Figure 6 indicates that the system gradually crosses over into this solitary diffusive regime. Yet, a scaling of the curves with the rotor mass does not lead to a single master curve because the diffusive regime only governs the final stage of the unthreading process. We have further simulated the rotor release from backbones of various lengths Nb (inset in Figure 6). Here, the initial rotor density was held constant at σr = 0.44, and the grafted chains contained N = 32 monomers. Once again, during the final stages of the rotor release, the scaling behavior Nthreaded ∼ (t − F


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application of simple transport models which could accurately represent the dynamics of this process. Computer simulations are therefore indispensable during additional in depth studies of the complex dynamics of rotor release. Fully quantitative results may also demand to go beyond the Rouse approximation and to include hydrodynamic effects, which were not considered in our present study. Other extensions of the simulations should include the implementation of electrostatic interactions or thermal solvent conditions.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (H.M.). *E-mail [email protected] (C.-X.W.).

Figure 8. Average washing time τ10 as a function of backbone length. The dashed line is shown to guide the eye; its slope ≈1.5. Here, the initial rotor density was kept constant at σr = 0.44, with grafted chains of length N = 32. Inset: washing times as a function of the degree of polymerization N. Initial number of grafted rotors: Nr = 80.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partly supported by the National Science Foundation of China under Grant No. 11274258. We thank Gerhard Wenz for helpful comments regarding this manuscript.

of immobile solvent and neglecting all effects of hydrodynamic interaction. These may turn out to be considerable because the rotor release triggers a coordinated motion of a large amount of molecules, which should lead to entrainment effects of the solvent and an enhanced speed of unthreading. In a similar manner, the sliding motions of individual rotors along the backbone might become coupled through solvent flow. It is therefore desirable to analyze the impact of these hydrodynamic effects on the dynamics of unthreading. Computationally, this would be a demanding task which has to be left for future studies.

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REFERENCES

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V. SUMMARY Grafted polyrotaxanes are interesting because of their rich chemical and mechanical properties: The grafting allows to hide backbone or rotor properties from the environment, making the molecule solvable under conditions in which the bare backbones would coagulate. Furthermore, the polyrotaxanes can be made mechanically unstable: Grafted rotors are then released as free starlike molecules, which could serve as drug deliverer or as catalytic agents. Our combined theoretical and simulation study shows that most of the basic properties of such polyrotaxanes are easily understood with the help of scaling theory of polymer brushes. We have demonstrated that properties like brush extension, backbone tension, and longitudinal rotor mobility are following simple power laws as functions of the rotor density and chain length (Figure 3). The molecules studied here differ from conventional bottle brushes because the (mobile) anchor points are not evenly distributed along the backbone, which in turn delivers a uniform backbone tension (Figure 2). These differences would diminish asymptotically toward the limit of very long backbones or short chain lengths of the grafted polymers,11,13,14 but the sliding rotors are also suspected to add another relaxation mode to the viscoelastic properties of grafted polyrotaxanes.9 The breakup of these polyrotaxanes and the dynamics of rotor release unfold a rather complex process, which we have separated into three stages: a rapid backbone retraction, followed by a driven expulsion of rotors, and a final diffusive stage. The simulation results (summarized in Figure 6) indicate that additional factors like crowding of released rotors and backbone coiling are influential and thereby hampering the G


Macromolecules

Article

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Grafted Polyrotaxanes - American Chemical Society

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