Water Diffusion in Amorphous Hydrophilic Systems: A Stop and Go

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Water diffusion in amorphous hydrophilic systems: a stop and go process Karol Kulasinski, Robert Guyer, Dominique Derome, and Jan Carmeliet Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b03122 • Publication Date (Web): 21 Sep 2015 Downloaded from http://pubs.acs.org on September 22, 2015

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Figure 1. a) Snapshots of the modelled MD structures in dry state: (a) amorphous cellulose, (b) galactoglucomannan, a hemicellulose, (c) cellulose microfibril, composed of crystalline cellulose (grey) and hemicellulose (red). 76x92mm (300 x 300 DPI)

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Figure 1. b) Snapshots of the modelled MD structures in dry state: (a) amorphous cellulose, (b) galactoglucomannan, a hemicellulose, (c) cellulose microfibril, composed of crystalline cellulose (grey) and hemicellulose (red). 238x289mm (96 x 96 DPI)


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Figure 1. c) Snapshots of the modelled MD structures in dry state: (a) amorphous cellulose, (b) galactoglucomannan, a hemicellulose, (c) cellulose microfibril, composed of crystalline cellulose (grey) and hemicellulose (red). 264x289mm (96 x 96 DPI)


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Figure 3. a) Output of the model: (a) microscopic diffusion coefficient and (b) tortuosity. 998x879mm (96 x 96 DPI)


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Figure 3. b) Output of the model: (a) microscopic diffusion coefficient and (b) tortuosity. 998x879mm (96 x 96 DPI)


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Figure 4. a) (a) Porosity and (b) free volume as related to the total volume, as a function of H2O content. 70x61mm (600 x 600 DPI)


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Figure 4. b) (a) Porosity and (b) free volume as related to the total volume, as a function of H2O content. 70x61mm (600 x 600 DPI)


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Figure 5. a) Model of a binding site: (a) Fourier transform of MSD during waiting periods, after background removal at s→0, (b) estimated binding energy at a binding site. 998x879mm (96 x 96 DPI)


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Figure 5. b) Model of a binding site: (a) Fourier transform of MSD during waiting periods, after background removal at s→0, (b) estimated binding energy at a binding site. 998x879mm (96 x 96 DPI)


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Water diffusion in amorphous hydrophilic systems: a stop and go process Karol Kulasinski1,2, Robert Guyer3,4, Dominique Derome2, Jan Carmeliet1,2 * 1

Chair of Building Physics, Swiss Federal University of Technology ETH Zurich, Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland 2 Laboratory for Multiscale Studies in Building Physics, Swiss Federal Laboratories for Materials Science and Technology Empa, Überlandstrasse 129, 8600 Dübendorf, Switzerland 3 Solid Earth Geophysics Group, Los Alamos National Laboratory, MS D446, Los Alamos, New Mexico 87545, United States 4 Department of Physics, University of Nevada, Reno, Nevada 89557, United States * corresponding author: [email protected]

Abstract

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The diffusion of H2O in three amorphous polymer-H2O systems is studied as a function of H2O content using molecular dynamics. A picture of H2O molecule motion comprising alternating steps of being bound at an adsorption site (‘stop’) and moving (‘go’) emerges. This picture is made quantitative. The bound time, frequency of stop-go steps and tortuosity all decrease with H2O content. Fourier analysis of particle motion during bound time segments provides a measure of an attempt frequency that is connected quantitatively to the bound time and an activation energy of a hydrogen bond. On increase in H2O content, the polymer-H2O systems swell leading to an increase in diffusion coefficient and porosity, and a decrease in activation energy.

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Cellulosic materials are the subject of intense investigation. They are basic to numerous applied physical systems, e.g., the paper of Leonardo's self-portrait 1, a T-shirt 2, a hydro-actuated plant response 3. Consequently these materials are studied in the laboratory at the thermodynamic level (adsorption isotherms, elastic constants, etc. 4,5), at the mesoscopic level (n-scattering from a "stick" of a wood cell wall 6) and at the microscopic level (X-ray scattering, NMR, etc. 7–9). These materials are ideal systems for exploration by modeling with molecular dynamics. Cellulosic materials are of interest in the crystalline or amorphous state, dry or in contact with H2O, e.g. the adsorption of water causes the swelling of wood, leading to an increase in the ability of wood to transport water 10–13.

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Phenomena like hygro-actuated plant response, e.g. the timely opening of a Pine cone for seed dispersal that depends on the coupling of H2O to a cellulosic-like material, the closing of the resurrection plant to protect the plant during a drought 14, focus attention on the dynamics of H2O in these materials. Several analytical models for the diffusion of H2O in various polymeric system have been proposed 15–17.

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Molecular dynamics (MD) studies of H2O diffusion in a number of complex systems, e.g. amorphous polyamide, have been undertaken 18–22. These studies, modest in scope, yield results in qualitative accord with a number of phenomenological models. There is a number of attempts that explain the diffusion of fluid at the molecular scale, e.g. 23–27. Jepps et al. 24 consider two diffusive regimes, at low and high saturation, and attribute to them two different flow models. A similar approach has recently been used to describe the behavior of adsorbed molecules at a larger scale 28 or, by a subdivision of molecular trajectories, to quantify the generalized fast-exchange diffusion equation 27.

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In this paper, we describe the results of an extensive MD study of the transport of H2O in several amorphous cellulosic systems. The systems we study have no simple geometry where concepts like substrate, interface, vicinal would apply. They are polymer-H2O systems in which the polymer, fully engaged with the H2O, evolves in structure as the H2O molecules move within it at fixed H2O content and evolves further in structure as the H2O content evolves. We observe a stop-and-go behavior and introduce a model to organize the analysis of our findings that is tied quantitatively to a microscopic model of H2O motion. This model involves a picture of H2O movement that holds promise for

Introduction

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application beyond the domain of our study, e.g. a modification of diffusion constants. The efficacy of this model justifies the expectation that the style of analysis described here has general validity.

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Materials and Methods In this paper we investigate MD models of H2O in three polymeric systems: amorphous cellulose 29 (AC), amorphous hemicellulose 29 (HC) and a heterogeneous structure comprising crystalline cellulose and galactoglucomannan, a model for a microfibril (μF) 8,30 (Fig. 1). Both polymers are present in large quantities in wood tracheids, being building units of the S2 layer structure of a wood cell wall. Amorphous cellulose and hemicellulose are strongly hydrophilic due to their open porous structure and the presence of three exposed hydroxyl groups per glucose unit with which H2O molecules easily form hydrogen bonds 31,32.

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The MD simulations are carried out using Gromos 53a6 united-atom force field 33,34 with leap-frog algorithm for integration of Newton’s equations of motion using Single Point Charge water molecules in isobaric-isothermal ensemble at 300 K and zero pressure, controlled by Velocity-Rescaling thermostat 35 and anisotropic Berendsen barostat 36. The cut-off radius of the Van der Waals interactions between atoms, implemented as Lennard-Jones potential, is set to 1 nm, the long-range electrostatic interactions are implemented with Particle-Mesh Ewald summation, the energy and pressure are corrected for dispersion forces, and the distance between oxygen and hydrogen in OH bonds is constrained. The simulations are integrated with a time step of 1 fs and periodic boundary conditions are applied in all directions. Prior to relaxation runs, the systems are energy-minimized with steepest-descent and then conjugated-gradient algorithms. The obtained structures, nominally having volume 37, 95, 94 nm3, respectively, and containing 180, 432, 470 basic units, respectively, are formed using the temperature/annealing protocol described previously 29,37,38. As a result, they are characterized by a density of 1.33 (AC), 1.22 (HC) and 1.35 g cm-3 (microfibril) 29.

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Figure 1. Snapshots of the modelled MD structures in dry state: (a) amorphous cellulose, (b) galactoglucomannan, a hemicellulose 29, (c) cellulose microfibril, composed of crystalline cellulose (grey) and hemicellulose (red).

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Molecules of H2O are introduced one by one into the void space between the polymer chains, each single insertion being followed by a 10 ps relaxation run, as described in details in 38. For each system (AC, HC, μF), the water content (number of H2O molecules) is characterized by s=N/Nmax where N is the number of adsorbed water molecules and Nmax is the maximum number of the molecules that can be adsorbed in each system. Nmax is determined from experimentally measured adsorption isotherms and equal to 800, 2900 and 1000, for AC, HC and μF, respectively. As a preface to the studies of H2O diffusion, the systems have been studied in the range 0≤s≤1 as equilibrium 2 ACS Paragon Plus Environment

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thermodynamic systems; the chemical potential, elastic constants and porosity have been determined 29,37,38. Porosity is understood as the volume of the voids, when all water is removed, divided by the total volume, whereas the free volume is the remaining void volume when water molecules are still present. Both quantities are determined as the acceptance ratios, measured using an H20 molecule as the probe, an obvious choice in a H20-polymer interactions study. Typically, a water molecule is inserted 106 times per structure and per s, therefore the free space is properly sampled. This common approach was used for example by Lourenco et al. 39 using the Van der Waals radii suggested by Bondi 40. Additionally, the behavior of the hydrogen bond structure has been studied 29,37.

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To calculate the diffusion coefficient a system is brought to thermal equilibrium at s by energy minimization and equilibration and the Mean Square Displacement (MSD) of N(s) H2O molecules, followed over the subsequent tmax=2 ns, is averaged

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= lim → ∑

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The diffusion coefficient is obtained from the slope of the linear part of an MSD curve, typically after 600 ps. We note that within 2 ns the molecules travel on average a distance of 2 nm which is almost 2/3 of the size of the simulation box and, by running longer 10 ns simulations, it was found that 2 ns is sufficient to extract a valid diffusion coefficient (Fig. 2a). We complement the determination of D with the examination of the motion of individual molecules. We observe that the motion of an H2O molecule is built up of alternating segments of moving (‘go’) and ‘standing still’ (‘stop’ or ‘waiting’). The relative amount of time spent in the two states evolves as s evolves. We quantify the description of H2O motion using these two states. We define a criterion for ‘stop’: an H2O molecule is waiting at time t, ti≤t≤ti+1, if the deviation of the displacement from its average value is smaller than 0.1 nm (OH bond length) and (b) the rate of change of its squared displacement is smaller than 8×10-8 nm2 ns-1, in order to eliminate very slow but progressive movement. This value is selected by an observation of H2O molecules movement at low saturation (Fig. 2b). The waiting time is then τwi= ti+1-ti. As waiting and moving succeed one another, a trajectory is a sequence of alternating segments (Fig. 2b). We define two average times, waiting and moving, at each s:

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= ∑ ∑ , ! = ∑ ∑ ! (2)

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where Pi is the number of changes of state of an H2O molecule. The frequency of switching between states during the simulation time is ν, given by

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" =

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We necessarily have

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$ = $ + $! = ∗ "$ + ! ∗ "$, " = + !

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We define a microscopic diffusion coefficient, Dµ, related to the time during which a particle is moving, tm=τmνt, schematically,

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= lim

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and the related tortuosity ξ as

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! " = * ! " (5)

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where Dbulk is the diffusion coefficient of bulk liquid H2O. The microscopic diffusion coefficient describes the molecular diffusion in the porous material as if the molecules did not wait at the sorption sites, by “removing” the waiting time. We employ the definition of tortuosity as related to the complexity of the path an H2O molecule follows during its trajectory, as defined for example by Johnson et al. 41. With this definition, at large water content a water molecule is thought of simply be in bulk water. The separation of particle motion into two distinct states is qualitatively similar to the treatment of H2O in the rather different context of bulk/vicinal water in a pore studied by Levitz et al.

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Finally we suggest that a quantitative understanding of τw can be provided by the transition-state formula 42 for escape over a barrier

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= 23 4

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where fa is an attempt frequency and ϵB is an activation energy of an H2O to an adsorption site. The adopted physical picture is that the motion of a molecule through the system has no reason to be characterized by a particular frequency except during those periods when it is effectively stopped at a binding site where a molecule experiences locally an approximately harmonic potential. Occurrences of this harmonic potential are then found in the analysis of time series of particles displacement.

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Results We present the results for three different hydrophilic polymers in order to clarify that the model is not case-dependent. The diffusion coefficient, which we show in Fig. 2c for each case, although differing slightly among the systems, displays a similar behavior for all, i.e. an increase of about one order of magnitude as s evolves from 0 to 1. We note that the associated relative error on the data ranges between 5-10% and decreases with s. The values of D at s=1 are in agreement with measurements 6,7,10,11,18 and a similar exponential relation is reported by Topgaard and Söderman 26 that used NMR to characterize the water adsorbed in cellulose fibers. For ν, τw and τm we find the results shown in Fig. 2d-f. At s→0 the amount of time spent at an adsorption site is comparable to the amount of time spent moving, τw/τm≈1, whereas as s→1 the waiting time is small compared to the moving time, τm/τw≈40. Qualitatively, a similar picture has been found by Malani and Ayappa 43 where, by studying the dynamics of rotational jumps of water molecules confined in a hydrophilic pore system, they demonstrated that the water in the contact layer possesses a ‘residence time’ that is approximately 10 times larger than that of bulk water. The longer waiting time is a consequence of the stronger hydrogen bonds and stronger electrostatic interactions. The waiting time at saturation obtained in this study is larger than the measured residence time of a bulk water, 2-4 ps 43,44, as bulk water is obviously devoid of solid-water interactions.

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We note that the nature of sorption sites of cellulose and hemicellulose, that causes molecules to wait, is their electrostatic attractiveness to water molecules as the sorption sites are strongly polarized and form hydrogen bonds. For example, in the applied force field, the difference of the charge between the oxygen and the hydrogen of a hydroxyl group is 1.051e, whereas, for comparison, the corresponding difference in a SPC model of water molecules amount to 1.23e. These large charge shifts result in a strong attractive forces between the water molecules and sorption sites, which leads to formation of hydrogen bonds.

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, 89 = :9 ; ln23 (7)

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Figure 2. (a) MSD curves in log-log plot for different s. (b) Example of a single water molecule trajectory with highlighted waiting periods. Basic results: (c) diffusion coefficient, (d) average frequency of stop&go periods, (e) average waiting time, (f) average moving time.

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After an extraction of Dµ from D, we find the result shown in Fig. 3a and the associated tortuosity shown in Fig. 3b. We find Dµ to monotonically increase with s. The tortuosity, a measure that uses D in bulk H2O as reference (determined by MD, Dbulk=4.17), decreases by a factor of about 10 in going from s=0 to s=1. We note that we documented in 29 that the corresponding porosity of AC and HC increases by about a factor of 10 in going from s=0 to s=1. We note that Topgaard and Söderman attributed the difference between the measured diffusion coefficient and that of bulk water to tortuosity and that they found tortuosity to decrease with H2O content 26. In a similar way as used in the fast-exchange diffusion model 27, the obtained diffusion coefficient D(s) may be interpreted as the fraction of H2O molecules found instantaneously in the mobile state multiplied by the microscopic diffusion coefficient Dµ. (a) (b)

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Figure 3. Output of the model: (a) microscopic diffusion coefficient and (b) tortuosity.

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The decrease in tortuosity can be understood by analyzing the system porosity (Fig. 4a). The porosity at a given H2O content is estimated by probing the pore space with a water molecule and calculating the corresponding acceptance ratio 29. As a result of adsorption, porosity increases in a linear manner (s>0.2 for μF) and this increase is accompanied by a decreasing tortuosity, as the pores merge and expand 29. A diffusion coefficient increasing with pore size has been also found by Jepps et al. 23. The free volume, i.e. the pore volume minus the volume occupied by H2O molecules is estimated at different H2O content and presented in Fig. 4b. The free volume related to the total volume decreases as the adsorbed amount increases, in particular for s0.5, the percentage of the free volume remains at a constant level, below 10%. For this reason, the increasing diffusion coefficient cannot be explained on the basis of the free volume theory 45, attributing an increase in diffusion to an increase in the free volume available on average for a water molecule, which justifies the choice of our approach.

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Figure 4. (a) Porosity and (b) free volume as related to the total volume, as a function of H2O content.

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To employ the transition state formula, Eq. (7), with τw in hand, we need fa or ϵB. In order to find fa, we go to s→0 and examine the Fourier transform of the displacement of H2O molecules during the times that are defined as waiting times. This Fourier transform has a single peak against a background that we remove by considering the motion of a H2O molecule in bulk water, i.e. a system without adsorption sites. What results from this procedure is the Fourier spectrum shown in Fig. 5a.

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We take fa to be the frequency of oscillatory motion of an H2O molecule at an adsorption site to be 2.4×1013 s-1 for AC, 1.5×1013 s-1 for HC and 2.6×1013 s-1 for μF. Frequencies of this order are sensibly related to the expected behavior of H2O, e.g. the Debye temperature of water corresponds to a frequency of order 1013 s-1 46. Employing these frequencies at all s leads to the "barrier height" or activation (binding) energy, ϵB, as a function of s shown in Fig. 5b. The activation energy decreases slightly as s increases. The activation energy at s→0, e.g. ϵB(0)=22.6 kJ·mol-1 for AC, also found by other methods 18, is close to the corresponding difference in chemical potential at s→0, which amounts to 21.4 kJ mol-1 38. This implies that, in the limit s→0, the change in chemical potential is the activation energy of hydrogen bonds. It is assumed that at a given saturation the water molecules are uniformly distributed in the system and the adsorption sites have similar energies. However, the calculated energy is an average energy of the binding sites at a given saturation and does not have to be identical for all sorption sites.

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The experience of a molecule in the system is approximated as made up of two states. The molecule is trapped at a binding site or free to move diffusively in the bulk. The energy ϵB is the activation energy that must be provided by thermal fluctuations to promote the molecule from a binding site (where it waits) into the bulk (where it is able to move diffusively).

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Figure 5. Model of a binding site: (a) Fourier transform of MSD during waiting periods, after background removal at s→0, (b) estimated binding energy at a binding site.

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Dry amorphous cellulosic material comprises a "backbone structure", whose integrity is partially enforced by H-bonds, and that is entangled with itself, the entanglement being partially supported also by H-bonds. The introduction of H2O molecules, having an affinity for H-bonds, into such a system results in breaking of the H-bonds leading to an unfolding of the self-entanglement and a modification of the structure. As H2O molecules are introduced, the system swells. The pore space, that necessarily accompanies the disorganization associated with amorphous packing, evolves, becomes larger, as the porosity of HC and AC increases by an order of magnitude in going from a degree of saturation s=0 to s=1, and is re-arranged leading to a decrease in tortuosity. The resulting re-arrangement feeds back on the H2O molecules offering them different/additional opportunities to bond. Added to the dynamics conferred by this geometry is that from thermal fluctuations. Thus the H2O molecules, whose diffusion coefficient we have measured, move through a porosity that is a dynamic entity changing with degree of saturation.

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Conclusion This qualitative picture is made quantitative with the MD simulation. The changes in the diffusion coefficient, as s increases, result from changes in the geometry of the system and in the encounters of H2O molecules with this geometry. At s→0 the H2O molecules spend a large fraction of their time bound, in essential isolation, and when liberated move in a tortuous environment characterized by: small porosity (φ≈0.1), large waiting time (τm/(τw+τm)≈0.4), activation energy close to the chemical potential (ϵB≈∆µ(0)) and high tortuosity (ξ≈100). The H2O molecules bound at adsorption sites are not diffusing and therefore the diffusion coefficient at low s is low. On increase in s, the H2O molecules, in competition with one another for binding sites that are less attractive due to superposed Coulomb interactions, spend more time moving in an environment opened up by swelling and less tortuous. At s→1, a system is characterized by large porosity (φ≈10·φ(0)), negligible waiting time (τm/(τw+τm)≈1), smaller activation energy (ϵB≈0.85ϵB(s→0)) and lower tortuosity (ξ≈10). As a consequence the diffusion coefficient increases with a factor 100 compared to that of the dry material. We note that, although the presented model does not involve porosity directly, it is implicitly involved in the tortuosity, as the increase in porosity is accompanied by an opening of the pore network. The decrease in tortuosity is a result of unfolding chains, an increase in porosity and pore size, as well as merging of water clusters 29,38. This quantification description rests importantly on the two-state picture of the H2O molecules experience in the system. This picture is given support by the connection we have established between the waiting time τw, the attempt frequency fa and the activation energy ϵB. We have shown that water diffusion in hydrophilic systems can be 8 ACS Paragon Plus Environment

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understood by the notion of ‘stop&go’ periods. This model can be used to predict a change in diffusion coefficient resulting from a potential modification of the activation energy of adsorption sites. It may lead to an improved design of functionalized polymer systems, tracking their structural change, swelling and diffusion upon increase of H2O content.

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FOR GRAPHICAL TABLE OF CONTENT ONLY

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Figure 2. (a) MSD curves in log-log plot for different s. (b) Example of a single water molecule trajectory with highlighted waiting periods. Basic results: (c) diffusion coefficient, (d) average frequency of stop&go periods, (e) average waiting time, (f) average moving time. 69x61mm (600 x 600 DPI)


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Water Diffusion in Amorphous Hydrophilic Systems: A Stop and Go

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