Molecular Dynamics of Polyrotaxane in Solution Investigated by Quasi

May 15, 2019 - However, the sliding dynamics of the rings in PR have not yet been fully understood. ... PR with deuterated PEG (Mw = 35000, Cambridge ...
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Article Cite This: J. Am. Chem. Soc. 2019, 141, 9655−9663

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Molecular Dynamics of Polyrotaxane in Solution Investigated by Quasi-Elastic Neutron Scattering and Molecular Dynamics Simulation: Sliding Motion of Rings on Polymer Yusuke Yasuda,†,∥ Yuta Hidaka,†,∥ Koichi Mayumi,*,† Takeshi Yamada,‡ Kazushi Fujimoto,§ Susumu Okazaki,§ Hideaki Yokoyama,† and Kohzo Ito*,†

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†

Department of Advanced Materials Science, School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwa-noha, Kashiwa, Chiba 277-8561, Japan ‡ Neutron Science and Technology Center, Comprehensive Research Organization for Science and Society (CROSS), IQBRC Building, 162-1 Shirakata, Tokai, Naka, Ibaraki 319-1106, Japan § Department of Materials Chemistry, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan S Supporting Information *

ABSTRACT: In this study, we investigated the molecular dynamics of polyrotaxane (PR), composed of α-cyclodextrins (CDs) and a poly(ethylene glycol) (PEG) axial chain, in solution by means of quasi-elastic neutron scattering (QENS) measurements and full-atomistic molecular dynamics (MD) simulations. From QENS experiments, we estimated the diffusion coefficients of CD and PEG monomers in PR, which are in quantitative agreement with those obtained by MD simulations. By analyzing the simulation results, we succeeded, for the first time, in observing and quantifying the sliding motion of CD along a PEG chain. The diffusion coefficient for the sliding motion is almost 6 times lower than that of the translational diffusion of CD in PR at room temperature. The retardation of the sliding motion is caused by the energy barrier on PEG produced by molecular interactions between CD and PEG. We propose a simple equation to describe the diffusion coefficient of the sliding dynamics in PR by combining the Einstein−Stokes diffusion model and a one-dimensional jump diffusion model. This work provides a general strategy for the molecular designs to control the sliding motion in PR.

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INTRODUCTION Polyrotaxane (PR), a topologically connected supramolecule on whose axial chain ring molecules are threaded,1−8 has attracted much attention due to the unique molecular dynamics of the rotation and sliding motions of the ring molecules with topological confinement by the axis. The ring molecules in PR can slide along the axial chain, and the sliding motion enables us to fabricate functional molecular devices and materials based on PR.9−11 A typical example is a molecular shuttle, of which the ring position on the axial chain is controlled by external stimuli such as temperature,12 pH,13 and light.14 Recently, PR composed of α-cyclodextrins (CDs) as rings and poly(ethylene glycol) (PEG) as an axial polymer has been utilized to develop bulk polymeric materials such as gels15,16 and elastomers17,18 with mechanical softness and toughness. By cross-linking rings on different PRs in solution, Okumura and Ito fabricated “slide-ring gels” in which the polymer chains are connected by figure-eight-shaped crosslinks consisting of two rings.19 The sliding motion of the crosslinked molecules on the polymer chains yields mechanical properties unique to the slide-ring gels, such as the pulley effect19 and large crack resistance.20,21 More recently, Ikejiri et © 2019 American Chemical Society

al. developed photoresponsive actuators by introducing crosslinked daisy chains, double-threaded rotaxanes.22 As shown above, the “slidability” of the rings along the axis in PR is a dominant factor in characterizing the unique functions of PRbased devices and materials. However, the sliding dynamics of the rings in PR have not yet been fully understood. There are limited numbers of works on the dynamics of PR consisting of CD and PEG in solution. Beckham et al. conducted 2D DOSY NMR experiments on PR in solution and found that the measured diffusion coefficients of CD and PEG in PR are identical.23 They observed a translational diffusion of PR as a whole; however, they did not access the local motions in PR. Zhao et al. carried out dynamic light scattering (DLS) measurements on PR solutions and observed cooperative diffusion modes.24 The accessible length scale of DLS is 100 nm to 1 μm, which is much larger than the radius of gyration (Rg) of PR used (about 20 nm) and the size of CD (about 1 nm). To observe the molecular internal motion in PR, we require a nanometer-scale spatial resolution. Received: April 9, 2019 Published: May 15, 2019 9655

DOI: 10.1021/jacs.9b03792 J. Am. Chem. Soc. 2019, 141, 9655−9663

Article

Journal of the American Chemical Society

CD, PEG, hPR, and dPR were dissolved in dimethyl sulfoxide-d6 (DMSO-d 6) (purchased from Fujifilm Wako Pure Chemical Industries, Ltd.). The concentrations of hPR and dPR were both 10 wt %. The concentrations of CD and PEG were 4.3 and 6.2 wt %, respectively, to match the concentrations of CD and PEG in the hPR solution. The solution sample was set into a concentric double cylinder aluminum cell having a 14 mm inner diameter of the outer cylinder and 12 mm outer diameter of the inner cylinder, under atmospheric exposure and sealed with indium. QENS measurements were conducted using a time-of-flight nearbackscattering spectrometer DNA35 BL02, at Materials and Life Science Experimental Facility (MLF) at the Japan Proton Accelerator Research Complex (J-PARC) in Tokai, Japan. The energy resolution was 3.6 μeV for a Si(111) analyzer. Dynamic structure factors S(Q, ω) over an energy transfer range, −0.04 < ΔE/meV < 0.1, were obtained. The amplitude of the scattering vector, Q, was in the range of 0.125− 1.825 Å−1. All of the measurements were conducted at 303 K for ∼6 h, with a 500 kW proton beam power. To extract the dynamics of the solute, the scattering from the solvent was removed by subtraction based on the volume fraction. Full-Atomistic Molecular Dynamics (MD) Simulation. Fullatomistic MD simulations on CD, PEG, and PR in DMSO were conducted using GROMACS version 2016.5,36 with the revised CHARMM 35 force field for ethers37 and CHARMM 36 force field for carbohydrates.38 The PR used in the MD consists of PEG having 80 monomers and 3 CDs, which corresponds to the same coverage of CD on PEG as PRs for QENS. To prevent the dethreading of the CDs from the PEG axis, Lennard−Jones radii of the hydrogen atom at the end of PEG was set larger than the inner radii of CD. For the simulations of PR, we placed 1 PR in a 15 × 15 × 15 nm cubic simulation box. Subsequently, we filled the box with DMSO molecules. In the initial state, PEG in PR is fully stretched, and three CDs are arranged at regular intervals of 20 PEG monomers in the head-to-head/tail-to-tail alignment. Next, we conducted the energy minimization of the system with the steepest descent method, followed by NVT equilibration for 100 ps at 300 K and NPT equilibration for 200 ns at 300 K. The temperature and pressure were controlled by the velocity rescale method39 and Parrinello−Rahman method,40 respectively. Short-range van der Waals (VDW) forces and electrostatic forces in real space were switched to zero between 10 and 12 Å. For the long-range calculation of electrostatic forces, we used the particle mesh Ewald (PME) method.41 For accelerating the calculation, we used the LINCS algorithm42 to constrain bonds. After that, the conformation of PEG completely relaxed as shown in Figure S1. A snapshot of PR visualized by visual molecular dynamics (VMD)43 is shown in Figure 1a. After NPT equilibration for 200 ns at

Quasi-elastic neutron scattering (QENS) has Å to nanometer length scale and picosecond to nanosecond time scale resolutions25−27 and has been used to detect monomer diffusion, segmental motion, and collective diffusion of polymers in solution.28 Mayumi et al. conducted neutron spin echo (NSE) measurements on PR solutions.29 Compared with that of native PEG in solution, the segmental motion of PR in solution is retarded by the formation of the PR structure. To reveal the internal motion in complex systems such as PR, a selective deuterated labeling is a powerful tool for QENS. Using deuterated PEG in PR, we can separate the dynamics of CD and PEG in PR. Furthermore, molecular dynamics (MD) simulation is an effective method for evaluating the internal dynamics in molecular assemblies. Using MD simulation, we can simulate the dynamics within a time scale of 1 ps to 1 μs and a spatial scale of 1 Å to 100 nm, which cover similar time and spatial scales of QENS. Actually, QENS experiments and MD simulations have been combined to reveal the internal dynamics of complex systems. Rosenbach et al. performed QENS and MD to investigate the dynamics of methane in metal organic frameworks (MOFs); they have found a tunnellike diffusion trajectory of methane in MOFs.30 Harpham et al. studied the dynamics of water in binary mixtures of water and dimethyl sulfoxide using QENS data and MD simulations.31 Furthermore, Brodeck et al. reported that dynamics of poly(ethylene oxide) (PEO) in a mixture of poly(ethylene oxide) and poly(methyl methacrylate) (PMMA) are strongly influenced by a PMMA matrix, based on NSE experiments and MD simulations.32 In this paper, we reveal the molecular dynamics of PR composed of CD and PEG in solution, determined by the combination of QENS experiments and full-atomistic MD simulations. For simplification, we prepared PR in which PEG is sparsely covered with CDs. The QENS experiment using PRs with hydrogenated and deuterated PEGs enables us to evaluate separately the translational diffusion coefficients of the CDs and PEG monomers in PR solutions. For the MD simulations, we first obtain the translational diffusion coefficient of the CDs and PEG monomers and compare the MD results with the QENS data to confirm that the simulation model is valid enough to reproduce the molecular dynamics of PR in solution. Subsequently, we investigate the sliding dynamics in PR and estimate the diffusion coefficient for the sliding motion. The sliding motion of CDs on PEG is a onedimensional jump diffusion behavior, and it is suppressed by an energy barrier between CD and PEG. From the temperature dependence of the diffusion coefficient for the sliding motion, we propose a simple equation to describe the sliding dynamics of PR, as a combination of the one-dimensional jump diffusive motion with an energy barrier along the axial chain, and the Einstein−Stokes-type diffusion influenced by the solvent viscosity.

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Figure 1. Snapshots of (a) PR, (b) free CD, and (c) free PEG in DMSO obtained from simulations and visualized by VMD.43 The DMSO molecules are hidden for the sake of visibility.

EXPERIMENTAL SECTION

Quasi-Elastic Neutron Scattering (QENS). The PR used for the QENS experiments was synthesized from polyethylene glycol (PEG) (Mw = 35000, Fluka) and α-cyclodextrin (CD) (Nihon Shokuhin Kako Co. Ltd.), according to our previous method.33 To extract the scattering component from the CD in PR, PR with deuterated PEG (Mw = 35000, Cambridge Isotope Laboratories, Inc.) was synthesized additionally. These two PRs with hydrogenated and deuterated PEGs are named hPR and dPR, respectively. The PEG chains with 795 monomers contained about 27 CDs; the coverage of CD on PEG was 6.9%, since a CD covers two PEG monomers.34

300 K, we conducted a 200 ns production run at 300 K with a time step of 2 fs, and the positions of all the atoms were outputted every 1 ps. For comparison, we also simulated free CD and PEG in DMSO (Figures 1b,c). The number of CD/PEG in a simulation box was the same as that in the PR solution. To estimate the energy barrier for the sliding motion, we performed simulations at different temperatures (300−420 K in 30 K temperature steps). We calculated the mean squared displacement (MSD) of each CD and PEG monomer using the simulated trajectories of hydrogen 9656

DOI: 10.1021/jacs.9b03792 J. Am. Chem. Soc. 2019, 141, 9655−9663

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Journal of the American Chemical Society

Figure 2. (a) Fitting result of S(Q, ω) for free CD by a single Lorentzian function. (b) Fitting result of S(Q, ω) for free PEG by a single Lorentzian function. (c) Q2 dependence of the half width of the half-maximum Γ for free CD (red markers) and free PEG (blue markers), obtained from the fitting. The black lines are the fitting results by Γ = DQ2.

Figure 3. (a) QENS profiles S(Q, ω) of hPR (green line) and dPR (red line) solutions at Q = 0.925 Å−1. The difference between hPR and dPR is also shown (green line). (b) Fitting result of dPR by a single Lorentzian function and a flat background. (c) Fitting result of the difference in S(Q, ω) between hPR and dPR by double-Lorentzian functions.

Figure 4. (a) Q2 dependence of the half width of half-maximum Γ for CD in PR (red filled markers) and the slow (blue filled markers) and fast modes (blue open markers) for PEG in PR. The black lines are the fitting results by Γ = DQ2. (b) Ratio of the slow (blue filled markers) and fast (blue open markers) mode of PEG in PR. The black lines are guides for eyes. (c) Schematic illustration showing two PEG monomers covered by CD, and other monomers in front and behind CD move with CD cooperatively (monomers inside the red circles); monomers far from CD have relatively high mobility, just as PEG monomers in free PEG. There are 26 monomers between two CDs on average.

atoms because the dynamics of hydrogen atoms were observed from QENS due to the large incoherent scattering cross section of the

hydrogen atom. First, we deduced the MSDs of all of the hydrogen atoms (MSDH) using the following equation: 9657

DOI: 10.1021/jacs.9b03792 J. Am. Chem. Soc. 2019, 141, 9655−9663

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Figure 5. (a) Time evolution of MSD for the free CD, free PEG, and CD in PR at 300 K, calculated from the MD simulations. Fitting lines for calculation of diffusion coefficients are shown in black lines. (b) Schematic illustration of the definition of inclusion position, Nmon, and index. (c) Nmon dependence of the diffusion coefficients of the PEG monomers in PR, DPEG. The black dotted line denotes the diffusion coefficient of the free PEG monomer, and the red solid line corresponds to the diffusion coefficient of CD in PR. MSDH (t ) = ⟨(rH(t + t0) − rH(t0))2 ⟩t0

diffusion coefficient of CD in PR (10.9 ± 0.1 Å2/ns), while the larger is almost equal to the translational diffusion coefficient of a free PEG monomer (39.9 ± 0.5 Å2/ns). Therefore, the PEG monomers near CDs slowly diffuse due to the existence of CDs, whereas the PEG monomers far from CD have almost the same mobility as those in free PEG. Figure 4b shows the ratio of these two components. The ratio of the PEG monomers showing the slow diffusive mode is ∼30%. This is much larger than the covered ratio of the PEG monomers by CD (6.9%). This means that the dynamics of not only two PEG monomers covered by a CD molecule but also other monomers in the neighborhood of CD are suppressed by the CD (Figure 4c). Conversely, the PEG monomers far from CD are not affected by CD but have high mobility like free PEG. The inhomogeneity of the PEG monomer dynamics in PR suggests that the sliding mode of CD along PEG is slower than the observed translational diffusion modes of CD/PEG monomer within the covered time scale of the QENS measurements. If the sliding mode of CD is faster than the translational diffusion modes of the CD and PEG monomer, the translational diffusion of PEG monomers should be retarded homogeneously. To evaluate the sliding dynamics in PR, we conducted full atomistic MD simulations of the PR solutions. First, the validity of our simulation model was confirmed by comparing the diffusion coefficients of CD and PEG monomers obtained from the MD simulations with those evaluated by the QENS experiments. Figure 5a shows the time evolutions of the MSDs for free CD, free PEG, and CD in PR in the time range 0 < t < 400 ps. The CDs in PR show smaller MSD than free CDs show, which is consistent with the QENS result. From the slope of MSD against time, we estimated the diffusion coefficients, D = ΔMSD(t)/6Δt, in the length scale corresponding to the Q range for the QENS experiments (4.0 Å2 < MSD < 14.2 Å2). As shown in Table 1, the obtained diffusion coefficients for the free PEG, free CD, and CD in PR

(1)

Here, rH(t) indicates the position of the hydrogen atom at time t. Subsequently, we take an ensemble average of MSDH for each CD and PEG monomer to deduce MSDCD and MSDPEG:

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MSDx = ⟨MSDH ⟩x

(x = CD, PEG)

(2)

RESULTS AND DISCUSSION The QENS profiles S(Q, ω) of both CD and PEG were fitted by the following equation with the Lorentzian function (Figures 2a,b): | lA Γ o o Sexp(Q , ω) = m o π ω2 + Γ 2 } o ⊗ R(Q , ω) + BG n ~

(3)

Here, R(Q,ω) is the resolution function, ⊗ is the convolution operation, and BG is the flat background. The measurement results of vanadium were utilized for the resolution function. As shown in Figure 2c, the obtained half width of the halfmaximum, Γ, of CD is proportional to Q2, and it indicates that CD exhibits the translational Brownian diffusion in solution. The slope, Γ/Q2, represents the translational diffusion coefficient, D, of CD, which was evaluated to be 13.05 ± 0.05 Å2/ns. Moreover, for free PEG, Γ increases proportionally with Q2, and the Brownian motion is attributed to the monomer diffusion since the measured Q range corresponds to length scales smaller than the Kuhn segment length of PEG: 20 Å,44 which corresponds to Q = 0.31 Å−1. The PEG monomer diffusion coefficient was calculated to be 39.9 ± 0.5 Å2/ns. Figure 3a shows the QENS profiles of hPR and dPR and the difference between them (hPR − dPR). The QENS profile of hPR contains the incoherent scattering of both CD and PEG, while the incoherent scattering from CD is dominant for dPR. Therefore, the dynamics of CD and PEG are obtained from S(Q, ω) of dPR and the difference between the profiles of hPR and dPR, respectively. The profile of dPR was well fitted with a single Lorentzian function (Figure 3b). In contrast, the difference of the profiles for hPR and dPR did not show a single Lorentzian profile, but a double-Lorentzian one (Figure 3c). The obtained Q2 dependence of the half width of halfmaximum Γ is shown in Figure 4a. From the equation Γ = DQ2, the translational diffusion coefficient of CD in PR was obtained as 10.9 ± 0.1 Å2/ns, which is slightly smaller than the value of free CD. The translational diffusion coefficient of the PEG monomer in PR afforded two values: 10.6 ± 0.4 and 45.0 ± 0.8 Å2/ns. The smaller one is close to the translational

Table 1. Diffusion Coefficients Obtained by QENS Experiments (Dexp) and MD Simulations (Dsim) for the Free PEG, Free CD, and CD of PR in Solution component free PEG Dexp (Å2/ns) Dsim (Å2/ns) 9658

39.9 27.4

free CD 13.1 12.3

CD in PR 10.9 8.57

DOI: 10.1021/jacs.9b03792 J. Am. Chem. Soc. 2019, 141, 9655−9663

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Figure 6. (a) Trajectories of the inclusion position of CDs in PR. (b) Time evolution of MSDslide for the sliding motion of CDs along PEG. The dotted line shows the fitting result to estimate the diffusion coefficient for the sliding motion in PR.

Figure 7. MSD of (a) free CD, (b) free PEG, and (c) sliding motion of CDs along the PEG axis at different temperatures. (d) Normalized diffusion coefficient ηD/T against T−1 for the translational diffusion of the free CD and free PEG and the sliding motion of CDs along PEG axis in PR (T = 300, 330, 360, 390, and 420 K).

estimated (Figure 5c). When Nmon is zero and the PEG monomer is covered with CD, DPEG is 8.17 Å2/ns, which is almost the same as the diffusion coefficient of CD in PR. DPEG increases gradually with Nmon and saturates to 25.9 Å2/ns at Nmon = 13. Figure 5c suggests that the CD coverage suppresses the dynamics of not only the PEG monomers at the inclusion position but also those near the CD. From the MD simulation, the average diffusion coefficient of whole the PEG monomers in PR DPEGinPR was estimated to be 20.6 Å2/ns. The ratio of DPEGinPR to the diffusion coefficient of free PEG DfreePEG was 0.75. On the other hand, from the QENS result that 30% PEG monomers have the diffusion coefficient of CD in PR and 70% PEG monomers have that of free PEG, the average diffusion

are close to those from the QENS experiment. From this result, the validity of the simulation model is confirmed. Next, the dynamics of the PEG monomer in PR were analyzed. The QENS results indicated that the mobility of the PEG monomer in PR depends on the proximity of the PEG monomer to CD along the chain. The inclusion position of a CD on PEG is defined as the index of the nearest neighbor PEG monomer to the center-of-mass (CoM) of the CD (Figure 5b). The distance of each PEG monomer from the CD along the chain is expressed by the PEG monomer number from the inclusion position, Nmon. The MSD of each PEG monomer with the distance, Nmon, was calculated (Figure S2), and the diffusion coefficient of the PEG monomer, DPEG, was 9659

DOI: 10.1021/jacs.9b03792 J. Am. Chem. Soc. 2019, 141, 9655−9663

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Journal of the American Chemical Society coefficient of PEG in PR DPEGinPR was 34.7 Å2/ns, and the ratio DPEGinPR/DfreePEG was 0.87, which is close to that from the MD simulation. These findings from the MD simulations are consistent with the QENS experimental results, indicating that the simulation model and condition are valid enough to reproduce the dynamic properties of PR in solution. From the MD simulations, we evaluated the sliding dynamics of CDs along PEG in PR. First, all of the PEG monomers were given indexes (from 0 to 79) from one end of the PEG to the other (Figure 5b). Inclusion positions are determined by the CoM of the CDs and PEG monomers. We traced the positions of the three different CD molecules in the one-dimensional coordinate along the PEG axis, as shown in Figure 6a. From the trajectory, we calculated the mean squared displacement for the sliding motion of the CD along the PEG, MSDslide (t), from the equation MSDslide(t) = ⟨(i(t) − i(0))2⟩, where i(t) is the inclusion position at time t. As shown in Figure 6b, MSDslide(t) increases rapidly in a short time scale (MSDslide (t) < 0.4 [index2]), after which it increases linearly with time in a long time range (MSDslide (t) > 0.4 [index2]). This result indicates that CDs in PR exhibit a jump diffusion motion with an energy barrier between CD and PEG. The fast motion on the short time scale reflects the translational vibration of CDs trapped in the energy barrier on PEG. In the long time range, CDs show the slow-motion corresponding to a diffusive sliding mode. From the slope of MSDslide(t) in the range of 1 index2 < MSDslide < 2 index2, the diffusion coefficient of the sliding motion, Dslide, was estimated using equation Dslide = ΔMSDslide(t)/2Δt. Here, the sliding diffusion dynamics are one-dimensional; therefore, MSD is divided by 2Δt. Consequently, Dslide is calculated to be 0.105 (index2/ns). This value can be converted to 1.42 Å2/ns along the axis, based on the assumption that the monomer length of PEG is 3.68 (Å/index) determined by the distance between neighboring oxygen atoms of PEG with all-trans conformation. This distance was calculated from the angle and bond parameters in the force field. The obtained Dslide is six times smaller than the diffusion coefficient of CD in PR in the absolute coordinate, 8.57 Å2/ns, suggesting that the sliding motion in PR is suppressed by the energy barrier on PEG for CD. To evaluate the energy barrier for the sliding motion on PEG, the temperature dependence of the sliding dynamics was simulated. Parts a−c of Figure 7 show the MSDs of the free CD, free PEG, and the sliding motion of CD along PEG, respectively. The Brownian diffusions of free CD and PEG monomers in the absolute coordinate exhibit a continuous increase in MSD with temperature. On the other hand, the increase of MSDslide in Figure 7c is accelerated with temperature. A dominant factor for the temperature dependence of MSD should be the change in the solvent viscosity η(T) with absolute temperature T. For a normal Brownian motion, the diffusion coefficient is described by Einstein− Stokes equation D=

kBT 6πR hη(T )

viscosity. For the free CD and PEG, the normalized diffusion coefficient η(T) D/T is found to be independent of T, indicating that the motions of the free CD and PEG monomers obey the normal Einstein−Stokes diffusions. From eq 4, the hydrodynamic radii Rh for the free CD and free PEG monomer are deduced as 9.08 and 3.74 Å, respectively. These values are reasonable for the translational diffusion of CD and monomer diffusion of PEG, respectively. In contrast, the normalized diffusion coefficient of the sliding motion, η(T) Dslide/T, is not constant but shows an Arrhenius-type temperature dependence: log (η(T)Dslide/T) ∝ − 1/T, which suggests the jump diffusion motion with energy barrier.47−49 Thus, Dslide is described by the following simple equation

Ea zyz kBT ji expjjj− zz j z η (T ) (5) k NAkBT { where Ea is an activation energy corresponding to the energy barrier on PEG and NA is the Avogadro number. This equation is consistent with our previous coarse-grained MD simulation work on the sliding dynamics of a ring molecule in rotaxane.49 By fitting the simulation data with eq 5, Ea for the sliding dynamics in PR is estimated to be 8.92 kJ/mol. Liu et al. conducted a free energy calculation based on MD simulations and estimated the energy barrier for CD on PEG to be about 2 kcal ≑ 8.37 kJ/mol,50 which is almost the same as the obtained Ea. Equation 5 is given from some previous theoretical works on the one-dimensional diffusion coefficient of a Brownian particle subjected to a periodic potential.51−53 Equation 5 for the sliding dynamics in PR can be understood from analogy with the ion diffusion in an electric field.48 The one-dimensional diffusion coefficient for the jump diffusion of the ions under periodic electric potential is described as below:48 Dslide ∝

Dj =

DB

(

ϕ(x)

exp − k T B

)

exp x

( ) ϕ(x) kBT

(6)

x

Here, ϕ(x) is the electric potential as a function of the x coordinate, and DB is the diffusion coefficient for the Brownian motion with no potential barrier, ϕ(x) ≡ 0, and is given by eq 4. We define the difference between the maximum and minimum values of ϕ(x) as Eb. When Eb is much larger than

( )

kBT, the denominator of eq 6 is proportional to exp

Eb kBT

independent of temperature. Then eq 6 is reduced to eq 5. For the sliding motion of CDs along PEG, Eb = Ea/NA = 1.48 × 10−20 J is actually much larger than kBT = 4.14 × 10−21 J at 300 K. This means that the sliding motion obeys the jump diffusion of CD in a periodic potential based on the internal interaction between CD and PEG axis.

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CONCLUSION We performed QENS experiments and full-atomistic MD simulations on PR, in which PEG is sparsely covered with CDs to reveal the molecular dynamics, particularly the sliding dynamics in PR solutions. From the QENS with deuteration labeling of PEG in PR and MD simulations, we separately estimated the translational diffusion coefficients of CDs and PEG monomers in PR. The diffusion coefficients for CDs in PR obtained from the QENS experiments were almost the same as those from the MD simulations. Both MD and QENS results demonstrate that PEG monomers inside CD diffuse

(4)

where Rh is the hydrodynamic radius of the solute. In Figure 7d, the diffusion coefficients of the free CD, free PEG, and the sliding motion are normalized by T/η(T) based on the Einstein−Stokes equation. The viscosity of DMSO η(T) was estimated from experimental data from ref 45 and the Andrade equation,46 a well-known semiempirical formula for fluid 9660

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Journal of the American Chemical Society together with the CD as a whole and that those far from the CD exhibit high mobility just as free PEG. These results proved that the simulation model was valid enough to reproduce the dynamics of PR in solution. From the trajectory analysis of the MD simulations, we evaluated the sliding dynamics of CD on PEG. At 300 K, the one-dimensional diffusion coefficient of the sliding motion, Dslide, is six times lower than that of the translational diffusion of CD in PR at the absolute coordinate. The retardation of the sliding motion suggests the energy barrier on PEG for CD. From the temperature dependence of Dslide, it is found that the sliding motion of CD on PEG does not follow the Einstein− Stokes equation for Brownian diffusion. We proposed a simple relationship for the sliding motion (eq 5) composed of the Einstein−Stokes part for the Brownian diffusion and the Arrhenius part for one-dimensional jump diffusion with the energy barrier. The simple model for the sliding motion given by eq 5 is valid when the interior of the rings interacts strongly with the axial polymer and when Ea is much larger than NAkBT. Apart from PR consisting of CD and PEG, PRs for molecular shuttles contain station sites on their axis which interacts with their rings. To control the sliding motion in PR, tuning Ea in eq 5 is required. Ea is determined by the interaction between the interior of the rings and the axial polymer. As far as we know, this is the first work to reveal the sliding dynamics of PR by the combined technique of QENS and full-atomistic MD simulation. The obtained simple equation for the sliding dynamics is quite useful for the molecular design of PR and PR-based materials to control the sliding speed of the ring on the axis.

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ACKNOWLEDGMENTS

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REFERENCES

We thank Prof. Shinji Tsuneyuki for the many helpful comments and discussions. This work was supported by the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), JST-Mirai Program Grant No. JPMJMI18A2, and the Materials Education Program for the Future Leaders in the Research, Industry, and Technology (MERIT). The neutron experiment at the Materials and Life Science Experimental Facility of the JPARC was performed under a user program (Proposal No. 2018A0235). The calculation was performed by a supercomputing system at the Institute for Solid State Physics (ISSP) (Project No. H30-Ba-0017) and by Oakforest-PACS in Supercomputing Division, Information Technology Center, The University of Tokyo (Project code: pk0033)

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/jacs.9b03792. Simulation conditions, system information for MD simulation, and MSD of PEG monomer in PR estimated from MD simulation (PDF)

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Article

AUTHOR INFORMATION

Corresponding Authors

*[email protected] *[email protected] ORCID

Yusuke Yasuda: 0000-0002-2870-8524 Koichi Mayumi: 0000-0002-1976-3791 Takeshi Yamada: 0000-0001-5508-7092 Kazushi Fujimoto: 0000-0001-9769-8414 Hideaki Yokoyama: 0000-0002-0446-7412 Present Address

(K.M. and K.I.) Department of Advanced Materials Science, School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwa-noha, Kashiwa, Chiba 277-8561, Japan. Author Contributions ∥

These authors contributed equally.

Notes

The authors declare no competing financial interest. 9661

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Journal of the American Chemical Society

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