Sub-diffusion and population dynamics of water confined in soft environments

Samuel Hanot∗

Institut Laue-Langevin - 71 Avenue des Martyrs - CS 20156 - 38042 GRENOBLE CEDEX 9

Sandrine Lyonnard† and Stefano Mossa‡

Univ. Grenoble Alpes, INAC-SPRAM, F-38000 Grenoble, FranceCNRS, INAC-SPRAM, F-38000 Grenoble, France and

CEA, INAC-SPRAM, F-38000 Grenoble, France(Dated: October 11, 2018)

We have studied by Molecular Dynamics computer simulations the dynamics of water confinedin ionic surfactants phases, ranging from well ordered lamellar structures to micelles at low andhigh water loading, respectively. We have analysed in depth the main dynamical features in termsof mean squared displacements and intermediate scattering functions, and found clear evidencesof sub-diffusive behaviour. We have identified water molecules lying at the charged interface withthe hydrophobic confining matrix as the main responsible for this unusual feature, and provided acomprehensive picture for dynamics based on a very precise analysis of life times at the interface. Weconclude by providing, for the first time to our knowledge, a unique framework for rationalising theexistence of important dynamical heterogeneities in fluids absorbed in soft confining environments.

I. INTRODUCTION

Nanoconfinement and interfacial interactions pro-foundly alter thermodynamical, structural and transportproperties of confined fluids. A molecular-level under-standing of these modifications is essential in a diver-sity of fields, ranging from biology [1] to nanoscience [2].A situation of paramount interest is that of wateradsorbed in natural or synthesized soft media, suchas biomolecules, macromolecular gels, polymers, self-assembled and active materials. In particular, it is be-lieved that the function-structure relationship in thesesystems is primarily correlated to the amount of the ad-sorbed fluid and to its peculiar properties. Water in thevicinity of soft charged walls is expected to show sig-nificant deviations from the bulk behaviour, e.g., dis-tortions of the hydrogen bonds network and anomalousdiffusion character, which might in turn profoundly im-pact the final morphology and functional behaviour ofthe confining media. A deeper insight of such complexfluid-matrix interplay can be obtained by using numeri-cal tools. Here, we use Molecular Dynamics simulationsof a self-assembling model system to demonstrate and ra-tionalize the general existence of spatially heterogeneousdynamics in a fluid adsorbed in soft confining environ-ments.

The generic situation of soft mobile confining mediahas been less explored than that of hard confining mate-rials. Detailed studies on confinement in highly stylisedimmobile matrices have been described, which aim atunderstanding the universal phenomena originating fromboth the detailed nature of the fluid/matrix interactionsand the restricted character of the available space. These

∗ hanot@ill.fr† sandrine.lyonnard@cea.fr‡ stefano.mossa@cea.fr

include, together with many other systems, simulationsof water in carbon nanotubes [3], confined between twosmooth hydrophobic plates [4] or patterned walls [5], orin the case of patterned hydrophilic and hydrophobicwalls [6]. In contrast, most studies on water confinedin (disordered) soft matrices are very specific, and focuson particular confining macromolecules, without any con-cern for generality. These investigations include, amongothers, neutron scattering and molecular dynamics simu-lations of tau and MPB proteins [7], dynamics of water inreverse surfactant micelles [8], or simulation of polymerelectrolyte membranes [9, 10]. These molecules, be theyof biological or synthetic origin, are very complex, andone finds extremely difficult to separate properties thatare specific to the particular chemistry of the confiningmatrix, from those that are general and emerge imme-diately when water is in contact with any soft confiningmedia. Still, in general we expect that the main fea-tures of water transport in these systems must conformto some universal description, dictated by the principlesof statistical mechanics.

It is well understood that water transport around pro-teins is highly anomalous [11–14], with sub-diffusive wa-ter molecules playing a crucial role in the function of theprotein. Notwithstanding this well-established observa-tion, the existence of sub-diffusion in synthetic materi-als is scarcely discussed. Surprisingly, most studies onsynthetic materials focus on very complex simulations,with very detailed force fields, but they generally do notcapitalise on this modelling accuracy to provide detailedand up-to-date analysis of the transport properties of theab(d)sorbed fluid. For instance, the dynamics of waterin surfactants is treated only in the reverse micelle regionof the phase diagram [15], and most studies consider wa-ter adsorbed in proton-conducting fuel cells ionomers asa Fickian diffusive system [9, 16]. Yet, the functionalproperties of these materials are also determined by themobility of water confined in soft ionic nanochannels.

arX

iv:1

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0494

1v1

[co

nd-m

at.s

oft]

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Aug

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FIG. 1. Typical simulation snapshots of the ionic surfactants model studied here, at hydration levels λ = 4, 16, 32, fromleft to right. Lamellar, cylindrical and micellar self-organised phases are evident. The hydrophobic section of the surfactantmacromolecules is represented by the grey beads, the charged hydrophilic units by the green ones. Water molecules are indicatedin blue, the hydronium cations in red. These configurations are examples drawn from the larger set of matrix templates wehave considered in our investigation of the dynamics of water molecules confined in self-assembled soft environments.

In addition, from the point of view of the elementarywater-matrix interactions, ionomers, for instance, sub-stantially resemble biological matter: i) they both con-tain hydrophilic and hydrophobic parts, ii) neutral andcharged groups, iii) water is confined at the nano-scale,and iv) it evolves in the presence of soft charged inter-faces. From the point of view of the function, these mate-rials are also alike. In one case we must understand howparticular species reach specific active sites on proteinsor lipidic membranes, and how this relates to the sur-rounding water. In the other, we aim at grasping themechanisms behind transport of ions within the ionicchannels, which depend on the shuttling efficiency of thewater molecules.

Here, we try to elucidate these issues by investigat-ing the transport properties of water molecules confinedin self-organised ionic surfactants phases, across a largerange of hydration levels. We have selected a perfluo-rosulfonic surfactant molecule to insure perfectly phase-separated morphologies with well-defined sharp inter-faces, arising from the extremely hydrophobic characterof the tail and important super-acidity of the head group.We have considered the numerical model for ionic surfac-tants of Ref. [17], which is able to stabilise well-orderedself-assembled phases, ranging from lamellae to reversemicelles at low and high water content, respectively. Wehave systematically followed the constrained evolutionof water adsorbed in these mobile soft charged environ-ments, on extended time scales. On the basis of datasets of unprecedented statistical accuracy, we demon-strate that it is possible to rationalise in a comprehensivepicture both the average sub-diffusive character of the dy-namics, due to the subtle interplay between confinementat the nano-scale and direct interaction with interfaceswith different topological properties, and the spatially-resolved heterogeneous behaviour in different regions of

the ionic channels [18]. We also believe that our workboth highlights the similarities in transport properties ofsystems of biological interest and synthetic materials, andprovides physical insight for a convincing interpretationof phenomenological modelling of experimental data.

II. THE MODEL

In Ref. [17], we described a coarse-grained numeri-cal model for (explicit) water confined in ionic surfac-tants phases. We considered a three-components mix-ture, formed by the surfactant macro-molecules, the po-lar solvent (water), and the solvated counter-ions (hydro-nium). We chose a united-atoms representation for thesurfactant molecule, inspired by the model for the side-chain of Nafion of Ref. [19]. The hydrophobic part ofthe molecule is represented by a chain of seven Lennard-Jones (LJ) beads of neutral electric charge, each onecoarse-graining an entire CF2 group. Similarly, the headgroup is schematised by two charged LJ beads, one forthe sulphur atom S and one for the O3 group, with atotal charge q = −e. The covalent bonds are modeled byquadratic potentials. The non-bonded interaction forceswere accurately parametrised, in order to generate an ex-tremely neat phase separation between the hydrophobicmatrix and the ionic channels, at all investigated watercontents. The popular SPC/E model was used for thewater molecules. All details can be found in [17].

The water content λ, i.e., the number of watermolecules per ionic surfactant macromolecule, influencesthe macroscopic self-assembly and generates a sequenceof phase changes, encompassing lamellar, cylindrical andmicellar phases by increasing hydration. In Fig. 1 weshow typical snapshots of the simulated systems, atthe indicated values of λ, increasing from left to right.

3

Also, in Fig. 2 we show the λ-variation of the aver-age confining size and the surface of the generated hy-drophilic/hydrophobic interfaces. (This last quantityhas been calculated by determining a polyhedral mesharound the beads pertaining to the surfactants [20], andis normalised to the total volume it contains.) Bothdata sets generally increase with λ following, however,non-trivial patterns which also highlight the boundariesbetween the different phases. The behaviour of theconfining size can be attributed to the system swellingupon increasing water loading. A more subtle argument,based on an increasing curvature radius of the interfaces,is needed to fully explain the details of the interfacesgrowth [17].

The interest of this model in the present context istwo-fold. First, it constitutes a very effective tool toautonomously generate very different confining environ-ments, modulating both the typical confinement sizesand the extent and topological features of the formedinterfaces. Second, the extremely neat phase separationprovides us with the possibility of precisely identify andcharacterise the hydration layer adjacent to the interfacewith the hydrophobic matrix, complementary to the vol-ume occupied by water of relatively higher bulk-like char-acter, far from the boundaries. Both points are crucialfor rationalising the dynamical behaviour of the adsorbedfluid, as it will be clear below.

III. SIMULATION DETAILS

We have used the parallel high-performance computingtool for Molecular Dynamics simulations LAMMPS [21]to generate the trajectories. In order to assure an opti-mal thermalisation of the systems at all hydrations, wereinitialised the simulation boxes starting from the self-assembled configurations analysed in [17]. We consid-ered 9 values of the hydration λ ranging in the inter-val [1, 32], which encompass all interesting regions of thephase diagram. A pure water system (bulk) trajectorywas also generated, as a reference. Note that the mobil-ity of the surfactant macro-molecules lifts the need for anadditional average over multiple realisations of the con-fining matrix. We verified in a few cases, however, thatequivalent results are found when preparing the systemin different initial conditions.

At each λ, we produced a trajectory spanning a to-tal time τmax = 10 ns, a dynamical window sufficientlyextended to allow comfortably observing the very slowdynamics in the most confined cases. The time step usedfor the numerical integration of the equations of motionis δt = 1 fs. We worked in the constant temperatureand pressure (NPT) ensemble, with T = 300 K andP = 1 atm, with the characteristic time of both ther-mostat and barostat set to τT = τP = 1 ps. We dumpedcomplete systems configuration with a frequency of 10 ps,for subsequent analysis.

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acta

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idth

[Å]

λ

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FIG. 2. Variation with the water content, λ, of the av-erage size of the self-assembled ionic channels and the ex-tension of the generated hydrophobic/hydrophilic interfaces,normalised to the volume occupied by the surfactant macro-molecules. Data for the confining size are similar to thosereported in [17], re-evaluated from the new system snapshots.The arrows indicate the axis relevant for each data set.

IV. RESULTS

In this Section we describe our computer simulationdata. We have fully characterised the dynamics of wa-ter molecules adsorbed in the surfactants nano-phasesat the different hydration levels, in terms of mean-squared displacements and intermediate scattering func-tions. We have next completed our analysis by probingwell-designed correlation functions associated to the typ-ical life-times of water molecules in the hydration layer,in contact with the confining hydrophobic matrix.

A. Evidence of sub-diffusive behaviour

By using the produced system trajectories, we firstcomputed the water molecules mean-squared displace-ment (MSD)

〈r2(t)〉 =1

Nw〈Nw∑i=1

|ri(t)− ri(0)|2〉, (1)

where Nw is the number of water molecules, and ri(t)is the position vector of the oxygen atom pertaining tomolecule i at time t. In Fig. 3 (top) we show our dataat the investigated various hydration levels (higher ontop), together with the results for bulk water. We ob-serve two interesting features: first, by decreasing thehydration, the mean-squared displacement at fixed timedecreases, as generically expected due to an increasingcrowding associated to the surfactant units. Second, and

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MSD

(t)

[Å2]

t [ps]

bulk32

2412

108

64

21

α

λ

FIG. 3. Main panel: Mean-squared displacements 〈r2(t)〉,calculated from the time-dependent coordinates of the wa-ter oxygen atoms. The solid lines are the power-law fitsdescribed in the main text, the dashed lines indicates thelimiting Fickian (top) and highly sub-diffusive (bottom) lim-its, corresponding to α = 1 and 0.4, respectively. Inset: λ-dependence of the exponent α, determined from 〈r2(t)〉 ∝ tα.

more intriguing, while the data-points for the bulk fallon a straight line of unitary slope (the linear depen-dence expected for water at ambient conditions), thisslope progressively decreases with the hydration, indi-cating a power-law behaviour with an exponent less thanone. This is characteristic of sub-diffusion [22].

To quantify this feature, we fit the MSD at longtimes to a power-law of the form 〈r2(t)〉 ∝ tα, with0 < α = α(λ) ≤ 1. We show the results of the fit aslines in Fig. 3, and the values of the exponent α(λ) inthe inset of the same figure. We observe that α variesslowly from a value close to 1 at λ = 32, correspond-ing to the expected Fickian diffusion, to α ' 0.85 atλ = 5, close to the boundary of the lamellar phase. Thisbehaviour seems to correlate with the corresponding de-crease of the available interface shown in Fig. 2. Next,α abruptly decreases to a value ' 0.4 at λ = 2, cer-tainly controlled by the extreme confining sizes at al-most constant available interface. Note that this valueis even lower than α = 0.5, corresponding to the caseof single-file diffusion [15], where molecules are organisedin uni-dimensional configurations, with a vanishing posi-tion swap probability for adjacent molecules. Altogether,these data demonstrate that our model allows to contin-uously tune the degree of sub-diffusion by varying thehydration level, interpolating between the extreme casesof Fickian and single-file-like diffusion.

In this context, the fluctuation-dissipation Einstein re-lation relating the mean-squared displacement to the dif-fusion coefficient, does not hold in the usual form. Wecan, however, define a time and length-scale dependent

generalised (effective) diffusion coefficient, Deff, from thelocal slope of the mean squared displacement as [24],

∂t〈r2(t)〉 =α

t〈r2(t)〉 = 6Deff. (2)

The effective diffusion coefficients and the corresponding〈r2(t)〉 are plotted parametrically in Fig. 4 (top). Con-sistently with the data of Fig. 3, the most diluted casesexhibit diffusion coefficients that are rather constant andclose to the value of the bulk. In contrast, upon decreas-ing hydration, Deff is strongly suppressed, with this ef-

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[10−5cm

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NMRQENS

r2 = 6Å2

r2 = 1000Å2

FIG. 4. Top: The effective diffusion coefficient, Deff, com-puted from Eq. (2), plotted parametrically against 〈r2(t)〉.The solid black line is the result in the bulk for the consideredwater model, corresponding to D ' 2.74 × 10−5 cm2/s. Thedashed vertical lines, indicate the (squared) length scales ' 6and ' 1000 A2 used to select the values of Deff shown in thebottom panel, as discussed in the text. Bottom: The valuesfor Deff selected by the two verticals lines of the above panelare shown as closed symbols. We also show the diffusion coef-ficients obtained by PFG-NMR and QENS spectroscopy [23],which probe the dynamics on long and short length scales,respectively.

5

fect becoming increasingly evident as the measured 〈r2〉increases.

An important remark is in order at this point. Thedata shown in Fig. 4 (top) provide us with a unique ratio-nalisation of the diffusion properties probed by differentexperimental spectroscopic techniques. We illustrate thispoint in Fig. 4 (bottom), where we compare our resultswith the values obtained by Pulsed-Field Gradient Nu-clear Magnetic Resonance (PFG-NMR ) [25] and Quasi-Elastic Neutron Scattering (QENS) experiments on gelsof PFOSA surfactants [23]. We recall that PFG-NMR(open circles) measures the self-diffusion of water on largedistances, of the order of the micrometer (µm), witha typical time scale of a few milliseconds, while QENS(open squares) probes proton displacements of the orderof a few angstroms (A), on time scales ranging from 1 totypically a few hundreds picoseconds.

Following an uncommon protocol, we sample our dataat fixed values of 〈r2〉 = (l∗)2, compatible with theselength scales (vertical dashed lines in the top panel ofFig. 4). Obviously, the µm range explored by PFG-NMRis far beyond the limit of our simulations. We note how-ever, that at least for the highest hydration levels, Deff

is almost length-scale independent for 〈r2(t)〉 ≥ 103 A2.This allows us to safely fix an observation threshold atl∗ ' 32 A. Similarly, we have determined a short length-scale value of Deff with l∗ ' 2.4 A, which is compatiblewith the typical nano-scale distances probed by QENS.Interestingly, the simulation points nicely reproduce thehydration dependence of both experimental data sets,without resorting to any additional consideration aboutdifferent dynamical processes on different time scales.Our model therefore seems to provide not only an ade-quate dynamical behaviour compared to the real system,but also a framework to interpret experimental results,without needing any fitting procedure (beside the deter-mination of α). In the following we will demonstratethat it also is an efficient tool to pinpoint the origin ofthe sub-diffusive behaviour described here.

B. Existence of two dynamical populations

Additional information on the dynamics of the watermolecules is carried by the (self-)intermediate scatteringfunction,

Fs(q, t) =1

Nw

Nw∑i=0

⟨eiq·(ri(t)−ri(0))

⟩|~q|=q

, (3)

where 〈〉|~q|=q is the spherical average over wave vectorsof modulus q. A strong connection exists in the bulkbetween the Fs(q, t) and the diffusion coefficient. Weexpect it to partially survive, even if in some modifiedform, in confinement conditions. The Fourier transformof Eq. (3) is the dynamical structure factor, S(q, ω), mea-sured in the QENS experiments mentioned above. Herewe focus on the value q∗ = 0.7 A−1 [26]. We show our

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FIG. 5. The self-intermediate scattering functions, Fs(q, t),calculated from the oxygen atoms of the water molecules viaEq. (3) (symbols). These data correspond to q∗ = 0.7 A−1, atthe indicated values of λ. We also show as solid lines the best-fits to the data, according to Eq. (4). These results stronglysupport the hypothesis of the existence of two dynamical pop-ulations, as discussed in details in the main text.

data at all the indicated values of λ in Fig. 5, togetherwith the reference case of the bulk. We observe that thedecay of Fs(q, t) becomes progressively slower as the hy-dration is decreased, and that the general shape of thesefunctions is quite complex. In the following we quantifythese observations.

The long-time behaviour of Fs(q, t) for bulk molecularliquids is often represented as a stretched exponential of

the form ∝ e−(t/τ)β . Here τ is the typical time scale as-sociated to the observed relaxation, and 0 < β ≤ 1 is thestretching parameter. The case β = 1 corresponds to thesimple exponential behaviour, β < 1 describes the casewhere non-trivial processes concur to the average relax-ation of the correlation function. As one could expectfrom the extremely complex behaviour of 〈r2(t)〉, thissimple functional form alone is not able to accurately re-produce our data, especially at low λ where the degreeof sub-diffusivity is higher.

The simplest alternative choice is to assume the exis-tence of two molecular populations characterised by dif-ferent dynamical properties [23, 27, 28]. In this quitecrude representation, a subset of the total number of wa-ter molecules evolve diffusively, not very differently thanthose in bulk conditions, on a relatively fast time scale.The remaining part, in contrast, include all the complexeffects coming from confinement in the surfactant phase,evolving on slower time scales. The precise features ofthe two populations should intuitively depend on the de-tails of the confinement environment and, ultimately, onλ. Given the additivity of Eq. (3), we can therefore con-sider the very simple model,

Fs(q, t) = fSD e−(

tτSD

)β+ (1− fSD) e

−(tτD

), (4)

6

where a simple and a stretched exponentials encode thedynamics of the diffusive (D) and sub-diffusive (SD) wa-ter molecules, respectively [29]. Since this latter popu-lation is the slowest, we can expect τD < 〈τSD〉, wherethe average time scale associated with the sub-diffusivepopulation is 〈τSD〉 = (τSD/β) Γ (1/β), with Γ(x) theGamma function. We can also obtain the fraction of dif-fusive molecules, as fD = (1− fSD).

We were able to obtain excellent fits to our data withthis model, that we show as solid lines in Fig. 5. Theobtained values of fSD span the entire possible variationrange, assuming a value of 0.05 at the highest λ and of0.95 at the lowest available hydration level. This varia-tion is followed by β-values ranging from 0.6 to 0.3. Theseobservations point to an increasing modification of thedynamics compared to the bulk unconstrained case, andto a very probable connection between the sub-diffusiveaverage character of the dynamics and the existence ofmolecular populations, characterised by heterogeneousdynamical properties. We postpone a much more elab-orated discussion of the calculated values for both thetime scales and fSD to the following (see Fig. 9).

C. The interface as the source of sub-diffusion

The self-assembled confining environments consideredhere are characterised not only by variable size of theionic domains available for transport (Fig. 2), but also bythe variable topological features of the formed interfaces,which evolve from the almost perfect planar configura-tions of the lamellar phase, to configurations of increasingconvexity in the cylindrical and micellar phases. It is nat-ural to expect that beside the obvious role played by theextremely tight confining sizes at very low λ, these well-developed interfaces significantly perturb the dynamicsof the adsorbed fluid molecules, participating to the gen-eration of the observed sub-diffusion. More specifically,the surfactant sulfonic acid groups are hydrophilic andnegatively charged, resulting in a strong attractive po-tential between the interface and fluid molecules withinthe range of these interactions. This leads to the hy-pothesis that the fluid molecules in the vicinity of theinterface can be trapped at the interface, which reducestheir possibility of diffusing at the normal rate. Thisconstrain would next be transferred to successive fluidlayers, with a strength reduced as the distance from theinterface increases.

On this basis, we now focus on the dynamical prop-erties of the fluid adsorbed at the interface only, by se-lecting the sub-set of fluid molecules comprised in thefirst hydration layer of the surfactant heads. We there-fore define the binary presence function, p(r), for a watermolecule at position r: p(r) = 1 if it exists a chargedsurfactant head at r0, such that |r0 − r| ≤ rc, and zerootherwise. In what follows we have fixed rc = 4 A. Ap-plying this operator to the position of the oxygen atomsfor every snapshot of our trajectory, we have obtained Nw

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FIG. 6. Top: Probability distributions of the residence timesobserved during the 10ns simulation run. The long-time be-haviour is reminiscent of a power-law, and in the inset we plotthe exponent of power-law fits. Bottom: Distribution of theresidence times of the water molecules present at the inter-face a the end of the simulation run. The black arrow pointsto the long-time bump in the distribution that is particularlyvisible at low hydration (λ ≤ 6).

series of Ns binary values, {pi,n}. Here, Ns is the numberof system snapshots, each corresponding to a simulationtime tn = n δt, with n = 0, . . . , Ns−1, and i = 1, . . . , Nw.

From the lists {pi,n} we can in principle determine theprobability distribution of the residence times at the in-terface of all molecules, by simply counting the number ofnon-zero segments of given size (for each i). This shouldallow us to define an average survival time at the inter-face. Unfortunately this is problematic, as it is clear fromthe data sets shown in Fig. 6 top, at the indicated valuesof λ. These distributions show significant power-law tails∝ t−x (solid lines in the main panel and points in the in-set), with even x < 2 at the lowest hydrations (λ ≤ 2)which hinders a consistent definition of average values,and 2 < x < 3 for 2 < λ ≤ 10, which results in an un-

7

defined variance. This also implies that there is alwaysa significant probability to observe life-times correspond-ing to the total time length of the simulation, makingextremely hard adequate statistical sampling. This con-clusion is verified from the data of Fig. 6 (bottom), wherewe plot the same probability distributions as above, withthe additional constraint that the considered non-zerosegments must include τmax [30]. Clearly, at low hy-dration values the most likely residence time of watermolecules present at the interface at the end of the runis the duration of the run, as indicated by the arrow.

These observations are of particular interest, becauselong-tailed distributions of survival times are known togive rise to sub-diffusive random walks. This is thecase, for instance, of the continuous-time random walk(CTRW), where particles are allowed to move after hav-ing been arrested for a waiting time, tw, drawn from apower-law distribution. Evidences of CTRW have beenobserved in simulations of water confined in lipid bi-layers [11, 12], among other cases.

The difficulties described above can be partially cir-cumvented by determining the time scale associated tothe variation of well-designed correlation functions. Wefirst consider the correlation function of the survivaltimes of water molecules laying within the first hydra-tion layer of the sulfonated groups as

C1(tn) =1

1− n/Ns

⟨∑Ns−1−ns=0 (1− pi,n+s)

∏n+s−1m=s pi,m

⟩i⟨∑Ns−1

s=0 pi,s

⟩i

.

(5)Here, 〈〉i is the average over all water molecules i =1, . . . , Nw, and tn = tn+s − ts. Note that the productat the numerator is non-zero only if particle i survivesat the interface at all times ts ≤ t < tn+s, subsequentlydetaching exactly at tn+s. We plot our data in the mainpanel of Fig. 7 as symbols, at the indicated values of λ.

Already by visual inspection it is clear that the typi-cal lifetime of water molecules at the interface increasesby lowering λ. Interestingly, we also find that two dif-ferent mechanisms seem to concur at the relaxation ondifferent time scales. Our data can be represented at all

hydrations by a modified exponential, C1(t) ∝ e−(t/τs)βs

at short times, crossing-over to a power-law behaviour,C1(t) ∝ t−αs , at longer times. Note that the cross-overpoint depends on λ. The fitting curves are shown as solidand dashed lines in Fig. 7.

In our understanding, the faster process is associatedto the exchange mechanism of water molecules which de-tach from the interface and transit in higher-order hydra-tion layers, on a (well-defined) typical time scale τs(λ).Indeed, the functional form used is suggested by an anal-ogy with the decay of a subset (population) of a givensystem, characterised by some particular feature. Simi-larly to the case of radioactive decay, for instance, if theprobability to disintegrate (or to detach from the inter-face in our case) is uniform, then C1(t) ≈ e−t/τs . In con-trast, if the probability to detach is not a constant, then

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FIG. 7. Main panel: Survival-time correlation functions ofwater at the interface, C1(t), computed via Eq. (5), at theindicated values of λ. Insets: λ-dependence of the two fittingparameters βs and αs, obtained by representing the data witha modified exponential decay at short-times and a long-timepower-law. The rationale for the fits shown as solid lines inthe main panel is discussed in details in the main text.

C1(t) can relax both faster or slower than a simple expo-nential. The mildest deviation from the pure exponential

is C1(t) ≈ e−(t/τs)βs

, with the exponent βs quantifyingthe degree of the deviation: for βs < 1 (stretched) thedecay is slower than exponential, faster for βs > 1 (com-pressed). In more extreme cases, scale free power-lawbehaviour have also been observed [11, 12].

Our analysis suggests that τs increases from 60 to22000 ps as the hydration λ decreases from 32 to 1 (seeFig. 9), meaning that the exchange mechanism at the in-terface is activated by hydration. Also, we have foundthat βs varies from 0.6 at λ = 1 to 0.8 at λ = 32(inset of Fig. 7). These observations therefore seem todemonstrate that hydration controls both typical time-scale and statistical features of the postulated exchangemechanism. In addition, the self-similar behaviour ob-served at longer times clearly is the echo of the power-lawtails of the static distribution functions of the {pi,n} ofFig. 6. It is natural to expect that this relaxation stemsfrom trapping at the interface of water molecules on ex-tremely extended time scales. This would also directlyallow to identify interfaces as the main source of watersub-diffusion, rationalising this mechanism in terms of aCTRW approach.

It is not possible, however, to associate a well-definedtime scale to this dynamics in terms of Eq. (5). Indeed,in order to highlight the dynamics most controlled by theinterface, beyond the exchange process quantified above,one should lift the excessive constraint of water moleculesuninterruptedly bounded to the interface during the timewindow tn implicit in that definition. We therefore define

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

101 102 103 104

C2(t)

t [ps]

3224

1210

86

42

1

FIG. 8. Time autocorrelation functions of {pi,n} (symbols),computed via Eq. (6), at the indicated values of λ. The con-tinuous lines are the results of the fits to our data accordingto Eq. (7). A comprehensive discussion of these results isprovided in the main text.

the time autocorrelation function of {pi,n},

C2(tn) =

⟨∑Ns−1−ns=0 pi,s+npi,s

⟩i⟨∑Ns−1

s=0 p2i,s

⟩i

. (6)

At variance with the case of Eq. (5), with this defini-tion water molecules that are at the interface at timests and ts+n contribute to C2(tn), notwithstanding theirfaith between ts and ts+n, and at ts+n+1. This func-tion therefore still selects molecules that detach from theinterface, but also those that return to the interface, atsome later time after leaving. We expect, in general, thatthis dynamics takes place on time scales comparable toor longer than τs. We plot our data as symbols in Fig. 8,at the indicated values of λ.

The data show a two-relaxation pattern, that we de-scribe as,

C2(t) = c e−(t/τi)βi

+ (1− c) e−(t/τ∞)β∞ . (7)

We consider here two points. First, the initial decay ofC2(t) takes place on a quite large range of time scales τi,ranging from ' 100 ps at λ = 32 to ' 1 ns at λ = 1.This quantifies dynamics at the interface, as we will seebelow. Second, the functions exhibit an abrupt singularbehaviour as t approaches the total time window of oursimulation, τmax, and this cutoff is more pronounced atlower hydrations. We indicate this limit time scale asτ∞, which characterises molecules that are permanentlyattached to the interface on the time scale of our simu-lation. We postpone a comprehensive discussion of thesedata to the next Section.

V. DISCUSSION

We now describe the implications of the data reportedin the previous Section, and provide a consistent uniquepicture of our observations. From the analysis above,we have been able to extract five time-scales relevant forthe confined dynamics of the water molecules. In par-ticular, τD and τSD are associated to the relaxation ofthe intermediate scattering function, τs characterises thesurvival-time correlation function C1(t), τi and τ∞ havebeen determined from the binary presence autocorrela-tion function C2(t). We plot all data as a function of thehydration level λ in the top panel of Fig. 9.

We see that the long-time component of the autocorre-lation of the binary presence, τ∞, is λ-independent, andof the order of the total simulation time. We associatethis time-scale to the water molecules that stay in theinterfacial layer for times that are compatible with theduration of the entire simulation run. As already antic-ipated, this is an artificial feature which originates fromarbitrarily fixing a bound from above on the available ob-servation time. As a consequence, all dynamics on timescales larger than τ∞ cannot be probed in equilibrium inour simulation.

Interestingly, the time-scales τi of the short-time com-ponent of C2(t) and τs of the initial decay of C1(t) are, atvariance, strongly reduced as the water content increases.We can associate these relaxations to an exchange mech-anism between the interfacial region and the rest of theaqueous domains, which is activated by the increase ofthe hydration. This idea is strongly supported by a sur-prising observation: τi and τs are very similar, especiallyat low-λ, to the time scale τSD that we extracted fromthe Fs(q

∗, t), and associated with the sub-diffusive frac-tion of water molecules. These therefore seem to be thosepertaining to the population with a dynamics of primaryinterface-dominated character.

An other observation additionally strengthen our pic-ture. In Fig. 9 (bottom) we show two quantities: thestatic fraction over the total number of water moleculeslying at the interface, fi, which we obtain by simplycounting the number of molecules comprised in the hy-dration layer; and the dynamic fraction of sub-diffusiveparticles, fSD, obtained from Eq. (4). The two datasets exhibit very similar variation on the whole hydra-tion range, and the two curves actually coincide at thelowest hydration ranges, λ ≤ 2. At higher hydration lev-els, fSD is close to, although always higher than, fi. Weinfer from the similarities of the typical time scales andassociated fractions (over the total number) that inter-facial and sub-diffusive populations coincide, or at leastthat the sub-diffusive population is a superset of the in-terfacial one.

We propose a schematic overview of our discussion inFig. 10, where we partition into three main areas thesurfactant phase environment, with both the confiningmatrix and the complementary space available for trans-port in the ionic channel, of typical size l: i) the solid-

9

100

101

102

103

104

105

0 5 10 15 20 25 30 35

t[p

s]

λ

τDτSD

τiτ∞

τsbulk

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25 30 35

frac

tion

ofm

olec

ules

λ

hydration water, fisub-diffusive water, fSD

FIG. 9. Top: λ-dependence of the relevant time-scales asso-ciated to the dynamics of the confined water molecules: τDand τSD are associated to the relaxation of the intermediatescattering function, τs characterises the survival-time corre-lation function, C1(t), τi and τ∞ have been determined fromthe binary presence autocorrelation function, C2(t). A com-prehensive discussion of these data is included in the maintext. Bottom: Static fraction of water molecules residing atthe interface, fi, together with the dynamic fraction of sub-diffusive molecules, fSD. The two data-sets coincides at lowhydration. This point, together with the discrepancy at high-λ are described in details in the main text.

like interface (bottom), which comprises the surfactanthydrophilic heads; ii) the hydration layer (centre), thatcovers the interface and comprises all water moleculeswithin rc = 4 A from the the nearest surfactant head;and iii) the bulk region (top) consisting of all remainingwater molecules and extending over a typical length scalelb, which depends on λ. Obviously, l = 2(lh + lb/2).

The hydration layer is characterised by the fraction ofmolecules it comprises, fi, together with the characteris-tic time of their interface-dominated dynamics, τi. Thefraction of water at the interface coincides at low-λ (andhas a very large overlap at high-λ) with the fraction of

sub-diffusive water (fi ≈ fSD), and one finds τi ≈ τSD.The molecules at the boundary with the region extendingbeyond the first hydration shell and comprising the re-maining fraction, 1−fi ≈ fD, participate to the exchangeprocess with the molecules pertaining to the bulk-likereservoir. This takes place on a time scale τs, which alsocorrelates to the sub-diffusive time scale of the interme-diate scattering function, with τs ≈ τSD.

The sub-diffusive character of the mean-squared dis-placements (Fig. 3) also finds a clear rationalisationin this picture. 〈r2(t)〉 grasps an average dynamics,which results from a λ-dependent interplay of interface-dominated and bulk-like behaviour. The source of theobserved sub-diffusion is trapping of water molecules atthe interface, with a power-law distribution of trappingtimes, as demonstrated in Fig. 7. This breaks the va-lidity of the central limit theorem in a fashion remi-niscent of a continuous time random walk, immediatelyproviding a rigorous justification of the observed non-linearity of 〈r2(t)〉 at long times. Also note that in gen-eral fi ≤ fSD, and at high hydrations (above λ = 6)τs and τi are slightly larger than τSD. This indicatesthat water molecules exiting the hydration shell retaina certain degree of sub-diffusivity, with this anomalouscharacter progressively disappearing when moving awayfrom the interface.

FIG. 10. Schematic overview of the interplay between thestructure of the surfactant phase and the transport propertiesof the confined water molecules. This sketch complements thedetailed discussion contained in the main text.

10

FIG. 11. Joint probability distribution of the total displacement, l, of water molecules on the entire time scale of the run, τmax,and the total time, th, they have spent (even discontinuously) in the hydration layer of a surfactant head, on the same timewindow. The probability is represented as colour maps, for the indicated values of λ: yellow indicates high probability, indigolower. In the insets we show the projection on the xy-plane of the trajectory followed by two molecules chosen at random atthe corresponding λ, with the incremental distance travelled at each time indicated by the displayed colour code.

To further clarify this point, we have quantifiedthe correlation between the total displacement, li =∑Ns−1n=1 |ri(tn) − ri(tn − 1)| , of water molecule i on the

entire time scale of the run (τmax ' 10 ns) and the to-tal time, thi , it has spent (even discontinuously) in thehydration layer of a surfactant head, on the same timewindow. In Fig. 11 we show the joint probability distri-butions of these two quantities as colour maps, for theindicated values of λ; yellow indicates high probability,indigo lower. We observe a clear trend as the hydrationincreases.

At very low water content (λ = 1), almost all watermolecules remain in the hydration layer for most of theduration of the simulation run (th ≈ τmax), and the totaldistance travelled does not strongly depend on the timespent at the interface, fluctuating around l ≈ 750 A. Inthe inset we show the projection on the xy-plane of thetrajectory followed by two molecules chosen at random,

with the distance from the nearest O3 bead indicated bythe displayed colour code. The motion is clearly very con-strained (probably bi-dimensional) in both cases, with avery limited portion of space explored. For λ = 4 and 8,however, the probability to observe very high values ofth progressively decreases, while the distribution of theth becomes wider. This effect is accompanied by longerl travelled at shorter th, with a degree of variation in-creasing with λ, as confirmed by the data in the insets.At λ = 32, eventually, the probability accumulates inthe high-l/low-th region, indicating that practically allmolecules transit between the interface and the bulk-likeregion (the trajectory of one molecule is shown in theinset, and clearly shows segments of high-speed travelfar from the interface, and “sticky” regions that trap themolecule).

The conclusion arising from the above arguments isthat the picture based on two well-identified dynami-

11

cal population, that fully accounts for the behaviour inthe low-λ regime, must be modified at high water con-tent, where we observe an increasing rate of the exchangemechanism and of the number of molecules participatingto it. Indeed, it is not possible to account for the complexdistributions of Fig. 11 in terms of an average τi only, andone should rather consider a λ-dependent set of popula-tions, characterised by a distribution of τi (τND).

VI. CONCLUSIONS

In this work we have followed numerically the dy-namics of water molecules constrained in self-organisedionic surfactant phases. By varying the water content,we have considered an extended range of soft mobileconfining environments, characterised by variable con-fining scales and topological properties of the generatedhydrophilic/hydrophobic interfaces. These two featurescontribute in a highly non-trivial fashion to transport ofthe adsorbed fluid, that we have managed to characterisein terms of a space-dependent dynamical behaviour.

More in details, we have demonstrated that both theobserved sub-diffusive character of the average meansquared displacements, increasing by decreasing the

length scale of the confinement, and the details of therelaxation of the intermediate scattering function, canbe rationalised in terms of two dynamical populations.The first is predominantly interface-dominated, with amarked non-diffusive behaviour, the second has a bulk-like character. The average dynamical behaviour of theadsorbed fluid is primarily determined by the exchangemechanism between the two dynamical groups, whoseprecise features depend on the water content. This sim-ple picture allows for a complete quantitative character-isation of dynamics at low-λ, while at high-λ we haveproposed that an entire λ-dependent set of populationsmust be considered to obtain the same accuracy.

Overall, we have managed to provide, to the best ofour knowledge, the first comprehensive picture of the dy-namics of water molecules in soft confining environments,ranging from the case of extremely tight channels, to thatwhere the interface is in contact with an infinite reser-voir of bulk water. These two limits cover, we believe,most part of the typical situations encountered in hy-drated natural and synthesised materials. Our model,as a consequence, could play for soft materials a criticalparadigmatic role similar to that of carbon nanotubes forconfinement in rigid ordered matrices.

[1] John WR Schwabe, “The role of water in protein-dnainteractions,” Current opinion in structural biology 7,126–134 (1997).

[2] Gerhard Hummer, Jayendran C Rasaiah, and Jerzy PNoworyta, “Water conduction through the hydrophobicchannel of a carbon nanotube,” Nature 414, 188–190(2001).

[3] Yingchun Liu, Qi Wang, Tao Wu, and Li Zhang, “Fluidstructure and transport properties of water inside carbonnanotubes.” The Journal of chemical physics 123, 234701(2005).

[4] Pradeep Kumar, Sergey V. Buldyrev, Francis W. Starr,Nicolas Giovambattista, and H. Eugene Stanley, “Ther-modynamics, structure, and dynamics of water confinedbetween hydrophobic plates,” Phys. Rev. E 72, 051503(2005).

[5] Nicolas Giovambattista, Peter J. Rossky, and Pablo G.Debenedetti, “Phase transitions induced by nanoconfine-ment in liquid water,” Phys. Rev. Lett. 102, 050603(2009).

[6] Nicolas Giovambattista, Pablo G. Debenedetti, andPeter J. Rossky, “Hydration behavior under con-finement by nanoscale surfaces with patternedhydrophobicity and hydrophilicity,” The Journalof Physical Chemistry C 111, 1323–1332 (2007),http://dx.doi.org/10.1021/jp065419b.

[7] Giorgio Schiro, Yann Fichou, Francois-Xavier Gallat,Kathleen Wood, Frank Gabel, Martine Moulin, MichaelHartlein, Matthias Heyden, Jacques-Philippe Colletier,Andrea Orecchini, Alessandro Paciaroni, Joachim Wut-tke, Douglas J. Tobias, and Martin Weik, “Translationaldiffusion of hydration water correlates with functional

motions in folded and intrinsically disordered proteins,”Nature Communications 6, 6490 (2015).

[8] Michael R Harpham, Branka M Ladanyi, Nancy ELevinger, and Kenneth W Herwig, “Water motion inreverse micelles studied by quasielastic neutron scatter-ing and molecular dynamics simulations,” The Journalof chemical physics 121, 7855–7868 (2004).

[9] Ram Devanathan, Arun Venkatnathan, and MichelDupuis, “Atomistic simulation of nafion membrane. 2.dynamics of water molecules and hydronium ions,” TheJournal of Physical Chemistry B 111, 13006–13013(2007).

[10] Ram Devanathan, Arun Venkatnathan, and MichelDupuis, “Atomistic simulation of nafion membrane: I.effect of hydration on membrane nanostructure,” TheJournal of Physical Chemistry B 111, 8069–8079 (2007).

[11] Eiji Yamamoto, Takuma Akimoto, Masato Yasui, andKenji Yasuoka, “Origin of subdiffusion of water moleculeson cell membrane surfaces,” Scientific reports 4 (2014).

[12] Eiji Yamamoto, Takuma Akimoto, Yoshinori Hirano,Masato Yasui, and Kenji Yasuoka, “Power-law trappingof water molecules on the lipid-membrane surface induceswater retardation,” Physical Review E 87, 052715 (2013).

[13] Anna Rita Bizzarri and Salvatore Cannistraro, “Molecu-lar Dynamics of Water at the ProteinSolvent Interface,”The Journal of Physical Chemistry B 106, 6617–6633(2002).

[14] Francesco Pizzitutti, Massimo Marchi, Fabio Sterpone,and Peter J Rossky, “How protein surfaces induce anoma-lous dynamics of hydration water.” The journal of phys-ical chemistry. B 111, 7584–90 (2007).

[15] Karger Hahn, J Karger, and V Kukla, “Single-file dif-

12

fusion observation,” Physical review letters 76, 2762(1996).

[16] Anurag Prakash Sunda and Arun Venkatnathan, “Molec-ular dynamics simulations of side chain pendants ofperfluorosulfonic acid polymer electrolyte membranes,”Journal of Materials Chemistry A 1, 557–569 (2013).

[17] Samuel Hanot, Sandrine Lyonnard, and Stefano Mossa,“Water confined in self-assembled ionic surfactant nano-structures,” Soft Matter 11, 2469–2478 (2015).

[18] Daiane Damasceno Borges, Alejandro A Franco, KouroshMalek, Gerard Gebel, and Stefano Mossa, “Inhomoge-neous transport in model hydrated polymer electrolytesupported ultrathin films,” ACS nano 7, 6767–6773(2013).

[19] Elshad Allahyarov and Philip L Taylor, “Simulationstudy of the correlation between structure and conductiv-ity in stretched nafion,” The Journal of Physical Chem-istry B 113, 610–617 (2008).

[20] Alexander Stukowski, “Computational analysis methodsin atomistic modeling of crystals,” JOM 66, 399–407(2014).

[21] Steve Plimpton, “Fast parallel algorithms for short-rangemolecular dynamics,” Journal of computational physics117, 1–19 (1995).

[22] Ralf Metzler, Jae-Hyung Jeon, Andrey G Cherstvy, andEli Barkai, “Anomalous diffusion models and their prop-erties: non-stationarity, non-ergodicity, and ageing at thecentenary of single particle tracking,” Physical ChemistryChemical Physics 16, 24128–24164 (2014).

[23] Quentin Berrod, Sandrine Lyonnard, Armel Guillermo,Jacques Ollivier, Bernhard Frick, and Gerard Gebel,“Qens investigation of proton confined motions inhydrated perfluorinated sulfonic membranes and self-assembled surfactants,” in EPJ Web of Conferences,

Vol. 83 (EDP Sciences, 2015) p. 02002.[24] Mike P Allen and Dominic J Tildesley, Computer simu-

lation of liquids (Oxford university press, 1989).[25] Quentin Berrod, Sandrine Lyonnard, Armel Guillermo,

Jacques Ollivier, Bernhard Frick, Abdelatif Manseri,Bruno Ameduri, and Gerard Gebel, “Nanostructure andtransport properties of proton conducting self-assembledperfluorinated surfactants: a bottom-up approach to-wards pfsa fuel cell membranes,” Macromolecules (inpress).

[26] At this q-value the effect of confinement is clearly ob-served in the QENS spectra analysis. More precisely,the Half Width at Half Maximum (HWHM) of theLorentzian-shaped fitting line starts to deviate from aclassical random jump diffusion behaviour and enters intoa plateau-like regime, attributed to spatial restrictions.

[27] Ferdinand Volino, Jean-Christophe Perrin, and SandrineLyonnard, “Gaussian model for localized translationalmotion: application to incoherent neutron scattering,”The Journal of Physical Chemistry B 110, 11217–11223(2006).

[28] Jean-Christophe Perrin, Sandrine Lyonnard, and Fer-dinand Volino, “Quasielastic neutron scattering studyof water dynamics in hydrated nafion membranes,” TheJournal of Physical Chemistry C 111, 3393–3404 (2007).

[29] Note that for molecular liquids the functional form ofEq. (4) is sometimes used for fitting Fs(q, t) in the entiretime range, with the simple and stretched exponentialwell representing the short (caging) and long (structuralrelaxation) time behaviour, respectively. Here, these twoterms are needed for convincingly reproducing the struc-tural relaxation alone, for times t ≥ 10 ps.

[30] Note that the two probability distributions must coincidein the infinite-time limit.