# A Molecular Interpretation of the Dynamics of Diffusive Mass Transport of Water within a Glassy Polyetherimide

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.1.1. Relevant Results on Equilibrium Thermodynamics of a PEI–Water System

^{−1}), which were isolated by Difference Spectroscopy (DS), upon elimination of the polymer matrix interference [11]. Using this approach, it was also possible to determine the evolution with sorption time of the ν(OH) profile. The complex, partially resolved pattern, suggesting the occurrence of more than one species of penetrant, was interpreted with the aid of two-dimensional correlation analysis [10]. It was concluded that two couples of signals are present, each belonging to a distinct water species. In particular, the sharp peaks at 3655–3562 cm

^{−1}were assigned to isolated water molecules interacting via H-bonding with the PEI backbone (cross-associated or first shell water molecules). The first shell adsorbate was found to have a 2:1 stoichiometry, with a single water molecule bridging two carbonyls (i.e., –C=O∙∙∙H–O–H∙∙∙O=C–). A second doublet at 3611–3486 cm

^{−1}was associated with water molecules self-interacting with the first shell species through a single H-bonding (self-associated or second-shell water molecules). Analysis of the substrate spectrum revealed that the active sites (proton acceptors) on the polymer backbone are the imide carbonyls, while the involvement of the ether oxygens is negligible, if any.

_{2}O system, evidenced that the predictions on HB formation at equilibrium agree very well with the experimental results obtained by in situ infrared spectroscopy and with the theoretical results obtained by MD simulations [10]. This is evident in Figure 2, where this comparison is reported with reference to the HB interactions actually established in the system, i.e., water self-interactions (indicated by “11” subscript) and water–PEI (carbonyl group) cross-interactions (indicated by “12” subscript) as a function of mass fraction of water, ω

_{1}. In Figure 2 n

_{11}/m

_{2}and n

_{12}/m

_{2}stand for the mmol of H-bond formed per gram of polymer.

**Figure 2.**Comparison of the predictions of the NRHB-NETGP model for the amount of self and cross-HBs with the outcomes of FTIR spectroscopy and of MD simulations. Reprinted with permission from the authors of [10]. Copyright (2017) American Chemical Society.

_{2}O in glassy PEI. This expression will be used in the present contribution to evaluate the thermodynamic factor that appears in the theoretical expression of PEI/H

_{2}O mutual diffusivity, as detailed in the following section.

#### 1.1.2. Mutual vs. Intra-Diffusion Coefficients

_{i}, ${\rho}_{i}$, ${\omega}_{i}$, and $\overline{{\upsilon}_{i}}$ represent, respectively, the molar concentration of component i, the mass of component i per volume of mixture, the mass fraction of component i, and the partial molar volume of component i. The symbol ${\underset{\xaf}{u}}_{i}$ represents the velocity of molecules of component i, referred to a lab fixed frame of reference. In the present context, we deal with the specific case of penetrant–polymer mixtures and we will refer to the penetrant with subscript 1 and to the polymer with the subscript 2. From the previous Equations (1) and (2) it is readily derived that, as in Equations (5) and (6),

_{12}is the mutual diffusivity of the “12” system. Note that, as (see Equation (9))

_{12}, is defined intrinsically by Equation (7) for both components. In fact, as we will see in the following, this coefficient is a property of the binary system and is a function of temperature and concentration.

_{0}= 0/0.6), the weight fraction of water within the polymer is always lower than 0.01. This implies that no relevant stresses develop as consequence of water sorption. The low amount of penetrant absorbed combined with the absence of polymer swelling allows also the assumption of a constant mixture density. In addition, the bulk velocity of the polymer/water mixture can be considered to be negligible (i.e., ${\underset{\xaf}{u}}^{M}\cong 0;{\underset{\xaf}{u}}^{V}\cong 0$), in view of the low intrinsic mobility of polymer (i.e., ${\underset{\xaf}{u}}_{2}\cong 0$), that is the largely prevailing component. Moreover, Equation (7) can be taken as a constitutive expression of the diffusive mass flux as, in the case at hand, other driving forces beside the composition gradient can be ruled out. In fact, (i) the driving force related to the difference of the body forces acting per unit of mass of each component is equal to zero since in this case they are only associated to the gravitational field, (ii) the driving force related to the gradient of temperature is zero in view of the isotheral condition, and (iii) the driving force related to the gradient of pressure is zero in view of the uniformity of the state of stress. Finally, note that the transport of fluids in polymers is slow enough to assure that also the inertial contributions can be neglected. Therefore, as in Equation (11),

_{12}independent of composition, takes the form of the so-called Fick’s second law [15], as in Equation (13):

_{i}represents the activity coefficient of component i in the binary system. The equality of the two mutual volumetric diffusion coefficients appearing in Equations (14) and (15), follows from the Gibbs–Duhem equation and from the definition of volume average velocity.

_{i}represents the molecular molar weight of component i.

_{1}and D

_{2}. Expressions have been proposed relating the mutual diffusivity in a binary mixture, D

_{12}, or, similarly, any other type of mutual-diffusion coefficient referred to a different frame of reference, to the intra-diffusion coefficients of the two components.

_{12}, D

_{1}, and D

_{2}—can be expressed in terms of the molecular friction coefficients (in the case at hand, penetrant–penetrant, polymer–polymer, and penetrant–polymer friction coefficients, respectively, denoted by ζ

_{11}, ζ

_{22}, and ζ

_{12}) [16], as in Equations (18)–(20).

_{A}is the Avogadro’s number.

_{1}and D

_{2}, that do not need the knowledge of friction coefficients. However, since three friction coefficients appear in Equations (18)–(20) in general, it is not possible to express D

_{12}, only in terms of D

_{1}and D

_{2}(i.e., with no friction coefficients). This is, however, possible if special circumstances occur, e.g., if one is able to write a relationship linking the three friction coefficients, or if one considers the limit of trace amount of penetrant, or in the cases where D

_{1}> D

_{2}. For instance, assuming that ζ

_{12}is the geometric mean of ζ

_{11}and ζ

_{22}[18,19] or, alternatively, assuming that the ratio between the friction coefficients is constant [16] it is possible to obtain the following relationship, as in Equation (21):

_{i}represent, respectively, the molar chemical potential and the molar fraction of component i, and P and T represent, respectively, the spatial uniform pressure and temperature of the binary mixture. In the present context D

_{1}> D

_{2}and the water molar fraction range is approximately 0.94–0.97, thus assuring that it is also ${D}_{1}{x}_{2}>{D}_{2}{x}_{1}$. Therefore, the relationship (21) reduces to an explicit relationship relating the measured mutual-diffusion coefficient ${D}_{12}$ just, to the intra-diffusion coefficient of water D

_{1}, as in Equation (22):

_{12}from Equation (22), one then needs to know the expressions of D

_{1}and μ

_{1}. In the present investigation, the value of the intra-diffusion coefficient has been retrieved from MD simulations of a PEI/H

_{2}O system with uniform concentration, by averaging the statistics of the evolution of the diffusion path with time of each single water molecule. The estimate of μ

_{1}as a function of concentration has been instead obtained by using the NRHB-NETGP thermodynamic model. The parameters of this for the water/PEI system model are available in a previous publication by our group [10]. The set of equations involved in the calculation of μ

_{1}according to the NRHB-NETGP must be solved numerically, so that only an implicit expression for the penetrant molar chemical potential as a function of concentration at a given pressure and temperature is available. Therefore, the derivative of the NRHB-NETGP penetrant molar chemical potential appearing in Equation (22) has been evaluated numerically. In particular, it has been estimated assuming a centered difference finite scheme with a variable concentration step equal to ${10}^{-6}{c}_{1}$. This step has provided an excellent compromise between the accuracy of the approximated numerical scheme adopted and the round-off error deriving from the finite digit arithmetic associated to the calculator used.

_{12}, obtained from Equation (22) for the PEI/H

_{2}O system, based on information provided by MD calculation for D

_{1}and by NRHB-NETGP model of mixture thermodynamics for μ

_{1}, will be compared with the experimental values obtained independently by in situ time-resolved infrared spectroscopy.

_{12}, and intra-diffusion coefficient of the penetrant, D

_{1}, converge to the same value [18].

## 2. Results and Discussion

#### 2.1. Determination of Mutual Diffusivity of the Water–PEI System from Vibrational Spectroscopy

^{−1}). In fact, difference spectra can be collected in this range as a function of time, providing an accurate evaluation of the sorption/desorption kinetics.

^{−1}) and L is the sample thickness. Equation (23) has been used to best fit experimental sorption kinetics data using D

_{12}as fitting parameter, assumed to be independent on concentration.

^{−8}cm

^{2}/s, that is in good agreement with previous literature reports on commercial polyimides [21].

_{0}interval from 0 to 0.6 performing “differential sorption tests”, i.e., increasing stepwise by a 0.1 increment the relative pressure of H

_{2}O vapor. The related Fick’s diagrams are reported in Figure 4. Consistently with the assumption of a constant diffusivity, the D

_{12}values obtained from the fitting of kinetics data using Equation (23) are rather independent of concentration (average value: D

_{12}= 1.55 × 10

^{−8}± 0.03 × 10

^{−8}cm

^{2}/s); only at p/p

_{0}= 0.1 the diffusivity is appreciably lower (D

_{12}= 1.37 × 10

^{−8}± 0.03 × 10

^{−8}cm

^{2}/s).

#### 2.2. Molecular Dynamics Simulations

System | Box (nm^{3}) | $\mathit{\omega}$ | Water Molecules | Total Particles | Simulation Time (ns) |
---|---|---|---|---|---|

I | 6.31000 | - | 0 | 22,680 | 120 |

II | 6.31308 | 0.0042 | 86 | 22,938 | 198 |

III | 6.31358 | 0.0048 | 100 | 22,980 | 200 |

IV | 6.31589 | 0.0057 | 120 | 23,040 | 200 |

V | 6.31980 | 0.007 | 150 | 23,130 | 200 |

VI | 6.32867 | 0.01 | 220 | 23,340 | 240 |

^{−8}cm

^{2}/s at a vanishingly small water concentration, thus confirming the consistency of MD simulations.

#### 2.3. Comparison of Theoretical Predictions with Results of Vibrational Spectroscopy

_{12}, determined experimentally by FTIR spectroscopy and discussed in Section 2.1, and the values of this coefficient predicted using in Equation (22) the values of D

_{1}estimated by MD calculations and the values of ${\left(\frac{\partial {\mu}_{1}}{\partial \mathrm{ln}{x}_{1}}\right)}_{T,P}$ estimated using the NRHB-NETGP model. In order to compare experimental results with theoretical findings, values of mutual diffusivity estimated from the experimental differential sorption steps by fitting sorption kinetics using Equation (23) are reported as a function of the average water mass fraction present within the polymer during the test, calculated as the arithmetic average of the uniform initial and final water mass fraction.

_{12}seemingly approach the experimental values when water concentration tends to zero. We remind that, in this limit, the theoretical values of D

_{12}and D

_{1}tend to the same value. Conversely, as the concentration increases, a gradually increasing departure of the theoretical values of D

_{12}from the experimental values is evident. This mismatch could be attributed to the fact that, as reported in literature [22], the MD approach implemented here is reliable at quite low penetrant concentration while it provides a progressively increasing overestimation of the dependence of intra-diffusion coefficient as the water concentration increases and, in turn, an increasing overestimation of D

_{12}values.

#### 2.4. H-Bond Lifetimes

_{ij}(t) equals unity if the particular tagged pair of molecules is hydrogen bonded and is zero otherwise. The sums are over all pairs and t

_{0}is the time at which the measurement period starts (C(0) =1).

_{ij}variable is allowed to make just one transition from unity to zero when the H-bond is first observed to break, but is not allowed to return to unity should the same bond reform subsequently. On the basis of the results reported in [10], we confined our analysis to one kind of HB acceptor that is the carboxylic group (defined as AC1 in [10]). In Figure 9, C(t) corresponding to system II (the one containing 86 water molecules) and system VII (the one containing 220 water molecules) is reported. Interestingly, no reasonable fitting of C(t) correlation functions was obtained using a single exponential decay (see dashed lines in both figures). Instead, a better agreement has been obtained using a sum of two different exponential decays, i.e., in Equation (26):

**Table 2.**HB Lifetime values and relative population weights (A

_{1}and A

_{2}) obtained fitting the HB autocorrelation functions by Equation (26).

System | τ_{1} | τ_{2} | A_{1} | A_{2} |
---|---|---|---|---|

II | 4.4 ps | 2362 ps | 0.62 | 0.38 |

VI | 3.9 ps | 118 ps | 0.67 | 0.33 |

## 3. Experimental

#### 3.1. Materials

#### 3.2. FTIR Spectroscopy

_{0}) were collected in the transmission mode, monitoring the characteristic signature of the penetrant up to the attainment of sorption equilibrium. The sorption experiments were performed in a custom designed, vacuum-tight cell positioned in the sample compartment of the spectrometer. This cell was connected through service lines, to a water reservoir, a turbo-molecular vacuum pump, and pressure transducers. Full details of the experimental setup are reported in [29]. Before each sorption measurement, the sample was dried under vacuum overnight at the test temperature in the same measuring apparatus. The FTIR spectrometer was a Spectrum 100 from PerkinElmer (Norwalk, CT, USA), equipped with a Ge/KBr beam splitter and a wide-band deuterated triglycine sulfate (DTGS) detector. Parameters for data collection were set as follows: resolution = 2 cm

^{−1}, optical path difference (OPD) velocity: 0.5 cm/s, and spectral range: 4000−600 cm

^{−1}. A single spectrum collection took 2.0 s to complete under the selected instrumental conditions. Continuous data acquisition was controlled by a dedicated software package for time-resolved spectroscopy (Timebase from PerkinElmer, Norwalk, CT, USA). The Absorbance spectrum of the penetrant was obtained by use of the single-beam spectrum of the cell containing the dry sample as background [10].

#### 3.3. MD Simulations

#### 3.3.1. Polymer Model

#### 3.3.2. Simulation Details

^{-−1}, a timestep of 1 fs, and density field density update performed every 0.1 ps. For more details see in [36,37,39].

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mensitieri, G.; Iannone, M. Modeling accelerated ageing in polymer compsites. In Ageing of Composites; Martin, R., Ed.; CRC Press: Boca Raton, FL, USA, 2008; pp. 224–282. [Google Scholar]
- Li, D.; Yao, J.; Wang, H. Thin films and membranes with hierarchical porosity. In Encyclopedia of Membrane Science and Technology; Hoek, E.M.V., Tarabara, V.V., Eds.; John Wiley & Sons, Inc.: Chichester, UK, 2013. [Google Scholar]
- Bernstein, R.; Kaufman, Y.; Freger, V. Porosity. In Encyclopedia of Membrane Science and Technology; Hoek, E.M.V., Tarabara, V.V., Eds.; John Wiley & Sons, Inc.: Chichester, UK, 2013. [Google Scholar]
- Brunetti, A.; Barbieri, G.; Drioli, E. Analytical applications of membranes. In Encyclopedia of Membrane Science and Technology; Hoek, E.M.V., Tarabara, V.V., Eds.; John Wiley & Sons, Inc.: Chichester, UK, 2013. [Google Scholar]
- Koros, W.J.; Fleming, G.K.; Jordan, S.M.; Kim, T.H.; Hoehn, H.H. Polymeric Membrane Materials for Solution-Diffusion Based Permeation Separations. Prog. Polym. Sci.
**1988**, 13, 339. [Google Scholar] [CrossRef] - Christie, S.; Scorsone, E.; Persaud, K.; Kvasnik, F. Remote Detection of Gaseous Ammonia Using the near Infrared Transmission Properties of Polyaniline. Sens. Actuators B. Chem.
**2003**, 90, 163. [Google Scholar] [CrossRef] - Scherillo, G.; Petretta, M.; Galizia, M.; La Manna, P.; Musto, P.; Mensitieri, G. Thermodynamics of Water Sorption in High Performance Glassy Thermoplastic Polymers. Front. Chem.
**2014**, 2, 25. [Google Scholar] [CrossRef] [PubMed][Green Version] - Scherillo, G.; Galizia, M.; Musto, P.; Mensitieri, G. Water Sorption Thermodynamics in Glassy and Rubbery Polymers: Modeling the Interactional Issues Emerging from Ftir Spectroscopy. Ind. Eng. Chem. Res.
**2013**, 52, 8674. [Google Scholar] [CrossRef][Green Version] - Scherillo, G.; Sanguigno, L.; Galizia, M.; Lavorgna, M.; Musto, P.; Mensitieri, G. Non-Equilibrium Compressible Lattice Theories Accounting for Hydrogen Bonding Interactions: Modelling Water Sorption Thermodynamics in Fluorinated Polyimides. Fluid Phase Equilib.
**2012**, 334, 166. [Google Scholar] [CrossRef] - De Nicola, A.; Correa, A.; Milano, G.; La Manna, P.; Musto, P.; Mensitieri, G.; Scherillo, G. Local Structure and Dynamics of Water Absorbed in Poly(Ether Imide): A Hydrogen Bonding Anatomy. J. Phys. Chem. B
**2017**, 121, 3162–3176. [Google Scholar] [CrossRef] - Mensitieri, G.; Scherillo, G.; Panayiotou, C.; Musto, P. Towards a Predictive Thermodynamic Description of Sorption Processes in Polymers: The Synergy between Theoretical Eos Models and Vibrational Spectroscopy. Mater. Sci. Eng.
**2020**, 140, 100525. [Google Scholar] [CrossRef] - Panayiotou, C.; Pantoula, M.; Stefanis, E.; Tsivintzelis, I.; Economou, I.G. Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 1. Pure Fluids. Ind. Eng. Chem. Res.
**2004**, 43, 6592. [Google Scholar] [CrossRef] - Panayiotou, C.; Tsivintzelis, I.; Economou, I.G. Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 2. Multicomponent Mixtures. Ind. Eng. Chem. Res.
**2007**, 46, 2628. [Google Scholar] [CrossRef] - Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena; John Wiley & Sons: New York, NY, USA, 2007. [Google Scholar]
- Crank, J. The Mathematics of Diffusion; Claredon Press: Oxford, UK, 1975. [Google Scholar]
- Bearman, R.J. On the Molecular Basis of Some Current Theories of Diffusion1. J. Phys. Chem.
**1961**, 65, 1961–1968. [Google Scholar] [CrossRef] - Caruthers, J.M.; Chao, K.C.; Venkatasubramanian, V.; Sy-Siong-Kiao, R.; Novenario, C.R.; Sundaram, A. Handbook of Diffusion and Thermal Properties of Polymers and Polymer Solutions; Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York, NY, USA, 1998. [Google Scholar]
- Vrentas, J.S.; Duda, J.L. Diffusion in Polymer–Solvent Systems. Ii. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecular Weight. J. Polym. Sci.
**1977**, 15, 417. [Google Scholar] [CrossRef] - Vrentas, J.S.; Vrentas, C.M. Predictive Methods for Self-Diffusion and Mutual Diffusion Coefficients in Polymer–Solvent Systems. Eur. Polym. J.
**1998**, 34, 797. [Google Scholar] [CrossRef] - Musto, P.; La Manna, P.; Cimino, F.; Mensitieri, G.; Russo, P. Morphology, Molecular Interactions and H
_{2}O Diffusion in a Poly(Lactic-Acid)/Graphene Composite: A Vibrational Spectroscopy Study. Spectrochim. Acta A Mol. Biomol. Spectrosc.**2019**, 218, 40–50. [Google Scholar] [CrossRef] - Musto, P.; Ragosta, G.; Mensitieri, G.; Lavorgna, M. On the Molecular Mechanism of H2o Diffusion into Polyimides: A Vibrational Spectroscopy Investigation. Macromolecules
**2007**, 40, 9614. [Google Scholar] [CrossRef] - Milano, G.; Guerra, G.; Müller-Plathe, F. Anisotropic Diffusion of Small Penetrants in the Δ Crystalline Phase of Syndiotactic Polystyrene: A Molecular Dynamics Simulation Study. Chem. Mater.
**2002**, 14, 2977–2982. [Google Scholar] [CrossRef] - Rapaport, D.C. Hydrogen Bonds in Water. Mol. Phys.
**1983**, 50, 1151–1162. [Google Scholar] [CrossRef] - Chowdhuri, S.; Chandra, A. Dynamics of Halide Ion−Water Hydrogen Bonds in Aqueous Solutions: Dependence on Ion Size and Temperature. J. Phys. Chem. B
**2006**, 110, 9674–9680. [Google Scholar] [CrossRef] [PubMed] - Luzar, A. Resolving the Hydrogen Bond Dynamics Conundrum. J. Chem. Phys.
**2000**, 113, 10663. [Google Scholar] [CrossRef][Green Version] - Luzar, A.; Chandler, D. Hydrogen-Bond Kinetics in Liquid Water. Nature
**1996**, 379, 55–57. [Google Scholar] [CrossRef] - Luzar, A.; Chandler, D. Effect of Environment on Hydrogen Bond Dynamics in Liquid Water. Phys. Rev. Lett.
**1996**, 76, 928–931. [Google Scholar] [CrossRef] [PubMed] - Mijović, J.; Zhang, H. Molecular Dynamics Simulation Study of Motions and Interactions of Water in a Polymer Network. J. Phys. Chem. B
**2004**, 108, 2557. [Google Scholar] [CrossRef] - Cotugno, S.; Larobina, D.; Mensitieri, G.; Musto, P.; Ragosta, G. A Novel Spectroscopic Approach to Investigate Transport Processes in Polymers: The Case of Water–Epoxy System. Polymer
**2001**, 42, 6431. [Google Scholar] [CrossRef] - Jorgensen, W.L.; Nguyen, T.B. Monte Carlo Simulations of the Hydration of Substituted Benzenes with Opls Potential Functions. J. Comput. Chem.
**1993**, 14, 195. [Google Scholar] [CrossRef] - Jorgensen, W.L.; Maxwell, D.S.; TiradoRives, J. Development and Testing of the Opls All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids. J. Am. Chem. Soc.
**1996**, 118, 11225. [Google Scholar] [CrossRef] - Price, M.L.P.; Ostrovsky, D.; Jorgensen, W.L. Gas-Phase and Liquid-State Properties of Esters, Nitriles, and Nitro Compounds with the Opls-Aa Force Field. J. Comput. Chem.
**2001**, 22, 1340. [Google Scholar] [CrossRef] - Milano, G.; Muller-Plathe, F. Cyclohexane;Benzene Mixtures:Thermodynamics and Structure from Atomistic Simulations. J. Phys. Chem. B
**2004**, 108, 7415. [Google Scholar] [CrossRef] - Tironi, I.G.; Sperb, R.; Smith, P.E.; van Gunsteren, W.F. A Generalized Reaction Field Method for Molecular Dynamics Simulations. J. Chem. Phys.
**1995**, 102, 5451. [Google Scholar] [CrossRef] - Berendsen, H.J.C.; Postma, J.P.M.; van Gunsteren, W.F.; Hermans, J.; Pullman, B. Intermolecular Forces. In Proceedings of the Fourteenth Jerusalem Symposium on Quantum Chemistry and Biochemistry, Jerusalem, Israel, 13–16 April 1981; p. 331. [Google Scholar]
- Milano, G.; Kawakatsu, T. Hybrid Particle-Field Molecular Dynamics Simulations for Dense Polymer Systems. J. Chem. Phys.
**2009**, 130, 214106. [Google Scholar] [CrossRef] - Milano, G.; Kawakatsu, T. Pressure Calculation in Hybrid Particle-Field Simulations. J. Chem. Phys.
**2010**, 133, 214102. [Google Scholar] [CrossRef] - Zhao, Y.; De Nicola, A.; Kawakatsu, T.; Milano, G. Hybrid Particle-Field Molecular Dynamics Simulations: Parallelization and Benchmarks. J. Comput. Chem.
**2012**, 33, 868. [Google Scholar] [CrossRef] - De Nicola, A.; Kawakatsu, T.; Milano, G. Generation of Well-Relaxed All-Atom Models of Large Molecular Weight Polymer Melts: A Hybrid Particle-Continuum Approach Based on Particle-Field Molecular Dynamics Simulations. J. Chem. Theory Comput.
**2014**, 10, 5651. [Google Scholar] [CrossRef] [PubMed] - Van der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A.E.; Berendsen, H.J.C. Gromacs: Fast, Flexible, and Free. J. Comput. Chem.
**2005**, 26, 1701. [Google Scholar] [CrossRef] [PubMed] - Berendsen, H.J.C.; Postma, J.P.M.; Gunsteren, W.F.v.; DiNola, A.; Haak, J.R. Molecular Dynamics with Coupling to an External Bath. J. Chem. Phys.
**1984**, 81, 3684. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The water species identified spectroscopically, with indication of the signals they produce.

**Figure 3.**Fick’s plot (A(t)/A(∞) vs √t) for the sorption test at p/p

_{0}= 0.6 at T = 303.15 K. The inset displays the time-evolution of the analytical band.

**Figure 4.**Fick’s plots (A(t)/A(∞) vs √t) for the “differential sorption tests” performed in the p/p

_{0}interval 0–0.6.

**Figure 5.**Values of water–PEI mutual diffusivity, D

_{12}, determined from FTIR spectroscopy reported as a function of relative pressure of H

_{2}O vapor at 303.15 K.

**Figure 6.**Intra-diffusion coefficient of water calculated from mean square displacement of molecular dynamics simulations at different water concentration.

**Figure 7.**Distribution of self-diffusion coefficients of water molecules for two compositions: (

**A**) ω = 0.01 and (

**B**) ω = 0.0057. (

**C**) Distribution of water self-diffusion coefficients weighted by the time spent by each water molecule forming hydrogen bonds with acceptor AC1 (only first shell) for both compositions ω = 0.01 (blue points) and ω = 0.0057 (green points).

**Figure 8.**Values of water intra-diffusion coefficient determined from MD simulations, ${D}_{1}^{MD}$, of water–PEI mutual diffusion coefficient determined from Equation (22), ${D}_{12}^{theory}$, and of water–PEI mutual diffusion coefficient determined experimentally from FTIR spectroscopy, ${D}_{1}^{\mathrm{exp}}$.

**Figure 9.**Time behavior of continuous HB correlation functions for System II (

**A**) and System VII (

**B**).

**Scheme 1.**Chemical structure of PEI repeating unit. For sake of clarity, all hydrogen atoms are omitted. Non-bonded interactions over two consecutive bonds are excluded.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Correa, A.; De Nicola, A.; Scherillo, G.; Loianno, V.; Mallamace, D.; Mallamace, F.; Ito, H.; Musto, P.; Mensitieri, G. A Molecular Interpretation of the Dynamics of Diffusive Mass Transport of Water within a Glassy Polyetherimide. *Int. J. Mol. Sci.* **2021**, *22*, 2908.
https://doi.org/10.3390/ijms22062908

**AMA Style**

Correa A, De Nicola A, Scherillo G, Loianno V, Mallamace D, Mallamace F, Ito H, Musto P, Mensitieri G. A Molecular Interpretation of the Dynamics of Diffusive Mass Transport of Water within a Glassy Polyetherimide. *International Journal of Molecular Sciences*. 2021; 22(6):2908.
https://doi.org/10.3390/ijms22062908

**Chicago/Turabian Style**

Correa, Andrea, Antonio De Nicola, Giuseppe Scherillo, Valerio Loianno, Domenico Mallamace, Francesco Mallamace, Hiroshi Ito, Pellegrino Musto, and Giuseppe Mensitieri. 2021. "A Molecular Interpretation of the Dynamics of Diffusive Mass Transport of Water within a Glassy Polyetherimide" *International Journal of Molecular Sciences* 22, no. 6: 2908.
https://doi.org/10.3390/ijms22062908