Unraveling insights for ion transport enhancement by metal–organic framework-5 in polyethylene oxide and polyvinylidene fluoride electrolytes: classical molecular dynamics and metadynamics simulations

Abstract

The structural and dynamical properties of simulation systems consisting of metal organic frameworks (MOFs), particularly MOF-5, lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) salt, and two well-known polymers, poly(vinylidene fluoride) (PVDF) and poly(ethylene oxide) (PEO), in different compositions were investigated using classical molecular dynamics (cMD) and well-tempered metadynamics (WT-MD) simulations. The cMD simulations revealed that the PEO-based hybrid electrolyte system containing LiTFSI salt and MOF-5 (System D) exhibits the highest diffusion coefficient for Li⁺ ions and comparatively weaker interactions between Li⁺ ions and the surrounding TFSI⁻ anions than the other investigated systems. The enhanced ionic mobility is attributed to the favorable local environment generated by the incorporation of MOF-5, which facilitates efficient Li⁺ ion transport while reducing ion-trapping effects, as evidenced by structural properties in terms of radial distribution functions, structure factors, spatial distribution functions, three-dimensional density distributions, ion pairing, and contact number analysis obtained from cMD. In addition, consistent with the structural properties, the dynamical and thermodynamic properties, evaluated through mean squared displacement and potential of mean force analysis, also indicate a high diffusion coefficient for Li⁺ ions and favorable free-energy of Li⁺–PEO interactions in System D. Furthermore, WT-MD simulations demonstrated that System D possesses a relatively lower free-energy barrier for Li⁺ ion mobility, indicating energetically favorable ion transport pathways. The combined findings from cMD and WT-MD simulations highlight the critical role of MOF-5 in regulating the dynamics and transport behavior of Li⁺ ions. Therefore, this study provides valuable design guidelines for engineering hybrid MOF–polymer electrolytes with optimized ion transport properties for solid-state battery applications.

MOF-5 incorporated PVDF- and PEO-based electrolytes were studied using cMD and WT-MD simulations to examine Li⁺ coordination and migration, providing molecular level insights into Li⁺ transport within the polymer-based electrolytes.

1. Introduction

Rechargeable batteries are currently in high demand due to their excellent performance and environmentally friendly nature, especially for applications in portable electronic devices and electric vehicles. Among these energy storage technologies, lithium-ion batteries (LIBs) are predominant owing to their high energy density and superior energy storage efficiency. The exponential growth in the use of LIBs over the past few decades has increased the urgency for advancements in the materials used for both electrodes and electrolytes. In addition to the significant role of electrodes, high-performance electrolytes must possess not only electrochemical stability over a wide range of voltages and temperatures but also high Li⁺-ion conductivity coupled with excellent electrode compatibility.^1,2 Although conventional liquid electrolytes are still extensively used in commercial devices, research efforts are progressively shifting toward solid-state electrolytes, which offer a safer alternative to conventional electrolytes. Solid-state electrolytes provide several advantages, such as non-flammability, improved thermal stability, and a significantly reduced risk of leakage and combustion compared with their liquid counterparts.^1,3

Solid-state electrolytes (SSEs) can generally be classified into polymer-based electrolytes, inorganic electrolytes, and composite solid electrolytes. Among these, polymer-based electrolytes are of several distinct types, including solid polymer electrolytes (SPEs), gel or plasticized polymer electrolytes, and composite polymer electrolytes.^1,4 Materials such as polyvinylidene fluoride (PVDF) and polyethylene oxide (PEO) belong to the category of SPEs, which have attracted significant attention because of their low cost, ease of processing, and excellent compatibility with electrode surfaces; however, their ionic conductivity at room temperature remains limited, typically ranging from 10⁻⁷ to 10⁻⁶ S cm⁻¹.^5–7 This limitation can be mitigated by incorporating inorganic nanoparticles into SPEs which in turn helps improve properties like ionic conductivity, mechanical integrity, thermal resistance, and electrochemical stability.^8,9 Furthermore, the addition of nanoparticles such as metal–organic frameworks (MOFs), silica, and alumina has been shown to substantially enhance the performance of electrolytes.^10–12 These additives enhance the Lewis acid-base dual functionality of the materials, facilitating ionic dissociation and thereby improving ionic conductivity.^13,14 Moreover, the large surface area and porous structure of these materials contribute to improved interfacial stability with lithium metal.¹⁵

Metal–Organic Frameworks (MOFs) are crystalline porous materials formed through the coordination of metal ions or clusters with organic ligands, offering tunable pore architectures, large surface area, and superior thermal and structural durability.^16,17 These structural characteristics provide several advantages, rendering MOFs exceptionally adaptable for diverse applications including catalytic processes, selective gas adsorption, and advanced electrochemical systems.^18–20 Among various MOFs, MOF-5, commonly referred to as IRMOF-1 stands out as one of the extensively studied materials because of its well-defined porous framework, high surface area, excellent thermal stability, and readily tunable chemical functionality.^21–23 The structure of MOF-5 consists of zinc oxide tetrahedral (Zn₄O) clusters bridged by 1,4-benzenedicarboxylate (BDC) linkers, forming a highly stable cubic lattice.^19,24,25 The incorporation of MOFs into the polymer matrices disrupts the crystalline structure of the polymer, increasing its amorphous content and thereby facilitating lithium-ion transport.^13,23,26 Through interactions with polymer chains, MOFs can hinder segmental motion of chains, enhancing the thermal stability of the SPEs due to their rigid structures, which resist thermal disturbances.^23,27,28 For instance, the incorporation of 5 wt%Al-MOF nanorods into a PEO:LiTFSI electrolyte system led to a marked increase in ionic conductivity, which was attributed to the effective dissociation of Li⁺ ions from TFSI⁻ anions via Lewis acid-base interactions.²⁸ A similar enhancement was observed upon the addition of 10 wt% of Mg-BTC MOF to a PEO system, resulting in a substantial increase in conductivity by a two-order-of-magnitude at 20 °C. This increase was attributed to weaker ion pairing and enhanced Li⁺ ion mobility.²⁷

Additional improvements in the ionic conductivity of polymer-based electrolytes have been reported through the introduction of functionalized MOFs. For instance, UiO-66 funtionalized with amino groups (NH₂-BDC) significantly enhance the ionic conductivity of PEO-based electrolyte formulations. This was inferred as the occurrence of strong electron-donating ability of the amine groups toward Li⁺ ions, which promotes salt dissociation and reduces ion pairing.²⁹ Angulakshmi et al. found that incorporating UiO-66-NH₂@SiO₂ into the PEO:LiTFSI electrolyte boosted the ionic conductivity by two orders of magnitude at approximately 70 °C. This increase in the ionic conductivity was owing to the microporous structure of the MOF, which enabled efficient Li⁺ conduction.²⁰ Similarly, Yuan et al. demonstrated that PEO-LiTFSI electrolyte (EO : Li = 10 : 1) containing 10 wt% MOF-5 achieved an ionic conductivity of 3.16 × 10⁻⁵ S cm⁻¹ at 25 °C, where MOF-assisted formation of Li⁺ channels played a central role in enhancing ion transport.²²

ZIF-8, a distinct member of the MOF family, possesses a highly porous structure, excellent adsorption and catalytic performance, and high thermal stability. Due to these characteristic properties, it has been extensively investigated as a photocatalyst for various applications, including energy systems.³⁰ ZIF-8 has also demonstrated promising performance when incorporated into PVDF-based electrolyte systems, achieving ionic conductivity of 1.5 × 10⁻⁴ S cm⁻¹ and lithium-ion transference number of 0.833.³¹ Density functional theory (DFT) calculations have revealed that Li⁺ ions encounter a lower migration barrier in ZIF-8 (∼0.07 eV) compared to PVDF (∼0.15 eV), indicating that the MOF framework supports more efficient ion-conduction pathways. Furthermore, the porous ZIF-8 structure enhances ion transport by preferentially confining TFSI⁻ anions, thereby facilitating Li⁺ migration. Zheng's group synthesized HKUST-1 MOF nanocrystals in situ on a three dimensional (3D) polyimide (PI) fibrous scaffold, which was subsequently incorporated into a PVDF matrix. This strategy led to the fabrication of high-performance solid-state electrolytes (HKUST@PI/PVDF SSEs) that exhibited improved physicochemical properties.³² Similarly, a hybrid electrolyte was designed employing inter-linked polymer matrix comprising poly(vinylidene fluoride-co-hexafluoropropylene (PVDF-HFP) and poly(diallyldimethylammonium chloride-TFSI (PDADMA-TFSI), embedded with MOF-based materials.³³ This structural modification has introduced a dual-confinement interface that suppresses TFSI⁻ diffusion while facilitating efficient Li⁺ ion transport through the ordered MOF structure. In another study, Ma et al. designed a hybrid structure termed PPM (PAN-PVDF-MOF), integrating heat-treated PAN fibers coated with UiO-66-NH₂ MOF particles and embedded within a PVDF matrix. This architecture effectively addressed the problem of dendritic lithium growth in PVDF-based solid electrolytes. The MOF-coated fibers interacted with DMF solvent molecules, weakening the Li⁺–O coordination and facilitating Li⁺ migration through interconnected fiber networks, resulting the electrolyte to achieve an enhanced conductivity of 1.03 mS cm⁻¹.

Despite the promising role of MOFs in improving Li⁺ transport within solid electrolytes, the molecular mechanism responsible for this improvement remain insufficiently understood. In particular, the influence of MOF-5 on the local solvation environment, ion mobility, and Li⁺ migration within different polymer matrices has not been fully established. Therefore, a detailed molecular level investigation is needed to clarify how MOF-5 modifies the structural and dynamical properties of polymer electrolytes and how these changes are related to ion transport. In the present study, the structural and dynamical properties of MOF-5 containing PVDF and PEO polymers with LiTFSI salt were systematically investigated using cMD simulations. The behavior of the pure polymers and LiTFSI containing systems was compared with their corresponding MOF-5 containing systems to determine the effect of the MOF-5 framework on polymer structure, density, ion coordination, self diffusion, ion transport, and other relevant properties. In addition, WT-MD simulations were employed for the MOF-5 containing systems to determine the free energy profiles associated with Li⁺ movement and to provide a molecular level description of the energetic factors governing ion migration. The main contribution of this work is the combined analysis of structural, dynamical, and thermodynamic properties to establish a well-defined relationship between the incorporation of MOF-5 and Li⁺ transport in different polymer systems. The comparative investigation of PEO and PVDF further provides insight into how the nature of the polymer matrix influences the ability of MOF-5 to facilitate ion transport. These findings provide a molecular level understanding of the role of MOF-5 in polymer electrolytes and offer useful guidance for the rational design of advanced solid polymer electrolytes for next generation LIBs.

2. Methods

2.1. Polymer modeling and refinement

The two polymers, polyvinylidene fluoride (PVDF) and polyethylene oxide (PEO) were modeled, each consisting of 30 monomer units and denoted as PVDF₃₀ and PEO₃₀, respectively. The polymer structures were generated using the xLEaP module of AmberTools_version, which was particularly used to build the polymer backbone followed by structure optimization.^34,35 Along these polymers, MOF-5 and bis(trifluoromethanesulfonyl)imide (TFSI⁻) anions were optimized using the GFN2-xTB (Geometry, Frequency, Noncovalent, eXtended Tight-Binding), a semi-empirical approach known for providing a favorable balance between accuracy and computational efficiency.^36,37 Performing optimization before MD simulation is an essential step, as it ensures structural stability, generates realistic starting geometries, and improves simulation reliability by minimizing unfavorable forces and positioning atoms in energetically favorable configurations.

The optimized PVDF₃₀ and PEO₃₀ structures exhibited elongated, tube-like conformations. To obtain more realistic configurations, molecular dynamics (MD) simulations were performed in vacuum using the AMBER software suite (version 21.0) for a total simulation time of 1 ns. Initially, energy minimization was carried out to eliminate unfavorable contacts and reduce the potential energy of the system, thereby providing a stable initial configuration for subsequent simulations. The minimization was carried out for 10 000 cycles, with the first 1000 steps consisted of steepest descent minimization and the remaining 9000 steps were based on conjugate gradient method. Following minimization, the systems were gradually heated from 0 to 298 K over 1 ns using Langevin thermostat with a time step of 1 fs, allowing the polymer chains to relax. Subsequently, the systems were equilibrated at 298 K using the same thermostat to maintain the desire temperature. The SHAKE algorithm was employed to constrain bonds involving hydrogen atoms, thereby ensuring numerical stability during the simulations. Finally, a 1 ns production run was performed under the same conditions. The resulting trajectories were analyzed to evaluate the conformational stability and reliability of the polymer structures. During the simulations, the initially extended polymer chains gradually relaxed and adopted more bent conformations due to thermal fluctuations, providing realistic structural models for subsequent studies (Fig. S1).

2.2. Classical molecular dynamics simulations

The coordinates of the polymers obtained from the vacuum simulations served as the starting point for the modeling and simulation of polymer-based electrolyte systems. For this purpose, the polymer structures were packed into a simulation box of (56 × 56 × 56 ų) using the PACKMOL program to generate the initial configuration for the electrolyte systems.³⁸ Classical MD (cMD) simulations were performed using the LAMMPS package (29 Oct 2020) to investigate MOF-5 based polymer electrolytes comprising PVDF₃₀ and PEO₃₀ in the presence of LiTFSI salt.³⁹ Seven systems were investigated, consisting of PVDF₃₀ and PEO₃₀ polymers, MOF-5, LiTFSI salt in different compositions, as listed in Table 1. Representative visualizations of these systems are shown in Fig. 1.

Table 1. Number of ion pairs and other components, and their density of the investigated systems obtained from simulations at 298 K.

Composition	Li+	TFSI−	PVDF	PEO	MOF-5	Density (g cm−3)
Pure PVDF	0	0	30	0	0	1.49
Pure PEO	0	0	0	30	0	1.13
System A	90	90	30	0	0	1.67
System B	90	90	30	0	1	1.59
System C	90	90	0	30	0	1.41
System D	90	90	0	30	1	1.29
Reference	90	90	0	0	1	1.49

Fig. 1. (a) Different components of simulation systems and (b) representative snapshot of the equilibrated simulation systems.

In the present work, a hybrid force field strategy was employed because no single transferable classical force field has been established that can reliably describe both polymer electrolytes and MOFs with comparable accuracy. The OPLS-AA force field was adopted for the polymer components and has demonstrated an excellent agreement with experimental and computational studies including density and RDF,^40–42 while the parameters for LiTFSI were adopted from the literature.^43,44 Furthermore, Ozlek et al. successfully employed MD simulations to investigate PEO/LiTFSI/MOF-5 composite electrolytes and demonstrated that the selected force field methodology reliably captured the structural characteristics and lithium ion transport behavior of MOF-containing polymer electrolytes.⁴⁵ The MOF-5 framework was represented using Consistent Valence Force Field (CVFF) parameters for the organic linker and Dreiding force field was applied to the inorganic cluster. Both CVFF and Dreiding are well-established force fields that have been successfully applied in MD simulations of MOF and have been shown to reproduce the structural integrity and geometrical characteristics of MOF-5.^46,47 To ensure compatibility between the polymer electrolyte and the MOF, all nonbonded interactions between atoms belonging to different force fields were calculated using the standard Lorentz–Berthelot mixing rules together with coulombic electrostatic interactions. Because no covalent bonds were introduced between the MOF and the surrounding electrolyte, the interfacial interactions are governed entirely by van der Waals and electrostatic terms, for which the adopted mixing rules provide a consistent and widely accepted treatment. The atomic partial charges of MOF-5 were obtained from GFN2-xTB quantum mechanical calculations using the default Mulliken population analysis implemented in the xTB program. No empirical adjustment or manual reassignment of atomic charges was performed. The calculated atomic charges were directly used in the molecular dynamics simulations to provide a self-consistent electrostatic description of MOF-5. Since the charges were derived from the electronic structure of the complete system, slight variations in the atomic charges reflect differences in the local chemical environments. The use of GFN2-xTB-derived partial charges has been demonstrated to provide reliable electrostatic descriptions for molecular and materials simulations while maintaining a significantly lower computational cost than conventional density functional theory calculations.^37,48–50 The initial configurations of the simulation systems were first subjected to energy minimization using the conjugate gradient method to remove unfavorable interactions. The minimized systems were then equilibrated at 5 K for 10 ns. Subsequently, the temperature was gradually increased from 5 to 298 K over an additional 20 ns in the isothermal isobaric (NPT) ensemble, allowing the simulation box volume to adjust and expand following the addition of other components. The densities obtained for PVDF and PEO systems showed good agreement with experimental values of 1.77 and 1.13 g cm⁻³, respectively, as well as with previously reported simulation results.^40,41,51 Following NPT equilibration, production MD simulations of each simulation system were carried out for 100 ns using a time step of 1 fs in the canonical (NVT) ensemble. Other parameters including temperature and pressure were maintained at 298 K and 1 bar, respectively, by employing the Nosé–Hoover thermostat and barostat. Long-range non-bonded interactions, including electrostatic and van der Waals (vdW) interactions, were treated using the particle–particle particle-mesh (pppm) method with a cut-off distance of 12 Å.

2.3. Well-tempered metadynamics simulations

Well-tempered metadynamics (WT-MD) is an enhanced sampling technique widely used in MD simulations to reconstruct the free energy surface along selected collective variables (CVs).⁵² The selection of CV plays a significant role in metadynamics, as it determines the regions of the system's configuration space that are actively explored, thereby facilitating efficiently overcoming of free energy barriers. Metadynamics applies a bias potential based on the chosen CVs in order to explore rare events and thus obtains the free energy landscape along the selected variables. It is also essential for CV to capture the slow degrees of freedom relevant to the process of interest since neglecting important motions can lead to incomplete or biased free energy surfaces. This is quite concerning in electrolyte systems, where ion coordination, complex solvation, and ion transport often occur on timescales that are inaccessible to conventional cMD simulations. Well-chosen CVs are also expected to distinguish between relevant states of the system, enable the system to overcome energy barriers, and represent the full atomic coordinates using a small number of physically meaningful descriptors.⁵² The investigation of Li⁺ ion migration requires the consideration of several physical parameters such as, ion coordination behavior, and the distance between the Li⁺ ion and a relevant structural reference, which capture ion mobility and changes in the solvation environment. A distance-based CV enables the free-energy profile to track the movement of Li⁺ ions between different coordination environments while overcoming solvation energy barriers. This approach has been applied in the investigation of lithium solvation and ion transport in ionic liquids and electrolyte materials, where the distance between lithium ions and surrounding ions or solvent molecules is monitored to identify local minima and transition states within the free energy landscape.

Following the (cMD) simulations, equilibrated structures extracted from each trajectory were used as the starting configurations for WT-MD simulations. Prior to applying the bias potential, a 5 ns equilibration was performed using PLUMED (version 2.7.3) coupled with LAMMPS (version 2 Aug 2023) to ensure the stability of the selected CVs.⁵³ Because the simulations focused on MOF-based systems, careful selection of the CVs was critical for accurate exploration of the free-energy landscape. To avoid artifacts arising from periodic boundary conditions (PBCs), the center of mass (COM) of MOF-5 (MOF-5_COM) was calculated without applying PBCs and was paired with a randomly selected lithium ion located near a MOF-5 coordination site. The WT-MD simulations employed Gaussian hills deposited every 500 steps, with an initial height of 1.0 kJ mol⁻¹ and a bias factor of 20. The Gaussian width represent as SIGMA was set to 0.25 Å, and all simulations were conducted at 298 K. A two-dimensional bias grid was defined with boundaries ranging from 0.0 to 5.0 Å in each direction and discretized into 500 bins. This setup was designed to capture meaningful free-energy variations as Li⁺ ions explored different local environments. After equilibration, the final frame of each simulation was used to initiate a 30 ns WT-MD production run. To improve sampling efficiency and statistical confidence, replica-based WT-MD simulations were performed using different initial velocity seeds. A total of three replicas were generated for each of the three systems, resulting in nine independent simulations, each with a duration of 30 ns. Analysis of the resulting trajectories focused on free-energy profiles, and the time-dependent evolution of the distance between Li⁺ ions and their local environments.

2.4. Analysis of the simulations

The equilibration of cMD systems was assessed by monitoring potential energy, density, and simulation box volume during the 20 ns NPT equilibration at 298 K. As shown in Fig. S2, these properties reached stable average values and fluctuated around their equilibrium values without systematic drift, indicating that the systems had reached an equilibrated state before the production simulations. Following equilibration, 100 ns NVT trajectories were generated and used for the structural and transport analyses. Detailed analyses of the trajectories were carried out to evaluate the structural and dynamical properties of the simulated systems. Trajectory processing and data analysis were performed using different tools, including TRAVIS, and MDAnalysis (version 2.9.0).^54,55 Python-based scripts were extensively developed and applied for statistical data plotting, whereas molecular illustrations were generated using Chimera,⁵⁶ and Visual Molecular Dynamics (VMD) software.⁵⁷ The trajectory analysis included several key properties, such as the radial distribution function (RDF), spatial distribution function (SDF), order parameter (P₂), structure factor (S(q)), 3D-density distribution function (3D-DDF), ion pairing analysis, contact analysis, mean squared displacement (MSD), self-diffusion coefficient, free energy landscape (FEL) and distance time analysis, which provided deeper insights into the structural and dynamical properties of Li⁺ ions in these systems.

To quantify how particle density varies as a function of distance from a reference particle, the radial distribution function (RDF) is a fundamental analysis tool in cMD simulations. It helps in understanding the local structural arrangements within liquids, polymers, and complex materials such as MOFs loaded with guest molecules.

1

Additionally, the number density of atoms was computed using a reference particle at a distance (r) and is mathematically formulated as:

2

where NI(r) represents the number integral evaluated up to a distance r_∞, ρ_B is the number density of the surrounding particles, and g(r) denotes the RDF.

The order parameter provides insight into how the electrolyte environment influences polymer-chain conformations and segmental alignment, both of which are directly linked to ion transport properties. The order parameter was evaluated using the second-order Legendre polynomial:

3

where θ is the angle between a bond vector and the principal axis of the polymer chain. Per-chain and global averages were obtained by averaging over all bonds and chains, respectively.

The static structure factor, (S(q)), was evaluated for different atomic pairs in the electrolytes based on their RDFs and was calculated according to

4

where ρ represents the number density of the selected atom type. In the equation, g(r) describes the spatial distribution of atoms, and the Fourier transformation projects this information into reciprocal space.

The lifetime of Li⁺–TFSI⁻ ion pairs was calculated using the dimer autocorrelation function (DACF) approach implemented in TRAVIS.⁵⁴ For each Li⁺–TFSI⁻ pair, the occurrence function was used to describe whether the pair satisfied the selected distance criterion at each time step. The resulting DACF, (C_DACF(τ)), was normalized to its value at zero time lag and described the persistence of the ion pair as a function of the time lag, (τ). Following the aggregate lifetime formalism described by Brehm et al.,⁵⁴ the characteristic lifetime for the ion pair was calculated as

5

where (C_DACF(τ)) is the normalized DACF and (τ) is the time lag. In the present analysis, the lifetime was obtained by numerical integration of the DACF occurrence data using the trapezoidal integration method. The exponential fitting functions provided in the TRAVIS output were not used for the lifetime calculation. The same procedure was consistently applied to all investigated systems.

The mean squared displacement (MSD) is a fundamental quantity used to analyze the diffusive behavior of particles in MD simulations. It quantifies the mean squared distance that a particle travels over time and is mathematically defined as;

6

where N is the total number of atoms or molecules to be averaged, and r_j(t) and r_i(0) represent the reference position and the position at time t of the ith atom and molecule, respectively.

The self-diffusion coefficient (D) was determined using the Einstein relation:

7

where d represents the number of dimensions (here d = 3), r(t) is the duration of the simulation time in which the property is to be measured, r(0) is the reference position and D stands for the self-diffusion coefficient expressed in cm² s⁻¹ unit.

TRAVIS automatically identifies the time interval over which the MSD exhibits linear behavior and performs a least-squares fit to determine the slope. Consequently, the fitted MSD curves, plotted alongside the raw MSD data, represent the Fickian (long-time) regime used for estimating D. This approach ensures a consistent and reliable evaluation of particle mobility across all simulated systems.

The potential of mean force (PMF), (W(r)), was calculated to evaluate the interaction strength and free-energy profile between different atomic pairs in the simulated systems. The PMF was calculated from the RDF, (g(r)), obtained from MD simulations using the relation:

W(r) = −RT ln g(r) 8

where R is the universal gas constant (8.314 J mol⁻¹ K⁻¹), T is the simulation temperature (298K), and g(r) is the RDF as a function of intermolecular distance (r).

3. Results and discussion

3.1. Structural properties

Detailed insights into the structural and dynamical properties of these simulation systems were obtained based on the interactions between their different components. Prior to evaluating these interactions, the densities of the simulation systems were calculated and are listed in Table 1. The density values for the pure polymer systems, PVDF and PEO, were calculated to be 1.49 and 1.12 gm cm⁻³, respectively, which correlated well with values reported in the literature.^40–42 After the addition of LiTFSI salt to pure PVDF and PEO polymers, resulting in respective Systems A and C, the densities were found to be higher than those of the corresponding pure polymer systems. The trend in the observed densities exhibits an obvious dependence on both composition and structural characteristics, indicating denser packing due to enhanced interactions between the salt and the polymers. For instance, the high density of System A (1.67 gm cm⁻³), combining LiTFSI with PVDF, indicates strong ion–polymer interactions. In contrast, the pure PEO-based system exhibits a significantly lower density (1.13 gm cm⁻³), which can be attributed to its greater flexibility and comparatively loose packing. The addition of LiTFSI salt to PEO, resulting in System C, shows a similar increase in density (1.41 gm cm⁻³); however, this value remains lower than that of the corresponding PDVF–salt system (System A). Furthermore, the incorporation of MOF-5 into Systems A and C, yielding Systems B and D, results in a consistent reduction in density with respective values of 1.59 and 1.29 gm cm⁻³. This decrease in density can be attributed to the intrinsic porosity of MOF-5, which disrupts the efficient packing observed in the densely packed LiTFSI–PVDF and LiTFSI–PEO systems. Nevertheless, comparison of the densities for the MOF-5-containing Systems B and D indicates that the relatively greater flexibility of PEO makes its structure being largely affected due to addition of MOF-5.

The interactions between the polymers and the salt were evaluated using RDFs, which provide atomic-level insights by analyzing the correlations between the physical and structural properties of the systems at the microscopic scale. Initially, RDFs were computed between Li⁺ ions and the characteristic atoms of the PVDF and PEO polymers, namely, Li⁺–F_PVDF and Li⁺–O_PEO, as shown in Fig. 2. Based on the Li⁺–F_PVDF RDFs obtained for Systems A and B, the average distance between Li⁺ ions and the fluorine atoms of PVDF polymer was found to be 2.27 Å. Integration of RDFs up to the first peak minimum yielded coordination numbers of 2 and 1 for system A and B, respectively (Fig. 2a). Additionally, analysis of the relative intensities of the first RDF peaks revealed that the interactions between Li⁺ ions and the fluorine atoms in System A were weaker than that in System B. This observation suggests that the presence of MOF-5 influences the structural and spatial organization of the polymer chains, thereby forming a local environment that favors interactions between Li⁺ ions and the fluorine atoms of PVDF. Similarly, the Li⁺–O_PEO RDFs were calculated for Systems C and D, which produced the first RDF peak at 2.02 Å, agreeing well with neutron diffraction experiments (2.10 Å) and previous simulation studies (2.12 Å) (Fig. 2b).^40–42,58 Although, the peak positions were closely similar, a slight difference in peak intensity was observed between the System C and System D. Based on the peak intensities, System D, which contains MOF-5, exhibits stronger and more stable interactions between Li⁺ ions and the PEO chains. These results from the Li⁺–F_PVDF and Li⁺–O_PEO RDF analysis indicate that the incorporation of MOF-5 enhances the structural arrangement of the polymer matrix, which is favorable for the optimal interactions involving Li⁺ ions. Furthermore, the Li⁺–F_PVDF and Li⁺–O_PEO RDFs were superimposed, as depicted in Fig. 2c, to compare the relative strengths of the interactions between Li⁺ ions and the two polymers. The overlapped RDFs revealed that the Li⁺–O_PEO distance is shorter than the corresponding distance between Li⁺ ions and PVDF chain, as deduced from the positions of the first peak maxima. Moreover, the peak intensities revealed the stronger preference of Li⁺ ions for coordination with PEO. The Li⁺–polymer RDFs exhibit a strong dependence on the nature of the coordinating atoms. For instance the Li⁺–O_PEO RDF displays a sharp first peak followed by a near-zero-minimum and a broader second peak, indicating strong and stable coordination between Li⁺ ions and the ether oxygen atoms of PEO. In contrast, the Li⁺–F_PVDF RDF exhibits a non-zero minimum between two broad peaks, suggesting weaker and more transient interactions between Li⁺ ions and PVDF. These observations were inferred as the interactions in the PEO systems are dominated by strong Lewis acid-base interactions between Li⁺ ions and ether oxygen atoms, resulting in localized solvation and structured coordination environments. In contrast, PVDF primarily serves as a structural framework, with fluorine atoms contributing weakly only to lithium-ion coordination.

Fig. 2. Radial distribution functions (RDFs) of Li⁺ ions with respect to (a) the fluorine atom of PVDF, (b) the ether oxygen atom of PEO, and (c) superimposed plots of Li⁺–F_PVDF and Li⁺–O_PEO RDFs.

The presence of polymers, both without and with MOF-5, was envisaged to perturb the interactions between the cations and anions in LiTFSI, as demonstrated by Li⁺–F_PVDF and Li⁺–O_PEO RDFs displayed in Fig. 2. To further analyze these perturbations, Li⁺–TFSI⁻ RDFs were computed for Systems A-D. The resulting RDFs exhibited a first peak located at 2.02 Å for all systems (Fig. 3a). Despite that the mean distance between Li⁺ ions and TFSI⁻ anions remained unchanged, the peak intensities and the positions of the first peak minima in all cases were distinct. These variations indicate that the Li⁺–TFSI⁻ interactions in the PVDF-based Systems A and B are stronger compared to those in the PEO-based Systems C and D, corroborating the findings based on the Li⁺–F_PVDF and Li⁺–O_PEO RDFs. The results further demonstrate that the incorporation of MOF-5 influences the cation–anion interactions, particularly, Li⁺ ions in the System D which displayed weaker interactions with TFSI⁻ anions compared to those in the System B (Fig. 3b). This observation indicates that the combined effects of the polymer matrix and MOF-5 play a significant role in governing the structural properties. For comparison, a Reference system consisting of MOF-5 and LiTFSI, without any polymer, was also evaluated alongside the polymer-based systems. The Li⁺–TFSI⁻ interactions exhibit contrasting behavior depending on the polymer environment. In PEO-based systems, the strong coordination of Li⁺ ions with the ether oxygen atoms of PEO reduces direct Li⁺–TFSI⁻ association, resulting in weaker ion pairing and a greater degree of ionic dissociation. In contrast, the relatively weak interactions between Li⁺ ions and PVDF allows Li⁺ ions to interact more strongly with TFSI⁻ anions, leading to more pronounced ion pair formation and a well-defined coordination shell in the RDFs. This behavior reflects stronger Li⁺–TFSI⁻ association and a reduced contribution of polymer within the local environment of Li⁺ ions. To gain deeper insights into the Li⁺–TFSI⁻ interactions, atom-to-atom pair correlation functions, namely Li⁺–O_TFSI and Li⁺–N_TFSI were also calculated. These RDFs exhibited trends similar to those observed for the overall Li⁺–TFSI⁻ RDFs across all systems; however, the peak structure for the Li⁺–O_TFSI RDF was sharper and more intense (Fig. S3) than for Li⁺–N_TFSI RDF (Fig. S4) that has slighter broader peak. The peak trends in these RDFs indicate that oxygen atoms contribute more directly to coordinate with Li⁺ ions than the nitrogen atoms within TFSI⁻ anions.

Fig. 3. Radial distribution functions (RDFs) of Li⁺ ions with TFSI⁻ anions in (a) PVDF-based and (b) PEO-based systems.

Fig. 4 illustrates the possible interactions between Li⁺ ions and different components of MOF-5, including the organic linkers and zinc-clusters, as the framework was envisaged to facilitate Li⁺ ions to optimally interact with the polymer matrices. The interactions of Li⁺ ions with the linkers and zinc-clusters were evaluated through Li⁺–O_linkers and Li⁺–Zn_{zinc-clusters} RDFs, along with those of the Reference system comprising only MOF-5 and LiTFSI. A well-structured first peak was observed at 2.07 Å for both the Reference and System B; however, the corresponding peak in System D appeared at a slightly larger distance of 2.12 Å (Fig. 4a). Although the peak positions for the Reference system and System B were identical, the higher peak intensity for the latter system indicates stronger interactions between Li⁺ ions and the linkers. This enhancement is facilitated by PVDF matrix, which promotes a more favorable spatial arrangement of Li⁺ ions near the linker oxygen atoms. In contrast, System D exhibited a lower peak intensity, which can be attributed to the presence of PEO, whose flexible nature allows Li⁺ ions to interact more dynamically with other components of the system. Additionally, the Li⁺–Zn_{zinc−clusters} RDF was computed to evaluate the spatial distribution of Li⁺ ions with respect to the metal centers of MOF-5. System B exhibited a well-defined first peak at 2.52 Å, indicating spatial correlation and localization of Li⁺ in the vicinity of the zinc-clusters (Fig. 4b). This apparent correlation arises from interactions between Li⁺ ions and Zn atoms associated with the Zn-based secondary building units, as direct interactions between Li⁺ ions and Zn atoms are inherently unlikely to occur. In comparison, the corresponding RDFs for the Reference system and system D displayed first peaks at 2.72 and 2.77 Å, respectively, with contrasting intensities. The peak intensity for System B was even higher than that of the Reference system, indicating a greater local density of Li⁺ ions near the zinc clusters. Conversely, the low-intensity peak observed in System D at a similar distance reflects the influence of PEO, which facilitates more dynamic interactions of Li⁺ with other species in the system. It is important to note that the Li⁺–O_linkers and Li⁺–Zn_{zinc-clusters} RDFs across all investigated systems exhibit similar peak positions, indicating comparable characteristic distances; however, the differences in peak intensities reflect variations in the degree of spatial correlation and the relative strength of interactions involving Li⁺ ions and MOF-5. The observed correlation between Li⁺ ions and the zinc-clusters is primarily governed by the intrinsic topology of MOF-5. Li⁺ ions preferentially interact with the oxygen atoms of the organic linkers, which are structurally connected to the Zn-based secondary building units, rather than forming direct interactions with Zn centers themselves. Consequently, an apparent yet non-bonding spatial relationship is observed between Li⁺ ions and the Zn atoms.

Fig. 4. Radial distribution functions (RDFs) of Li⁺ ions with respect to (a) organic linker and (b) zinc-clusters in MOF-5 framework.

The structure factor, S(q), in such multicomponent polymer–salt systems characterizes the spatial correlations among ions and other components arising from the combined effects of Coulomb interactions and thermal motion. The positions and intensities of the peaks in the calculated S(q) profiles provide information about interionic/interatomic distances and the degree of structural organization within the systems, which is significant for understanding the relationship between ion ordering and electrolyte performance. Fig. 5 illustrates the S(q) profiles of the polymer-salt systems and provides insight into the distinct structural organization of Li⁺ ions within these electrolytes. Systems A and B exhibit relatively sharp-narrow but low-intensity peaks at q= ∼1, indicating weak ionic correlations and largely disordered ionic distributions (Fig. 5a). In contrast, Systems C and D display more pronounced and intense peaks, as shown in Fig. 5b, reflecting stronger ionic correlations and enhanced spatial ordering arising from well-defined intermolecular interactions. The incorporation of MOF-5, as observed in Systems B and D, results in the reduction of peak intensity, suggesting that the spatial correlations between Li⁺ ions and TFSI⁻ anions are partially disrupted. This behavior indicates that MOF-5 offers additional interaction sites for strong Li⁺–MOF-5 interactions that directly influence cation–anion associations within the electrolyte. Consequently, the ionic distribution becomes more localized, while the long-range structural ordering of the ions is reduced within the electrolyte matrix. A comparison of the S(q) profiles further reveals that System D, comprising PEO, LiTFSI, and MOF-5 exhibits S(q) value approaching unity, indicating the formation of a more homogeneous ionic distribution in the presence of MOF-5. This observation suggests that MOF-5 facilitates stronger cation–polymer associations while simultaneously suppressing ion aggregation. Such a more uniform ionic environment may contribute to improved electrolyte performance by promoting efficient ion transport and reducing the extent of ion aggregation.

Fig. 5. Structure factor (S(q)) of the (a) Li⁺–PVDF interactions in Systems A and B, and (b) Li⁺–PEO interactions in Systems C and D.

The orientational order parameter (P₂) is a significant descriptor for understanding the structural organization and mobility of polymer chains in polymer-electrolyte systems. It provides information regarding the degree of alignment and ordering of polymer segments, which, in turn, affect the local dynamics of the polymer chains and influence ion transport behavior. To evaluate the structural ordering of the polymer systems, the orientational order parameter was calculated for all investigated cases. This parameter provides a quantitative measure of the chain organization and offers insights into how the incorporation of LiTFSI salt and MOF-5 affects the structural arrangement of the polymer matrix. The calculated order parameters are reported as mean values with their standard deviation in Table S1. The pure PVDF system exhibits a relatively higher order parameter averaged at 0.081, indicating a greater degree of chain ordering. This value decreases to 0.071 upon addition of the salt (System A), suggesting that the presence of the salt induces structural perturbations within the polymer matrix. A further reduction in the order parameter is observed in System B, reaching a value of 0.069 as depicted in Fig. 6a. This reflects the combined effect of the salt and MOF-5, which promote structural disorder through interactions among ions, polymer chains and MOF-5, thereby disrupting the alignment of the polymer chains. In the PEO-based systems, the pure PEO polymer exhibits a very low order parameter of 0.004, indicating a highly disordered and flexible chain configuration (Fig. 6b). Upon addition of the salt, resulting in System C, the order parameter further decreases to 0.003, demonstrating that the incorporation of the salt increases the configurational randomness in the system. The subsequent addition of MOF-5 in System D leads to an even greater degree of disorder, with the order parameter decreases to 0.002, the lowest among all investigated systems. This suggests that the combined presence of salt and MOF-5 significantly enhances chain mobility and reduces structural ordering in PEO. These findings clearly demonstrate that PVDF maintains a comparatively higher degree of structural ordering than PEO, whereas the addition of the salt and MOF-5 progressively reduces ordering in both polymer systems. This contrast behavior highlights the strong influence of ionic and the framework interactions on polymer chain organization.

Fig. 6. Orientational order parameter (P₂) of (a) PVDF in System A and B, and (b) PEO in System C and D, in comparison to pure PVDF and PEO.

To complement the RDF results, SDF analysis was performed to visualize the three-dimensional structural attributes of Li⁺ ions surrounding the polymers, specifically with respect to the fluorine atoms of PVDF and the oxygen atoms of PEO in the investigated systems. As illustrated in Fig. S5, System A exhibits a more pronounced density of Li⁺ ions around the fluorine atoms, revealing that the polymer environment provides a favorable spatial arrangement for Li⁺ ions. In contrast, in System B, the density of Li⁺ ions around PVDF is perturbed due to the presence of MOF-5, where additional coordination sites are available. A similar trend was observed in the System C, where Li⁺ ions show a higher density around PEO owing to the strong coordination between Li⁺ ions and ether oxygen atoms, as shown in Fig. S5. After the incorporation of MOF-5 (System D), Li⁺ ions are also attracted by the framework through coordination with its linker oxygen atoms, which compete with the polymer and consequently reduce the probability of finding Li⁺ ions near the PEO chains.

Moreover, SDF analysis of Li⁺ ions around TFSI⁻ anions was performed for all the investigated systems. As demonstrated in Fig. S6, the Reference system exhibits the most intense and extensive isosurfaces around the anions, reflecting a high density of Li⁺ ions in the proximity of the anions. This indicates strong ion pairing and effective coordination, likely driven by favorable electrostatic attraction between Li⁺ ions and the delocalized negative charge of TFSI⁻ anions. Such strong association may promote local ordering and reduce ionic mobility due to ion pairing. On the other hand, Systems A and B, also show substantial Li⁺ ion density near TFSI⁻ anions, although lower than that observed in the Reference system. The presence of PVDF, a semi-crystalline polymer with relatively low polarity, likely reduces Li⁺–polymer coordination, thereby allowing greater interactions between Li⁺ ions and either MOF-5 or the anions. In contrast, Systems C and D exhibit the least pronounced isosurfaces around the TFSI⁻ anions, corroborating the RDF analysis. Systems B and D, which contain MOF-5, show minimal Li⁺ ion density near both TFSI⁻ anions and MOF-5, suggesting that Li⁺ ions exist in a more loosely coordinated state. This behavior may arise from competitive; however, weak interactions among Li⁺, polymers, and MOF-5, resulting in enhanced Li⁺ mobility rather than confinement within the framework. The combined RDF and SDF analysis reveal that the chemistry of MOF-5, through its metal nodes and organic linkers, together with polymer selection, plays a significant role in shaping the spatial distribution and coordination dynamics of Li⁺ ions.

MOF-5 provides a three dimensional interconnected porous framework constructed from Zn₄O secondary building units linked by terephthalate ligands. Its cubic structure contains interconnected pore cavities with relatively large pore dimensions and narrow windows, providing accessible pathways for ionic species within the framework.^59,60 In particular, the combination of pore size and interconnected windows can facilitate the movement of solvated Li⁺ species while the framework can also influence the local distribution of the counterions. Previous experimental work has shown that incorporation of MOF-5 into polymer electrolytes can modify the porous environment and improve lithium ion transport, supporting its role as an ion conducting filler.⁶¹ To further examine the possible Li⁺ migration pathways associated with MOF-5, the temporal distribution of Li⁺ ions relative to MOF-5 was analyzed at 0, 25, and 50 ns (Table S2). The results show that the average Li⁺ to MOF-5 distance and the number of Li⁺ ions located within 3 Å of MOF-5 vary with the polymer environment, indicating different degrees of Li⁺ interactions with MOF-5 surface. In the Reference system, the Li⁺ population remains relatively close to MOF-5 throughout the simulation, whereas the polymer containing MOF-5 shows a larger change in the average Li⁺ to MOF-5 distance with time. This behavior suggests that Li⁺ ions do not remain permanently localized at MOF-5 surface but can exchange between the MOF-5-associated regions and the surrounding electrolyte environment. Such dynamic ion redistribution is consistent with the proposed role of porous MOFs in modifying Li⁺ transport pathways in polymer electrolytes, where the MOF surface and pore environment can provide additional regions for ion coordination and migration.^61–63 The corresponding structural visualization is provided in Fig. S7, where the Li⁺ distribution around MOF-5 at different stages of the trajectory is shown. These results provide a trajectory-based view of Li⁺ movement near MOF-5 and complement the SDF analysis. Importantly, the distance analysis should also be interpreted as evidence of the spatial association and redistribution of Li⁺ ions around MOF-5 rather than as a direct determination of a unique migration pathway.

To further evaluate the density distribution of Li⁺ ions around the polymers, three-dimensional density distributions of the ions were computed in Systems A–D as shown in Fig. 7. The density distribution of Li⁺ ion density around MOF-5 was also calculated in all systems, including the Reference system. Fig. S8 displays the density distribution for the Reference system, depicting a relatively low mean density (Table S3) and limited spatial distribution of Li⁺ ions around the framework. This behavior reflects weak electrostatic interactions between Li⁺ ions and the inorganic nodes of MOF-5. In the System B, the mean density of Li⁺ ions around the framework increased, suggesting enhanced ionic confinement, as shown in Fig. 7b. Although PVDF is chemically inert and lacks strong donor atoms, its semi-crystalline nature and electronegative fluorine atoms can influence the polymer free volume and dielectric environment, thereby restricting Li⁺ ion mobility and leading to tighter packing around MOF-5. The 3-D density distribution of system D displayed in Fig. 7c exhibits broader distribution of Li⁺ ions. This behavior can be attributed to inherent properties of the ether oxygen atoms in PEO, which provide multiple interaction sites for Li⁺ ions and facilitate solvation as well as dynamic exchange of Li⁺ ions between interaction sites in polymer and MOF-5. These findings highlight the synergistic role of MOF-5 structural chemistry, which offers confined pore spaces and open metal sites, together with the identity of the polymer matrix, in governing the extent and nature of Li⁺ interactions and spatial organization within hybrid electrolytes.

Fig. 7. Three-dimensional density distribution of Li⁺ ions around PVDF polymer (a) System A and (b) System B, and PEO polymer (c) System C and (d) System D.

Based on the 3D-density distribution in System A, mean Li⁺–PVDF distance was determined as 24.05 Å as mentioned in Table S4, indicating a heterogeneous distribution of Li⁺ ions, with the polar C–F bonds in PVDF contributing to modest interactions. Upon the addition of MOF-5 (System B), the Li⁺–PVDF distance slightly decreased to 23.97 Å, suggesting that MOF-5 provides additional interaction sites, resulting in competition between the framework and the polymer for Li⁺ coordination. This competing behavior slightly redistributes the Li⁺ ion density without significantly enhancing confinement. In contrast, System C exhibits a lower mean Li⁺–PEO distance compared to the PVDF-based systems (Fig. 7), indicating that Li⁺ ions are less concentrated and more distributed throughout the PEO matrix. This behavior can be attributed to the presence of ether oxygen atoms in PEO, which offers flexible and transient interaction sites for Li⁺ ions. These interactions enable Li⁺ ions to dynamically get associated with the polymer chains, preventing excessive localization while promoting moderate mobility. Nevertheless, the broad distribution and relatively high maximum observed in the density profile suggest that although Li⁺ ions do not form dense and localized clusters, occasional spatial grouping may occur due to transient ion pairing or local polymer chain conformations. On the other hand, the case of System D exhibits a more confined Li⁺ distribution, as illustrated in Fig. 7d, indicating that Li⁺ ions are more uniformly dispersed and spatially regulated throughout the system. The cooperative presence of PEO and MOF-5 contributes to this behavior, as the ether oxygen atoms of PEO chains provide a flexible solvation environment, while the Lewis acidic metal within MOF-5 offers coordination sites. Consequently, these interactions are evaluated to promote a balanced distribution of Li⁺ ions, potentially enhancing ion transport while maintaining structural stability within the hybrid electrolyte.

To further characterize the spatial distribution of ions within the MOF-5 pore, the pore occupation of Li⁺ and TFSI⁻ was quantified over the complete 50 ns trajectories. The quantitative pore occupation data are summarized in Table S5. Both Li⁺ and TFSI⁻ are able to access the MOF-5 pore; however, their spatial distributions are distinct. Li⁺ ions exhibit relatively localized high density regions within the accessible pore space, whereas the larger TFSI⁻ anions show a broader distribution throughout the pore. The comparison between the systems also reveals a strong dependence of pore occupation on the polymer environment. The Reference system shows moderate occupation of the pore by both ionic species, while System B exhibits lower Li⁺ occupation but retains appreciable TFSI⁻ presence within the pore. In contrast, System D shows substantially greater occupation of the MOF-5 pore by both Li⁺ and TFSI⁻, indicating enhanced penetration of both ionic species into the accessible pore region. These differences suggest that the surrounding polymer environment influences the balance between ion confinement and redistribution within MOF-5. The distinct Li⁺ and TFSI⁻ density patterns further indicate that the two ionic species do not experience the pore environment in the same manner, which can arise from differences in their size, mobility, and interactions with the MOF surface and surrounding electrolyte. Together, these results provide structural evidence that the MOF-5 pore creates an additional region for ion redistribution and may contribute to the modification of ion transport pathways in the polymer electrolyte. This interpretation is also consistent with previous studies showing that porous MOF environments can alter ion confinement, local coordination, and transport behavior in polymer based electrolytes.^60,62,63

To further examine the structural organization of ions within the MOF-5 pore, the orientation of TFSI⁻ relative to MOF-5 was analyzed together with the three dimensional density distributions. As shown in Table S6, the mean orientation angles and their corresponding standard deviations are broadly similar among the Reference, System B, and System D, indicating that TFSI⁻ anions do not adopt a strongly preferred orientation in any of the systems. The P₂ values also remain close to zero, suggesting considerable orientational freedom and weak directional ordering of TFSI⁻ anions. Nevertheless, the broader orientational distribution observed in System B indicates greater orientational flexibility, which may facilitate local molecular rearrangement within the MOF-5 pore. In contrast, System D shows a slightly more restricted orientational distribution, suggesting stronger local interactions with its surrounding chemical environment. The similar average orientation observed in System B and System D indicates that the polymer environment does not substantially alter the overall orientation of TFSI⁻, although differences in local interactions may still influence its spatial organization. This behavior can be related to the distinct chemical environments provided by PEO and PVDF. The ether oxygen atoms of PEO provide favorable coordination sites for Li⁺ ions and contribute to LiTFSI solvation, whereas PVDF lacks equivalent donor oxygen sites for direct Li⁺ coordination. Consequently, the local interactions among Li⁺, TFSI⁻, the polymer matrix, and the MOF-5 surface can differ between the two polymer environments. Previous studies have shown that polymer chemistry and ion–polymer interactions can strongly influence ion coordination, speciation, and mobility in solid polymer electrolytes.⁶⁴

The orientation analysis is also consistent with the three dimensional density distributions obtained within the MOF-5 pore. The density analysis shows that Li⁺ and TFSI⁻ do not occupy the pore in an identical manner, with Li⁺ exhibiting more localized high density regions, whereas TFSI⁻ shows a more extended spatial distribution. This difference indicates that the MOF-5 pore provides a heterogeneous environment in which the cations and anions experience different spatial constraints and intermolecular interactions. The broad TFSI⁻ orientation distribution further indicates that the anions can adopt multiple orientations within the pore rather than being restricted to a single preferred configuration relative to the MOF. Thus, the combined density and orientation results indicate that the MOF-5 pore modifies the local spatial organization of Li⁺ and TFSI⁻, while the polymer chemistry contributes to the nature of their surrounding coordination environment. These findings are consistent with previous reports showing that MOF incorporation can modify ion distribution, local interactions, and transport behavior in polymer-based electrolytes.^61–63

The environment of Li⁺ ions was further elucidated through ion pairing analysis, with the corresponding values reported in Table S7. System B exhibits the highest degree of ion pairing, with 270 ion pairs, followed by the Reference system with 260 ion pairs and System A with 250 pairs, as illustrated in Fig. S9. The high degree of ion pairing in System B suggests strong electrostatic interactions resulting from the reduced solvation capability of PVDF and the confinement by MOF-5, both of which promote ion association. In contrast, System D exhibits weaker ionic interactions and enhanced Li⁺ coordination with the ether oxygen atoms of PEO. In the absence of MOF-5, System C possesses a slightly higher ion pairing capacity than System D; however, it remains significantly lower than that in the Reference and the PVDF-based systems. The reduced ion pairing capacity observed in System D indicates greater ionic dissociation, which is favorable for Li⁺ mobility. Considering both ion pairing and solvation behavior, System D emerges as the most promising candidate for Li⁺ transport among the MOF-5-based systems. The presence of both PEO and MOF-5 provides a balance between effective Li⁺ solvation by PEO and spatial confinement or structural ordering by MOF-5 due to its porous framework, thereby facilitating enhanced Li⁺ mobility while minimizing ion clustering.

The ion pairing analysis further reveals significant variations in the coordination environment of Li⁺ ions across the studied systems. Specifically, the Reference system, System B, and System D, each representing a unique combination of MOF-5 and polymer, display distinct ion association behaviors. To quantify these differences, Li⁺ ions were classified into contact ion pairs (CIP), solvent-separated ion pairs (SSIP), and free ions based on distance criteria between Li⁺ and the oxygen atoms of TFSI⁻ anions, as shown in Fig. S10. The distance criteria used for ion pairing classification was characterized using the minimum Li⁺–O_TFSI distance. A contact ion pair (CIP) was defined when the Li⁺–O_TFSI distance was ≤ 3.0 Å. A solvent separated ion pair (SSIP) was defined for distances > 3.0 Å and ≤ 5.0 Å, while Li⁺ ions at distances > 5.0 from TFSI⁻ were classified as free ions. These cutoff values were applied consistently to all systems. Mean populations and standard deviations for ion pairs, CIP, SSIP, and free ions are also provided in Table S8. Among all systems, System A contains the lowest SSIP population, suggesting that Li⁺ ions in this system tend toward either strong association (CIP) or complete dissociation (free ions). System B exhibits the highest CIP population, signifying strong ionic interactions likely stabilized by a local environment that favors tight ion association. This behavior points toward a more compact electrolyte structure, which may hinder Li⁺ mobility. In terms of SSIP, System C exhibits the highest fraction, implying moderate ion association potentially mediated by polymer–ion interactions. In contrast, System D shows the lowest number of CIPs, indicating greater ion dissociation and weaker electrostatic interactions that are favorable for enhanced ion transport. Based on the ion pairing analysis, System D possesses the highest number of free Li⁺ ions, reinforcing the presence of a highly dissociative environment likely associated to favorable interactions between MOF-5 and PEO. On the other hand, the Reference system contains the lowest fraction of free ions, suggesting stronger ion binding and/or limited polymer-assisted solvation.

To further characterize the dynamic behavior of Li⁺ and TFSI⁻ association, ion pair lifetimes were evaluated using time-dependent autocorrelation function. Here, ion pair lifetimes were calculated using time correlation function of the ion pair contact state. The autocorrelation function was used to monitor the persistence of Li⁺–TFSI⁻ contacts within the specified cutoff distance and to determine the characteristic lifetime of the ion pairs. The resulting lifetimes provide a dynamic measure of ion association and complement the static ion pairing populations. Mean ion-pair lifetimes and standard deviations are given in Table S9 and the DACF results reveal differences in the persistence of Li⁺–TFSI⁻ association among the investigated systems. The Reference system, System A, and System B exhibit comparable mean occupancies and relatively long DACF lifetimes, indicating a relatively persistent ion association in these electrolytes. In contrast, Systems C and D show lower mean occupancies and shorter DACF lifetimes, indicating a lower persistence of Li⁺–TFSI⁻ association. These differences suggest that the local chemical environment in Systems C and D promotes more frequent disruption and reformation of Li⁺–TFSI⁻ contacts, providing greater opportunity for Li⁺ ions to exchange between TFSI⁻ anions and other coordinating sites within the electrolyte. Such dynamic ion association is relevant to Li⁺ transport because Li⁺ mobility depends on continuous changes in its coordination environment rather than on a permanently fixed association with a single anion. Previous studies have shown that Li⁺–TFSI⁻ association and changes in the Li⁺ coordination environment can influence ion mobility in polymer electrolytes.^65–67 Nevertheless, the shorter ion pair lifetime should be interpreted as evidence of more dynamic ion association rather than direct evidence of enhanced Li⁺ transport, since Li⁺ mobility is also governed by its coordination with the polymer matrix, polymer segmental dynamics, and interactions with MOF-5.

To understand the coordination environment and spatial proximity of Li⁺ ions within these hybrid electrolyte systems, contact analysis between Li⁺ ions and MOF-5, as well as TFSI⁻ anions and MOF-5, was performed using distance-based criterion of 3.5 Å. This criterion was used to calculate the number of MOF-5 atoms in direct contact with Li⁺ and TFSI⁻ throughout the simulation trajectory. Mean contact numbers and their standard deviations for Li⁺ ions and MOF-5, as well as TFSI⁻ anions and MOF-5 are reported in Tables S10 and S11. For Li⁺ and MOF-5 interactions, the Reference system shows the highest degree of coordination, as displayed in Fig. S11, reflecting strong electrostatic binding of Li⁺ ions to MOF-5. System B exhibits moderate Li⁺–MOF-5 interactions, indicating that PVDF does not hinder the accessibility of MOF-5 to Li⁺ ions. In contrast, System D demonstrates minimal Li⁺–MOF-5 contact, consistent with competitive solvation of Li⁺ by the ether oxygen atoms of PEO, which dynamically shield Li⁺ ions from the MOF-5 surface. The contact analysis for TFSI⁻–MOF-5 interactions revealed a reverse trend as the Reference system exhibits the highest number of TFSI⁻–MOF-5 contacts, as shown in Fig. S12. System B displays the lowest contact numbers, whereas System D records comparatively higher TFSI⁻–MOF-5 interactions. This behavior suggests that PEO alters the coordination between TFSI⁻ and MOF-5; however, does not completely suppress the accessibility of TFSI⁻ anions to MOF-5. In fact, the weaker Li⁺–MOF-5 interactions observed in System D may provide TFSI⁻ with greater freedom to interact with the Lewis acidic sites of MOF-5 compared to the more competitive and spatially restricted environments present in PVDF-based systems.

These contact analysis results are consistent with the TFSI⁻ density and orientation distributions discussed above. The broader spatial distribution and diverse orientations of TFSI⁻ anions within the MOF-5 pore are reflected in its ability to maintain contact with MOF-5 across the different polymer environments. In particular, the higher TFSI⁻ and MOF-5 contacts in System D are consistent with the greater TFSI⁻ pore occupation observed from the three dimensional density distribution, whereas the lower contacts in System B correspond to its lower overall TFSI⁻ interaction with MOF-5. Thus, the combined density, orientation, and contact analyses indicate that TFSI⁻ anions retain substantial spatial and orientational freedom within the MOF-5 pore, while its interaction with the framework is influenced by the surrounding polymer environment. This redistribution of TFSI⁻, together with the different Li⁺ coordination environments, provides a molecular basis for the differences in Li⁺ mobility observed between the PEO- and PVDF-based systems.

The end-to-end distance was analyzed to examine changes in polymer chain extension following incorporation of MOF-5. The results in Table S12 and Fig. S13 show that the structural response depends on the polymer matrix. The PVDF-based systems exhibit relatively modest changes in chain extension, with System A showing a more compact average configuration, whereas System B remains comparatively close to the pure PVDF system. In contrast, a more pronounced reduction in chain extension is observed for System C and System D, which exhibit smaller average end-to-end distances than pure PEO. These results indicate that incorporation of the corresponding components produces a stronger change in the overall dimensions of PEO chains than in PVDF chains. The different responses of PEO and PVDF polymers can be related to their distinct chemical environments. PEO contains ether oxygen atoms that can coordinate to Li⁺ and participate directly in the local solvation environment of LiTFSI, whereas PVDF has a fluorinated backbone and provides a different interaction environment for the ions. Previous experimental and computational studies have shown that polymer chemistry and chain dynamics can influence ion coordination and the structural environment experienced by mobile ions in polymer electrolytes.^68,69 In particular, ion transport in PEO-based electrolytes is closely related to the local segmental environment, although this relationship can vary depending on the electrolyte composition and structural organization.^68,70 The end-to-end distance should therefore be interpreted primarily as a structural descriptor rather than a direct measure of ionic transport. A change in chain dimensions can modify the local environment and accessibility of coordinating sites; however, Li⁺ transport also depends on ion pairing, coordination dynamics, segmental motion, and the spatial distribution of accessible transport regions. Thus, the end-to-end distance provides complementary structural information for interpreting the dynamical properties of the electrolyte systems.

The radius of gyration provides an independent measure of the overall spatial dimensions of the polymer chains. The radius of gyration follows the general trend observed for the end-to-end distance (Table S13). As shown in Fig. S14, The PVDF-based systems show relatively modest changes, with the reduction being more apparent for System A, whereas System B remains comparatively close to pure PVDF. The PEO systems show a clearer reduction in radius of gyration, particularly for System C, consistent with the shorter end-to-end distances observed for Systems C and D. The agreement between these two structural descriptors indicates that the reduced dimensions in Systems C and D are not limited to the separation of the chain termini but reflect a more compact overall spatial distribution of the polymer chains. A similar interpretation of the radius of gyration as a descriptor of polymer chain expansion and structural response has been reported in molecular simulations of PEO-based electrolytes.⁷¹ The temporal standard deviations remain relatively small compared with the differences between the polymer systems, indicating that the average chain dimensions remain reasonably stable over the analyzed trajectories. The larger variation between individual chains is expected for flexible polymer systems because different chains can adopt different conformations within the same electrolyte environment. Therefore, the observed differences in average end-to-end distance and radius of gyration reflect changes in the overall conformational state rather than a complete restructuring of the polymer chains.

The conformational response was further examined using backbone torsional angle distributions, with the corresponding values summarized in Table S14. As shown in Fig. S15, all systems exhibit broad torsional distributions extending across a wide angular range, with a substantial contribution near ± 180°, characteristic of trans-like backbone configurations. The broad distributions indicate that the polymer chains retain substantial conformational flexibility during the simulations. The PVDF-based systems show broader torsional distributions than the PEO-based systems, whereas the differences between the pure and modified systems within each polymer family remain relatively modest. The relatively similar torsional distributions between the corresponding pure and modified systems indicate that incorporation of MOF-5 does not substantially alter the overall range of backbone conformations sampled by either polymer. This observation is important when considered together with the end-to-end distance and radius of gyration results. For PVDF, the modest changes in chain dimensions occur without a major redistribution of the backbone torsional states. For PEO, the decrease in both end-to-end distance and radius of gyration occurs while the overall torsional distribution remains broadly similar to that of pure PEO. This suggests that the more compact PEO configurations arise mainly from changes in the spatial arrangement of the chains rather than from a complete change in the accessible backbone conformational states.

The three structural descriptors provide complementary information on the polymer response. The end-to-end distance describes chain extension, the radius of gyration characterizes the overall spatial distribution of the chain, and the torsional angle distribution describes local backbone conformations. Their combined results indicate that the incorporation of MOF-5 produces a more noticeable change in the average dimensions of the PEO chains, while the overall range of backbone conformations remains largely preserved. These structural changes should be considered together with the ion coordination, ion pairing, and diffusion analyses when assessing their influence on Li⁺ transport.

3.2. Dynamical properties

The structural properties provide evidence that the System D can potentially facilitate lithium transport, which is essential for the performance of SPE. To further establish the potential of this system relative to the other investigated systems, the evaluation of dynamical properties can augment the initial findings obtained from the structural analysis. For this purpose, MSD was calculated to gain insights into the mobility and transport properties of the species within the systems. In general, a linear increase in MSD with time indicates normal diffusive behavior, allowing the calculation of the self-diffusion coefficients via the Einstein relation. Therefore, MSD analysis is widely used to understand ion transport, molecular motion, and dynamics in various materials and complex fluids. To assess the statistical reliability of the diffusion coefficients, the final 50 ns of each 100 ns production trajectory were divided into five consecutive 10 ns blocks. Diffusion coefficients for Li⁺ ions and TFSI⁻ anions were calculated independently for each block from the slope of MSD curve in the linear regime. The reported diffusion coefficients are presented as the mean ± standard deviation values calculated from the five blocks. The linearity of MSD in the fitting region was quantified using the coefficient of determination (R²), which ranged from 0.966 to 0.998 across all systems, indicating a high degree of linearity in the selected fitting regions. The block-averaged diffusion coefficients, standard deviations, and corresponding R² values for all systems are summarized in Table S15. MSD profiles were generated for both Li⁺ and TFSI⁻ ions to investigate ionic mobility in these hybrid electrolyte systems. The MSD curves reveal distinct mobility trends across the different systems. Fig. 8a shows that the mobility of Li⁺ ions follows the order System D > System C > System A > System B > Reference system, indicating that Systems D facilitates enhanced Li⁺ ion transport compared to the other investigated systems. An identical mobility trend was observed for TFSI⁻ anions, as shown in Fig. 8c, further highlighting the combined influence of polymer type and the presence of MOF-5. The consistently lower MSD slopes observed for the Reference system, which lacks a polymeric component, suggest restricted ion dynamics, most likely due to stronger electrostatic confinement within the MOF-5 pores.

Fig. 8. Mean square displacement (MSD) as function of time lag for (a) Li⁺ ions along with (b) linear fit curve, and for (c) TFSI⁻ anions along with (d) linear fit curve, representing diffusive regime, from which the self-diffusion coefficient D was determined.

Based on the MSD profiles, self-diffusion coefficients (D) were calculated from the linear region of the MSD curves to provide a quantitative measure of ionic mobility. Among all systems, System D exhibits the highest diffusion value with D_Li⁺ = 18.20 × 10⁻⁷ cm² s⁻¹ and D_TFSI⁻= 76.49 × 10⁻⁷ cm² s⁻¹, as shown in Table 2. This enhanced ionic transport in System D can be attributed to the flexible, ether-rich structure of PEO, which enables dynamic coordination with Li⁺ ions and facilitates their efficient transport. The inclusion of MOF-5 likely introduces interconnected nanoscale channels and reduces local crystallinity within the polymer matrix, thereby lowering the barriers to ion diffusion. The diffusion coefficients of Li⁺ and TFSI⁻ obtained from the simulations are reported in units of 10⁻⁷ cm² s⁻¹. Notably, the highest values were observed for the ions in System D, which contains both MOF-5 and the PEO-based polymer electrolyte. Although the systems modeled in the present study differ from those reported previously in terms of polymer chain length, total monomer count, and salt concentration, the diffusion coefficients for Li⁺ ions generally fall within the range of ∼1 × 10⁻⁷ to 1 × 10⁻⁸ cm² s⁻¹, providing a useful benchmark for comparison.^40,45 Furthermore, difference in force field parameters, such as the use of OPLS-AA in the present work compared with OPLS2005 in previous studies, may also contribute to variations in the calculated diffusivity.⁴⁰ Therefore, this comparison is intended to provide qualitative validation of the present results, rather than establish a direct numerical correlation. System B exhibits a moderate diffusion coefficient for Li⁺ ions. This observation is consistent with the lower polarity of PVDF and the more confined, yet organized, transport environment provided by MOF-5. In contrast, the Reference system, which lacks a polymeric component, exhibits the lowest Li⁺ mobility, likely due to strong Li⁺–TFSI⁻ ion pairing and restricted motion within the MOF-5 pores. In all systems, the diffusion of TFSI⁻ anions is faster than of Li⁺ ions. This result is consistent with previous experimental and computational studies reported in the literature.⁷² The higher mobility of TFSI⁻ anions can be attributed to the significantly stronger interactions between Li⁺ ions and the oxygen atoms of the PEO polymer, whereas TFSI⁻ anions interact more weakly with its surrounding environment. Consequently, TFSI⁻ anions diffuse more freely than Li⁺ ions. In PVDF-based systems, Li⁺ ions do not interact strongly with TFSI⁻ anions, indicating that PVDF play limited role in Li⁺ solvation compared with PEO.⁷³ This explains why Systems C and D exhibit higher diffusion coefficients than Systems A and B, even after the incorporation of MOF-5. Nevertheless, the addition of MOF-5 to PVDF-based electrolytes still contributes to a moderate enhancement in ionic diffusivity.

Table 2. Self-diffusion coefficients of Li⁺ and TFSI⁻ ions (10⁻⁷ cm² s⁻¹).

Systems	Li+ ions	TFSI− anions
Reference	8.4261	23.798
System A	13.8311	42.050
System B	13.8303	42.694
System C	10.8650	48.020
System D	18.2009	76.486

The higher Li⁺ ion diffusion observed in Systems C and D can also be attributed to the combined effects of Li⁺ coordination, ion association, and polymer dynamics. PEO contains regularly distributed ether oxygen sites that provide favorable coordination environments for Li⁺ and allow exchange between neighboring coordination sites. In the present simulations, the PEO-based systems show higher Li⁺ mobility together with weaker and more dynamic Li⁺–TFSI⁻ association, suggesting that Li⁺ ions can more readily exchange between TFSI⁻ anions and PEO coordination sites. This interpretation is also consistent with recent computational studies showing that Li⁺ coordination with PEO oxygen atoms and ion association strongly influence Li⁺ transport in PEO–LiTFSI electrolytes.^74,75

In contrast, PVDF lacks the regularly distributed ether oxygen sites as present in PEO and therefore provides a different chemical environment for Li⁺ transport. The lower Li⁺ diffusion observed in the PVDF-based systems, together with stronger Li⁺–TFSI⁻ association, suggests less frequent exchange between Li⁺ coordination environments. This agrees with recent all-atom simulations of PVDF-based electrolytes, which showed that the local chemical environment and ion dissociation strongly affect Li⁺ distribution and transport.⁷⁶ Experimental studies of PEO–LiTFSI have also directly measured lithium self-diffusion together with ionic conductivity and polymer dynamics, supporting the connection between polymer motion, local ion environment, and Li⁺ transport.⁷⁷ The higher Li⁺ diffusion in the PEO systems is therefore attributed to the favorable ether oxygen coordination environment, more dynamic Li⁺–TFSI⁻ association, and greater flexibility of the PEO matrix rather than to ion pairing alone.

3.3. Thermodynamic properties

PMF profiles were generated for different interacting species based on the RDFs and were used to estimate the thermodynamics of Li⁺ interactions with other components of the systems in terms of free energy. The PMFs obtained for the Li⁺–F_PVDF and Li⁺–O_PEO interactions are illustrated in Fig. 9. For the Li⁺–F_PVDF interactions in Systems A and B, the corresponding free-energy values were calculated to be −0.12 and −0.20 kJ mol⁻¹, respectively, as deduced from the minima in the PMF profiles (Fig. 9a). These values indicate stronger attractive interactions between Li⁺ ions and the PVDF matrix in the presence of MOF-5 (System B) than System A, which does not contain MOF-5. This observation clearly demonstrates that MOF-5 enhances the stability of Li⁺–polymer interactions by improving the local structural organization of the system. A similar trend was observed for the Li⁺–O_PEO interactions, which exhibited free energy values of −7.91 and −8.07 kJ mol⁻¹ for Systems C and D, respectively (Fig. 9b). The more negative free energy value observed for System D indicates significantly stronger interactions between Li⁺ ions and the ether oxygen atoms of PEO, which can be attributed to the high electron-donating ability of the ether oxygen atoms, facilitating Li⁺ coordination with the polymer. The stronger Li⁺–O_PEO interactions also suggest enhanced ionic dissociation of LiTFSI and improved stabilization of Li⁺ ions within the polymer matrix. Furthermore, the PMF results corroborate the RDF findings, indicating that PEO may contribute to superior ionic transport properties in PEO-based systems compared to PVDF-based electrolytes.

Fig. 9. Potential of mean force (PMF) profile as a function of (a) Li⁺–F_PVDF and (b) Li⁺–O_PEO radial distance.

The present PMF results are also consistent with recent computational studies showing that the chemical nature of the polymer strongly affects the local Li⁺ solvation environment. Recent all-atom MD simulations of PVDF-based LiFSI electrolytes demonstrated that direct Li⁺ interaction with the PVDF matrix is relatively limited, while the surrounding electrolyte species play an important role in Li⁺ solvation.⁷⁶ Similarly, recent ab initio MD simulations of PVDF-based electrolytes showed that Li⁺ ions weakly interact with the PVDF matrix, with the Li⁺ solvation environment being mainly determined by the electrolyte species.⁷³ In contrast, computational and experimental studies of PEO-based LiTFSI electrolytes have established the important role of ether oxygen atoms in Li⁺ coordination and solvation.⁷⁸ Thus, although the absolute PMF values obtained here should not be directly compared with values from studies using different models and reaction coordinates, the stronger Li⁺ interaction with PEO than with PVDF is qualitatively consistent with the reported molecular level differences in Li⁺ coordination within these two polymer environments.

WT-MD simulations were performed to investigate the interactions of Li⁺ ions with different site in MOF-5 and to examine their behavior within the framework in different electrolyte systems. For this purpose, the collective variable (CV) was defined as the distance between Li⁺ ions and MOF-5_COM. This CV was chosen because it directly describes the position and migration behavior of Li⁺ ions relative to the porous MOF-5 structure. The Li⁺–MOF-5_COM distance serves as an important reaction coordinate, as it enables monitoring of Li⁺ localization near MOF-5 as well as its migration toward the polymer matrix and TFSI⁻ anions. Distance-based CVs between ions or molecular species and the center of mass of a surface or framework are well established for investigating adsorption, ion association, and free energy profiles in heterogeneous and porous environments.^79,80 Smaller Li⁺–MOF-5_COM distances indicate stronger confinement of Li⁺ ions within the MOF-5 structure, whereas larger distances suggest that Li⁺ ions move away from MOF-5 and interact more strongly with the surrounding polymer chains or anions. Therefore, this CV provides valuable insight regarding Li⁺ transport pathways, adsorption behavior, and the influence of polymer matrices on ion mobility. The free energy surfaces obtained from WT-MD simulations revealed multiple minima and maxima along the Li⁺–MOF-5_COM distance coordinate, indicating the presence of several energetically favorable and unfavorable regions for Li⁺ ion transport.

In addition, the Li⁺–MOF-5_COM distance was selected as the CV because the primary objective of the metadynamics simulations was to investigate the free energy associated with Li⁺ approach to and association with MOF-5. This distance-based CV provides a direct measure of the Li⁺-MOF-5 separation. Nevertheless, it represents a one dimensional projection of the underlying free energy landscape and does not explicitly capture other coupled molecular degrees of freedom. Convergence of WT-MD simulations was assessed by comparing the reconstructed free energy profiles at different simulation times. The profiles showed progressively smaller changes during the later stages of the simulations, indicating stabilization of the calculated free energy profiles (Fig. S16).

In Reference system comprising only MOF-5 and LiTFSI salt, free energy minima and maxima were observed at distances of 15.96, 16.56, 19.41, 20.93, and 25.10 Å, with corresponding free-energy values of 6.73, 6.78, 4.87, 5.79, and 1.29 kJ mol⁻¹. The presence of minima at relatively larger Li⁺–MOF-5_COM distances, as illustrated in Fig. 10, suggests that Li⁺ ions are distributed around MOF-5 without strong confinement at shorter distances. The comparatively moderate free energy barriers indicate that Li⁺ ions experience limited stabilization near MOF-5 and can migrate between different coordination environments with moderate resistance. Since no polymer component is present in the Reference system, the behavior of Li⁺ ions is mainly governed by interactions with MOF-5 and TFSI⁻ anions. In System B, several minima and maxima were observed at distances of approximately 2.31, 2.67, 3.30, 3.58, 7.51, 8.14, 11.01, 11.92, 13.05, 17.54, 17.82, 23.22, 23.43, 25.74, and 25.95 Å, with corresponding free energy values of 12.17, 12.19, 12.05, 12.06, 7.50, 8.13, 0.26, 0.50, 0.24, 4.08, 4.08, 9.57, 9.55, 11.18 and 11.17 kJ mol⁻¹. The appearance of minima at shorter Li⁺–MOF-5_COM distances indicates that Li⁺ ions can approach MOF-5 more closely in the presence of PVDF. Accordingly, the relatively high free energy values in certain regions suggest the existence of stronger energetic barriers for Li⁺ migration. The broader distribution of minima along the reaction coordinate reflects multiple Li⁺ coordination environments arising from competition among Li⁺–MOF-5 interactions, Li⁺–TFSI⁻ ion pairing, and polymer effects. Since PVDF possesses relatively weak coordinating ability toward Li⁺ ions, the polymer primarily acts as a passive matrix, allowing Li⁺ ions to interact more strongly with MOF-5 and TFSI⁻ anions. The very low free energy minima observed at larger distances indicate energetically stable Li⁺ configurations in which ions are partially dissociated and distributed throughout the electrolyte system. In case of System D, minima and maxima were located at approximately 6.8, 7.79, 13.4, 13.61, 16.55, 17.61, 19.29, 19.71, 20.97 and 24.55 Å along the free energy profile, corresponding to free energy values of 9.67, 9.88, 3.56, 3.57, 1.90, 2.17, 1.48, 1.49, 2.05 and 0.4 kJ mol⁻¹. Compared with the PVDF-based system, the PEO-based electrolyte exhibits lower free energy barriers and more energetically favorable regions for Li⁺ transport. This behavior can be attributed to the strong coordinating ability of the ether oxygen atoms in PEO chains, which effectively interact with Li⁺ ions and facilitate ion mobility throughout the polymer matrix. The lower free energy regions observed in this system suggest easier Li⁺ hopping between MOF-5 coordination sites and reduced resistance for ion migration. Interactions between Li⁺ ions and PEO chains weaken Li⁺–TFSI⁻ ion pairing and promote greater ion dissociation, which is beneficial for enhanced ionic conductivity in the MOF-5-containing system. This interpretation is consistent with previous MD simulation studies showing that Li⁺ transport in PEO-based electrolytes involves dynamic exchange between ether oxygen coordination sites and anion coordination environments.^65,66

Fig. 10. Free energy landscape (FEL) of Li⁺ ion diffusion within the MOF-5 framework, obtained from WT-MD simulations in the (a) reference system, (b) System B and (c) System D; the free energy profile (ΔG) is plotted as a function of the chosen collective variable (CV), based on the distance between the Li⁺ ion and MOF-5_COM.

The distance-time analysis between Li⁺ ions and MOF-5_COM were also obtained from WT-MD simulations, which provide important insights into the strength of Li⁺ interactions and mobility within different polymer electrolyte systems. The mean distance were calculated to be 23.82 ± 2.99, 13.38 ± 2.40, and 19.94 ± 3.01 Å for the Reference, System B and System D, respectively (Fig. 11 and Table 3). In the Reference system, the largest mean distance indicates relatively weak coordination between Li⁺ ions and MOF-5. This observation demonstrates the effect of polymer absence, as Li⁺ ions are more freely distributed throughout the system and experience reduced confinement near the MOF-5 surface. The standard deviation of 2.99 Å suggests consistent ionic fluctuations during the simulations. This behavior may arise from weaker electrostatic attractions between Li⁺ ions and MOF-5 atomic sites, while TFSI⁻ anions also contribute to ion dissociation and enhanced ionic mobility. For System B, the Li⁺–MOF-5 distance decreases significantly, indicating stronger localization of Li⁺ ions near MOF-5. PVDF contains highly electronegative fluorine atoms that generate strong dipolar interactions within the electrolyte environment. These interactions can promote Li⁺ ion association near the MOF-5 pores. The smaller standard deviation of 2.40 Å indicates restricted motion of Li⁺ ions and a more stable coordination environment. Such reduced fluctuations imply that Li⁺ ions remain strongly trapped or coordinated within specific regions, leading to comparatively lower ion mobility. This observation is chemically consistent with the semi-crystalline nature of PVDF, where limited segmental motion can hinder long-range Li⁺ ion transport.⁶ In the case of System D, the mean distance lies between those of the Reference system and System B, while the relatively large standard deviation suggests greater fluctuations in the position of Li⁺ ions relative to MOF-5. This behavior reflects the highly dynamic nature of the PEO matrix, as the ether oxygen atoms strongly coordinate with Li⁺ ions and facilitate continuous ion hopping along the polymer chains through segmental motion. As a result, Li⁺ ions repeatedly move toward and away from the MOF-5 surface during the simulations, producing larger positional variations. These increased fluctuations also suggest enhanced ion mobility and more efficient Li⁺ transport pathways within the PEO-based electrolyte. The distance analysis also demonstrates that the nature of the polymer environment strongly influences Li⁺ ion distribution and mobility near the framework. The PVDF-based system exhibits stronger Li⁺ confinement and reduced fluctuations, whereas the PEO-based system displays greater dynamic behavior and enhanced ion transport due to polymer chain flexibility and ether oxygen coordination. In contrast, the Reference system shows the weakest confinement effect, resulting in the largest mean Li⁺–MOF-5 separation.

Fig. 11. Distance profiles as a function of simulation time for the (a) reference system, (b) System B and (c) System D; the black line shows the mean value obtained from three independent simulation replicas, whereas shaded regions correspond to standard deviations.

Table 3. Mean Li⁺–MOF-5 distances along with their standard deviation in different MOF-based systems.

System	Mean distance (Å)	Average std. dev. (Å)
Reference	23.82	2.99
System B	13.38	2.40
System D	19.94	3.01

The WT-MD results, therefore provide a consistent molecular picture in which the polymer chemistry influences how Li⁺ ions interact with and move relative to MOF-5. The stronger localization observed in the PVDF-based system contrasts with the more dynamic Li⁺ redistribution observed in the PEO-based system. This interpretation is consistent with previous computational studies of PVDF- and PEO-based electrolytes, which have reported stronger and more dynamic Li⁺ coordination with ether oxygen atoms of PEO than with the PVDF backbone.^65,73,76 Experimental studies of PEO-based lithium salt electrolytes also support the importance of ether oxygen coordination in determining the local Li⁺ environment and its dynamics.⁶⁷ Thus, the present WT-MD results are in qualitative agreement with the molecular level picture established by both simulation and experimental studies. However, the CV for the Li⁺–MOF-5_COM distance describes radial association with the framework and should be considered together with the three dimensional density and trajectory analyses rather than as a direct determination of a single Li⁺ migration pathway.

4. Conclusion

The simulation results provide molecular level insights into the role of MOF-5 in regulating Li⁺ transport in PVDF- and PEO-based LiTFSI electrolytes. The combined use of cMD and WT-MD simulations enabled a systematic investigation of the polymer environment, ion pairing, polymer-ion coordination, MOF-5 interactions, and energetic aspects of Li⁺ migration. The results also reveal that Li⁺ transport is not only controlled by the incorporation of MOF-5 or Li⁺–MOF-5 interactions, but also by the interplay between polymer-mediated solvation, ion pairing, the framework interactions, and the availability of energetically favorable migration pathways. Among the investigated systems, System D containing PEO, MOF-5, and LiTFSI showed the most favorable overall transport properties. The lower Li⁺–TFSI⁻ association, more uniform Li⁺ distribution, and enhanced Li⁺ mobility suggest that PEO provides a favorable environment for ion transport. The relatively fewer Li⁺–MOF-5 contacts in System D further suggest that Li⁺ ions are not strongly confined to the framework and can dynamically exchange between PEO coordination sites and their surrounding environment. Meanwhile, the interaction of TFSI⁻ with MOF-5 may help maintain the framework environment without imposing strong restrictions on Li⁺ movement. The free energy results provide further support for this behavior, as System D exhibits accessible free energy minima and relatively low barriers for Li⁺ migration, indicating energetically favorable pathways for ion movement.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data supporting this study are provided within the article and its supplementary information (SI) files. Supplementary information is available. See DOI: https://doi.org/10.1039/d6ra06179f.