Thermodynamic driving forces in contact electrification between polymeric materials

Contact electrification, or contact charging, refers to the process of static charge accumulation after rubbing, or even simple touching, of two materials. Despite its relevance in static electricity, various natural phenomena, and numerous technologies, contact charging remains poorly understood. For insulating materials, even the species of charge carrier may be unknown, and the direction of charge-transfer lacks firm molecular-level explanation. Here, we use all-atom molecular dynamics simulations to investigate whether thermodynamics can explain contact charging between insulating polymers. Based on prior work suggesting that water-ions, such as hydronium and hydroxide ions, are potential charge carriers, we predict preferred directions of charge-transfer between polymer surfaces according to the free energy of water-ions within water droplets on such surfaces. Broad agreement between our predictions and experimental triboelectric series indicate that thermodynamically driven ion-transfer likely influences contact charging of polymers. Furthermore, simulation analyses reveal how specific interactions of water and water-ions proximate to the polymer-water interface explain observed trends. This study establishes relevance of thermodynamic driving forces in contact charging of insulators with new evidence informed by molecular-level interactions. These insights have direct implications for future mechanistic studies and applications of contact charging involving polymeric materials.


Introduction
Contact electrification, or contact charging, is a widely observed phenomenon that results in static charges present on materials based on their touching (1,2,3,4,5,6,7).In nature, such charging manifests in dust storms, which generate substantial charge via collisions of sand particles (8,9), and in ash plumes of volcanic eruptions, which accumulate and release charge in the form of volcanic lightning (10).In modern technology, contact charging enables xerographic printing (11,12) and energy generation in wearable devices (13,14).Undesirable charging also underlies issues in several industrial applications (15,16), such as wall-sheeting in reactors (17), disruption of particle mixing (18) and hazardous electrostatic discharge (19).Despite this prevalence, precisely how and why contact charging occurs in many scenarios remains ambiguous.Therefore, understanding contact charging is of interest to advance fundamental science and to enhance technological processes (20,21,22).
The mechanism of contact charging strongly depends on the nature of the charge carriers, the materials, and the environment.Three modes of charging include electron transfer (23,24,6,25) wherein surface work functions direct charge transfer, ion transfer (3,26) wherein intrinsic or acquired mobile ions transfer between materials, and material transfer (27) wherein charged pieces of material physically move between surfaces.While electron transfer dominates charging of metals (2) and semiconductors with small band-gaps, the presence of insulating layers atop materials can obfuscate understanding predicated solely on work functions (7).Moreover, contact charging of insulating materials themselves, such as polymers (28,29), likely requires other charge-carrier species.One compelling hypothesis is that unequal transfer of cations and anions between materials results in sustained, asymmetric charge accumulation on surfaces (3).This mode requires that materials must either natively possess or otherwise acquire mobile ions, raising questions as to what ions are present.
Water-ions-hydronium (H 3 O + ) and hydroxide (OH − )-are viewed as potential chargecarriers underlying contact charging of insulating materials (3,30).Water is almost ubiquitously present, in real-world and experimental systems alike, having been detected across diverse chemical surfaces and a broad range of conditions (31,32,33,34,35,36,38,37). Mosaic patterns of charge on polymer surfaces following contact have been attributed to the presence of water patches (39), as water has been observed to only partially cover surfaces, forming patches or islands (38,37).Effects of relative humidity on electrostatic charging highlight a potential role of water and its ions (30,37,28,40) as do numerous studies related charging phenomena directly at liquid-solid interfaces (41,42,43,44,45).Furthermore, there are existing correlations between water-related properties and contact charging of polymers, such as acid/base dissociation constants (46), Lewis acidity or basicity of polymers (47), and zeta potentials of non-ionic polymers (48,3).While such work establishes a potential role of water and associated ions in many circumstances, why water-ions should concentrate on a certain material after contact with another is unclear.Various theoretical and conceptual frameworks have been constructed to explain waterion transfer as a mechanism for contact charging of polymers.For example, a lattice model introduced by Grosjean et al. (50) quantitatively accounts for mesoscale spatial correlations that might explain contact charging between polymer surfaces of the same chemistry.Jaeger and coworkers examined the role of surface hydrophilicity on charging, finding consistency with models premised on OH − diffusion between adsorbed water patches with asymmetric coverage on the contacting surfaces (51,33).Nevertheless, these models generally lack nanoscopic attributions to specific molecular-level underpinnings.Although molecular simulation techniques, such as density-functional theory and ab initio molecular dynamics, have been deployed to unravel complex nanoscale phenomena of contact charging in systems comprised of crystalline minerals, MXenes, oligomers, and water (26,52,53,54,55), studies involving polymers are nascent.
In this study, we employ molecular dynamics (MD) simulations to investigate whether thermodynamic driving forces for water-ion transfer can feasibly impact contact charging of insulating polymers.We hypothesize that polymer surfaces present distinct nanoenvironments for water molecules and water-ions that result in chemical-potential differences, which govern asymmetric transfer of ions between surfaces upon contact.To test this hypothesis, we utilize thermodynamic integration (56) to extract relative free energies of H 3 O + and OH − on polymers of varying hydrophilicity (57).These free energies, which are sensitive to polymer chemistry and underlying molecular interactions, provide a basis to predict the direction of ion-transfer between polymer surfaces.Such predictions enable construction of a triboelectric series based entirely on thermodynamic driving forces, which intriguingly illustrates good agreement with experimental triboelectric series.Further simulations that directly probe ion partitioning between two surfaces illustrate similar trends.This consistency establishes the viability of thermodynamically driven waterion transfer in contact charging of polymers.Furthermore, the methodology highlights molecular-level nuances that may hold other implications for contact charging and general understanding of water-polymer interfacial interactions.

Hypothesis of thermodynamically driven water-ion transfer
The possibility of contact charging as a process driven by the relative ion-surface affinities has been considered since at least the 1950s (58), although molecular evidence is scarce.Here, we consider whether the free energies of H 3 O + and OH − within droplets on different polymer surfaces (Fig. 1a) are predictive of contact charging (Fig. 1b).The posited mechanism of charging is that (i) water droplets on surfaces contain H 3 O + and OH − with chemical potentials that depend principally on surface chemistry but also other factors (e.g., preexisting ion concentration, humidity, electric fields, etc.), (ii) water-ions can diffuse between surfaces when they are sufficiently close, and (iii) the relative abundance of water-ions on two surfaces following diffusion events is biased by the relative chemical potentials.Here, water ions may arise from ambient water, as suggested by previous experimental studies (59,60), but all calculations are agnostic to their precise origin.
Fig. 1b reflects a difference in relative partitioning between surfaces of the ions.As such, contact charging can arise even if both ions favor the same surface given disparity in transfer free energies.Consequently, ∆F +− AB predicts the direction of charge-transfer between contacting surfaces if the charge-carrier species are H 3 O + and/or OH − and populations are thermodynamically controlled and charge-transfer events are independent.
The hypothesis is evaluated by comparison to experimental triboelectric series, which organize materials according to their relative propensity to acquire charges during contact charging (3).Conventionally, triboelectric series are represented in a one-dimensional progression based on relative propensity to acquire positive/negative charge, although results do not always neatly and consistently organize in this manner.We reference three previously reported triboelectric series that feature the polymers in this study as 'S1' (3), 'S2' (64), and 'S3' (62).These three series provide relatively consistent expectations, although there are some differences and/or omissions.In S1, the ordering, from more positive to negative, is N66, PVA, PMMA, PE, PVC, PTFE.In S2 and S3, PVA is absent, the positions of PVC and PTFE are switched in S2, and the positions of N66 and PMMA are switched in S3.Less complete polymer triboelectric series can be formulated from elsewhere and display overall similar trends (see SI Appendix Fig. S1).AB values obtained from MD simulations.To first order, the matrix is organized by material (6 × 6 matrix), and results are further resolved for each A-B into a 5 × 5 sub-matrix based on water-droplet size; color intensity reflects the magnitude of thermodynamic driving force.Compared to experimental triboelectric series (SI Appendix, Fig. S1), the simulation results broadly align with the direction of charging observed in S1, S2, and S3.In comparison to S1, simulation predictions agree with nine of fifteen material combinations, while three pairs yield inconclusive results or depend on droplet size, and three pairs exhibit opposite trends.However, when compared to S2 and S3 (which lack data for PVA), the agreement improves, as simulations predict PVC acquires negative charge over PTFE (as in S2) and N66 acquires negative charge over PMMA (as in S3).Thus, the thermodynamically informed predictions capture general trends in contact charging between polymers of different chemistry.

Consistency of free-energy trends and contact charging
The few disparities between simulation predictions and empirical charging results arise in material pairings that also demonstrate experimental variability.For PVC-PTFE, S1 and S3 (and other series, see SI Appendix Fig. S1) suggest that PTFE exhibits a strong tendency to acquire negative charge.However, our previous study on polymer hydrophobicity (57) indicates that water structuring and dynamics are relatively more similar between PTFE and PE than with PVC.These prior observations align with our current free-energy results, showing a vanishing ∆F +− AB for PE-PTFE and consistent behavior between PE-PVC and PTFE-PVC, and the experimental outcome reported via S2.Consequently, results involving PTFE may be sensitive to experimental conditions, potentially related to mechanisms not captured by simulations, such as the presence of acid and base groups post polymerization, bond breaking (26), or minor inaccuracies in molecular models.For N66-PMMA, S1 and S3 differ, with the latter aligning with the thermodynamic predictions.Lastly, several inconsistent or inconclusive combinations involve PVA; the aqueous solubility of PVA poses an experimental challenge and is also a notable factor in our previous study (57).Considering the substantial agreement for many material pairings and the technical challenges encountered with others, we conclude that thermodynamically driven water-ion transfer can plausibly influence polymer-polymer contact charging.

Role of water-surface interactions
Analysis of the polymer-water interface provides nanoscale insights into the trends of water-ion free energies.To first order, we note general correlation with metrics of polymer hydrophobcity (57).Overall, hydrogen-bonding polymers (PMMA, N66, PVA) tend to acquire positive charge more easily than non-hydrogen-bonding polymers (PE, PTFE, PVC).Furthermore, within those respective groups, increasing hydrophobicity tends to correlate with more positive-charging.To further understand these trends and how they manifest, we examine the ion-water-polymer interactions.Fig. 3a compares how water, H 3 O + , and OH − distribute in the vicinity of chemically distinct, amorphous polymer surfaces.
Relative to OH − , H 3 O + tends to reside closer to the polymer-water interface, orienting its oxygen atom to maximize hydrogen-bond donation to water (SI Appendix, Fig. S2).Surfaces lacking hydrogen bonds, such as PE, PTFE, and PVC, allow easy access for H 3 O + to the interfacial layers, explaining the similar free energy values (F + S ) observed in Fig. 2b.However, H 3 O + is relatively more stable (lower F + S ) in proximity to hydrogenbonding polymers (PMMA, N66, and PVA).The stronger interfacial interactions with PMMA, N66, and PVA also explain the apparent insensitivity of F + S to droplet size (Fig. 2b), as the preferred nanoenvironment of H 3 O + remains relatively consistent as droplet size increases.Notably, H 3 O + is predominantly excluded from the interfacial layer of PVA, the most hydrophilic polymer, aligning with its higher F + S compared to PMMA and N66.This highlights an intriguing interplay between ion-polymer interactions and competing water interactions, such that ion chemical potential is not a monotonic function of hydrophilicity.
Although OH − predominantly situates in secondary interfacial layers or the bulk of water droplets, its trends also correlate with hydrophobicity and hydrogen-bonding behavior.The nearly equivalent F − S between PE and PTFE reflects consistency in OH − distribution, which derives from their similarity in hydrophobicity and contact angles (57).Water-ions are not notably stabilized in PE and PTFE relative to free water droplets (SI Appendix, Fig. S5), likely because PE and PTFE create typical hydrophobic interfaces that do not significantly impact water structure (57).This implies that charging trends of PE and PTFE are mostly dictated by the other polymer in the contact-pair.In other words, the propensity for PTFE to acquire negative charge over N66, for example, is not due to its affinity for OH − but rather the affinity of N66 towards H 3 O + .Free-energy trends among N66, PVA, and PMMA align with hydrogen-bonding behavior.While N66 and PVA offer stabilizing interactions that lower F − S , PMMA only functions as a hydrogen-bond acceptor, disturbing the hydrogen-bonding network of OH − (Fig. 3b) and effectively excluding OH − from the interfacial layer of water, resulting in higher F − S (57).In contrast to PMMA, water in proximity to PVC orients its oxygen atoms towards the surface because of the strong attraction of chlorine atoms (57), which allows water molecules to readily form hydrogen bonds with OH − in the second water layer (Fig. 3b).Thus, distinct nanoenvironments for H 3 O + and OH − arise from the hydrophobicity and hydrogen-bonding behavior of the polymer surfaces, largely explaining trends in F + S and F − S .To further explore the sensitivity of F + S and F − S to interfacial interactions, we assess the role of nanoscale polymer surface morphology, which can influence hydrophobicity and hydrogen-bonding behaviors.Fig. 3c shows the difference in F + S and F − S between amorphous and crystalline surfaces (for PE, PVC, and N66) and between atactic and isotactic amorphous surfaces (for PVA).Overall, the simulations capture some sensitivity of F + S and F − S to surface morphology, but the extent depends on polymer chemistry.The transition from PE to PE * has no notable effect, as water structuring near PE * remains similar to that of PE, resulting in nearly equivalent nanoenvironments for H 3 O + and OH − and correspondingly indistinguishable free energies.However, for PVA, PVC, and N66, F + S or F − S can shift on scales relevant for charging predictions in Fig. 2a.Increased intra-chain hydrogen bonding and reduced hydrogen bonding with water for PVA † (57) permits more favorable water-structuring around OH − , thereby increasing its stability.In N66 * , the crystalline structure similarly reduces hydrogen bonding with water and results in a more hydrophobic surface, creating a less favorable nanoenvironment for H 3 O + within the interfacial layer.In PVC * , enhanced chain interactions diminish interfacial water structuring, subsequently weakening interactions with OH − in secondary water layers.These findings underscore the importance of polymer-water interactions in water-ion free energies and indicate how surface heterogeneities and semicrystallinity may subtly influence water-ion transfer and contact charging.

Connections to other charging phenomena
Although thermodynamic driving forces for ion transfer are most significant when considering different surfaces, Fig. 2b shows that the free energy of water-ions is also influenced by droplet size, and Fig. 3c illustrate sensitivty to surface heterogeneities.The former effect is evident in the internal color variation within the diagonal material squares in Fig. 2a.Notably, for more hydrophilic polymers (PMMA, N66, and PVA), the thermodynamic driving forces are comparable to those for chemically distinct surfaces (off-diagonal squares of Fig. 2a); Fig. 3c also conveys non-trivial differences that exceed 5 kcal/mol.These findings may have implications for contact charging of chemically identical materials (66).If water exists on polymer surfaces as droplets of varying sizes (37) or the surfaces vary in crystallinity/patterning, these results suggest that those variabilities could create additional thermodynamic driving forces for ion redistribution and subsequent contact charging.Considering that relative humidity likely influences the distribution of droplet sizes on a surface, resulting differences in water-ion chemical potentials might account for certain humidity effects on contact charging.It is notable that the free energy of H 3 O + appears less sensitive to droplet size compared to OH − , particularly for hydrophilic polymers.In addition, as polymer surfaces become increasingly wet, we anticipate that any thermodynamic driving force for ion-transfer between surfaces will diminish since the contribution of the water-polymer interface will comprise an overall lesser fraction of the total ion free energy; such an effect could relate to observations of decreased contact charging at high relative humidity.(40) Although the present work does not thoroughly analyze the implications of droplet or surface heterogeneities or the precise connection between droplet size and humidity, such factors could be considered in future work.

Validation by two-surface simulations
In the preceding analysis, calculating ∆F +− AB involved simulating a water droplet containing a single ion above isolated polymer surfaces.As a more stringent test of these predictions, we conduct simulations with both H 3 O + and OH − present between distinct polymer surfaces and assess preferential partitioning.Fig. 4a illustrates the simulation setup wherein a water bridge (N w = 4000) containing a H 3 O + /OH − pair forms between surfaces A (top) and B (bottom) separated by distance d.The propensity for surfaces to acquire specific charges is measured via the free energy F AB (p z ) where the collective variable p z = z H 3 O + − z OH − is the dipole of the ionic pair in z-direction.As a collective variable, p z reports the relative positioning of water ions with respect to the two surfaces: more positive p z indicates H 3 O + is closer to surface A and OH − is closer to B, more negative p z indicates the opposite, and small p z suggests little to no asymmetric affinity.Similar to ∆F +− AB , we examine the change in free energy when the dipole is flipped: AB represents a pseudo-equilibrium constant for the exchange process scenarios for K AB (p z ) are depicted in Fig. 4b.For example, if K AB (p z ) > 1, H 3 O + should preferentially partition towards A, with the expectation that A becomes relatively positive and B negative.The free energy F AB (p z ) is computed using umbrella sampling and the weighted histogram analysis method (67); further details regarding the calculation and formulation of K AB (p z ) are in 'Methods.'Results of the two-surface simulations align well with the expectations from ∆F +− AB (Fig. 2a) and the structural analysis (Fig. 3).Fig. 4b displays K AB (p z ) for different pairs of materials, with row labels corresponding to surface A and column labels corresponding to surface B. For PE-PTFE, K AB (p z ) ∼ 1, which is consistent with prior discussion on the similarity of water/ion nanoenvironments.In PVA-PTFE and PVA-PE, for which results from single-surface calculations (Fig. 2b) were mixed and dependent on droplet size, K AB (p z ) < 1 indicating that OH − prefers PVA over the more hydrophobic PTFE and PE.This preference arises mainly from the recruitment of water towards the more hydrophilic surface (SI Appendix, Fig. S4) rather than surface-specific interactions.The remaining pairs yield K AB (p z ) > 1, indicating enhanced thermodynamic stability of H 3 O + closer to surface A (row) and for OH − to be closer to B (column) than the reverse situation.Thus, the two-surface simulations provide valuable validation for the overall thermodynamic framework and offer more direct support of thermodynamically driven water-ion transfer as a mechanism of contact charging.

Conclusions
Molecular dynamics simulations were used to investigate thermodynamically driven waterion transfer as a mechanism of contact charging between insulating polymers.The ubiquity of water, correlations with hydrophobicity, and importance of humidity inform a specific hypothesis: distinct nanoenvironments for water proximate to polymer surfaces generate chemical-potential gradients that govern asymmetric transfer of water-ions upon contact (Fig. 1a,b).To investigate this hypothesis, we calculated free energies of waterions in water droplets on chemically and structurally distinct polymer surfaces; these were subsequently used to predict the thermodynamically preferred direction of contact charging between various commodity polymers (Figs.2a and 3c).Despite the simplicity of the calculations, which technically relate to the first ion-transfer event on a pristine surface and ignore kinetic factors, the predictions align remarkably well with many results of experimental triboelectric series (Fig. S1).Subsequent simulations that directly examine partitioning of H 3 O + and OH − between two surfaces offer further support (Fig. 4).The molecular resolution afforded by the simulations importantly reveals key interactions and properties, such as surface hydrophobicity and hydrogen-bonding capabilities, that underlie relative affinities of ions to specific surfaces (Figs.3a,b).While other contactcharging mechanisms should not be disregarded, these results emphasize the plausibility of thermodynamic driving forces with well-defined molecular underpinnings in contact charging between insulating materials, such as polymers.
The findings offer valuable insights into the complex phenomenon of contact electrification and highlight opportunities to explore further implications across scientific and technological domains.Coupling molecular simulation with free-energy calculations can be extended to explore other aspects of contact charging, including the role of humidity (26,36,68,40), temperature (33), external electric fields, (36) ion correlations, and local geometry (65,40).Additionally, there are potential implications for contact charging between chemically identical materials, particularly regarding variations in free-energy due to differences in droplet sizes and surface morphology, though further investigation is required to ascertain their precise relevance.Moreover, future study could explore kinetic factors like asymmetric ion diffusion (51) and their interplay with thermodynamic considerations, such as ion distribution within a droplet or free-energy barriers formed during material contact.These kinetic factors could influence results based on the choice of reference probe materials and would not be explainable within a thermodynamic framework.Lastly, molecular simulations of the kind used here can provide chemically specific parameters for macroscopic models of contact charging, enabling quantitative comparisons with experiments and enhanced understanding.

Molecular Dynamics Simulations
All MD simulations were conducted using the LAMMPS simulation package (version 3, Mar 2020) (69).Polymers were described by parameters from the all-atom Optimized Potentials for Liquid Simulations (OPLS-AA) force field (70, 71), while water was described using the extended simple point charge model (SPC/E) (72,73).The water ions were modeled using a non-polarizable force-field designed to be used in conjunction with the SPC/E water model and parameterized to reproduce experimental solvation free energies and activities of H 3 O + -Cl − and Na + -OH − salt solutions (74).Preparation of polymerwater systems followed methodology of our previous work (57), with the addition of either H 3 O + or OH − at the center-of-mass of the water droplet as required.PTFE is terminated with a trifluoromethyl group, while all other polymers are terminated with methyl groups.We note that the polymer structures are idealized in the sense that they do not reflect realistic synthetic procedures, which may result in branching structures, cross-linking, and acidic/basic terminal end groups.Simulation cells were periodic in the x and y directions but non-periodic in z; Ewald summation was accomplished via the approach of Yeh and Berkowitz (75) with extension to non-neutral systems by Ballenegger et al. (76).After initial preparation, systems were simulated for 20 ns to generate initial configurations.Subsequently, trajectories of 40 ns were run to analyze the distribution of ions and water near polymer interfaces.More detailed information regarding simulation procedures and calculations are provided in the SI.

Single-surface Free-energy Calculations
The free energy associated with adding an ion of type α to a water droplet on surface S, F α S , was calculated using thermodynamic integration (TI).TI was practically implemented using 12-point Gauss-Legendre quadrature for each ion, following the approach of Ref. 57, which calculates the excess chemical potential of water.Simulations at each quadrature node started from the final configuration of the 20-ns equilibration trajectory.Each simulation was run for 6 ns, of which the last 5 ns were used to estimate ensemble averages.

Two-surface Free-energy Calculations
The free energy as a function of ionic dipole within a water bridge between surfaces A and B, F AB (p z ), was calculated using umbrella sampling with statistical re-weighting via the weighted histogram analysis method (67).Two-surface systems were prepared by combining two previously equilibrated polymer-water systems, mirroring the coordinates of one system across the xy-plane and shifting it vertically by a specified distance d, which was set as the average distance between polymer interfaces.Data was collected across 36 windows that each employ a harmonic biasing potential on p z .The biasing potentials utilized spring constants of 47.8011 kcal/mol and equilibrium positions at -35 to 35 Å in 2 Å increments.To prevent pairing of H 3 O + and OH − at small p z , the force-field interaction between oxygen atoms on H 3 O + and OH − was adjusted to ensure that the two ions would not bind (SI Appendix, Fig. S6).This modification focused analysis on ion affinity for surfaces without conflation from ionic attraction, which was not the primary focus here, and also outside the realm of application of the force-field, which does not describe recombination into neutral water species.Consequently, F AB (p z ) is conditional on the ions remaining separate species.
For all calculations, simulations are first run for 10 ns to equilibrate the surface-watersurface geometry.Biasing potentials were subsequently imposed for each window, and trajectories were run for 15 ns.Trajectories for windows with |p z | < 10 Å were extended for an additional 7.5 ns to enhance convergence.Initially, calculations were performed at d = 25 Å for all surfaces.However, for some pairings (N66-PE, N66-PTFE, N66-PVC, PE-PTFE, PMMA-PTFE, and PVC-PTFE), the resulting F AB (p z ) was relatively flat because ions could readily interact with both surfaces.For these surfaces, additional calculations were conducted at d = 40 Å to better distinguish surface affinities between H 3 O + and OH − ; calculations at greater separations yielded similar results (see SI Appendix, Fig. S3).76.Ballenegger, V., Arnold, A. & Cerdà, J. J. Simulations of non-neutral slab systems with long-range electrostatic interactions in two-dimensional periodic boundary conditions.The Journal of Chemical Physics 131, 094107 (2009).

Comparison to Triboelectric Matrices Sourced from Literature Data
Triboelectric series were collected from ten publications from 1898 to 2022 that featured at least a subset of the six polymers investigated in this work.The reported triboelectric series, which are a common representation of contact charging experiments, were then formulated as a matrix as introduced in the main text.Because many experiments utilize a "probe" material rather than direct material contact, we note that this matrix is an idealistic representation.Nevertheless, in the framework of thermodynamics, even if the charging were completely mediated by a third "probe" material, the equilibrium condition would reflect the same charging pattern as if the target pair had been in direct contact.Fig. S1 summarizes all the data.The figure overall conveys some level of inconsistency in experimental settings even for the same reported materials; however, there are also some broadly conserved trends.For example, most of the top-right corner and bottom-left appear consistently colored for the experimental data, and these trends are further reflected in the predictions made by the free-energy calculations.

Comparison of Ion Distributions in Proximity to PVA
In Fig. 3A, the distribution of H 3 O + and OH − appear similar along the single dimension relative to the polymer interface.However, Fig. S2 illustrates that the two ions do exhibit differences when resolving positioning relative to both the polymer interfaces and water interfaces.In Fig. S2, positioning along the diagonal but away from the origin indicates residence resides toward the center of the droplet, as it is simultaneously distanced from both interfaces.Meanwhile, positioning in a lower horizontal band towards y = 0 suggests the ion is close to a water interface; however, it need not be close to a polymer interface (moving right), implying that the ion is closer to a water-vapor interface.Therefore, the simulations indicate that H 3 O + displays some preference to reside near the water-vapor interface, while OH − is always found in the interior of the water droplet.This behavior is generally preserved across all surfaces.

Distance Analysis of Two-surface Free-energy Calculations
To guide selection of distances for the two-surface free-energy calculations, a set of preliminary simulations at varying distances (d = 15, 25, 40, 55 Å) for PMMA-PVC pairings.PMMA-PVC were selected based on the predictions to acquire the most positive/negative charge from the single-surface thermodynamic integration calculations; the span of distances allowed for the formation of a water bridge between the two surfaces without any direct contact of PMMA and PVC atoms.Fig. S3A provides the corresponding free-energy profiles as a function of ionic dipole.At the small separation of d = 15 Å, preference for either ion to specific surfaces is not clearly evident.At larger separations (d = 25, 40, 55 Å), relative affinities become statistically discernible, with all separations yielding qualitatively similar interpretations.Consequently, a first group of simulations amongst all surface pairs were performed with d = 25 Å (Fig. S3).For a subset of pairs (N66-PE, N66-PTFE, N66-PVC, PE-PTFE, PMMA-PTFE, and PVC-PTFE), relative surface affinities were not obvious at d = 25 Å, and additional simulations were run  S4), which resulted in all water residing on PVA and no stable water bridge.For all polymer pairs, a 10 ns equilibrium simulation was performed.The preliminary simulations used 7.5 ns of simulation in each biasing window.The first group of simulations were run for 15 ns for each biasing window.An additional 7.5 ns simulation are performed for |p z | < 10 Å to get better sampling of the small |p z | region.For pairs that exhibited flatter free-energy profiles, an additional 7.5 ns of simulation were run to assess convergence.The second group of simulations were performed for 15 ns in each biasing window.

Free-energy Calculations of Ion in The Free Water Droplet
To further understand our single free-energy results, we performed calculations involving a free water droplet.The simulation systems are generated by placing a water molecules in a spherical geometry at the center of simulation cell; subsequent simulation procedures follow those of the single-surface free-energy calculations.Fig. S5 shows that the free energy of adding an ion to a free water droplet are statistically indistinguishable from those of adding to droplets on PE and PTFE.This implies that the ions within free water droplets are stabilized to a similar extent as on hydrophobic surfaces.By extension, these results suggest that the free-energy trends of PE and PTFE are primarily induced by chemically specific effects moreso related to the other  (B) Configurational snapshot of water bridge forming between PE and PVA.Both PTFE and PE are substantially more hydrophobic than PVA.Consequently, the majority of water molecules in the system are recruited towards the PVA surface, which affects the accessible volume of ions.Both snapshots are rendered using OVITO (11).The atoms are colored such that carbon is gray, fluorine is green, hydrogen is white, oxygen in water and ion is red, and oxygen in PVA is pink.

Additional Simulation Details
All MD simulations were conducted using version 3 Mar 2020 of the LAMMPS simulation package (12).The polymer-water systems were prepared using a methodology similar to our previous work (13), except that water-ions were also embedded at the center-of-mass of the water droplets as needed.Periodic boundary conditions were applied in the x and y dimensions, while the z dimension was extended with fixed boundaries featuring repulsive walls that generated forces in a direction perpendicular to the wall.Polymers were described with parameters obtained from the all-atom Optimized Potentials for Liquid Simulations (OPLS-AA) force field (14, 15), while water was described using the extended simple point charge model (SPC/E) (16,17).The water ions were represented using a nonpolarizable force field that was parameterized to reproduce thermodynamic properties, such as solvation free energies in water (18).Real-space non-bonded interactions were truncated at 10 Å. Long-range electrostatics were handled using the particle-particle-particle-mesh Ewald summation method (19) with a convergence accuracy of 10 −5 ; this method was modified to accommodate the slab geometry with a non-periodic z dimension (20).Equations of motion were evolved using a velocity-Verlet integration scheme with a 1 fs timestep.A rigid geometry was maintained for all water and ion molecules using the SHAKE algorithm (21).Unless otherwise specified, temperature was controlled at 300 K using a Nosé-Hoover thermostat (22) with a damping constant of 100 fs.Following system preparation, 20 ns equilibrium simulations were conducted.Subse-quently, 20 ns production simulations were performed for all systems, and an additional 20 ns of simulations were conducted for N w = 2000 for structural analysis.Interfaces were identified according to the approach of Willard and Chandler; (23) these calculations were facilitated by the Pytim package (24) using the same settings as in our previous work (13).
Additional Description of Single-surface Free-energy Calculations Thermodynamic Integration (TI) was used to compute the free energy of adding one ion to the water droplet.Prior equilibrated configurations were used as the initial configuration for simulations used for TI: In Eq. ( 1), ⟨•⟩ λ denotes an ensemble-average obtained using λ, which is the thermodynamic path variable such that λ = 0 corresponds to a state with only water droplet and the polymer surface and λ = 1 corresponds to the state with a water droplet, a water ion, and a polymer surface.As shown, the integral is numerically approximated using 12-point Gauss-Legendre quadrature with λ ∈ {0.00922, 0.04794, 0.11505, 0.20634, 0.31608, 0.43738, 0.56262, 0.68392, 0.79366, 0.88495, 0.95206, 0.99078}.For the configurational potential energy with λ , U (λ, ⃗ q), we utilized a soft-core potential (25) for pairwise Coulombic and Lennard-Jones potential energy contributions involving the ion molecule: and U coul (r ij , q i , q j ) = λ q i q j 10(1 The utilization of soft-core potentials allows for Eq. ( 2) and Eq.(3) to possess the same λ, as opposed to performing TI in two stages (e.g., first handling U LJ terms and then U coul terms).
Because the simulation box is heterogeneous, we find this preferable for sampling efficiency as utilizing only Lennard-Jones interactions in the absence electrostatics causes the ion to predominantly explore the vast "vapor" phase.We note that the free energy depends explicitly on N w and also there exists a subtle finite-size effect based on our heterogeneous system construction.Detailed discussion of this finite-size effect can be found in our previous work; (13) however, the effect is inconsequential in the construction of our free-energy differences.

Additional Description of Two-surface Free-energy Calculations
Two-surface systems were prepared by flipping and adding one equilibrated polymer-water system with one ion type to another with the opposite ion.For each pair of polymers, one of three amorphous systems for one polymer was randomly chosen and paired with a randomly chosen system for the other polymer.The distances of two surfaces, d, was then set based on average surface interface position.The F AB (p z ) was calculated using umbrella sampling with statistical re-weighting via the weighted histogram analysis method (26).Data was collected across 36 windows that each employ a harmonic biasing potential on p z .The biasing potentials utilize spring constants of 47.8011 kcal/mol and equilibrium positions at -35 to 35 Å in 2 Å increments.Sampling was facilitated using version 2.8.1 of PLUMED (27).We note that the classical force field for H 3 O + and OH − was not parameterized (18) to handle possible recombination of ionic species into neutral water molecules.To focus sampling on configurations for which the ions are separate charged species and within the realm of applicability of the force field, we modified the non-bonded interaction between oxygen atoms on H 3 O + and OH − to be repulsive at distances less than approximately 4 Å.This is practically achieved by increasing ε in the Lennard-Jones potential to 1.0kcal/mol (Fig. S6).The net effect of this modification is that ions do not form unphysical hydrogen bonds, which would otherwise arise using the original parameters.Consequently, F AB (p z ) conditionally depends on the ions being separate charged species that are separated by approximately 3 Å.Formally, F AB (p z ) is biased by this modified potential, but Fig. S6 shows that this effectively negligible beyond 4 Å, and so re-weighting was not performed with respect to this bias.

Figure 1 :
Figure 1: Overview of hypothesis and systems.(A) Schematic depicting how the free energy of water-ions (H 3 O + and OH − ) may vary between two polymer surfaces.Differences in free energy result in a thermodynamic driving force for preferential partitioning of ions between surfaces.(B) A thermodynamic framework to predict the direction of contact charging.The free-energy difference ∆F +− AB determines whether a charge-separated pair is more stable in State II with free-energy F +− AB (H 3 O + near surface A and OH − near surface B) or State III with free energy F −+ AB (OH − near surface A and H 3 O + near surface B). (C) Summary of specific systems studied.The chemical structure of the constitutional repeat unit, internal reference name, and BigSMILES string of the six polymers studied are shown (top).In addition to three amorphous slabs per polymer, additional crystalline slabs of N66, PE, and PVC are studied as well as three amorphous PVA slabs comprising isotactic chains; these are respectively denoted as N66 * , PE * , PVC * , and PVA † (middle).For each polymer, simulations are run using water droplets comprised of N w = 2000, 1000, 500, 250, or 125 water molecules (bottom).Molecular renderings in panel B are produced using OVITO (49); carbon is gray, fluorine is blue, chlorine is green, oxygen is red, and hydrogen is white.The color-coding associated with polymer names in panel C is used throughout the text.

Fig. 2a depicts
Fig.2adepicts a triboelectric matrix derived from ∆F +− AB values obtained from MD simulations.To first order, the matrix is organized by material (6 × 6 matrix), and results are further resolved for each A-B into a 5 × 5 sub-matrix based on water-droplet size; color intensity reflects the magnitude of thermodynamic driving force.Compared to experimental triboelectric series (SI Appendix, Fig.S1), the simulation results broadly align with the direction of charging observed in S1, S2, and S3.In comparison to S1, simulation predictions agree with nine of fifteen material combinations, while three pairs yield inconclusive results or depend on droplet size, and three pairs exhibit opposite trends.However, when compared to S2 and S3 (which lack data for PVA), the agreement improves, as simulations predict PVC acquires negative charge over PTFE (as in S2) and N66 acquires negative charge over PMMA (as in S3).Thus, the thermodynamically informed predictions capture general trends in contact charging between polymers of different chemistry.The few disparities between simulation predictions and empirical charging results arise in material pairings that also demonstrate experimental variability.For PVC-PTFE, S1 and S3 (and other series, see SI Appendix Fig.S1) suggest that PTFE exhibits a strong tendency to acquire negative charge.However, our previous study on polymer hydrophobicity(57) indicates that water structuring and dynamics are relatively more similar between PTFE and PE than with PVC.These prior observations align with our current free-energy results, showing a vanishing ∆F +− AB for PE-PTFE and consistent behavior between PE-PVC and PTFE-PVC, and the experimental outcome reported via S2.Consequently, results involving PTFE may be sensitive to experimental conditions, potentially related to mechanisms not captured by simulations, such as the presence of acid and base groups post polymerization, bond breaking(26), or minor inaccuracies in molecular models.For N66-PMMA, S1 and S3 differ, with the latter aligning with the thermodynamic predictions.Lastly, several inconsistent or inconclusive combinations involve PVA; the aqueous solubility of PVA poses an experimental challenge and is also a notable factor in our previous study(57).Considering the substantial agreement for many material pairings and the technical challenges encountered with others, we conclude that thermodynamically driven water-ion transfer can plausibly influence polymer-polymer contact charging.

Figure 2 :
Figure 2: Results of free-energy calculations for amorphous polymers.(A) The thermodynamic driving force for water-ion transfer between surfaces A and B presented as a triboelectric matrix.The matrix is resolved 6×6 by material; each pair of materials is further resolved 5×5 accounting for differing droplet sizes.Droplet sizes (N w = 125, 250, 500, 1000, and 2000) increase left-toright and top-to-bottom.An approximate linear triboelectric series generated from the matrix simulation is shown for reference below the matrix results.(B) Results of thermodynamic integration calculations to extract F α S , the free energy of adding an ion of species α to surface S. Results for adding H 3 O + are shown at the left and OH − are at the right.Error bars reflect statistical uncertainties reported as the standard error of the mean calculated from independent thermodynamic integration trajectories.

Figure 3 :
Figure 3: Structural analysis of free-energy trends for H 3 O + and OH − across polymers.(A) Comparison of spatial distribution of water molecules, H 3 O + , and OH − in proximity to the polymer-water interface.(B) Simulation snapshots comparing OH − interactions in proximity to PMMA and PVC surfaces.Hydrogen-bonding interactions are indicated by thin blue bars when participating atoms are within 2 Å, except one lengthened bond in the PMMA system; interacting hydrogen atoms are labeled.Most surrounding water molecules are omitted for clarity.(C) Comparison of free energies for ion addition based on morphological changes to polymer slabs.Comparisons are made between amorphous-to-crystalline (denoted '*') PE, PVC, and N66 and amorphous atactic-to-isotactic (denoted ' †') PVA.Results on the left are for surfaces with N w = 2000 water molecules with bars grouped by material, data for H 3 O + to the left of that for OH − .Results on the right are for all droplet sizes; results between PE and PE * are statistically indistinguishable and not shown for clarity.Error bars reflect statistical uncertainties reported as the standard error of the mean calculated from independent thermodynamic integration trajectories.

Figure 4 :
Figure 4: Explicit partitioning of a H 3 O + /OH − pair between two polymer surfaces.(A) Simulation snapshot illustrating the general system setup for calculations.A water bridge of N w = 4000 water molecules forms between two polymer slabs positioned a distance d away, allowing for diffusion of H 3 O + and OH − between two surfaces, A and B. The relative positioning of H 3 O + and OH − with respect to the polymer surfaces can be monitored using p z , the ionic dipole in the z-direction.The specific system shown corresponds to PMMA (top) and PVC (bottom) positioned at d = 25 Å. (B) Interpretation of the exchange constant K +− AB (p z ).If K +− AB (p z ) > 1, H 3 O + exhibits more preference for A than OH − (A + B − ); if K +− AB (p z ) < 1, H 3 O + exhibits more preference for B than OH − (A − B + ); and if K +− AB (p z ) ∼ 1, there is no clear preference (A • B • ).(C) Results of K +− AB (p z ) for different pairs of materials.Results are for d = 25 Å except for pairs annotated with '**,' which use d = 40 Å to better characterize thermodynamic preference (see SI Appendix, Fig. S3).The shaded regions reflect statistical uncertainties reported as standard error of the mean calculated from bootstrap resampling.The black solid lines indicate the position of K +− AB (p z ) = 1.The gray dashed lines show exp[−β∆F +− AB /40] from single-surface calculations to compare the direction of ions; the scale factor is chosen to present all data on the same scale.

Figure S1 :
Figure S1: Different triboelectric matrices generated from previous published triboelectric series.The order of six polymer surfaces is in the order of this work.The linear versions of the series are listed below the matrices.Three series S1, S2, and S3 are labeled in the top left corner of the matrices.The origins of these triboelectric series are as follows: (A) This work (B) Ref. 1 (C) Ref. 2 (D) Ref. 3 (E) Ref. 4 (F) Ref. 5 (G) Ref. 6 (H) Ref. 7 (I) Ref. 8 (J) Ref. 9 (K) Ref. 10

Figure S3 :
Figure S3: Dependence of relative ion-surface affinities on distance between surfaces.(A) Comparison of relative ion partitioning between surfaces as a function of ionic dipole, K +− AB (p z ), as surfaces are separated at distances of d = 15, 25, 40, and 55 Å. (B) Summary of all K +− AB (p z ) for all pairs of surfaces separated by d = 25 Å.The surface pairs denoted with "**" are further simulated at d = 40 Å due to their relatively weak dependence on p z ; these results are shown in the main text.The shaded regions indicate statistical uncertainties that reflect the standard error of the mean as estimated from bootstrap resampling.

Figure S4 :
Figure S4: Effect of surface hydrophobicity on two-surface free-energy calculations.(A) Configurational snapshot of water bridge forming between PTFE and PVA separated by d = 25 Å.(B) Configurational snapshot of water bridge forming between PE and PVA.Both PTFE and PE are substantially more hydrophobic than PVA.Consequently, the majority of water molecules in the system are recruited towards the PVA surface, which affects the accessible volume of ions.Both snapshots are rendered using OVITO(11).The atoms are colored such that carbon is gray, fluorine is green, hydrogen is white, oxygen in water and ion is red, and oxygen in PVA is pink.

Figure S5 :
Figure S5: Results of free-energy calculations for a free water droplet and amorphous polymers.The free water droplet results are colored by cyan and other polymers are colored as in the main text.The markers of PE and PTFE are artificially shifted along the x-axis to better display the results.

Figure S6 :
Figure S6: Comparison of pairwise H 3 O + -OH − interaction energies with and without modified oxygen-oxygen interaction.The original model of Ref. (18) was not parameterized to directly capture OH − -H 3 O + interactions and does not address recombination into neutral species; conventional mixing rules results in an unphysical hydrogen bond between charged species at close separation distances.By increasing ε from 0.0478 kcal/mol to 1.0 kcal/mol, the H 3 O + and OH − are biased to maintain separation distances that avoid such behavior.Energies are computed as a function of distance between oxygen atoms d ion while the angle formed between O(H 3 O + )-H(H 3 O + )-O(OH − ) is 180 • .The inset schematic illustrates the specific geometry used.
illustrates contrasting scenarios of water droplets present on surfaces, A (blue) and B (red), that guide our calculations.In reference State I, droplets are neutral on both surfaces.In State II, contact yields a charge-separated pair where H 3 O + resides on A and OH − resides on B; the free energy of State II relative to State I is F +−