Cooperative and synchronized rotation in motorized porous frameworks: Impact on local and global transport properties of confined fluids

Molecules in gas and liquid as well as in solution, significant and Brownian motion. Molecules in the solid-state, although strongly immobilized, can still exhibit However, in most framework materials, these intramolecular dynamics are driven by temperature, and therefore are neither controlled nor spatially or temporarily aligned. In recent years, several examples of molecular machines that allow for a stimuli-responsive control of dynamical motion, such as rotation, have been reported. In this contribution, we investigate the local and global properties of a Lennard-Jones (LJ) fluid surrounding a molecular motor and consider the influence of cooperative and non-11 directional rotation for a molecular motor-containing pore system. This study uses classical 12 molecular dynamics simulations to describe a minimal model, which was developed to 13 resemble known molecular motors. The properties of an LJ liquid surrounding an isolated molecular motor remain mostly unaffected by the introduced rotation. We then considered an 15 arrangement of motors within a one-dimensional pore. Changes in diffusivity for pore sizes 16 approaching the length of the rotor were observed, resulting from rotation of the motors. We 17 also considered the influence of cooperative motor directionality on the directional transport 18 properties of this confined fluid. Importantly, we discovered that specific unidirectional rotation 19 of altitudinal motors can produce directed diffusion. This study provides an essential insight into molecular machine-containing frameworks, highlighting the specific structural arrangements that can produce directional mass transport.

1 Introduction 2 Transport properties of fluids in confinement are essential for both biological and 3 artificial systems. Enhancement of diffusion with the preservation of selectivity is essential for 4 the improvement of separation technologies 1 , particularly for membrane-based systems. 2, 3 In 5 biological systems, diffusion enhancement is primarily facilitated by pore shape agitation of the 6 channels embedded in soft matter. 4-8 However, similar effects can also be found in artificial 7 porous media such as carbon nanotubes in which phonon-induced oscillating friction is found 8 to enhance diffusion of fluids. [9][10][11] In both cases the nanoscopic oscillation of pore contraction 9 and expansion propagates cooperatively through the lattice allowing for accelerated transport 10 properties on a larger length scale.

19
In this contribution, we demonstrate the interactions and transport properties of fluids 20 in motorized porous frameworks and investigate the influence of cooperative synchronized and 21 non-cooperative rotation of molecular motors. This study is performed by molecular dynamics 22 simulations using a minimal toy-model, which resemble known molecular rotors. The 23 molecular machine design incorporates elements of existing molecular motors including their 24 geometry and hypothetical orientations in a one-dimensional pore, in an effort to establish a 1 close representation of a real-world system. Initially, the result of an isolated motor surrounded 2 by fluid was considered. Subsequently, different pore sizes were investigated so that the effects 3 of confinement on the transport properties of the fluid can be captured. We also probed the 4 influence of cooperativity on the transport properties of the confined fluid by computing 5 systems, where the rotation of two rotors are either correlated, in direction, or uncorrelated.
6 This allows us to investigate fundamental scenarios present in materials currently known in 7 literature 29 and compare this to proposed cooperatively working motorized frameworks. 8 This study provides a physical blueprint for molecular machine-containing frameworks 9 poised as new devices by which external stimuli can act to provide activated diffusion for 10 directional mass transport or nanoscale microfluidic devices, mimicking biological systems.
The energy coefficients , and were set to , and bond angle dihedral 500 300 100 7 , respectively. The specified geometry and thus equilibrium values of these potentials are 8 discussed in detail later. This provides a robust and customizable framework for modelling 9 molecular motifs.

10
The dynamics of the fluid, and dynamics of the MMs, were simulated using a Langevin 11 thermostat 38 in the ensemble, which effectively performs Brownian dynamics. 39 The 12 temperature of the fluid was controlled to using a damping parameter of . 1.0 * 1.0 13 Confinement was achieved using wall potentials that interact with fluid particles using only the 14 repulsive part of the LJ potential (Eqn. 1).

15
Representative input files for the simulation reported in this study are available online 16 in our data repository at https://github.com/jackevansadl/supp-data. 22 Depending on the molecular structure, rotary motion of MMs might involve multiple steps, 23 however for simplicity we will consider a bistable scenario in this work. 1 To dictate the photoactive motion displayed by this MM we define two potential states 2 of the system. This uses the dihedral potential between the particles A-B-C-D and A'-B-C-D' 3 noted in Eqn. 4. This potential has the parameter , which is set as 1 or -1. This can be changed 4 during the simulation to produce two distinct potential states as demonstrated in Fig. 3. These 5 potential states have a minimum at either or , which effectively switches the stable 0°180°6 configuration of particles A and A' to opposite sides of the molecule. Sequentially switching 7 the potential parameter during the simulation, after a set number of time steps, a stepwise 8 periodic rotation is achieved. This stepwise rotation is comparable to the type of motion 9 observed in the experimental system. The motion between the two states occurs over a few 10 simulation steps, which is comparable to the ultrafast dynamics reported for the excited-state 11 photoisomerization rotation found in the experimental system. 45 Throughout this study, the 12 switching period, between the two potentials, was chosen as . Currently, molecular rotary 50 13 motors exhibit MHz-scale rotational frequencies 46, 47 . If we consider the particles in this study 14 to have similar size and characteristics to Argon, the rotation rate is approximately GHz.

4.6
15 This is currently faster than is observed for MMs but represents the potential for such systems, 16 as with specific functionalisation these systems could in principle function in the GHz regime. 48 17 18 19 Figure 3. The potential energy surface for the dihedral interaction in the motor system 20 (a). Potential 1 and potential 2 refer to the two potential states and the arrow depicts 21 the effect of switching between these states. The stepwise periodic rotation observed 22 by switching the potential every .(b). This is demonstrated by the relative -23 position of the A particle, where the axle of the rotor is orientated in the -direction.

25
The simple switching of the dihedral potentials can effectively produce a rotation about 26 the axes of the molecule. However, because the potential switch sets this structural 1 configuration atop the symmetric maximum of the dihedral potential energy surface the system 2 can equally relax (by rotation) in either direction. This results in the non-directional rotation of 3 the motor as demonstrated in Fig. 4a. Unidirectional motion is a sought-after characteristic of 4 molecular rotors known to work in porous frameworks. 29 However, as with biological motor 5 systems, directionality is essential to perform mechanical tasks. 49 Synthetically this is achieved 6 by photo-responsive sterically overcrowded alkenes, which results in four distinct steps as 7 outlined in Fig. 1. 50 To achieve unidirectional rotation in our simulations we employ a torsional 8 bias force, orientated in the direction of the axle, thus mimicking the influence of sterically 9 crowded functionalisation. This use of applied torque avoids increasing complex descriptions 10 of the dihedral potential energy surface. While large bias forces result in continued, and 11 uncontrolled, rotation, subsequent to the potential switch event, a bias force of results 30 / 12 in singular and unidirectional rotation (Fig. 4b). Notably, the sign of this bias torque dictates   1 molecular rotor is equivalent to the size of the fluid particles this is perhaps not surprising. To 2 completely understand the effect of artificial molecular rotation in this system the local 3 properties of the rotor and fluid were investigated. As displayed in Fig. 5a, the rotation produces 4 a lower density of fluid in a spherical area directly surrounding the centre of the rotor (particle 5 B). This local defect in the structure is attributed to the constant reorganisation of the fluid due 6 to rotation. Moreover, the dynamics in this region also appear to be influenced by the motion.
7 The average velocity of spherical slices surrounding the rotor (Fig. 5b)  One appealing method to increase the effectiveness of molecular machines is their 19 positions within materials. 56 Especially confining a reaction media in close proximity to the 20 motion of rotors is expected to enhance dynamic effects. 27 We consider the effect of this by 21 arranging our motor system on the walls of a one-dimensional pore (Fig. 6a). The pore is 22 constructed using repulsive wall potentials that interact with only fluid particles. This 23 confinement arrangement is analogous to simulations of nanoconfined water, and these 24 repulsive walls are suggested to act like hydrophobic walls. 57 Two azimuthal rotors were 25 positioned on opposite walls and arranged such that the stable configuration of the rotor is 26 aligned with the one-dimensional pore. Unidirectional rotation is directed with the bias torque 27 in the same direction for each rotor. The LJ fluid was added to this system at a density of 0.09

28
. A lower density was employed to avoid over saturation. Importantly, two distinct pore sizes  10 We find that for the large pore system the global and local properties of the confined 11 fluid (Fig. 6b and Fig. 6c) show little influence from rotation. The transport of the fluid is 12 observed as the slope of the mean-squared displacement, also referred to as diffusivity (Fig.   13 6b). There appears to be little influence on the transport of the fluid for this larger pore structure.
14 Similarly, there is no local increases in the average motion (temperature) observed (Fig. 6c).

15
In contrast to the larger pore, the smaller pore structure of , which The synthetic design of molecular motors can hypothetically produce rotors that rotate 4 in distinctive directions. When immobilized on a surface these motors can exhibit different 5 orientations. 59 As a result, we also have considered if rotation of the motor is arranged in an 6 altitudinal fashion, 60 as illustrated in Fig. 2c. This arrangement can localize the specified motion 7 towards the centre of a porous structure where it may have a greater influence on the transport 8 properties. An equivalent one-dimensional pore system was constructed for investigating the 9 use of altitudinal rotors (Fig. 7a) which includes two motors orientated perpendicular to the 10 pore direction, such that the rotation is orientated in the direction of diffusion. The 11 unidirectional rotation for this case is also arranged with the bias torque in the same direction 12 for each rotor. The sampling and simulation details are analogous to that discussed for the 13 azimuthal case. 1 When the motors rotate, however, we observe no significant change in diffusivity, in 2 direct contrast to that observed for the azimuthal rotor system. This result suggests that only 3 specific orientations of rotation by molecular motors aligned in pore channels may lead to an 4 increase in transport properties.
5 Non-directional rotation in a one-dimensional pore 6 The unidirectional motion of molecular rotors is considered imperative for directing 7 movement at the molecular level. Thus, we sought to consider how non-directional motion 8 manifests and effects diffusion in small pore systems, for both the azimuthal and altitudinal 9 rotors. In fact, such a case reassembles the scenario of rotors frequently incorporated in the 10 backbone of porous frameworks but not governed by a stimulus, such as light. 28 To investigate 11 this, we conducted similar simulations as described in the previous two sections, but with the 12 absence of the bias torque. This resulted in non-directional motion of the rotors (Fig. 8).

21
The molecular simulations, described in this study, represents a minimalist and 22 qualitative picture of molecular motors, and the possible consequence of arranging them in 23 confined solid-state. Many approximations are made to describe the motion of rotors in this 24 study. For example, the rotation mechanism is significantly reduced to a two-state motion.
25 Notably, the rotation between the two-state occurs almost instantaneously, when the dihedral 26 potential is switch. In reality molecular motors show complex rotational dynamics with slow 27 and fast movements to produce a full rotation. 49 In future studies, it is possible to use more 28 complex potential energy surfaces and gradual potential changes to more realistically reflect 29 the intricate dynamics of real-world motors. Nevertheless, the potential switching mechanism 30 should provide the foundation to build new atomistic models of molecular machines. 27 locally and globally. We envision the guidelines, presented here, to be crucial for the design of 28 stimuli-responsive dynamic materials, capable of manipulating guest transport properties by 29 dynamic molecular machines. However, the currently reported porous materials that include 30 molecular machines 30 or rotors 28 do not meet the specified criteria defined by this investigation.

31
As discussed in the introduction, many transport phenomena are supported by activated 32 vibrations of the pore walls, 12 a property neglected in the present work. The combination and 33 alignment of local dynamics from a functional surface, by anchored groups, and global