Molecular Dynamics Simulation of Salt Diffusion in Polyelectrolyte

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Molecular Dynamics Simulation of Salt Diffusion in Polyelectrolyte Assemblies Ran Zhang, Xiaozheng Duan, Mingming Ding, and Tongfei Shi J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b02644 • Publication Date (Web): 05 Jun 2018 Downloaded from http://pubs.acs.org on June 5, 2018

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The Journal of Physical Chemistry

Molecular Dynamics Simulation of Salt Diffusion in Polyelectrolyte Assemblies Ran Zhang,* Xiaozheng Duan, Mingming Ding and Tongfei Shi* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, 130022, Changchun, JL, P. R. China

Email: [email protected]; [email protected]

1 Environment ACS Paragon Plus

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Abstract

The diffusion of salt ions and charged probe molecules in polyelectrolyte assemblies is often assumed to follow a theoretical hopping model, in which the diffusing ion is hopping between charged sites of chains based on electroneutrality. However, experimental verification of diffusing pathway at such microscales is difficult, and the corresponding molecular mechanisms remain elusive. In this study, we perform allatom molecular dynamics (MD) simulations of salt diffusion in polyelectrolyte (PE) assembly

of

poly

(sodium

4-styrenesulfonate)

(PSS)

and

poly

(diallyldimethylammonium chloride) (PDAC). Besides the ion hopping mode, the diffusing trajectories are found presenting common features of a jump process, i.e., subjecting to PE relaxation, water pockets in the structure open and close, thus the ion can move from one pocket to another. Anomalous subdiffusion of ions and water is observed due to the trapping scenarios in these water pockets. The jump events are much rarer compared with ion hopping but significantly increases salt diffusion with increasing temperature. Our result strongly indicates that salt diffusion in hydrated PDAC/PSS is a combined process of ion hopping and jump motion. This provides new molecular explanation for the coupling of salt motion with chain motion and the nonlinear increase of salt diffusion at glass transition temperature.


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1.

Introduction

Polyelectrolyte multilayers (PEM) and complexes (PEC) materials assembled by oppositely charged polyelectrolytes in solution are widely used in many fields in human society.1-3 The semipermeable nature of such products in membrane form plays an important role in many potential fields such as filtration, drug delivery, anticorrosion and sensors.4-7 The transporting of guest molecules is subjected to a series of factors. Porosity (inner structure) and thickness of the assembled films, for example, are related to the internal structure and the total travel length, and will thus greatly influence the diffusion in the material.8-9 During preparation the assembling ionic strength and temperature can greatly affect the pore size, hence affect the diffusion rate;10-11 as for film thickness, an increase of the number of polyelectrolyte layers will significantly decrease the diffusion rate.10,

12-13

Parameters such as pH, temperature and ionic strength during the processing also affect the final network structure, which influences permeability.14-18 What is more, Targeting diffusing species, the introduction of specific building block can increase/decrease its mobility and therefore promote molecular separation.19 In the last two decades, the focus of research has largely been paid to extracting mean diffusion coefficients of salt and small molecules in the PE assemblies. Techniques such as Electrochemical Impedance Spectroscopy (EIS), rotating disk electrode (RDE) and conductivity measurements are often used in experiments for the diffusion properties in PE complex.12,

20-25

At room temperature, the

diffusion coefficients of normal salt ions (Na+, I−, Br− and Cl−) normally fall in between 10−6 cm2/s and 10−7 cm2/s,10, 18, 25 while for high-valent redox ions like



ferricyanide a much lower value ranged from 10−9 cm2/s to 10−10 cm2/s is indicated.12, 24 Without the application of electrodes, Krasemann et al. found the PE assemblies subjected to only chemical potential difference rejected the permeation of high-valent ions, and the separation factors for ions rose up to 112.5 for Na+/Mg2+, and 45 for Cl−/SO42−.13 With fluorescence measurement of a charged probe molecule near PEM surfaces, Tauzin et al. detected a D at around 10−9 cm2/s during diffusion.26 Similar studies have been performed on the diffusion of probe molecules in PE brushes, and PE complex films.27-30 Ghostine et al. investigated the influence of temperature and salt concentration on transport of the triple charged redox ion, ferricyanide, through a PDAC/PSS multilayer.24 Moreover, the authors calculated the thermal activation energy for ferricyanide ions which was 98 kJ/mol.24 In comparison, normal ions (such as Cl−) and small molecular ions (such as amino acid cations) were characterized with activation energy cost 20~50 kJ/mol in ion exchange membranes,31-32 and the reported association strength of intrinsic PE charge compensation showed much lower value.18 As for the diffusion mechanism, it is commonly considered that the transport of small charged molecules is subjected to the confined water channels (pores) distributed in the PE complex, and follows the ion hopping mode from one polarized (charged) site to another.18, 26, 33 Moreover, the relaxation of PE chains is found strongly affecting the pathways in water channels and thus is coupled with the diffusion of these guest molecules.20, 24, 26, 34 The water pockets along the water channels may open and close due to thermal fluctuations of the PE matrix,35 so the salt ion is confined in a certain water pocket until it opens to a connecting pocket. This type of diffusion resembles the


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jump diffusion of gas molecules in amorphous polymers, and has been reported for water diffusing in hydrated polymers, 36-39 but not for the case here. Moreover, detection of the anomalous diffusion has been reported accompanying the transport of molecules in membranes,40-41 and this has not been clarified for the salt transport in polyelectrolyte assemblies either. Therefore, quantifying the mean value of diffusion coefficients can be only viewed as the first step to better understanding of the transport of guest molecules. Moreover, during ensemble averaging, the critical information such as individual probe trajectory and random intermittent behaviour is maybe lost.42 The diffusion modes of small charged molecules in PDAC/PSS, for instance, still remain elusive. For clarity of the following text, we refer the ion-hopping mode from one polarized (charged) site to another as mode 1, and the jump mode of ion moving from one pocket to another as mode 2. Experimental verification on such scales is difficult (the ion motions of mode 1 between PE sites occur in nanometer scale and on the order of nanoseconds or less). On one hand, the ideal diffusion mode 1 involves the presumption of discrete and recognizable PE charges as PE sites and charge neutrality of the diffusing ion and PE charge site.33 However in interpenetrating bulk structures, stoichiometric compensation is maintained globally without guaranteeing local charge neutralization with an ideal pair pattern43 at a higher resolution level. Hydrophobicity has also been reported to influence the distribution of PE chains and form local PSS-rich domains.44 On the other hand, the strong electrostatic interaction may influence the validity of mode 2, and chances are that PE charge clusters are functioning as the potential minima during ion diffusion and a salt ion is frequently subjected to control of such domains. Consequently the actual



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diffusing pathways can be complicated than that described by either of these two theories. Computer

modelling

serves

as

a

powerful

virtual

route

for

desired

thermal/mechanical information of PE complexes, focusing on the static/dynamic properties on the micro scale. Molecular dynamics (MD) simulation has already been applied in elucidating the thermal transition mechanisms of PE assemblies.45-46 With experimental literature as solid references, modelling could also be employed herein for investigating diffusing trajectories of salt ions, casting light on the corresponding transport mechanisms of PE assemblies. In this work, we are motivated to investigate the diffusion trajectories of simple ions (Na+ and Cl−) in interpenetrating PDAC/PSS assemblies. Interestingly, the trajectory analyses of salt ions indicate a diffusing way subjected to the combination of both diffusing modes.

2. Model

2.1 Model and system organization The transport properties of simple ions in hydrated PDAC/PSS complex are examined using the GROMACS 4.6.6 software.47-48 Figure 1 shows the detailed chemical structures of subunits forming the PSS and PDAC chains. The OPLS-aa force field49 is applied to describe the PE model. The parameters for PDAC and PSS sulfonate group follow the work of Qiao et al.50 Water is represented by the explicit TIP-4P water model,51 and the salt ions (Na+ and Cl−) by the standard OPLS-AA ion parameters. All PE chains are linear. The PSS is set as syndiotactic while the PDAC backbone is all in trans


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configuration. All covalent bonds are controlled by LINCS52 and the SETTLE algorithm is used for the water molecules.53 Validation of the PSS and PDAC models has been done in our earlier work.46 The system contains 10 PSS40 and 10 PDAC40 chains, 40 Na+ ions, 40 Cl− ions and finally 4030 water molecules, corresponding to a salt molar ratio of 0.1 and a water content of 37 wt %. The subscript 40 refers to the number of repeating subunits in each chain. The hydration and salt doping levels are based on PDAC/PSS extruded samples in the experimental work.54-55 The salt molar ratio (number of salt ions over the number of PE charges) is set to match samples soaked in ≈0.25 M NaCl solution.

Figure 1. Chemical structures of poly (sodium 4-styrenesulfonate) and poly (diallyldimethylammonium chloride) with partial charges assigned on the polar groups.

2.2 Simulation methods Molecular dynamics (MD) simulations are used to study the diffusion of salt ions in the PDAC/PSS assembly. For non-bonded interactions, a cutoff distance of 1.0 nm is used for the van der Waals interactions and the electrostatic contributions in the real space, while the reciprocal part of electrostatic interactions in the system is treated by the



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traditional PME method.56 The PDAC/PSS assembly is contained in a cubic simulation box, and 3D periodic boundary conditions are employed throughout the simulation study. In order to reduce the influence of individual initial configurations regarding ensemble averaging, 10 independent initial samples are prepared for each condition. The detailed steps for preparation of each initial configuration involve firstly packing the PSS and PDAC chains equilibrated in dilute solutions in a cubic box with the software Packmol

57

, then the PE chains

are solvated by generating a water layer around the PEs. Finally salt ions are added by replacing randomly chosen water molecules. An initial relaxation is performed by a 15 ns NPT simulation with a stochastic dynamics integrator58 at an inverse friction constant of 2 ps and a time step of 2 fs. The system pressure is maintained at 1 bar by employing the Berendsen barostat with a time coupling constant of 1 ps.59 During the relaxation the complex is annealed by raising the temperature T from an initial temperature of 300 K to 380 K and then back to desired temperature. Intertwined, mixed PE structures with relatively uniform distribution of water and salt are acquired during the sample preparation. With better lubrication at relatively high water content than earlier work, the system is much less disturbed by metastable conformational states.45-46 The production runs involve the standard leap frog integrator with a time step of 1 fs. The Nose-Hoover thermostat60-61 and Parrinello-Rahman barostat62 are used to control the temperature and the pressure with coupling constants of 0.1 ps and 1 ps, respectively. Simulations at discrete temperatures from 280 K to 360 K with an interval of 10 K are carried out. The corresponding simulation involves a


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further relaxation of 1 ns under the new thermostat and barostat at first, and then a production of 20 ns is carried out for analysis. In the data analysis, VMD63 is used for all molecular system visualizations. As for binding between two atoms, the cutoff distance for determining whether two atoms or groups are bonded is defined by the first local minimum of the radial distribution function if not otherwise stated. The average bonding lifetime is the contacting forward lifetime64 estimated by the GROMACS code g_hbond. The cutoffs are 0.66 nm for N-Cl−, 0.31 nm for OS-Na+, 0.24 nm for Water-OS, 0.32 nm for Na+-Water, 0.39 nm for Cl−-Water pairs. A cutoff of 0.35 nm and an additional angle cutoff at 30° for an acceptor-donor-hydrogen connection are used for Water-Water hydrogen bond. Here, N refers to the PDAC nitrogen atom, S the PSS sulfur atom and OS the PSS oxygen atom. The self-part of the van hove function estimated by the GROMACS code g_vanhove is employed to analyze the trajectories of salt ions.

3. Results and Discussion

3.1 Global diffusion properties In normal diffusion, the route of a particle can be described as a random walk, and with a large amount of steps the relationship between the mean squared displacement MSD and the diffusion coefficient D is MSD ~ Dt, where t stands for time. This is the fundamental scheme by which the diffusion coefficients are derived for characterization of tranport properties. However, in anomalous diffusion, the relation becomes MSD ~ Dtn, where the diffusion coefficient becomes time-dependent. This behavior is termed as super diffusion if n > 1, and subdiffusion if n < 1. The presence of such anomalous



diffusion is investigated at first for the salt ions (see Figure 2), and it turns out that MSD of salt ions shows a non-linear dependence on diffusing time, indicating the presence of anomalous subdiffusion. The power n, for instance, is roughly 0.68 and 0.69 for Cl− and Na+ at 300 K, and slowly increases with temperature. The diffuion of salt ions in PDAC/PSS shows anomalous subdiffusion features without clear signs of reaching the normal diffusion n = 1 even during longer simulation (see Figure S1). Similar anomalous subdiffusion has been observed, e.g., for perchlorate transport in polypyrrole membranes, polyethylene glycol in the agarose gel network and protein diffusing in crowded environments such as cytoplasm.40-41, 65 Subdiffusion of both water and ions in membranes for fuel cells has also been reported.66

Figure 2. Mean squared displacements (MSD) of salt ions as a function of simulation time in a log-log plot. The power n from a non-linear fit (based on MSD from 1 ns to 20 ns) equals 0.67, 0.68, 0.69, 0.70, 0.71 for the curves from 280 K to 360 K in (a) and 0.64, 0.69, 0.66, 0.68, 0.73 the same order in (b).

Unlike salt ions, the diffusion of water is characterized by a smaller degree of anomalous behavior, which bears the power n of around 0.87 at 300 K. Slowly the


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diffusion of water approaches normal diffusion (n = 1) on longer time scales (see Figure S1). Here, water mobility shows a lower degree of subdiffusion with a higher n, which can be attributed to the less strong interaction between water and the other components. For the salt ions, however, the degree of subdiffusion is higher with a lower n, which is believed to originate from the strong electrostatic attractions66 between salt and the charged polymer. As for the reason of returning to normal regime, study of water diffusing in a neutral polymer suggests that the polymer proportion in the media is responsible.67 If the strength of trapping induced by polymer increases, the returning of normal diffusion may be delayed and even not accessible. For instance, the anomalous behavior on shorter time scales has been detected for the diffusion of water and gas molecules in amorphous polymers,35, 67 but then return to normal diffusion on longer time scales. The returning of probe particles diffusing in phospholipid bilayers, however, depends on the structure composition.68 Even though the salt diffusion falls in the anomalous regime, the ion mobility can still be evaluated by their mean squared displacements, which signify the extent of diffusion during certain times. From MSD the effective diffusion coefficients Deff through nMSD = 6tDeff can be derived following the generalized form of the Einstein relation.66 For the mobility of a diffusing particle, Deff provides a scale-dependent (such as time or length) evaluation. In Figure 3, the time-dependent property is plotted for the diffusing ions and as a combined form25 for NaCl. Echoing the anomalous diffusing nature, Deff shows sharper decrements on shorter time scales and much smaller decrements approaching the long time limit. It was found with this scale-dependent relationship of Deff, the simulation results can be linked to the experimental D.66 In Figure 3a, Deff of NaCl at 20



ns equals 2.5×10−7 cm2/s, which is approaching the experimenntal value ≈2×10−7 cm2/s reported for NaCl in PDAC/PSS at similar salt molar ratio and hydration level.25

Figure 3. (a) The effective diffusion coefficient Deff versus simulation time. For NaCl, Deff is derived by the combination rule 2x1x2/(x1+x2), where x1 and x2 stand for the Deff of Cl− and Na+.25 (b) Natural log of the effective diffusion coefficient (at 20 ns) as a function of the inverse of temperature. The lines are linear fits following the Arrhenius law, Deff = A0exp(−Ea/(RT)), where A0 is the pre-exponential factor, Ea the activation energy and R the universal gas constant (8.314 J/(mol·K)). Ea can be collected from the slope −Ea/R, which yields 16.7 kJ/mol and 20.2 kJ/mol for


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Cl− and Na+ respectively. Then the activation energy is presented in (c) at different simulation times.

The natural log plot of Deff versus the reciprocal temperature exhibits validity of the Arrhenius law, with an example presented in Figure 3b for Cl− and Na+ at 20 ns. Consequently, Ea evolving with time can be extracted and shown in Figure 3c. The activation energies of both Cl− and Na+ increase with time in the range from 15 to 20 kJ/mol, with Ea for Cl− smaller than that of Na+. In literature there are few related reports except the activation energy (98 kJ·mol−1) calculated from mean diffusion coefficients for ferricyanide in PDAC/PSS multilayer.24 This higher value was attributed to the high-valent nature and larger volume size of the ferricyanide ions. However, there was no discussion on whether the anomalous diffusion of ferricyanide was present or not in the membrane, and diffusion mode 1 was assumed to be the fundamental transport mechanism of PDAC/PSS. Compared with our previous work,46 the bonding lifetimes shown in Figure 4a display significant reductions due to better lubrication at higher hydration level. In detail, we trace the lifetime of Na+-Cl−, water-ion, PE-ion, water-PE bonding as well as waterwater hydrogen bonding with temperature increasing from 280 K to 360 K. Water molecules exhibit faster relaxation which can be inferred from the much lower waterwater hydrogen bonding lifetime. The statistical bonding information provides an overview of the complicated environments during the diffusion of salt ions. Na+-Cl− and salt-PE bindings featured by the nature of electrostatic attraction present much larger values compared with other bindings involving water. The lifetimes for cation bonding with water/PE are apparently higher than those for anions. The bonding lifetime is



defined as 1/kb, where kb is the rate constant of bond breakage. The calculation procedure of kb is described in literature.64 For N-Cl− and OS-Na+ bindings, the breakage of bond means the two atoms/molecules fall out of the bonding cutoff. If we assume their breakages trigger ion hopping echoing the diffusion mode 1, the corresponding activation energy Ea derived from bond breakage can be obtained through the Arrhenius plot in Figure 4b, which yields Ea of 11.2 kJ/mol for N-Cl− and 14.9 kJ/mol for OS-Na+. Ea for breaking OS-Na+ bindings being larger agrees with the slower diffusion of Na+ mentioned above. However, the comparison of Ea derived from bond breakage and from Deff indicates the fundamental mechanism of ion mobility probably does not originate solely from the diffusion mode 1.

Figure 4. (a) Bonding lifetime of corresponding atom/molecule pairs as a function of temperature. (b) Natural log of the inverse of lifetime for OS-Na+ and N-Cl− bindings as a function of the reciprocal temperature. The linear fits of plotted data follow an Arrhenius law, kb = 1/lifetime = A0exp(−Ea/(RT)), which provides Ea of 11.2 kJ/mol and 14.9 kJ/mol for breaking the N-Cl− and OS-Na+ bindings, respectively.


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The snapshots in Figure 5 indicate that water molecules and salt ions travel in the channels formed by the scaffolding PE network, presented as the void space. The randomized distribution of PE chains (blue and green surfaces) presents the interpenetrating structure without large water pores or interruption of the network. We notice the distribution of PE charges do not compensate each other by ordered discrete PE charges43 but rather by randomly distributed small PDAC/PSS charge clusters. The relaxation of these networks can be clearly identified with varied distributions of PE chains at discrete time: 0 ns, 5 ns and 10 ns. Judging from the change of the PE chain surfaces, the relaxation can be attributed to the small-scale segmental motions. By setting water molecules invisible, the salt ions are observed travelling through the channels and preferring residing on opposite-charged PE surfaces.



Figure 5. Snapshots showing the evolution of the system with a time interval of 5 ns at 300 K. Chains are represented by polymer surfaces with blue for poly (sodium 4-styrenesulfonate) and green for poly (diallyldimethylammonium chloride). Salt ions are spheres with Na+ in red and Cl− in yellow, and water molecules are not presented for simplicity.

The origins of anomalous subdiffusion were often attributed to trapping of the diffusing molecules on different levels by the cavities, holes or crowding in the media.35, 41, 65-67 Therefore, the subdiffusion of salt ions in PDAC/PSS assembly is probably linked to the trapping of ions in the channels following concept of diffusion mode 2, and less likely to the binding with charged sites of PE. Thus the salt diffusion in PDAC/PSS is probably dominated by a mechanism of mixed diffusion modes. In the next subsection we will demonstrate the ion trapping following diffusion mode 2 and the combination of two basic diffusion modes through trajectory analysis. 3.2 Trajectory analysis In Figure 6a and 6b we present the 3D trajectories of Cl− and Na+ at 300K. The color lines demonstrate the total diffusing course of the ions in the PDAC/PSS.


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Several features can be then revealed. In some cases, the diffusing route of the ions displays frequently probed local places (such as the orange and green courses of Cl−, and the blue, black and orange ones of Na+). These places may correspond to the water pockets along the channels in the PE assembly, which provide steric trapping for the diffusing ions. The routes of Na+ are less expanded reflecting more thoroughly probed places. Some of them are probing only one very local space in the water channels which implies the diffusion of the Na+ is under stronger restriction. In other cases, the diffusing routes are not trapped in local places but show jumps from one local place to another (see Figure 6a for the blue, black and red courses of Cl−). Figure 6c and 6d show that through thermal activation (from 300K to 360K) the escaping from local trapping can be significantly accelerated except that some of Na+ still reside in the same local places. The diffusion mechanism of jump events indicates the presence of diffusion mode 2, and the diffusing routes clearly differ from those have been observed for pure water, which resemble a random walk.39, 67 At subresolution regions of the diffusing courses, evolving of ion positions is characterized by a shorter distance which may correspond to ion hopping following mode 1. By intuitive observation the jump distance between frequently probed places is clearly larger.



Figure 6. 3D trajectories of Cl− and Na+ ions sketched by colored lines at 300K and 360K. Six samples are presented in each plot.

Reviewing the nature of strong electrostatic interaction mentioned earlier, these frequently probed places may also correspond to the potential minima of the PE charge distribution. Therefore, electrostatic trapping by surrounding charges and steric trapping (water pockets) of chain segments along the water channel are probably combined. Next, we illustrate how the electrostatic trapping “works” with the steric trapping. We tracked the coordination number (y) and trajectories of individual ions, as well as their squared displacements (SD) for individual salt ions.


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In Figure 7 the 2D trajectory of a sample Cl− on the XZ plane clearly illustrates several jumps along the diffusing course following diffusion mode 2. The colored arrows mark out the escaping of this Cl− from one water pocket to another, and at the same time SD presents a jump-like increment. The coordination number y in Figure 7b describes the instant number of PDAC N atom and Na+ falling in the bonding cutoff of Cl−. The fluctuation of y mainly reflects the frequent hopping of ions (mode 1) among the PE charge sites in the pockets (see Figure S2 for an overall evaluation for N-Cl− binding events). Coinsidentally, the N-Cl− nonbonding stages (y = 0) echo the time intervals of the jump routes. This finds relevance with the jumping trajectory of a charged molecule on a PEM surface and those of water molecules in amorphous polymers.26,

67

In this sense, two

adjacent jumps may define the range of the electrostatic trapping cage (or water pocket). For instance, the second cage can be identified in between the blue and red arrows in Figure 7. There are maybe other electrostatic cages (water pockets) indicated by the purple and yellow circles, however the corresponding step change of SD is blurred, probably due to continuous diffusion under weaker electrostatic trapping strength.



Figure 7. (a) 2D trajectories of a Cl− ion (selected from Figure 7a, the trajectory marked red at 300K) projected on the XZ plane and (b) instant coordination number y for N-Cl− (black solid curve) and Na+-Cl− (red dashed curve) bonding. The inserted figure shows the squared displacement (SD) of this Cl− ion. Na+-Cl− bonding cutoff is 0.66 nm. The trajectory starts from (3.704, 2.465) and ends at (0.295, 0.971).

The local PDAC-rich domains may act as electrostatic cages capturing oppositecharged ions in the water channels (see the PE-salt Coulomb potential in Figure S3). However, small salt clusters may also effectively confine the diffusing ions. The coordination number of Na+-Cl− bonding (red dashed curve in Figure 7b) shows this Cl− is also interacting with several Na+ ions in the first water pocket. This scenario can be supported by the corresponding simulation snapshots. For instance, Figure 8a shows this Cl− ion is clustered by three Na+ ions dwelling in the PSS-rich domain. The snapshot during the next caging time is presented in Figure 8b. This Cl− is now staying in a PDAC local domain, while one of the Na+ ions binding with this Cl− in the first cage is now closely interacting with PSS.


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Figure 8. Snapshots reflecting the binding events of the sample Cl− at simulation time (a) 3 ns and (b) 10 ns. The same colors are assigned for the components as in Figure 5. At 3 ns this Cl− with index “4082” is clustered with three Na+, one of which bearing the index “4146” is also presented at 10 ns.

In Figure 9 the same combined analysis is performed on a sample Na+. The diffusion trajectory thoroughly explores a very local space in the PDAC/PSS and shows no signs of jumps of mode 2 in Figure 9a. In Figure 9b, the coordination number y of S-Na+ presents a relative narrow distribution around 3, characteristic of local overcharging. An overall evaluation for different S-Na+ binding events can be found in Figure S2. The change of SD at small scales corresponds to no jump motions but Na+ binding with different PSS charge sites in this cage. This implies this Na+ is trapped in a PSS charge cage due to strong PE-Na+ interaction (see Figure S3).



Figure 9. (a) 2D trajectory of a Na+ ion (from Figure 6b, the trajectory marked black at 300K) and (b) instant coordination number y for S-Na+ (black solid curve) and Na+-Cl− (red dashed curve) bindings. The inserted figure shows the squared displacement (SD) of this Na+ ion. The cutoff for Na+-Cl− bonding is 0.66 nm.

3.3 Diffusion mechanisms The revealed diffusing patterns suggest that salt diffusion in the PE matrix is broadly influenced by electrostatic/steric trapping along the water channels, which seems more complicated than simple steric trapping from pockets/cages.3536

Here the electrostatic trapping cages are found overlapping with the water

pockets strengthening the confining of the ion motion. Therefore the diffusing route of salt ions is featured by frequent ion hopping events of mode 1 and occasional jump motions of mode 2. The lifetime analysis and the activation energy of salt-PE binding can be employed to quantify mode 1 during salt


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diffusion, whereas a further investigation of mode 2 is still required to gain more insights of the diffusion mechanism. In order to distinguish these two diffusing modes and quantify the contribution of jump motion, we first provide the self-part of the van hove function Gs(r, t) of the diffusing ions in Figure 10, describing the probability of a salt ion moving distance r within time t. For Cl− at 280 K, two distinguished peaks of Gs can be located at around 0.1 nm and 0.3 nm (see Figure 10a), which imply the localized motions during Cl− binding to PE charges (or Na+) and the ion hopping of mode 1 between charged sites, respectively. In Figure 10c, Gs of Cl− shows the shift of the peak of localized motions to the one of hopping with temperature increase, with the latter peak ranged from 0.3 to 0.5 nm. No peak corresponding to the jump motion of mode 2 is indicated in these two panels. With a longer interval at t = 3 ns, the peak of hopping is probably combined with the correlation length of ion diffusing in the cage, and shifts to a much broader peak at larger distance. The broad peak relates to motions at larger distances and probably to the jump motion of mode 2. Moreover, this peak shift occurs at higher temperature with suppression on chain mobility (See Figure S4), which agrees with the underlying physics of diffusion mode 2. However, with crucial information such as the frequency/distance of jump still missing at each temperature, this correspondence still needs verification. As for Na+, Gs in Figure 10b shows only the significant peak ranged from 0 to 0.1 nm corresponding to the localized motions. The peak of hopping of mode 1 degrades to a formation of shoulder ranged from 0.2 to 0.3 nm. With larger time intervals (see Figure 10d and 10f), the jump motion of mode 2 fails to show itself



through clear peak shifts, which probably is convoluted with the peak of other motions. The analysis of Gs provides further information of diffusion mode 1 but still not enough for the jump motion of mode 2. Moreover, tracing the open and close of pockets (chain conformation change) has been reported difficult for identifying the jump motion.35

Figure 10. Self-part of the van hove function Gs(r, t) with (a), (c) and (e) calculated for the Cl− ions and (b), (d) and (f) for the Na+ ions. (a), (b) provide Gs at different times, and (c)-(f) present Gs at specific times with increasing temperature. Here r stands for the distance instead of a vector.


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Here, considering the correlation of electrostatic and steric trapping, we present a way to identify and quantify the diffusion mode 2 by tracing the individual trajectory (see SI for details). A diffusing move is defined as a jump motion if it satisfies (1) the salt ion’s center of mass moves over a certain distance d within a short time tjump, which has been employed in Ref. 35 and (2) the two local routes connected by this move have no correlation. For the value of d the bonding cutoffs of N-Cl− (0.66 nm) and S-Na+ (0.66 nm, the second minimum of RDF) were chosen. The choices of tjump at each temperature can be found in SI (see Figure S5). The application of the second criterion can effectively capture the major jump events and screen fake jumps in the diffusing routes (see Figure S6). Three jump events can be identified accordingly from the Cl− ion trajectory in Figure 7, including the two major jumps marked blue and red (see Figure S7). With this method we examined the trajectories of salt ions individually and extracted properties such as the proportion of salt performing jump moves and the average number of jumps per jumping ion during simulation. In Figure 11a the proportion of salt performing jumps significantly increases with temperature, suggesting more salt ions are released from trapping under thermal activation. The proportion of the Na+ ions is less than 0.4 before reaching 320 K, while for Cl− ions the proportion starts from ≈0.6 at 280 K and holds an advantage over Na+ by roughly 0.3-0.4 through the whole temperature range. The average number of jumps per jumping ion in Figure 11b shows an increase with temperature for both types of ions. The activation energy for performing jumps can be derived by the Arrhenius plot as well, which provides similar Ea (23.2



kJ/mol and 22.4 kJ/mol) for Cl− and Na+. These Ea of performing jumps are larger than those derived by Deff or by bonding lifetimes.

Figure 11. (a) The proportion of salt ions performing jump moves of mode 2 and (b) the average number of jumps per jumping ion. (c) Natural log of the total number of jumps NJ (number of ions performing jumps multiplied by the average number of jumps per jumping ion) as a function of the inverse of temperature. The linear fits of plotted data follow an Arrhenius law, NJ = A0exp(−Ea/(RT)), which provides Ea at 23.2 kJ/mol and 22.4 kJ/mol for the jump motions of Cl− and Na+, respectively.


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From the simulation time and the data in Figure 11b, the frequency of jump can be estimated. For instance, at 300 K, the frequency of jump for Cl− amounts to 0.22 ns−1 and for Na+ 0.13 ns−1, and at 340 K reaches 0.52 ns−1 and 0.24 ns−1, respectively. In this sense, the peak shift of Gs in Figure 10e can be attributed to the increasing of both the proportion of jumping ions and the jumping frequency. The lower proportion and less active jumping rate of Na+ somehow explain the missing representation of jump motion through Gs. When the chain motions are under suppression, both the proportion of jumping ions and the jumping frequency decrease (See Figure S8) due to the coupling of salt-chain motion, and this again interprets the peak shift of Gs at higher temperature in Figure S4. The averaged distance between local routes (calculated by center of mass) connected by such a jump ranges from 1.6 to 1.8 nm with considerable fluctuations (see Table S1). This somehow explains the broad peaks at higher temperatures in the analysis of Gs. Although the jump motion of mode 2 occurs less frequently compared with hopping of mode 1, they make considerable contribution to the ion mobility (comparison of SD profiles in Figure 7 and 9). Meanwhile, the contribution of hopping of mode 1 to effective diffusion is apparently restricted by the local steric/electrostatic trapping, which results in anomalous subdiffusion. In this sense, we can conclude that diffusion mode 2 rather than mode 1 mainly accounts for ion mobility.



4. Conclusions

In this work, the diffusion of simple salt ions (Na+ and Cl−) in PDAC/PSS assemblies is investigated by molecular dynamics (MD) simulation techniques. Salt diffusion in hydrated PDAC/PSS is finally interpreted by the combination of two diffusion modes, ion hopping of mode 1 and jump motion of mode 2. Ion hopping involves frequent binding with and breaking from the PE sites, however the trajectory is correlated because the trapping of ions by local water pockets/local potential minima in the PE structure. This is believed to be the cause of anomalous subdiffusion of salt ions. Ion jump enables the ion to break from the total control of such trapping, and frequent jump motion of mode 2 can significantly increase ion diffusion and the probability of jump benefits from thermal activation. This provides new molecular mechanism for the coupling of salt motion with chain relaxation and in turn the strong increase of salt diffusion at glass transition temperature. The successful jump motion of mode 2 depends on factors such as water hydration, the polarity of the PE charge groups and temperature. The study of simple salt ions diffusing in PE assemblies sheds light on the elusive diffusing modes of guest molecules in PE assemblies, which in a way completes our knowledge about the transport mechanisms.

Acknowledgements

This work is funded by Key Research Program of Frontier Sciences, CAS (QYZDYSSW-SLH027). We acknowledge the support of National Natural Science Foundation of China (21674114, 21604086) and the support from Special Program for Applied


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Research on Super Computation of the NSFC-Guangdong Joint Fund under Grant U1501501. We thank Dr. Maria Sammalkorpi and Dr. Piotr Batys for helpful discussions regarding the setup of the simulation.

Supporting Information (SI)

1) Mean squared displacement of water and salt. 2) Analysis of different polymer-salt binding events. 3) Potential development during salt diffusion. 4) Parallel simulation with position restraints on chains. 5) Method to identify the jump motion. This material is available free of charge via internet at http://pubs.acs.org/.

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Molecular Dynamics Simulation of Salt Diffusion in Polyelectrolyte

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