reorientation dynamics of nanoconfined acetonitrile: a critical examination of two-state models

Reorientation Dynamics of Nanoconfined Acetonitrile: A CriticalExamination of Two-State ModelsCassandra D. Norton and Ward H. Thompson*

Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, United States

ABSTRACT: Molecular dynamics simulations are used toinvestigate the reorientation dynamics of liquid acetonitrileconfined within a nanoscale, hydrophilic silica pore. The dynamicsare strongly modified relative to the bulk liquidthe time scale forreorientation is increased by orders-of-magnitude and the dynamicsbecome nonexponentialand these effects are examined at themolecular level. In particular, commonly invoked two-state (orcore−shell) models, with and without consideration of exchange ofmolecules between the states, are applied and discussed. A rigorousdecomposition of the acetonitrile reorientational correlationfunction is introduced that permits the approximations implicitin the two-state models to be identified and tested systematically.The results show that exchange is an important component of thenanoconfined acetonitrile reorientation dynamics and a two-statemodel with exchange can accurately describe the correlation. However, the faithfulness of the model is related to the separationof time scales in the two states, which exists for a wide range of definitions of the two states. This suggests that caution should beexercised when inferring molecular-level details from application of two-state models.

1. INTRODUCTIONNanoconfined liquids continue to attract attention for theirinteresting fundamental properties, the tunability of theconfining framework, and their range of potentially importantapplications.1,2 However, a full molecular-level picture of howthe structural and dynamical properties of a liquid are modifiedupon confinement is still lacking. Such an understanding wouldrepresent a key insight into the emergence of new phenomenaat liquid−solid interfaces, the effects of different frameworkproperties from size to surface chemistry, and the basis ofdesign principles for practical applications from catalysis tosensing.In this paper, we examine these issues in the context of the

reorientation dynamics of acetonitrile confined within ∼2.4 nmsilica pores. These dynamics are a critical component of themechanisms of important processes like electron and protontransfer reactions as well as spectroscopic probes such asinfrared and Raman spectra. Acetonitrile is of particular interestdue to its wide use as a solvent and its interesting vibrationalspectroscopy.3−5 In addition, acetonitrile in silicate glasses hasbeen the subject of numerous experimental and computationalstudies.3−17

Of particular note are experimental studies of acetonitrilereorientation in sol−gel pores using NMR6,7 and optical Kerreffect (OKE)9−13 spectroscopies. Jonas and co-workersinterpreted their NMR results in terms of a two-state, orcore−shell, model,6,7 perhaps the most commonly invokeddescription for nanoconfined liquids. In this model, themolecules in the liquid are assumed to consist of twocategories; those near the confining framework interface, the

“shell,” are assumed to have strongly modified properties, whilethe remainder, the “core”, are located further from the surfaceand are at best weakly modified. Frequently, the core is taken tohave the same behavior as the bulk liquid. This division ofmolecules assumes a distinct separation between these twocategories, typically based on the distance of a molecule fromthe interface. In analyzing their OKE results, Fourkas and co-workers extended this model to include the effect of exchangeof molecules between the two states.9−13

Two-state models provide a useful way to interpretexperimental measurements to yield molecular-level informa-tion.2,5−13,18−23 However, the underlying assumptions stronglyaffect the nature of the resulting inferences and it is thusimportant to clearly understand their accuracy. Moleculardynamics (MD) simulations can provide significant insight inthis regard as spectroscopic observables can be clearly related tothe molecular structure and dynamics to probe the two-statepicture.24−27 Indeed, simulations of acetonitrile reorientation insilica pores have been carried out by multiple groups (includingours).3,14,15,17 The results are in general agreement withexperimental measurements and each other, but analysis hasled to different conclusions regarding the validity of the two-state model.

Special Issue: James L. Skinner Festschrift

Received: February 7, 2014Revised: March 28, 2014Published: April 1, 2014

Article

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© 2014 American Chemical Society 8227 dx.doi.org/10.1021/jp501363q | J. Phys. Chem. B 2014, 118, 8227−8235

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The remainder of the paper examines this issue. Moleculardynamics simulation results for acetonitrile reorientationdynamics in the bulk liquid and silica pores are presented insection 2. The general features are described, and comparisonsare made with experimental data where possible. A systematicderivation of a two-state model, with and without core−shellexchange, is then presented in section 3 by introduction of arigorous decomposition of the reorientational correlationfunction; an appendix presents the models in the commonlyinvoked limit of single exponential dynamics for each state. Themodels are tested by analysis of the MD simulation data andthe individual approximations are identified and separatelyexamined. Finally, some conclusions are offered in section 4.

2. ACETONITRILE REORIENTATIONMolecular reorientation dynamics in liquids can be examined interms of the reorientational correlation functions

= ⟨ · ⟩C t P te e( ) [ (0) ( )] (1)

where P is the th Legendre polynomial, e(t) is the unit vectoralong a particular molecular axis at time t, and ⟨...⟩ indicates athermal average over all molecules in the liquid. In this work onthe rod-like molecule acetonitrile, this vector is taken to bealong the main molecular axis, e.g., the CCN direction.The correlation functions obtained from the present MD

simulations of bulk acetonitrile for = 1, 2, and 3 are shown inFigure 1. Each displays single-exponential dynamics following a

short-time inertial response. The C (t) can be fit with this formusing a Gaussian function to describe the inertial component.The resulting exponential time scales (τ ) are τ1 = 3.0 ps, τ2 =1.1 ps, and τ3 = 0.63 ps. For reasons that are unclear, theseresults differ somewhat from the values reported by Gee andGunsteren (τ1 = 1.93 ps; τ2 = 0.80 ps) when they presented theacetonitrile model.28 However, they are in good agreement withexperimental measurements of τ1 = 3.68 ps based on infraredspectral analysis,29 τ2 = 1.09 ± 0.10 ps from Ramanmeasurements,30 and ⟨τ2⟩ = ∫ C2(t) dt = 1.02 ps from NMRmeasurements31 (the present data give ⟨τ2⟩ = 0.95 ps). Thesedata show that the acetonitrile force field used here28 provides asatisfactory description of molecular reorientation in the liquid.It is interesting to note that these time scales are in

reasonable agreement with the Debye model for rotationaldiffusion, which predicts that τ1/τ2 = 3 and τ1/τ3 = 6; theresults here give τ1/τ2 = 2.7 and τ1/τ3 = 4.8. These values can

be compared to those for water, τ1/τ2 = 2.0 and τ1/τ3 = 3.1,which has been shown to reorient via jumps involving exchangeof hydrogen-bonding partners rather than by Debye rotationaldiffusion.32

When acetonitrile is confined in nanoscale silica pores, thereorientational dynamics are dramatically modified. In thefollowing, we will use C1(t) as a measure of these dynamics.Examination of the other correlation functions gives the sameessential behavior: while in section 3 we examine the accuracyof various two-state approximations for C1(t), we have carriedout the same analysis for C2(t) and the results are qualitativelythe same with quantitative errors that are slightly smaller due tothe faster decay of the C2(t) correlation function relative toC1(t). The C1(t) correlation function is plotted for acetonitrileconfined in ∼2.4 nm silica pores with different surfaceinteractions and compared with the bulk liquid result in Figure2. These results clearly show that the reorientational dynamics

are slowed by more than an order of magnitude uponconfinement and that this slowdown is sensitive to the surfacechemistry. Specifically, acetonitrile confined in a hydrophilicpore exhibits nearly the same behavior whether hydrogen-bonddonor groups, e.g., Si−OH, are present or modified so theycannot serve as donors (see section 5).In contrast, when a hydrophobic pore is modeled by setting

all pore atom charges to zero, the dynamics are bothsignificantly faster at long times (though still an order ofmagnitude slower than in the bulk liquid) and qualitativelydifferent. Namely, C1(t) for acetonitrile in the hydrophobicpore can be fit with a triexponential function with one timescale corresponding to inertial motion, 0.55 ps (16%), and twothat are longer than the bulk value: 6.2 ps (65%) and 27.0 ps(19%). The correlation function for the hydrophilic pore andthat with no hydrogen-bond donors exhibits a complex,nonexponential behavior; it is not well fit with a power-lawor stretched exponential over the entire span of the long-timedecay though each form reasonably describes part of thecorrelation function. These data improve on our initial report ofthe reorientational correlation function for nanoconfinedacetonitrile where C1(t) was calculated out to only 40 ps.3 Inthe remainder of this paper, we examine the underlyingmolecular-level acetonitrile dynamics while focusing on thehydrophilic pore case.

Figure 1. The C (t) reorientational correlation functions for bulkliquid acetonitrile. Results are shown for = 1 (black line), 2 (redline), and 3 (blue line).

Figure 2. The C1(t) reorientational correlation function for confinedacetonitrile in pores with hydrophilic (black line), hydrophobic (redline), and hydrogen-bonding “turned off” (blue line) surface chemistryis compared to the bulk result (violet line).

The Journal of Physical Chemistry B Article

dx.doi.org/10.1021/jp501363q | J. Phys. Chem. B 2014, 118, 8227−82358228

Jonas and co-workers used NMR relaxation measurements tocharacterize the reorientation dynamics of acetonitrile confinedin sol−gel pores of diameter 3.1, 4.0, 6.0, and 9.6 nm.6,7 Theyinterpreted their measurements in terms of a two-statemodelthe time scale of the NMR measurements is longcompared to any exchange between the core and shell. Fromthis analysis, they extracted rotational diffusion constants forbulk and “shell” acetonitrile, with the shell layer taken to be twomolecular diameters (8.2 Å), as ⟨τ2⟩ = 0.95 and 10.5 ps,respectively. The former is in excellent agreement with our bulksimulations, as noted above. The latter represents a time scalethat is fast relative to our simulation results for all poremolecules where we obtain ⟨τ2⟩ > 70 ps; it should be noted thatthe pore diameter used here is smaller than any they consideredand is only 3 times their shell thickness.Kittaka et al. probed acetonitrile reorientation in MCM-41

pores of diameter 2.04 and 3.61 nm using quasi-elastic neutronscattering (QENS).5 They found that the rotational timescalesτR = 2.5 ± 0.3 and 1.4 ± 0.3 ps for the smaller andlarger pores, respectivelywere not strongly modified fromthat of the bulk liquid (1.4 ± 0.2 ps). However, this is likelyindicative of the relative insensitivity of the QENS signal to thelonger-time reorientational dynamics. Tanaka et al. alsoextracted rotational time scales for acetonitrile confined inMCM-41 pores (3.2 nm in diameter) from an analysis of theinfrared line shape.8 They found that the reorientation timeswere nearly unmodified in the pores (1.6−1.7 ps) compared tothe bulk liquid (1.7 ps). However, we previously showed thatassumptions used in such line shape analysis, while reasonablefor the bulk, are not always valid for nanoconfined liquidswhere the relevant correlation function is multi- or non-exponential.3

Fourkas and co-workers have probed acetonitrile confined insol−gel glasses using OKE spectroscopy.9−13 These experi-ments measure the collective ( = 2) reorientational dynamicsof the confined liquid rather than the single-moleculereorientational dynamics examined by C2(t) in eq 1. Theyfound that the OKE signal, measured out to ∼60 ps, is bestdescribed by a triexponential decay.9,10 Recent simulations ofthe OKE signal by Milischuk and Ladanyi are generallyconsistent with the experimental results and analysis.17 Thefastest time scale, 1.66 ps at 290 K, matched that of bulkacetonitrile, while the additional time scales, 4.5 and 26 ps, weresignificantly slower. They interpreted these results in thecontext of a two-state model with exchange wherein moleculesin the interior of the sol−gel pore reorient like the bulk liquidand molecules at the pore interface reorient more slowly, givingrise to the longest time scale, but molecules can also exchangebetween these two populations. This exchange gives rise to theintermediate time scale as molecules move from the slowerinterfacial dynamics to rapid bulk-like dynamics throughdiffusion to the interior. Using a kinetic analysis, they showedthat this model predicts a triexponential decay for the OKEsignal, consistent with their results, and predicts the surfacelayer thickness as 4.7 Å, and the time scale for exchange as ∼6ps at 290 K.10

It is interesting to note that our single-molecule reorienta-tional correlation functions can also be fit to a triexponentialform for the first ∼60 ps or more, though clearly the long-timedynamics is distinctly nonexponential. Thus, our simulationssuggest that the reorientational dynamics near the poreinterface is too complex to be described by a single timescale.26 However, it is still interesting to investigate how the

dynamics of nanoconfined acetonitrile can be described, at themolecular level, by a two-state model (with or withoutexchange), which is perhaps the most invoked interpretationof experimental measurements in the literature. This is thesubject of the next section.

3. TWO-STATE MODELS3.1. A Decomposition of the Correlation Function.

Two-state models can be constructed in a way that makes theunderlying approximationsalong with their quality andoriginsclear. While this approach could be applied to anynanoconfined liquid property, here we consider the = 1reorientational correlation function in eq 1

= ⟨ · ⟩C t te e( ) (0) ( )1 (2)

where e(t) is a unit vector along the CN bond. Withoutapproximation, this correlation function can be decomposed byinserting unity in the form of a sum of projection operators:

= ϑ + ϑ + ϑ + ϑ→ →t t t t1 (0, ) (0, ) (0, ) (0, )s c s c c s (3)

Here, “s” indicates the “shell” or interfacial region and “c”denotes the “core” or interior, such that ϑs(0, t) = 1 for amolecule continually in the shell from time 0 to t and zerootherwise; ϑc(0, t) has the analogous meaning for molecules inthe core. Then, ϑs→c(0, t) = 1 if a molecule started in the shellregion at time 0 but left the shell before time t; note that themolecule can have any fate after leaving the shell and at time tmay be in the shell or the core. The final projection operator,ϑc→s(0, t) has an analogous interpretation for molecules initiallyin the core at time 0.Implicit in these definitions is the notion of a sharp division

between the two states, shell and core. Most often, this isassumed to be based on the distance from the confiningframework surface, d. In that case, the mathematical expressionfor the shell step function at time 0 would be

ϑ = Θ Δ − d(0, 0) [ (0)]s (4)

where Θ[x] is the standard Heaviside step function that is 0 forx < 0 and 1 for x ≥ 0, Δ is the width of the shell layer next tothe confining interface, and d(t) is the distance of a moleculefrom the interface at time t. The full projection operator canthen be written as

∏ϑ = Θ Δ − ′′=

t d t(0, ) [ ( )]t

t

s0 (5)

which naturally has a form that, in practice, is discretized intime when analyzing a molecular dynamics simulation. Thesedefinitions are used in the analysis below. Analogousexpressions can be written for the other projection operatorsin eq 3. In addition, other choices for differentiating the shelland core states are, of course, also possible.Inserting unity in this fashion into eq 2 gives

= ⟨ϑ · ⟩ + ⟨ϑ · ⟩

+ ⟨ϑ · ⟩ + ⟨ϑ · ⟩→ →

C t t t t t

t t t t

e e e e

e e e e

( ) (0, ) (0) ( ) (0, ) (0) ( )

(0, ) (0) ( ) (0, ) (0) ( )1 s c

s c c s(6)

We can note that, unlike C1(t), the correlation functions on theright-hand side are not normalized. Namely, at t = 0, e(0)·e(t)= 1 for all molecules and time origins, but

⟨ϑ ⟩ = f(0, 0)s s (7)



where fs is the equilibrium fraction of molecules in the shellregion. More generally,

⟨ϑ ⟩ =t f S t(0, ) ( )s s s (8)

where Ss(t) is the shell survival probabilitythe probabilitythat a molecule initially in the shell region at time 0 is still in theshell at time t. Likewise, ⟨ϑc(0, 0)⟩ = fc and ⟨ϑc(0, t)⟩ = fcSc(t),where Sc(t) is the core survival probability.Then, it is convenient to rewrite the first projected

correlation function in eq 6 as

⟨ϑ · ⟩

= ⟨ϑ ⟩⟨ϑ · ⟩

⟨ϑ ⟩

≡

t t

tt t

t

f S t C t

e ee e

(0, ) (0) ( )

(0, )(0, ) (0) ( )

(0, )

( ) ( )

s

ss

s

s s 1,s (9)

where C1,s(t) is now the normalized reorientational correlationfunction for molecules continually in the shell region. Byanalogy,

⟨ϑ · ⟩

= ⟨ϑ ⟩⟨ϑ · ⟩

⟨ϑ ⟩

≡

t t

tt t

t

f S t C t

e ee e

(0, ) (0) ( )

(0, )(0, ) (0) ( )

(0, )

( ) ( )

c

cc

c

c c 1,c (10)

where C1,c(t) is the normalized reorientational correlationfunction for molecules continually in the core region.The same approach can be taken for the correlation

functions involving ϑs→c(0, t) and ϑc→s(0, t) by noting that

⟨ϑ ⟩ = −→ t f S t(0, ) [1 ( )]s c s s (11)

a function that is initially zero and rises as molecules leave theshell region, reaching fs at long times. Then,

⟨ϑ · ⟩

= ⟨ϑ ⟩⟨ϑ · ⟩

⟨ϑ ⟩

≡ −

→

→→

→

→

t t

tt t

t

f S t C t

e ee e

(0, ) (0) ( )

(0, )(0, ) (0) ( )

(0, )

[1 ( )] ( )

s c

s cs c

s c

s s 1,s c (12)

where C1,s→c(t) is the normalized orientational correlationfunction for molecules that are initially in the shell region but

exit by time t; note that this includes molecules that return tothe shell region after exiting. Similarly,

⟨ϑ · ⟩

= ⟨ϑ ⟩⟨ϑ · ⟩

⟨ϑ ⟩

≡ −

→

→→

→

→

t t

tt t

t

f S t C t

e ee e

(0, ) (0) ( )

(0, )(0, ) (0) ( )

(0, )

[1 ( )] ( )

c s

c sc s

c s

c c 1,c s (13)

where C1,c→s(t) is the normalized orientational correlationfunction for molecules that begin in the core region but moveto the shell region before time t.Putting all of these pieces together gives the total

reorientational correlation function, without approximation, as

= + + −

+ −→

→

C t f S t C t f S t C t f S t C t

f S t C t

( ) ( ) ( ) ( ) ( ) [1 ( )] ( )

[1 ( )] ( )

1 s s 1,s c c 1,c s s 1,s c

c c 1,c s (14)

This expression is then a convenient starting point from whicha variety of two-state models can be obtained.

3.2. Two-State Model (with Slow Exchange). Thesimplest two-state model assumes that molecules in the shellregion reorient in the shell and molecules in the core regionreorient in the core, i.e., that exchange between the two states isslower than reorientation. Mathematically, this implies thatC1,s(t) decays much faster than Ss(t) and that C1,c(t) decaysmuch faster than Sc(t). Then, the terms involving exchange aseither 1 − Ss(t) or C1,s→c(t) are zero and likewise for thecorresponding core correlation functions. The reorientationalcorrelation function is thus approximated as

= +C t f C t f C t( ) ( ) ( )c1slow ex h

s 1,s c 1,c (15)

that is, a superposition of the reorientational correlationfunctions in the two regions. A simplified expression forC1slow exch(t) assuming single-exponential dynamics in the shell

and core regions is considered in the Appendix.This model fails to describe the reorientational dynamics in

nanoconfined acetonitrile. This is illustrated in Figure 3 wherethe full reorientational correlation function is compared to thetwo-state model with slow exchange, represented by eq 15. Theshell and core correlation functions, C1,s(t) and C1,c(t), are alsoshown. The results are shown for a core−shell dividing surfacebased on the distance from the nearest pore oxygen atom, d, setat Δ = 4.5 Å. Clearly, the slow exchange approximation predicts

Figure 3. Test of the two-state model with slow exchange for CH3CN in a ∼2.4 nm diameter hydrophilic silica pore. Left: The C1(t) reorientationalcorrelation function (black line) is compared with that for molecules continually in the shell, C1,s(t) (red line), that for molecules continually in thecore, C1,c(t) (violet line), and C1

slow exch(t) in eq 15 (dashed green line) using Δ = 4.5 Å. Right: C1slow exch(t) is compared to C1(t) (black line) for Δ =

4.0 (blue line), 4.5 (green line), 5.0 (violet line), 6.0 (red line), and 7.0 Å (magenta line).



reorientation that is much slower than the actual correlationfunction.Results of the two-state model for different choices of the

shell layer thickness from Δ = 4.0 to 7.0 Å are also shown inFigure 3. Clearly no choice of Δ yields a satisfactory agreementbetween the two-state model and the full simulation results.The approximation does improve consistently as Δ is increasedbut for the unsatisfying reason that more molecules areconsidered to be in the shell. Naturally, as fs approaches 1,C1,s(t) approaches the full correlation function C1(t) and themodel becomes exact, but also lacking in physical meaning.It is also useful to note that the shell correlation function has

an interesting (and clearly not single-exponential) form.Specifically, it decays for the first ∼50 ps before exhibiting aslow rise. Following this, C1,s(t) exhibits a long-time decay (notshown), leading to the tail in C1(t) shown in Figure 2. Thisbehavior is a consequence of exchange of molecules from theshell region. In particular, the molecules that remain in the shellfor a long time also reorient more slowly, so with increasing tthe population included in the average for C1,s(t) includes morerotationally restricted molecules as the more mobile moleculesexit the shell and are no longer included in the correlationfunction. This characteristic of C1,s(t) indicates that cautionshould be used when trying to characterize shell dynamics by asingle time scale.26

3.3. Two-State Model with Exchange. As illustratedabove, when exchange dynamics between the shell and core arerelevant to the reorientation, the simplified assumptions of theslow exchange approximation fail. In this case, Cheng et al. haveproposed a two-state model with exchange:15

= + + −C t f S t C t f C t f S t C t( ) ( ) ( ) ( ) [1 ( )] ( )1exch

s s 1,s c 1,c s s 1,c

(16)

They found that this expression was in excellent agreementwith the simulated C1(t) with the shell thickness taken to be 4.5Å, in good agreement with the 4.7 Å determined by Fourkasand co-workers based on OKE measurements.10 The same istrue in our simulations, which are presented for Δ = 4.5 Å inFigure 4 out to 250 ps. Specifically, C1

exch(t) in eq 16 agrees withthe full C1(t) within the (small) error bars of the latter over theentire time range. It is clear from Figure 4 that the exchange, asmeasured by Ss(t), is indeed taking place on time scalescomparable to the reorientation dynamics in the shell.However, this two-state model with exchange works for most

definitions of the shell layer, which is also illustrated in Figure 4.

Namely, the error in C1exch(t), eq 16, relative to the simulated

C1(t) is plotted for five different choices of the shell layerthickness ranging from 4.0 to 7.0 Å. In all cases, the error is lessthan 6.5% and it is significantly lower, less than ∼2%, for Δ ≥4.5 Å. This indicates that the interfacial thickness cannot beevaluated by varying Δ to give the best agreement with the fullreorientational correlation function. Indeed, the error generallydecreases as Δ increases simply by including more molecules inthe shell population.On the other hand, the data suggest that there may be

something special about Δ ≈ 4.5 Å, as the error does decreasesharply as Δ approaches this value. How this behavior could beuseful in terms of interpreting experimental data is not clear,particularly since experimental uncertainties can be bigger thanor on the order of the largest error observed here in C1

exch(t).Nevertheless, these results provide an impetus to take a deeperlook into the approximations implicit in eq 16 to gain greaterinsight into why it is accurate for such a wide range of shell−core definitions.The two-state model with exchange as expressed in eq 16

makes three approximations to the full decomposition, eq 14,each of which can be tested explicitly using the results of thepresent MD simulations. The first approximation is that

≈S t C t C t( ) ( ) ( )c 1,c 1,c (17)

that is, that molecules reorient before leaving the core region.This is examined in Figure 5 where the full contribution toC1(t) from molecules continually in the core region,fcSc(t)C1,c(t), is compared to the approximation, fcC1,c(t), fortwo values of the shell thickness, Δ = 4.5 and 7.0 Å. The errorin the resulting total reorientational correlation functionassociated with the approximation, namely, fcSc(t)C1,c(t) −fcC1,c(t), is also plotted. It can be seen from the results in Figure5 that the reorientation of molecules in the core is indeed fasterthan their exchange into the shell layer, as indicated by thecomparatively small magnitude of the error. The approximationintroduces error that peaks at a maximum of ∼0.035 around 1.6ps and then decays by 20−30 ps. These short time scales overwhich the error is significant are due to the rapid corereorientation itself, i.e., the decay of C1,c(t).The same physical approximation of faster reorientation than

exchange of core molecules leads to the second assumption

− ≈→f S t C t[1 ( )] ( ) 0c c 1,c s (18)

Figure 4. Left: Same as Figure 3, but the two-state model with exchange is tested. Thus, the shell survival probability, Ss(t) (blue line), is shownalong with C1

exch(t) in eq 16 (dashed green line) using Δ = 4.5 Å. Right: The error in the exchange model, [C1exch(t) − C1(t)]/C1(t), is plotted versus

time for Δ = 4.0 (blue line), 4.5 (green line), 5.0 (violet line), 6.0 (red line), and 7.0 Å (magenta line).



i.e., that molecules have fully reoriented before moving fromthe core region to the shell. The basis of these first twoapproximations was originally outlined by Loughnane et al. indeveloping their two-state-with-exchange model, where theyassumed that “molecules in the bulk reorient before they have achance to exchange into the surface layer”.10 This approx-imation is tested in Figure 6 where the fc[1 − Sc(t)]C1,c→s(t)

contribution is shown for Δ = 4.5 and 7.0 Å. As with the firstapproximation, the error, which here is represented by thecontribution itself, peaks at short times, 2−2.5 ps, with a valueof 0.035−0.04 before decaying. This decay is mostly completewithin 20−30 ps for Δ = 7.0 Å but persists for more than 100ps for Δ = 4.5 Å, though at a value of only ∼0.002. The longer-time dynamics associated with this approximation is presum-ably due to the incomplete reorientation of core moleculesbefore they exchange into the shell where they reorientconsiderably more slowly. This occurs slightly more if a thinnershell thickness is assumed. In any case, the overall errorassociated with this approximation is still quite modest and it isinteresting to note that it is of the opposite sign to the errorintroduced by the first assumption.

Finally, the third assumption is that molecules that begin inthe shell region but exit to the core region have reorientationdominated by the time in the core, which can be expressed as

≈→C t C t( ) ( )1,s c 1,c (19)

or alternatively

− ≈ −→f S t C t f S t C t[1 ( )] ( ) [1 ( )] ( )s s 1,s c s s 1,c (20)

These two versions of this approximation are probed in Figure7. It is clear in these results that C1,s→c(t) and C1,c(t) differsignificantly for smaller values of Δ such as 4.5 Å, while they arein quite good agreement for Δ = 7.0 Å. Specifically, in theformer case, C1,s→c(t) has a significant long-time decay beyond50 ps due to molecules that leave the shell and either stillreorient slowly or return promptly to the shell. Such dynamicsare less likely the larger the shell thickness. This indicates that itis not appropriate to view this third approximation in terms ofeq 19.On the other hand, the results in Figure 7 show that the

approximation as expressed in eq 20 is quite reasonable. Thefactor of 1 − Ss(t), which represents the time scale for arrival ofmolecules from the shell into the core region, significantlydampens the effect of the difference between C1,s→c(t) andC1,c(t) at intermediate times. A long-time tail is still present forthe Δ = 4.5 Å case in the full contribution to the correlationfunction and absent in the approximation, but the magnitude ofthis difference is comparatively small. Specifically, the errorpeaks at less than 0.05 around 16 ps. For the Δ = 7.0 Å case,the agreement is even betterthe error associated with thisapproximation is less than 0.01 for all times.It is interesting to note that the three approximations in the

two-state-with-exchange model, as represented in eq 16, allresult in error that peaks at relatively short times despite thelong-time decay of the full correlation function. This is clearlyrelated to the fact that the decomposition of C1(t) in eq 14involves only two components with significant long-timedynamics: those corresponding to molecules continually inthe shell, fsSs(t)C1,s(t), and to molecules that begin in the shellbut exchange to the core, fs[1 − Ss(t)]C1,s→c(t). The other twocomponents involve comparatively rapid reorientation in thecore which limits the error in their approximate forms to shorttimes. In this sense, the two-state model with exchangeprovides an excellent approximation to the full correlationfunction (Figure 4) due to the separation of time scalesbetween the core and shell dynamics more than the existence ofa sharp division between dynamics in the two regions. This isillustrated by the fact that changing Δ does not affect theaccuracy of the model; a separation of time scales is maintainedunless Δ is small enough that quite slow dynamics is includedin the core component.

4. CONCLUSIONS

Molecular dynamics simulations of nanoconfined acetonitrileshow strong modification of the reorientation dynamics relativeto the bulk liquid. The reorientation is considerably slowed in∼2.4 nm diameter silica pores, but the nature and degreedepends on the surface chemistry. The dynamics isnonexponential and slow in a hydrophilic porean effectattributable to the electrostatic interactions of the surface ratherthan the presence of hydrogen-bond donating OH moietiesbut considerably faster in a hydrophobic pore.

Figure 5. Test of the first approximation of the two-state model withexchange for CH3CN in an ∼2.4 nm diameter hydrophilic silica pore.The full contribution of the C1(t) correlation function, fcSc(t)C1,c(t)(solid lines), is compared to the approximate form, fcC1,c(t) (dashedlines), for Δ = 4.5 Å (black lines) and 7.0 Å (red lines). The errors,measured as the difference between these two results, are also shown(dot-dashed lines) in the upper panel.

Figure 6. Test of the second approximation of the two-state modelwith exchange. The contribution to the C1(t) correlation function, fc[1− Sc(t)]C1,c→s(t), is shown for Δ = 4.5 Å (black line) and 7.0 Å (redline).



To gain insight into the molecular-level origins of theobserved reorientation dynamics, a framework for deriving two-state, or core−shell, models has been developed. The approachis based on a rigorous decomposition of the reorientationalcorrelation function that serves as a starting point for variousapproximations. In this way, the underlying assumptions oftwo-state models, with and without inclusion of exchange ofmolecules between the shell and core states, can be isolated andanalyzed.A two-state model does not describe the reorientation

dynamics of acetonitrile confined within hydrophilic silicapores. This is due to the importance of exchange of moleculesfrom the shell to the core. A two-state model with exchangedoes accurately describe the reorientation dynamics. However,the approximation is excellent for a wide range of definitions ofthe shell state and thus does not suggest a clear, uniquedefinition of an interfacial layer. This is a consequence of theseparation of time scales between the slow reorientationdynamics in the shell and the fast dynamics in the core. Theseparation is maintained for any assumed shell thickness as longas it is large enough that no molecules included in the coreexhibit slow dynamics. The results thus suggest that cautionshould be exercised in inferring molecular-level behavior fromanalysis of data in terms of a two-state model (with or withoutexchange) and, further, that application of the model may beproblematic if the core and shell dynamics are closer in timescale. Looking forward, the present work indicates that moreneeds to be done in the development of strategies for extractingdetailed microscopic insight from experiments on andsimulations of nanoconfined liquids. This can be a significantchallenge due to the intrinsic information content ofexperimental observables combined with the complex dynamicspresent in these systems, and a combination of theoretical andexperimental approaches will likely be required.

5. SIMULATION METHODOLOGY

Classical molecular dynamics (MD) simulations of bulk andnanoconfined liquid acetonitrile were carried out using theDL_POLY_2 package.33 Bulk simulations involved 500acetonitrile molecules at a density of 0.764 g/cm 3 in theNVT ensemble using a Nose−́Hoover thermostat34,35 (1 pstime constant) and cubic periodic boundary conditions. Thelinear, three-site ANL model,28 with Lennard-Jones andCoulombic interactions, was used for acetonitrile. Interactions

were evaluated with a cutoff of 10.0 Å, and long-rangeelectrostatic interactions were included using three-dimensionalperiodic boundary conditions with an Ewald summation usingan Ewald parameter of α = 0.243 and a 6 × 6 × 6 k-point grid.Data were collected from a 2 ns simulation preceded by a 1 nsequilibration with a time step of 2 fs.Previously developed24,36 amorphous silica pore models were

used to simulate confined acetonitrile. The pores have a rigidsilica (SiO2) framework with surface silanol groups, SiOH andSi(OH)2, with fixed bond lengths but variable bond angles; 10pore models, prepared with the same procedure and nominaldiameter of ∼2.4 nm but different amorphous structure, wereexamined. Only quite modest variations between pores wereobserved in preliminary calculations so that the resultspresented here are obtained from a single pore. The poreatoms also interact with Lennard-Jones and Coulombicinteractions as described in detail elsewhere.3 The number ofacetonitrile molecules in each pore was determined in previousgrand canonical Monte Carlo (GCMC) simulations,37 152 forthe pore considered here. The simulation procedure was thesame as that for the bulk except the simulation cell was 44 × 44× 30 Å3 and the k-point grid was 10 × 10 × 8. For each pore,simulations were initiated with a 1 ns equilibration (startingfrom the results of GCMC simulations37) and data wascollected every 0.1 ps over a subsequent 20 ns simulation. Twotrajectories were propagated at a temperature of 298.15 K.In addition to the hydrophilic pores, two modified pore

surfaces were studied. The hydrophilic pores are terminatedwith hydroxyl groups present as silanols, SiOH, or geminals,Si(OH)2;

24,36 the pore considered here contained 42 silanolsand 7 geminals (though not all are sterically accessible to anacetonitrile molecule in the pore). Hydrophobic pores weremodeled by removing the charges on the pore atoms to give aneutral pore. Additionally, non-hydrogen-bonding pores weresimulated, generated by setting the silanol and geminalhydrogen charges to zero (compensated by a change in thecharge on the bonded oxygen); this yields a pore with nohydrogen bond donor groups but leaves the remaining liquid−pore electrostatic interactions intact. These models allow theeffects of surface chemistry to be studied independent of theeffective radius or the number of CH3CN molecules. For each,one 20 ns trajectory was run at T = 298.15 K.

Figure 7. Left: The C1,s→c(t) (solid lines) and C1,c(t) (dashed lines) correlation functions are compared for Δ = 4.5 (black lines) and 7.0 Å (redlines). Right: The full contribution to C1(t), fs[1 − Ss(t)]C1,s→c(t) (solid lines), is compared to the two-state model with exchange approximation, fs[1− Ss(t)]C1,c(t) (dashed lines), for Δ = 4.5 (black lines) and 7.0 Å (red lines). The error introduced in the correlation function, i.e., the difference inthe correlation functions, is shown in the upper panel (dot-dashed lines).



Error bars were calculated using block averaging with fiveblocks and reported at a 95% confidence level using the Studentt distribution.38

■ APPENDIX: TWO-STATE MODELS FORSINGLE-EXPONENTIAL DYNAMICS

It is interesting to consider the form of the reorientationalcorrelation function, C1(t), predicted by the different models ifsingle-exponential behavior is assumed for each process. Thisassumption is often made and can be used to effectively extracttime scale from experimental data, though it is frequently atodds with simulation results which show more complexdynamics,25,26 as in Figure 2.Two-State Model (with Slow Exchange)The two-state model with slow exchange predicts abiexponential form for confined liquid reorientation. Specifi-cally, if reorientation in the shell (core) region is characterizedby a particular time scale τs (τc), then the overall correlationfunction is given as12

= +τ τ− −C t f f( ) e ec t t1slow ex h

s/

c/s c

(21)

As indicated by the present simulations, while the coremolecule reorientation is well-described by a single timescale, the shell dynamics is not single-exponential.Two-State Model with ExchangeWhen exchange is considered, an additional time scale must beadded to the assumption of single exponential reorientationtimes τs and τc for the shell and core molecules. Namely, theshell survival probability is taken to be of the form Ss(t) = e−t/τe.(Note that because the shell and core populations are inequilibrium, detailed balance provides a relationship betweenthe time scales for exchange from the shell, τe, and core, τec, inthis single-exponential (kinetic) limit: τec = τe( fc/fs). However,τec does not enter in the two-state model with exchange.) Withthese assumptions, eq 16 becomes

= + −τ τ τ τ τ− + − − +C t f f( ) e e et t t1exch

s(1/ 1/ ) /

s(1/ 1/ )e s c e c

(22)

This indicates that the two-state model with exchange predictsa tri-exponential decay for the reorientational correlationfunction. It is interesting to note that the same expressionresults from the complete result for C1(t), eq 14, if it is assumedC1,s→c(t) ≈ C1,c→s(t) ≈ C1,c(t). Loughnane et al. also derived atri-exponential decay for the optical Kerr effect signal using akinetic version of this model.10 However, they obtaineddifferent time scalesτs, τc, and (1/τs + 1/τe)

−1than thosefound here. These differences arise from their assumption of anon-exchangeable population of acetonitrile molecules at theinterface. It is interesting to note that, if one further assumesthat the shell exchange time is long compared to the corereorientational time, this reduces to

= +τ τ τ− + −C t f f( ) e et t1exch

s(1/ 1/ )

c/e s c

(23)

a bi-exponential form that differs from the slow-exchange limitonly by the modification of the shell time scale due to exchangedynamics.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by the National Science Foundation(Grant CHE-1012661). W.H.T. thanks Dr. Damien Laage foruseful discussions.

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