Article pubs.acs.org/JPCB
Reorientation Dynamics of Nanoconfined Acetonitrile: A Critical Examination of Two-State Models Cassandra D. Norton and Ward H. Thompson* Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, United States ABSTRACT: Molecular dynamics simulations are used to investigate the reorientation dynamics of liquid acetonitrile confined within a nanoscale, hydrophilic silica pore. The dynamics are strongly modified relative to the bulk liquidthe time scale for reorientation is increased by orders-of-magnitude and the dynamics become nonexponentialand these effects are examined at the molecular level. In particular, commonly invoked two-state (or core−shell) models, with and without consideration of exchange of molecules between the states, are applied and discussed. A rigorous decomposition of the acetonitrile reorientational correlation function is introduced that permits the approximations implicit in the two-state models to be identified and tested systematically. The results show that exchange is an important component of the nanoconfined acetonitrile reorientation dynamics and a two-state model with exchange can accurately describe the correlation. However, the faithfulness of the model is related to the separation of time scales in the two states, which exists for a wide range of definitions of the two states. This suggests that caution should be exercised when inferring molecular-level details from application of two-state models. “shell,” are assumed to have strongly modified properties, while the remainder, the “core”, are located further from the surface and are at best weakly modified. Frequently, the core is taken to have the same behavior as the bulk liquid. This division of molecules assumes a distinct separation between these two categories, typically based on the distance of a molecule from the interface. In analyzing their OKE results, Fourkas and coworkers extended this model to include the effect of exchange of molecules between the two states.9−13 Two-state models provide a useful way to interpret experimental measurements to yield molecular-level information.2,5−13,18−23 However, the underlying assumptions strongly affect the nature of the resulting inferences and it is thus important to clearly understand their accuracy. Molecular dynamics (MD) simulations can provide significant insight in this regard as spectroscopic observables can be clearly related to the molecular structure and dynamics to probe the two-state picture.24−27 Indeed, simulations of acetonitrile reorientation in silica pores have been carried out by multiple groups (including ours).3,14,15,17 The results are in general agreement with experimental measurements and each other, but analysis has led to different conclusions regarding the validity of the twostate model.
1. INTRODUCTION Nanoconfined liquids continue to attract attention for their interesting fundamental properties, the tunability of the confining framework, and their range of potentially important applications.1,2 However, a full molecular-level picture of how the structural and dynamical properties of a liquid are modified upon confinement is still lacking. Such an understanding would represent a key insight into the emergence of new phenomena at liquid−solid interfaces, the effects of different framework properties from size to surface chemistry, and the basis of design principles for practical applications from catalysis to sensing. In this paper, we examine these issues in the context of the reorientation dynamics of acetonitrile confined within ∼2.4 nm silica pores. These dynamics are a critical component of the mechanisms of important processes like electron and proton transfer reactions as well as spectroscopic probes such as infrared and Raman spectra. Acetonitrile is of particular interest due to its wide use as a solvent and its interesting vibrational spectroscopy.3−5 In addition, acetonitrile in silicate glasses has been the subject of numerous experimental and computational studies.3−17 Of particular note are experimental studies of acetonitrile reorientation in sol−gel pores using NMR6,7 and optical Kerr effect (OKE)9−13 spectroscopies. Jonas and co-workers interpreted their NMR results in terms of a two-state, or core−shell, model,6,7 perhaps the most commonly invoked description for nanoconfined liquids. In this model, the molecules in the liquid are assumed to consist of two categories; those near the confining framework interface, the © 2014 American Chemical Society
Special Issue: James L. Skinner Festschrift Received: February 7, 2014 Revised: March 28, 2014 Published: April 1, 2014 8227
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be compared to those for water, τ1/τ2 = 2.0 and τ1/τ3 = 3.1, which has been shown to reorient via jumps involving exchange of hydrogen-bonding partners rather than by Debye rotational diffusion.32 When acetonitrile is confined in nanoscale silica pores, the reorientational dynamics are dramatically modified. In the following, we will use C1(t) as a measure of these dynamics. Examination of the other correlation functions gives the same essential behavior: while in section 3 we examine the accuracy of various two-state approximations for C1(t), we have carried out the same analysis for C2(t) and the results are qualitatively the same with quantitative errors that are slightly smaller due to the faster decay of the C2(t) correlation function relative to C1(t). The C1(t) correlation function is plotted for acetonitrile confined in ∼2.4 nm silica pores with different surface interactions and compared with the bulk liquid result in Figure 2. These results clearly show that the reorientational dynamics
The remainder of the paper examines this issue. Molecular dynamics simulation results for acetonitrile reorientation dynamics in the bulk liquid and silica pores are presented in section 2. The general features are described, and comparisons are made with experimental data where possible. A systematic derivation of a two-state model, with and without core−shell exchange, is then presented in section 3 by introduction of a rigorous decomposition of the reorientational correlation function; an appendix presents the models in the commonly invoked limit of single exponential dynamics for each state. The models are tested by analysis of the MD simulation data and the individual approximations are identified and separately examined. Finally, some conclusions are offered in section 4.
2. ACETONITRILE REORIENTATION Molecular reorientation dynamics in liquids can be examined in terms of the reorientational correlation functions CS(t ) = ⟨PS[e(0) ·e(t )]⟩
(1)
where PS is the S th Legendre polynomial, e(t) is the unit vector along a particular molecular axis at time t, and ⟨...⟩ indicates a thermal average over all molecules in the liquid. In this work on the rod-like molecule acetonitrile, this vector is taken to be along the main molecular axis, e.g., the CCN direction. The correlation functions obtained from the present MD simulations of bulk acetonitrile for S = 1, 2, and 3 are shown in Figure 1. Each displays single-exponential dynamics following a
Figure 2. The C1(t) reorientational correlation function for confined acetonitrile in pores with hydrophilic (black line), hydrophobic (red line), and hydrogen-bonding “turned off” (blue line) surface chemistry is compared to the bulk result (violet line).
are slowed by more than an order of magnitude upon confinement and that this slowdown is sensitive to the surface chemistry. Specifically, acetonitrile confined in a hydrophilic pore exhibits nearly the same behavior whether hydrogen-bond donor groups, e.g., Si−OH, are present or modified so they cannot 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 both significantly faster at long times (though still an order of magnitude slower than in the bulk liquid) and qualitatively different. Namely, C1(t) for acetonitrile in the hydrophobic pore can be fit with a triexponential function with one time scale corresponding to inertial motion, 0.55 ps (16%), and two that are longer than the bulk value: 6.2 ps (65%) and 27.0 ps (19%). The correlation function for the hydrophilic pore and that with no hydrogen-bond donors exhibits a complex, nonexponential behavior; it is not well fit with a power-law or stretched exponential over the entire span of the long-time decay though each form reasonably describes part of the correlation function. These data improve on our initial report of the reorientational correlation function for nanoconfined acetonitrile where C1(t) was calculated out to only 40 ps.3 In the remainder of this paper, we examine the underlying molecular-level acetonitrile dynamics while focusing on the hydrophilic pore case.
Figure 1. The CS (t) reorientational correlation functions for bulk liquid acetonitrile. Results are shown for S = 1 (black line), 2 (red line), and 3 (blue line).
short-time inertial response. The CS (t) can be fit with this form using a Gaussian function to describe the inertial component. The resulting exponential time scales (τS ) are τ1 = 3.0 ps, τ2 = 1.1 ps, and τ3 = 0.63 ps. For reasons that are unclear, these results differ somewhat from the values reported by Gee and Gunsteren (τ1 = 1.93 ps; τ2 = 0.80 ps) when they presented the acetonitrile model.28 However, they are in good agreement with experimental measurements of τ1 = 3.68 ps based on infrared spectral analysis,29 τ2 = 1.09 ± 0.10 ps from Raman measurements,30 and ⟨τ2⟩ = ∫ C2(t) dt = 1.02 ps from NMR measurements31 (the present data give ⟨τ2⟩ = 0.95 ps). These data show that the acetonitrile force field used here28 provides a satisfactory 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 rotational diffusion, which predicts that τ1/τ2 = 3 and τ1/τ3 = 6; the results here give τ1/τ2 = 2.7 and τ1/τ3 = 4.8. These values can 8228
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dynamics of nanoconfined acetonitrile can be described, at the molecular level, by a two-state model (with or without exchange), which is perhaps the most invoked interpretation of experimental measurements in the literature. This is the subject of the next section.
Jonas and co-workers used NMR relaxation measurements to characterize the reorientation dynamics of acetonitrile confined in sol−gel pores of diameter 3.1, 4.0, 6.0, and 9.6 nm.6,7 They interpreted their measurements in terms of a two-state modelthe time scale of the NMR measurements is long compared to any exchange between the core and shell. From this analysis, they extracted rotational diffusion constants for bulk and “shell” acetonitrile, with the shell layer taken to be two molecular diameters (8.2 Å), as ⟨τ2⟩ = 0.95 and 10.5 ps, respectively. The former is in excellent agreement with our bulk simulations, as noted above. The latter represents a time scale that is fast relative to our simulation results for all pore molecules where we obtain ⟨τ2⟩ > 70 ps; it should be noted that the pore diameter used here is smaller than any they considered and 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 neutron scattering (QENS).5 They found that the rotational time scalesτR = 2.5 ± 0.3 and 1.4 ± 0.3 ps for the smaller and larger pores, respectivelywere not strongly modified from that of the bulk liquid (1.4 ± 0.2 ps). However, this is likely indicative of the relative insensitivity of the QENS signal to the longer-time reorientational dynamics. Tanaka et al. also extracted rotational time scales for acetonitrile confined in MCM-41 pores (3.2 nm in diameter) from an analysis of the infrared line shape.8 They found that the reorientation times were nearly unmodified in the pores (1.6−1.7 ps) compared to the bulk liquid (1.7 ps). However, we previously showed that assumptions used in such line shape analysis, while reasonable for the bulk, are not always valid for nanoconfined liquids where the relevant correlation function is multi- or nonexponential.3 Fourkas and co-workers have probed acetonitrile confined in sol−gel glasses using OKE spectroscopy.9−13 These experiments measure the collective (S = 2) reorientational dynamics of the confined liquid rather than the single-molecule reorientational dynamics examined by C2(t) in eq 1. They found that the OKE signal, measured out to ∼60 ps, is best described by a triexponential decay.9,10 Recent simulations of the OKE signal by Milischuk and Ladanyi are generally consistent with the experimental results and analysis.17 The fastest time scale, 1.66 ps at 290 K, matched that of bulk acetonitrile, while the additional time scales, 4.5 and 26 ps, were significantly slower. They interpreted these results in the context of a two-state model with exchange wherein molecules in the interior of the sol−gel pore reorient like the bulk liquid and molecules at the pore interface reorient more slowly, giving rise to the longest time scale, but molecules can also exchange between these two populations. This exchange gives rise to the intermediate time scale as molecules move from the slower interfacial dynamics to rapid bulk-like dynamics through diffusion to the interior. Using a kinetic analysis, they showed that this model predicts a triexponential decay for the OKE signal, consistent with their results, and predicts the surface layer thickness as 4.7 Å, and the time scale for exchange as ∼6 ps at 290 K.10 It is interesting to note that our single-molecule reorientational correlation functions can also be fit to a triexponential form for the first ∼60 ps or more, though clearly the long-time dynamics is distinctly nonexponential. Thus, our simulations suggest that the reorientational dynamics near the pore interface is too complex to be described by a single time scale.26 However, it is still interesting to investigate how the
3. TWO-STATE MODELS 3.1. A Decomposition of the Correlation Function. Two-state models can be constructed in a way that makes the underlying approximationsalong with their quality and originsclear. While this approach could be applied to any nanoconfined liquid property, here we consider the S = 1 reorientational correlation function in eq 1 C1(t ) = ⟨e(0) ·e(t )⟩
(2)
where e(t) is a unit vector along the CN bond. Without approximation, this correlation function can be decomposed by inserting unity in the form of a sum of projection operators: 1 = ϑs(0, t ) + ϑc(0, t ) + ϑs → c(0, t ) + ϑc → s(0, t )
(3)
Here, “s” indicates the “shell” or interfacial region and “c” denotes the “core” or interior, such that ϑs(0, t) = 1 for a molecule continually in the shell from time 0 to t and zero otherwise; ϑc(0, t) has the analogous meaning for molecules in the core. Then, ϑs→c(0, t) = 1 if a molecule started in the shell region at time 0 but left the shell before time t; note that the molecule can have any fate after leaving the shell and at time t may be in the shell or the core. The final projection operator, ϑc→s(0, t) has an analogous interpretation for molecules initially in 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 is assumed to be based on the distance from the confining framework surface, d. In that case, the mathematical expression for the shell step function at time 0 would be ϑs(0, 0) = Θ[Δ − d(0)]
(4)
where Θ[x] is the standard Heaviside step function that is 0 for x < 0 and 1 for x ≥ 0, Δ is the width of the shell layer next to the confining interface, and d(t) is the distance of a molecule from the interface at time t. The full projection operator can then be written as t
ϑs(0, t ) =
∏ Θ[Δ − d(t′)] t ′= 0
(5)
which naturally has a form that, in practice, is discretized in time when analyzing a molecular dynamics simulation. These definitions are used in the analysis below. Analogous expressions can be written for the other projection operators in eq 3. In addition, other choices for differentiating the shell and core states are, of course, also possible. Inserting unity in this fashion into eq 2 gives C1(t ) = ⟨ϑs(0, t )e(0) · e(t )⟩ + ⟨ϑc(0, t )e(0) ·e(t )⟩ + ⟨ϑs → c(0, t )e(0) · e(t )⟩ + ⟨ϑc → s(0, t )e(0) ·e(t )⟩ (6)
We can note that, unlike C1(t), the correlation functions on the right-hand side are not normalized. Namely, at t = 0, e(0)·e(t) = 1 for all molecules and time origins, but ⟨ϑs(0, 0)⟩ = fs 8229
(7)
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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) reorientational correlation function (black line) is compared with that for molecules continually in the shell, C1,s(t) (red line), that for molecules continually in the exch exch (t) in eq 15 (dashed green line) using Δ = 4.5 Å. Right: Cslow (t) is compared to C1(t) (black line) for Δ = core, C1,c(t) (violet line), and Cslow 1 1 4.0 (blue line), 4.5 (green line), 5.0 (violet line), 6.0 (red line), and 7.0 Å (magenta line).
exit by time t; note that this includes molecules that return to the shell region after exiting. Similarly,
where fs is the equilibrium fraction of molecules in the shell region. More generally, ⟨ϑs(0, t )⟩ = fs Ss(t )
⟨ϑc → s(0, t )e(0) ·e(t )⟩
(8)
where Ss(t) is the shell survival probabilitythe probability that a molecule initially in the shell region at time 0 is still in the shell 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
= ⟨ϑc → s(0, t )⟩
≡ fc [1 − Sc(t )]C1,c → s(t )
⟨ϑs(0, t )e(0) ·e(t )⟩ ⟨ϑs(0, t )⟩
≡ fs Ss(t )C1,s(t )
C1(t ) = fs Ss(t )C1,s(t ) + fc Sc(t )C1,c(t ) + fs [1 − Ss(t )]C1,s → c(t ) (9)
+ fc [1 − Sc(t )]C1,c → s(t )
where C1,s(t) is now the normalized reorientational correlation function for molecules continually in the shell region. By analogy,
⟨ϑc(0, t )e(0) ·e(t )⟩ ⟨ϑc(0, t )⟩
≡ fc Sc(t )C1,c(t )
(10)
where C1,c(t) is the normalized reorientational correlation function 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 ⟨ϑs → c(0, t )⟩ = fs [1 − Ss(t )]
C1slowexc h(t ) = fs C1,s(t ) + fc C1,c(t )
(11)
⟨ϑs → c(0, t )e(0) ·e(t )⟩ ⟨ϑs → c(0, t )e(0) ·e(t )⟩ ⟨ϑs → c(0, t )⟩
≡ fs [1 − Ss(t )]C1,s → c(t )
(15)
that is, a superposition of the reorientational correlation functions in the two regions. A simplified expression for exch Cslow (t) assuming single-exponential dynamics in the shell 1 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 where the full reorientational correlation function is compared to the two-state model with slow exchange, represented by eq 15. The shell and core correlation functions, C1,s(t) and C1,c(t), are also shown. The results are shown for a core−shell dividing surface based on the distance from the nearest pore oxygen atom, d, set at Δ = 4.5 Å. Clearly, the slow exchange approximation predicts
a function that is initially zero and rises as molecules leave the shell region, reaching fs at long times. Then,
= ⟨ϑs → c(0, t )⟩
(14)
This expression is then a convenient starting point from which a variety of two-state models can be obtained. 3.2. Two-State Model (with Slow Exchange). The simplest two-state model assumes that molecules in the shell region reorient in the shell and molecules in the core region reorient in the core, i.e., that exchange between the two states is slower than reorientation. Mathematically, this implies that C1,s(t) decays much faster than Ss(t) and that C1,c(t) decays much faster than Sc(t). Then, the terms involving exchange as either 1 − Ss(t) or C1,s→c(t) are zero and likewise for the corresponding core correlation functions. The reorientational correlation function is thus approximated as
⟨ϑc(0, t )e(0) ·e(t )⟩ = ⟨ϑc(0, t )⟩
(13)
where C1,c→s(t) is the normalized orientational correlation function for molecules that begin in the core region but move to the shell region before time t. Putting all of these pieces together gives the total reorientational correlation function, without approximation, as
⟨ϑs(0, t )e(0)·e(t )⟩ = ⟨ϑs(0, t )⟩
⟨ϑc → s(0, t )e(0) ·e(t )⟩ ⟨ϑc → s(0, t )⟩
(12)
where C1,s→c(t) is the normalized orientational correlation function for molecules that are initially in the shell region but 8230
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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 shown exch along with Cexch 1 (t) in eq 16 (dashed green line) using Δ = 4.5 Å. Right: The error in the exchange model, [C1 (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).
Namely, the error in Cexch 1 (t), eq 16, relative to the simulated C1(t) is plotted for five different choices of the shell layer thickness ranging from 4.0 to 7.0 Å. In all cases, the error is less than 6.5% and it is significantly lower, less than ∼2%, for Δ ≥ 4.5 Å. This indicates that the interfacial thickness cannot be evaluated by varying Δ to give the best agreement with the full reorientational correlation function. Indeed, the error generally decreases as Δ increases simply by including more molecules in the shell population. On the other hand, the data suggest that there may be something special about Δ ≈ 4.5 Å, as the error does decrease sharply as Δ approaches this value. How this behavior could be useful in terms of interpreting experimental data is not clear, particularly since experimental uncertainties can be bigger than or on the order of the largest error observed here in Cexch 1 (t). Nevertheless, these results provide an impetus to take a deeper look into the approximations implicit in eq 16 to gain greater insight 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 the present MD simulations. The first approximation is that
reorientation that is much slower than the actual correlation function. 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 in Figure 3. Clearly no choice of Δ yields a satisfactory agreement between the two-state model and the full simulation results. The approximation does improve consistently as Δ is increased but for the unsatisfying reason that more molecules are considered to be in the shell. Naturally, as fs approaches 1, C1,s(t) approaches the full correlation function C1(t) and the model 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 a slow rise. Following this, C1,s(t) exhibits a long-time decay (not shown), leading to the tail in C1(t) shown in Figure 2. This behavior is a consequence of exchange of molecules from the shell region. In particular, the molecules that remain in the shell for a long time also reorient more slowly, so with increasing t the population included in the average for C1,s(t) includes more rotationally restricted molecules as the more mobile molecules exit the shell and are no longer included in the correlation function. This characteristic of C1,s(t) indicates that caution should be used when trying to characterize shell dynamics by a single time scale.26 3.3. Two-State Model with Exchange. As illustrated above, when exchange dynamics between the shell and core are relevant to the reorientation, the simplified assumptions of the slow exchange approximation fail. In this case, Cheng et al. have proposed a two-state model with exchange:15
Sc(t )C1,c(t ) ≈ C1,c(t )
(17)
that is, that molecules reorient before leaving the core region. This is examined in Figure 5 where the full contribution to C 1(t) from molecules continually in the core region, fcSc(t)C1,c(t), is compared to the approximation, fcC1,c(t), for two values of the shell thickness, Δ = 4.5 and 7.0 Å. The error in the resulting total reorientational correlation function associated with the approximation, namely, fcSc(t)C1,c(t) − fcC1,c(t), is also plotted. It can be seen from the results in Figure 5 that the reorientation of molecules in the core is indeed faster than their exchange into the shell layer, as indicated by the comparatively small magnitude of the error. The approximation introduces error that peaks at a maximum of ∼0.035 around 1.6 ps and then decays by 20−30 ps. These short time scales over which the error is significant are due to the rapid core reorientation 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
C1exch(t ) = fs Ss(t )C1,s(t ) + fc C1,c(t ) + fs [1 − Ss(t )]C1,c(t ) (16)
They found that this expression was in excellent agreement with the simulated C1(t) with the shell thickness taken to be 4.5 Å, in good agreement with the 4.7 Å determined by Fourkas and co-workers based on OKE measurements.10 The same is true in our simulations, which are presented for Δ = 4.5 Å in Figure 4 out to 250 ps. Specifically, Cexch 1 (t) in eq 16 agrees with the full C1(t) within the (small) error bars of the latter over the entire time range. It is clear from Figure 4 that the exchange, as measured by Ss(t), is indeed taking place on time scales comparable 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.
fc [1 − Sc(t )]C1,c → s(t ) ≈ 0 8231
(18)
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Finally, the third assumption is that molecules that begin in the shell region but exit to the core region have reorientation dominated by the time in the core, which can be expressed as C1,s → c(t ) ≈ C1,c(t )
(19)
or alternatively fs [1 − Ss(t )]C1,s → c(t ) ≈ fs [1 − Ss(t )]C1,c(t )
(20)
These two versions of this approximation are probed in Figure 7. It is clear in these results that C1,s→c(t) and C1,c(t) differ significantly for smaller values of Δ such as 4.5 Å, while they are in quite good agreement for Δ = 7.0 Å. Specifically, in the former case, C1,s→c(t) has a significant long-time decay beyond 50 ps due to molecules that leave the shell and either still reorient slowly or return promptly to the shell. Such dynamics are less likely the larger the shell thickness. This indicates that it is not appropriate to view this third approximation in terms of eq 19. On the other hand, the results in Figure 7 show that the approximation as expressed in eq 20 is quite reasonable. The factor of 1 − Ss(t), which represents the time scale for arrival of molecules from the shell into the core region, significantly dampens the effect of the difference between C1,s→c(t) and C1,c(t) at intermediate times. A long-time tail is still present for the Δ = 4.5 Å case in the full contribution to the correlation function and absent in the approximation, but the magnitude of this difference is comparatively small. Specifically, the error peaks at less than 0.05 around 16 ps. For the Δ = 7.0 Å case, the agreement is even betterthe error associated with this approximation 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, all result in error that peaks at relatively short times despite the long-time decay of the full correlation function. This is clearly related to the fact that the decomposition of C1(t) in eq 14 involves only two components with significant long-time dynamics: those corresponding to molecules continually in the shell, fsSs(t)C1,s(t), and to molecules that begin in the shell but exchange to the core, fs[1 − Ss(t)]C1,s→c(t). The other two components involve comparatively rapid reorientation in the core which limits the error in their approximate forms to short times. In this sense, the two-state model with exchange provides an excellent approximation to the full correlation function (Figure 4) due to the separation of time scales between the core and shell dynamics more than the existence of a sharp division between dynamics in the two regions. This is illustrated by the fact that changing Δ does not affect the accuracy of the model; a separation of time scales is maintained unless Δ is small enough that quite slow dynamics is included in the core component.
Figure 5. Test of the first approximation of the two-state model with exchange 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) (dashed lines), 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.
i.e., that molecules have fully reoriented before moving from the core region to the shell. The basis of these first two approximations was originally outlined by Loughnane et al. in developing their two-state-with-exchange model, where they assumed that “molecules in the bulk reorient before they have a chance to exchange into the surface layer”.10 This approximation is tested in Figure 6 where the fc[1 − Sc(t)]C1,c→s(t)
Figure 6. Test of the second approximation of the two-state model with 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 (red line).
contribution is shown for Δ = 4.5 and 7.0 Å. As with the first approximation, the error, which here is represented by the contribution itself, peaks at short times, 2−2.5 ps, with a value of 0.035−0.04 before decaying. This decay is mostly complete within 20−30 ps for Δ = 7.0 Å but persists for more than 100 ps for Δ = 4.5 Å, though at a value of only ∼0.002. The longertime dynamics associated with this approximation is presumably due to the incomplete reorientation of core molecules before they exchange into the shell where they reorient considerably more slowly. This occurs slightly more if a thinner shell thickness is assumed. In any case, the overall error associated with this approximation is still quite modest and it is interesting to note that it is of the opposite sign to the error introduced by the first assumption.
4. CONCLUSIONS Molecular dynamics simulations of nanoconfined acetonitrile show strong modification of the reorientation dynamics relative to the bulk liquid. The reorientation is considerably slowed in ∼2.4 nm diameter silica pores, but the nature and degree depends on the surface chemistry. The dynamics is nonexponential and slow in a hydrophilic porean effect attributable to the electrostatic interactions of the surface rather than the presence of hydrogen-bond donating OH moieties but considerably faster in a hydrophobic pore. 8232
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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 (red lines). 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 in the correlation functions, is shown in the upper panel (dot-dashed lines).
were evaluated with a cutoff of 10.0 Å, and long-range electrostatic interactions were included using three-dimensional periodic boundary conditions with an Ewald summation using an 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 ns equilibration with a time step of 2 fs. Previously developed24,36 amorphous silica pore models were used to simulate confined acetonitrile. The pores have a rigid silica (SiO2) framework with surface silanol groups, SiOH and Si(OH)2, with fixed bond lengths but variable bond angles; 10 pore models, prepared with the same procedure and nominal diameter of ∼2.4 nm but different amorphous structure, were examined. Only quite modest variations between pores were observed in preliminary calculations so that the results presented here are obtained from a single pore. The pore atoms also interact with Lennard-Jones and Coulombic interactions as described in detail elsewhere.3 The number of acetonitrile molecules in each pore was determined in previous grand canonical Monte Carlo (GCMC) simulations,37 152 for the pore considered here. The simulation procedure was the same 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 (starting from the results of GCMC simulations37) and data was collected every 0.1 ps over a subsequent 20 ns simulation. Two trajectories 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 terminated with hydroxyl groups present as silanols, SiOH, or geminals, Si(OH)2;24,36 the pore considered here contained 42 silanols and 7 geminals (though not all are sterically accessible to an acetonitrile molecule in the pore). Hydrophobic pores were modeled by removing the charges on the pore atoms to give a neutral pore. Additionally, non-hydrogen-bonding pores were simulated, generated by setting the silanol and geminal hydrogen charges to zero (compensated by a change in the charge on the bonded oxygen); this yields a pore with no hydrogen bond donor groups but leaves the remaining liquid− pore electrostatic interactions intact. These models allow the effects of surface chemistry to be studied independent of the effective radius or the number of CH3CN molecules. For each, one 20 ns trajectory was run at T = 298.15 K.
To gain insight into the molecular-level origins of the observed reorientation dynamics, a framework for deriving twostate, or core−shell, models has been developed. The approach is based on a rigorous decomposition of the reorientational correlation function that serves as a starting point for various approximations. In this way, the underlying assumptions of two-state models, with and without inclusion of exchange of molecules between the shell and core states, can be isolated and analyzed. A two-state model does not describe the reorientation dynamics of acetonitrile confined within hydrophilic silica pores. This is due to the importance of exchange of molecules from the shell to the core. A two-state model with exchange does accurately describe the reorientation dynamics. However, the approximation is excellent for a wide range of definitions of the shell state and thus does not suggest a clear, unique definition of an interfacial layer. This is a consequence of the separation of time scales between the slow reorientation dynamics in the shell and the fast dynamics in the core. The separation is maintained for any assumed shell thickness as long as it is large enough that no molecules included in the core exhibit slow dynamics. The results thus suggest that caution should be exercised in inferring molecular-level behavior from analysis of data in terms of a two-state model (with or without exchange) and, further, that application of the model may be problematic if the core and shell dynamics are closer in time scale. Looking forward, the present work indicates that more needs to be done in the development of strategies for extracting detailed microscopic insight from experiments on and simulations of nanoconfined liquids. This can be a significant challenge due to the intrinsic information content of experimental observables combined with the complex dynamics present in these systems, and a combination of theoretical and experimental approaches will likely be required.
5. SIMULATION METHODOLOGY Classical molecular dynamics (MD) simulations of bulk and nanoconfined liquid acetonitrile were carried out using the DL_POLY_2 package.33 Bulk simulations involved 500 acetonitrile molecules at a density of 0.764 g/cm 3 in the NVT ensemble using a Nosé−Hoover thermostat34,35 (1 ps time constant) and cubic periodic boundary conditions. The linear, three-site ANL model,28 with Lennard-Jones and Coulombic interactions, was used for acetonitrile. Interactions 8233
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Error bars were calculated using block averaging with five blocks and reported at a 95% confidence level using the Student t distribution.38
ACKNOWLEDGMENTS This work was supported by the National Science Foundation (Grant CHE-1012661). W.H.T. thanks Dr. Damien Laage for useful discussions.
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APPENDIX: TWO-STATE MODELS FOR SINGLE-EXPONENTIAL DYNAMICS It is interesting to consider the form of the reorientational correlation function, C1(t), predicted by the different models if single-exponential behavior is assumed for each process. This assumption is often made and can be used to effectively extract time scale from experimental data, though it is frequently at odds with simulation results which show more complex dynamics,25,26 as in Figure 2. The two-state model with slow exchange predicts a biexponential form for confined liquid reorientation. Specifically, if reorientation in the shell (core) region is characterized by a particular time scale τs (τc), then the overall correlation function is given as12 (21)
As indicated by the present simulations, while the core molecule reorientation is well-described by a single time scale, the shell dynamics is not single-exponential. Two-State Model with Exchange
When exchange is considered, an additional time scale must be added to the assumption of single exponential reorientation times τs and τc for the shell and core molecules. Namely, the shell survival probability is taken to be of the form Ss(t) = e−t/τe. (Note that because the shell and core populations are in equilibrium, detailed balance provides a relationship between the time scales for exchange from the shell, τe, and core, τec, in this single-exponential (kinetic) limit: τec = τe( fc/fs). However, τec does not enter in the two-state model with exchange.) With these assumptions, eq 16 becomes C1exch(t ) = fs e−t(1/ τe+ 1/ τs) + e−t / τc − fs e−t(1/ τe+ 1/ τc)
(22)
This indicates that the two-state model with exchange predicts a tri-exponential decay for the reorientational correlation function. It is interesting to note that the same expression results from the complete result for C1(t), eq 14, if it is assumed C1,s→c(t) ≈ C1,c→s(t) ≈ C1,c(t). Loughnane et al. also derived a tri-exponential decay for the optical Kerr effect signal using a kinetic version of this model.10 However, they obtained different time scalesτs, τc, and (1/τs + 1/τe)−1than those found here. These differences arise from their assumption of a non-exchangeable population of acetonitrile molecules at the interface. It is interesting to note that, if one further assumes that the shell exchange time is long compared to the core reorientational time, this reduces to C1exch(t ) = fs e−t(1/ τe+ 1/ τs) + fc e−t / τc
(23)
a bi-exponential form that differs from the slow-exchange limit only by the modification of the shell time scale due to exchange dynamics.
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Two-State Model (with Slow Exchange)
C1slowexc h(t ) = fs e−t / τs + fc e−t / τc
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