J. Phys. Chem. B 2009, 113, 8887–8897
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Local Structural Effects and Related Dynamics in Supercritical Ethanol. 1. Mechanisms of Local Density Reorganization and Residence Dynamics Ioannis Skarmoutsos and Elvira Guardia* Departament de Fı´sica i Enginyeria Nuclear, UniVersitat Polite`cnica de Catalunya, B4-B5 Campus Nord UPC, 08034 Barcelona, Catalonia, Spain ReceiVed: February 4, 2009; ReVised Manuscript ReceiVed: April 14, 2009
The length scale effects on the relaxation processes describing the local density reorganization and residence dynamics of pure supercritical ethanol have been systematically investigated by employing molecular dynamics simulation techniques. The calculated static local density augmentation and enhancement values of ethanol have been found to be comparable to those of methanol at similar conditions but significantly lower than those of the strong associating fluid water, indicating thus the effect of hydrogen-bonding interactions on the creation of local density inhomogeneities in supercritical fluids. The bulk density dependence of local density reorganization dynamics has been studied as a function of the shell cutoff radius, revealing a significant change at length scales higher than the position of the first maximum of the radial pair distribution function. At length scales higher than this cutoff, the local density inhomogeneities affect more significantly the reorganization of the local environment and this becomes even more apparent at length scales corresponding to the second solvation shell radius. Additionally, the residence dynamics for the region extending up to the first solvation shell of ethanol was investigated, and the results obtained reveal that the density dependence of local density dynamics exhibit differences in correspondence to local density dynamics, something which becomes also apparent at length scales higher than the position of the first peak of the radial pair distribution function. I. Introduction It is widely known nowadays that supercritical fluids (scfs) have been classified as very powerful solvents for a wide range of chemical and industrial applications, due to their characteristic physicochemical properties.1-4 According to several experimental and theoretical studies devoted to the properties of scfs so far, these special characteristics are closely related to some peculiar structural effects occurring in these fluids, which are absent in conventional phases such as liquid and gas. More specifically, a scf exhibits significant spatial density fluctuations, and these characteristic density inhomogeneities become even more evident at thermodynamic states close to the fluid’s critical point.5,6 Taking also into account that the compressibility factor of a fluid is closely related to the existence of density fluctuations, we may conclude that the presence of density inhomogeneities strongly affects the behavior of the compressibility in scfs. It is well-known that the high compressibility values in the near critical region are the main reason for one of the most characteristic properties of scfs, where significant variations of density may be achieved with very small pressure changes, causing not only corresponding changes in their dissolving capability and reaction dynamics but also in the eclectic dissolution of different categories of solutes in sc solvents. Consequently, these unique physicochemical properties of scfs are closely related with the existence of density inhomogeneities. Therefore, a deeper understanding of the nature of these microscopic phenomena taking place on molecular length and time scales might be used as a springboard toward a further development of scf technologies. * Corresponding author. E-mail:
[email protected].
In previous works,7,8 one of the authors investigated the local density dynamics for the first and second solvation shell in the case of several scfs, and the significant bulk density effects upon these dynamics has been revealed. These results have also shown that the local density redistribution mechanism around each molecule at short length scales (first shell) is different than in the case of local density dynamics corresponding to larger length scales. Therefore, a more systematic investigation of the length scale effects on the local environment reorganization processes could lead to a deeper understanding of fundamental issues related to the behavior of sc (supercritical) solvents. Moreover, the investigation of the possible interconnection between these local structural effects and related single and collective dynamic properties could provide some answers on many open questions regarding these issues. Among several categories of sc fluids, sc alcohols are often used as solvents in scf processes as an alternative for the widely used sc water, because they provide similar characteristics with water and moreover exhibit lower critical parameters.9-11 One of the most characteristic properties of all these polar associating fluids is that their dielectric constant could be appropriately tuned with small density and pressure changes, allowing thus the eclectic dissolution of polar or nonpolar compounds over a wide density range. Furthermore, the lower alcohols (methanol, ethanol, 2-propanol, etc.) have been widely used as cosolvents in several important applications in order to increase the solubility of polar and high molecular weight compounds in the most commonly used solvent, the nonpolar sc CO2.4,12 In this sense, our present research efforts have been focused on the systematic exploration of the length scale effects on the local density dynamics of one of the most commonly used sc alcohols, namely sc ethanol, by performing a series of molecular dynamics simulations. By investigating the length scale effects on local
10.1021/jp901020x CCC: $40.75 2009 American Chemical Society Published on Web 05/11/2009
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TABLE 1: Summary of the Simulated Thermodynamic State Points for Supercritical Ethanol at T ) 533 Ka simulated state points
F (g/cm3)
A B C D E F G H I
0.053 0.097 0.168 0.237 0.316 0.348 0.395 0.432 0.552
12.5 Å has been applied for all LJ interactions and long-range corrections have been also taken into account. For the cross interactions, the geometric combining rules, instead of the common Lorentz-Berthelot rules, were used. Moreover, to account for the long-range electrostatic interactions, the Ewald summation technique was used, based on the more exact approximation of the Newton-Gregory forward difference interpolation scheme.25
a The critical parameters for ethanol are Tc ) 516.2 K and Fc ) 0.276 g/cm3 (ref 11).
density reorganization dynamics, much useful information concerning the dissolving capability and reaction dynamics in sc solvents could be extracted, promoting thus the development of new powerful scf technologies. Moreover, it is well-known that many researchers in the past have investigated local aggregation effects and related dynamics in molecular fluids in terms of the residence dynamics.13-17 Therefore, the investigation of possible resemblances and differences between these relaxation processes will also provide very important information concerning these different ways of analyzing aspects of local aggregation effects in molecular fluids. This paper is organized as follows: the computational details of the performed simulations are presented in section II. The results obtained and the following discussion upon them is presented in section III. Finally, section IV contains the general conclusions and remarks drawn from the present study. II. Simulation Details The sc ethanol simulation runs were performed in the NVT ensemble along the sc isotherm of 533 K and for a series of densities in the range 0.2-2.0Fc. The simulated state points of sc ethanol are presented in Table 1. The simulations were carried out using 500 molecules. Each simulation was extended to 1.0 ns to achieve equilibrium, starting from an initial face-centered cubic configuration, and the properties of the system were evaluated in subsequent simulation runs with a total duration of 500 ps. In all simulations the equations of motion were integrated using a leapfrog-type Verlet algorithm and the integration time step was set to 1 fs. The Berendsen thermostat18 with a temperature relaxation time of 0.5 ps was also used to constrain the temperature during the simulations. The intramolecular geometry of the species was also constrained by using the SHAKE method.19 The OPLS-UA potential model20 was employed to describe the site-site interactions between the ethanol molecules. This four-site potential model has been successfully used in previous studies of liquid and sc ethanol,21-24 as well as in the case of sc CO2-ethanol mixtures.12 The intermolecular interactions are represented as pairwise additive with site-site Lennard-Jones (LJ) plus Coulomb interactions due to the dipole moment of ethanol molecules. Moreover, the intramolecular torsion around the central C2-O bond is expressed by using the following potential function:
1 1 V(θ) ) V0 + V1(1 + cos θ) + V2(1 - cos 2θ) + 2 2 1 V (1 + cos 3θ) 2 3
(1)
All the interaction potential parameters of this model can be found in details in ref 20. In our simulations, a cutoff radius of
III. Results and Discussion A. Static Local Structure. In the present treatment, we have focused on the investigation of local density inhomogeneities (LDIs) in sc ethanol and we present some information relative to the local density augmentation (LDA) and local density enhancement (LDE) effects in this fluid. In order to extract information relative to the LDIs and the local densities around the ethanol molecules in the fluid, we have calculated the effectiVe local densities, Feff,l,7,8 and subsequently the LDA, (Feff,l - F) and LDE (Feff,l/F) factors, respectively (F is the bulk density). The calculation of these properties is based upon the coordination numbers, Nc, which can be calculated from the corresponding center of mass (com) radial pair distribution functions (pdfs), g(r). An analytic description and justification of the use of this specific methodology applied to investigate LDI effects in scfs is presented in previous publications.7,8,26-28 The com g(r) functions of sc ethanol at the selected state points A, C, E, G, and I are shown in Figure 1. From the comparison of these functions, it is seen that they exhibit a first peak localized at a distance of 4.5 Å for state point A with a marginal shift to 4.7 Å at the highest system density studied (state point I). In contrast to the first peak positions, we observe a significant density dependence of the first peak intensity. Concretely, the intensity of this function decreases from the value 2.18 (state point A) to 1.37 (state point I). Furthermore, the first peak is followed by a distinct minimum and further by a strongly reduced second maximum, while in all cases the third peak is not apparent. The first minimum increases on going from the lowest system density (state A) to the highest one (state I). The position of the first minimum of this function at state point I is observed at 7.1 Å, and according to the method employed to calculate local densities, this distance has been taken into account in our local density calculations as the cut off one for the first shell at each bulk density of the fluid. In the case of the second shell, we have used as a reference cut off distance the value of 11.5 Å, corresponding to the second minimum of the com pdf at the highest density. As we have previously mentioned, to extract information about the LDA and LDE effects in sc ethanol we have calculated the average coordination numbers Νc(F,Rc), corresponding to a spherical shell around each molecule, as a function of the sphere radius (Rc) and the bulk density (F). The bulk density dependence of Νc(F,Rc) for the first solvation shell of the ethanol molecules is presented in Figure 1 also. By inspecting the predicted behavior of Νc from this figure, we may observe that Νc is clearly nonlinear density dependent. This behavior is similar with the behavior of the coordination number of several scfs, calculated in previous studies.7,8,26,27 In order also to have a picture of the change in the structure of the first solvation shell of ethanol as the density is increasing, we present in Figure 2 some snapshots of the solvation shell around an ethanol molecule at several densities along the isotherm under investigation. It should be also noted that this nonlinearity of Νc against density indicates the presence of LDA around the ethanol
Structural Effects and Related Dynamics in sc EtOH
Figure 1. The calculated center of mass radial pair distribution functions g(r) and corresponding coordination numbers for the first shell cutoff of sc ethanol, obtained for several representative state points (see Table 1) in the framework of the present study.
molecules and the extent of this deviation from linearity determines the amplitude of the LDA and LDE parameters.
J. Phys. Chem. B, Vol. 113, No. 26, 2009 8889 According to some trial calculations, for densities higher than 2Fc a linear density dependence of Νc has been observed. Therefore, the reference density (Fref) employed for the calculation of the effective local densities and corresponding quantities (see eq 2 in ref 7) has been set equal to 2Fc. The calculated LDA and LDE parameters for the first and second shell of ethanol as a function of the bulk density are presented in Figure 3. According to our results, the calculated static local density augmentation (LDA) and local density enhancement (LDE) values for sc ethanol have been found to be comparable to those of sc methanol at similar conditions but significantly lower than those of the more strongly associating hydrogen-bonded fluid water,28 indicating thus the effect of hydrogen-bonding interactions on the creation of local density inhomogeneities (LDIs) in sc fluids. Following previous treatments, a four-parameter Weibull line shape function7,8,28 and sigmoidal Boltzmann7,8,28 one have been employed to fit the simulated data, and the results obtained from these fits are presented in Table 2, in comparison with previously obtained results8 for sc methanol. Moreover, the density dependence of the extent of local density fluctuations in sc ethanol has been investigated in terms of the distribution of the local coordination numbers corresponding to the first shell of the molecules, which is also presented in Figure 3. By inspecting this figure, we may observe that at very low densities, which correspond to gaslike structures, and also at the liquidlike density of 2Fc, these distributions are much sharper than in the case of those obtained at the intermediate density range. Furthermore, as we reach the range of densities where the LDA approaches its maximum value, these distributions become even broader. This behavior signifies the fact that in this density range we observe a more significant extent of spatial local density fluctuations. In other words, in this specific density range the fluid becomes more inhomogeneous. B. Local Density Reorganization. The time-dependent redistribution of the local environment around a central ethanol molecule has been investigated in the present study in terms of the local density time correlation function (tcf) C∆Fl(t), where ∆Fl(t) is the local-density deviation, relative to the mean local one, and is defined as ∆Fl(t) ) Fl(t) - 〈Fl〉. This tcf has been
Figure 2. Representative snapshots of the solvation shell around a central ethanol molecule (in circle) as the density of the system is increasing. Depicted are solvation shells with 2, 4, 7, and 10 ethanol molecules, respectively.
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Skarmoutsos and Guardia TABLE 2: Fitted Parameters of the Weibull7,8 and Sigmoidal Boltzmann7,8 Functions Obtained from the MD Simulations of Supercritical Ethanol and Methanol EtOH (present study)
MeOH (ref 8)
Weibull parameters
first shell
second shell
first shell
second shell
a/Fc b/Fc c F0/Fc
0.178 0.971 2.060 0.700
0.080 0.952 2.292 0.739
0.163 0.986 1.957 0.683
0.067 0.967 2.104 0.723
EtOH (present study)
Figure 3. Density dependence of the local density augmentation and enhancement factors corresponding to the first and second shell of ethanol, together with the coordination number distribution for the first shell at all the investigated state points.
extensively presented and discussed in previous publications.7,8,28,29 The correlation time τ∆Fl of this tcf describes the time required for the local environment around a molecule to become uncorrelated with its initial value. In other words, it describes the time scale required for the local environment around a central particle to become reorganized. Previous studies7,8,28 have led to the conclusion that the density dependence of the obtained local density reorganization times for several scfs depends also on the length scale of the region taken into account in the calculations. It is also very interesting to point out that by extending this length scale at larger radius, the reorganization time exhibits maximum values in the density range where the
MeOH (ref 8)
sigmoidal Boltzmann parameters
first shell
second shell
first shell
second shell
a′ b′/Fc F0′/Fc
1.442 0.302 0.792
1.138 0.207 1.015
1.517 0.354 0.626
1.136 0.260 0.925
LDIs become more significant and the LDA values also exhibit a maximum. However, this behavior is not present at short intermolecular distances. This very important observation was a driving force for us to explore in more details the local density reorganization at very short intermolecular distances and to depict some possible reasons for this change, as we extend the size of the solvation shell. The calculated local density tcfs C∆Fl(t) corresponding to the first and second shell of ethanol are presented in Figures 4 and 5, and the density dependence of the corresponding local density reorganization times τ∆Fl is shown in Figure 6. The cutoff distances used to specify the first and second shell have been set to 7.1 and 11.5 Å, respectively, as we also pointed out in the previous section. The behavior of these calculated tcfs and corresponding correlation times exhibits, more or less, many resemblances with the results obtained for several scfs. More specifically, in the case of the first solvation shell in the range of densities up to about the critical one we may observe very small changes in the obtained τ∆Fl values, followed by a rapid decrease at higher densities. On the other hand, at larger length scales (second shell) we may observe an increase of the τ∆Fl values up to a range of densities where the LDA is maximized, followed also by a rapid decrease at higher densities. This is a first indication that the mechanism responsible for the local density reorganization at shorter intermolecular distances exhibits differences in comparison with the mechanisms responsible for the relaxation processes at larger length scales. One other indication of this fact is that at short intermolecular distances the time decay of the obtained tcfs may be described in terms of a biexponential decay function:
C∆Fl(t) ) C1(t) + C2(t) ) ce-t / t1 + (1 - c)e-t / t2
(2) The resulting local density reorganization time in this case may also be written as a sum of two different contributions, corresponding to the fast and slow decay of these tcfs:
τ∆Fl ) ct1 + (1 - c)t2 ) τ1 + τ2
(3)
The results obtained in the present study signify also, as in the case of previous studies,7,8 that the decay of these tcfs at larger time scales, reflected in the time τ2, is mainly responsible for the bulk density dependence of the obtained τ∆Fl whereas,
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Figure 4. The calculated local density tcfs corresponding to the first solvation shell of sc ethanol.
Figure 5. The calculated local density tcfs corresponding to the second solvation shell of sc ethanol.
the contribution of the fast component on these τ∆Fl values, reflected in the time τ1, remains more or less constant, as we may see in Figure 6. These observations led us to the conclusion that presumably the contribution of several collective effects may have some important role on the determination of the local density redistribution at different length scales. According to previous assumptions,29 at short length scales the contribution of more direct two-body interparticle interactions determines mainly the behavior of these relaxation processes. At larger length scales, the arise of more complicated collective effects, due to more complex interactions associated with the coupling of the fluctuations of high and low-density neighboring regions, seems to affect more significantly these relaxation processes. Furthermore, at large length scales the existence of long-range critical density fluctuations has a significant effect on the local density dynamics. This effect has been explained in previous publications29 in terms of a coupling between the local environment and the slow, long-length-scale density fluctuation dynamics, which are characteristic in scfs. At short length scales, where the effects of the intermolecular potentials are quite strong, presumably these direct intermolecular contributions are more important and possibly affect more directly the solvation and reaction dynamics in sc solvents. However, at larger length scales around a central molecule the coupling between extended high- and low-density domains, which is of course related with the correlation length of the fluid, has also an effect on local density reorganization.29 Therefore, the maximization of large
length scale critical fluctuations close to the critical point of the fluid, whose dynamic behavior is reflected on the wellknown long-range critical slowing down effects, seems to have also an effect on long-range local density dynamics. Of course, one point which has to be mentioned here is that it is quite difficult to define the exact boundaries where the more complex interaction mechanisms start to affect more strongly the local density reorganization processes. However, some clear indications concerning the change in the local environment reorganization mechanisms could still be extracted even in the case of not extremely long length scales, providing us therefore some useful information about these boundary effects. Therefore, we decided to investigate the behavior of the local density tcfs of the first solvation shell with respect to the contribution of the processes taking place at very short distances, where mainly binary direct intermolecular interactions are present, as well as in respect to the contributions of the processes taking place at regions where collective many-body contributions start to become significant. To do so, we defined a space boundary inside the first solvation shell, splitting thus the first shell in two different regions. The first region, which hereafter will be named region 1, corresponds to very short intermolecular distances up to the location of the first peak of the com pdf g(r) of the ethanol molecules. This treatment has been based upon the fact that up to these distances the behavior of these pdfs is mainly determined by the pair intermolecular potential {g(r) ∝ exp[-U(r)]}, and therefore, the contribution of these direct intermolecular interactions is more important. The second
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Figure 6. The density dependence of the calculated local density reorganization times τ∆Fl corresponding to the first and second solvation shell of sc ethanol. Depicted is also the density dependence of the fast and slow components, τ1 and τ2, of the overall τ∆Fl value corresponding to the first shell.
region, which hereafter will be named region 2, corresponds to the region from the position of the first peak of the com g(r) up to the first minimum of this function, which determines the size of the first solvation shell. The cutoffs used in this study have been set to the values corresponding to the pdf’s g(r) first maximum and minimum at the highest density (state point I) and have been kept constant for all the investigated state points, since these positions are not significantly affected by the density (see Figure 1). These values are 4.7 and 7.1 Å, respectively. The local density tcfs corresponding to region 1, region 2, and the whole first shell (total) are presented for some representative state points in Figure 7. Furthermore, the density dependence of the corresponding local density reorganization times for the two regions 1 and 2 is presented in Figure 8. From these figures it becomes very clear that the local density reorganization processes at very short length scales are different than those corresponding to larger length scales. We may also notice that the density dependence of the obtained local density reorganization times corresponding to region 1 and region 2 are substantially different, signifying thus the length scale effects on the mechanisms of local density reorganization. We may observe from Figure 8 that the local density reorganization time τ∆Fl corresponding to region 1 decreases monotonically by increasing the density. Since the molecular effects in this region are more directly related to binary intermolecular interactions,
Figure 7. The calculated local density tcfs corresponding to regions 1 and 2 and the total first solvation shell of sc ethanol for some representative state points.
this rapid decrease might be explained in terms of the increase of the contributions in the structure of the fluid arising from repulsive interactions at the higher (liquid like) densities, as is well-known from the literature.30 However, in the case of region 2, possibly due to the contribution of more collective effects, critical fluctuations seem to have an effect to the local density reorganization inside this region, causing the maximization of τ∆Fl in the range of densities where critical fluctuations are more important. This behavior of τ∆Fl with density in region 2 is also clearly reflected on the density dependence of τ∆Fl corresponding to the whole first solvation shell, and this contribution can be seen by carefully inspecting Figures 6 and 8. From these figures we may see that τ∆Fl of region 2 increases up to densities close
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J. Phys. Chem. B, Vol. 113, No. 26, 2009 8893 the density reorganization in the first solvation shell, we calculated the different contributions to the overall local density tcf following a procedure which we will describe below. In general, independent of the definition of the local density around a central molecule (for the several definitions one may look at refs 5, 7, 8, 26), it can be very easily proved that in any case the normalized tcf C∆Fl(t) is identical with the normalized tcf of the quantity δNc(t) ) Nc(t) - 〈Nc〉, i.e. the deviation of the instantaneous coordination number corresponding to a specified cutoff distance around a central molecule:
C∆Fl(t) ≡ CδNc(t) )
〈δNc(0) δNc(t)〉 〈δNc(0)2〉
(4)
It is also very well-known that the local coordination number Νc(Rc), corresponding to a specified cutoff distance Rc, may be written as
Nc(Rc) ) 4πF
∫0R r2 g(r) dr c
(5)
In the case of the first solvation shell of ethanol, if Rc is the location of the first minimum of the com g(r) functions and R1 is the position of the first peak, then eq 5 may be written in the form Nc(Rc) ) 4πF
∫
R1 2 r 0
g(r) dr + 4πF
∫
Rc 2 r R1
g(r) dr ) N1 + N2
(6) In this equation N1 and N2 are the coordination numbers corresponding to regions 1 and 2, respectively. If δN1(t) ) N1(t) - 〈N1〉 and δN2(t) ) N2(t) - 〈N2〉 then
δNc(t) ) Nc(t) - 〈Nc〉 ) N1(t) + N2(t) - 〈N1 + N2〉 ) N1(t) - 〈N1〉 + N2(t) - 〈N1〉 ) δN1(t) + δN2(t) (7) Figure 8. The density dependence of the calculated local density reorganization times τ∆Fl corresponding to regions 1 and 2 inside the first solvation shell of sc ethanol.
to 0.7Fc, whereas in region 1 we observe a constant monotonic decrease of τ∆Fl with density. However, the different contributions of these two relaxation processes in regions 1 and 2 have as a result the existence of a plateau-like behavior of the resulting τ∆Fl values for the overall first shell in the low-density region. The existence of this plateau of τ∆Fl corresponding to the first solvation shell has also been observed for other scfs,8 but this is the first time where it is revealed that this plateau is a result of the contributions of the local environment reorganization processes of two different regions inside the solvation shell. Of course, at higher densities, the constant monotonic decrease with density of τ∆Fl corresponding not only to region 1 but also to region 2 results in the decrease of the overall local density reorganization time of the first shell also. In general, we might say that due to the change in the local density reorganization process when going from region 1 to region 2, the overall behavior of the corresponding τ∆Fl of the first shell with density starts to become different than in the case of region 1. Of course, this difference in the density dependence of τ∆Fl is much more apparent at even more extended length scales, like in the second solvation shell, where the values of τ∆Fl tend to exhibit maximum values in the density range where critical fluctuations are also maximized. In order to have also a more clear picture of the contributions of regions 1 and 2 on the overall relaxation processes describing
In this case, eq 4 may be written in the form
CδNc(t) )
〈δN1(0) δN1(t)〉
+
〈δN2(0) δN2(t)〉
+ 〈δNc(0) 〉 〈δNc(0)2〉 〈δN2(0) δN1(t)〉 〈δN1(0) δN2(t)〉 + 2 〈δNc(0) 〉 〈δNc(0)2〉 2
(8)
This equation may be written also in the form:
CδNc(t) ) CR1(t) + CR2(t) + CCross(t)
(9)
In eq 9 CR1(t) and CR2(t) are the autocorrelation functions corresponding to the reorganization of regions 1 and 2 and CCross(t) is the sum of the two cross-correlation functions. Before proceeding any further, we have to mention that eqs 8 and 9 are substantially different from eq 2, since they present the different length scale contributions on the local density reorganization of the first solvation shell of sc EtOH, whereas the model function in eq 2 presents the different time scale contributions in this relaxation process. In other words, eqs 2, 8, and 9 present different ways of depicting and analyzing local density reorganization mechanisms in scfs. In Figure 9 we present these contributions on the total C∆Fl(t) tcf for the first shell of ethanol for some representative state points. From Figure 9 we can see that the most important contributions are those corresponding to region 2, as well as those corresponding to the cross tcfs. These cross tcfs reflect the coupling between the two neighboring regions 1 and 2, and we may notice that their behavior depends very significantly on the bulk density of the system.
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Figure 10. The density dependence of the calculated τR1, τR2, and τCross values.
CCross(t) appearing in eq 8, we have to say that they exhibit a very fast increase at very short time scales up to 0.35-0.4 ps, where they reach a positive maximum value and then they start to decay to zero slowly. This decay is even slower than the decay of CR2(t) at very low densities; however, with increasing the density the decay becomes faster. Furthermore, we may observe a significant shift of the initial values of the cross tcfs CCross(t) at lower values as the density increases and at the highest density (state point I) we may observe a initial time value close to -0.57, whereas the maximum of CCross(t) located at 0.35 ps has a very small positive value of 0.04 and then the function decays slowly to zero. Following eqs 4 and 9, the local density reorganization time of the first shell may be expressed as
τ∆Fl ≡ τδNc )
∫0∞CδN (t) dt ) τR1 + τR2 + τCross c
(10)
Figure 9. The calculated contributions to the local density tcf corresponding to the total first solvation shell of sc ethanol (see eq 7) for some representative state points.
By analyzing the behavior of the tcfs appearing in eq 9, some very important conclusions may be drawn. At the low density region, up to about 0.7Fc, the decay of the tcf CR2(t), corresponding to region 2, becomes slower as the density increases. This behavior is clearly related to the fact that the local density reorganization time of region 2 is maximized in this density region. At densities higher than 0.7Fc, these functions start to decay much faster. On the other hand, the tcfs CR1(t), corresponding to region 1, decay faster as the density increases in the whole density range investigated, something which is also in accordance with the fact that the local density reorganization time of region 1 monotonically decreases with the density. Finally, concerning the sum of the two cross-components
In eq 10, the times τR1, τR2, and τCross depict the contributions of regions 1 and 2 and the cross-contributions to the overall reorganization time, respectively. The density dependence of these times is presented in Figure 10, and from this figure we may clearly notice the maximization of τR2 close to 0.7Fc and the monotonic decrease of τR1 and τCross with density. We may also notice that while at very low densities the cross-contributions are quite significant, this contribution decreases quite rapidly at higher densities. So, at the highest density investigated, the time integral of CCross(t) arising from the very fast initial part (negative part of the tcf) and the slow decay from the positive maximum value of 0.04 (located at 0.35 ps) to zero is almost close to zero, signifying thus the insignificant cross contributions to the local density reorganization dynamics at liquid-like densities. In general, taking also into account the behavior of other scfs,7,8 we may say that the density dependence of the local density reorganization times exhibits three different behaviors at three different length scales. First, at very short intermolecular distances (up to 4-5 Å), we may observe a monotonic decrease of these times when the density is increasing. By slightly increasing this length scale up to about 6-7 Å (which in many cases correspond to the first solvation shell boundary for several scfs), due to the fact that collective effects seem to have more important contributions in the range of low up to intermediate densities we do not observe a very significant change in the
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behavior of τ∆Fl, which decreases significantly at higher densities. Finally, at higher length scales, we observe an increasing behavior of τ∆Fl with density in the low-density region, and we observe a maximum in the range of densities where the critical fluctuations are also maximized. At higher densities, we also observe a rapid decrease of τ∆Fl, even in this length scale. C. Residence Dynamics. In order to extract more information about the dynamic processes taking place inside the first solvation shell of ethanol, we decided to investigate also several features related to the residence dynamics inside this shell. Since residence dynamics provides an alternative way to investigate dynamic local aggregation effects in molecular fluids, we sought for possible resemblances and differences between this relaxation process and the local density dynamics. According to the literature, the residence tcf inside a solvation shell around a central particle i could be defined as
Cres(t) )
〈hij(0) hij(t)〉t* 〈hij(0)2〉
(11)
The corresponding residence time is defined as
τres )
∫0∞Cres(t) dt
(12)
The variable hij has been defined in the following way: hij(t) ) 1, if molecule j is inside the solvation shell of molecule i at times 0 and t and the molecule j has not left in the meantime the shell for a period longer than t*. Otherwise, hij(t) ) 0. Of course, using this definition, the calculation of Cres(t) depends upon the selection of the parameter t*. The two limiting cases arising from this definition are (a) if t* ) 0, which represents the so-called continuous definition, and (b) if t* ) ∞, which represents the intermittent definition. We have to mention that these two definitions describe very different aspects of residence dynamics, since according to the continuous definition the exits of molecule j outside the shell of molecule i during the time interval [0, t] are not allowed. On the other hand, in the intermittent case the persistence of molecule j in the solvation shell of i at time t is investigated, regardless of multiple exits and entrances of this molecule in the shell during the time interval [0, t]. One other case of intermittent-like dynamics, which is often used in the investigation of residence dynamics, is the case when t* ) τresC, where τresC is the continuous residence time. In the present work, we have calculated the continuous residence functions CresC(t) and the corresponding continuous residence times τresC, as well as the functions corresponding to a value of t* ) 2 ps, which is close to the calculated values of τresC for the investigated state points of ethanol. We have calculated also the functions corresponding to t* ) ∞, but their very slow convergence to zero even at large time scales (∼40 ps) did not allow us to find a suitable fitting function to represent this very slow decay, in order to present the corresponding residence times. Therefore, we considered as more appropriate ones for the case of intermittent-like dynamics the functions obtained using t* ) 2 ps. These functions will be called Cres(t) hereafter and their corresponding residence times τres. We have to mention that we performed our analysis for the first solvation shell of ethanol, as well as for the regions 1 and 2, defined in the previous section, in order to extract additional information about the relaxation processes taking place in these specific length scales, which have been previously related with some changes in the local density reorganization mechanisms. The change in the shape and decay of the obtained Cres(t) for the whole first shell (total), as well as for regions 1 and 2 for
Figure 11. The calculated residence tcfs (using t* ) 2 ps) corresponding to regions 1 and 2 and the total first solvation shell of sc ethanol for some representative state points.
some representative state points, is presented in Figure 11. Furthermore, the density dependence of the calculated residence times τresC and τres is presented for the aforementioned space regions in Figure 12. By inspecting these figures, we may see that the shape and decay of the functions corresponding to regions 1 and 2 and the one corresponding to the total first shell are essentially different. Furthermore, the density dependence of the obtained residence times is different, signifying thus also a change in the mechanism describing the residence dynamics; the shell volume is extending to larger length scales. We may notice also here that the most significant changes are observed in the case of the intermittent-like dynamics, whereas in the case of
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Skarmoutsos and Guardia possibly due to the density dependence of τres corresponding to region 2, because at short distances we may observe a decrease of τres at low densities and then a plateau behavior, followed by a small increase at liquidlike densities. This behavior signifies a change in the intermittent-like residence dynamic process at this length scale. Note also that this intermittent-like residence relaxation process, which is related with allowed entrances and exits of a molecule in and out of a space region, respectively, seems to be more closely related with the local density reorganization process than the continuous one. Therefore, this is a possible reason why this change in the density dependence of the calculated residence times at larger length scales is more reflected in the case of the noncontinuous dynamics. Here we have to mention also that this density dependence of τres for region 1 has been observed in the case of the first solvation shell of sc water,13 which is located more or less in the same length scale of the space boundary for regions 1 and 2 of ethanol. By comparing also these density dependences of the obtained residence and local density reorganization times, we can easily observe that they reflect two different ways of analyzing the dynamics of molecular aggregation effects in scfs. However, we may find some resemblances in the density dependence of the fast component of the local density reorganization time for the first shell of ethanol and on the continuous residence time. By observing previous studies devoted to the investigation of collision effects in liquids,31 we may see that at very short length scales, close to the repulsive core of the intermolecular potential, the continuous residence time τresC resembles the collision duration time. Furthermore, previous MD studies focused on solvation dynamics in scfs32 have also shown that the fast component of the solvation tcf, which reflects the contribution of repulsive interactions, does not also exhibit significant density dependence. All these observations signify thus a possible interconnection of binary collision effects with continuous residence and the fast components of local density tcfs at short length scales. On the other hand, at larger distances, the appearance of more complex interactions and collective effects related more with the existence of critical fluctuations seems to affect more significantly the behavior of the slow decay of the local density tcfs, which determines the density dependence of the overall local density reorganization time. IV. General Conclusions
Figure 12. The density dependence of the calculated residence times τresC(t* ) 0 ps) and τres(t* ) 2 ps) corresponding to regions 1 and 2 and the total first solvation shell of sc ethanol.
continuous dynamics, we do not observe significant density dependence especially for regions 1 and 2. Concerning the continuous residence times τresC we may observe that in general they remain constant in the case of regions 1 and 2, whereas for the whole first shell they exhibit a minimum close to the density region where critical fluctuations are maximized, and then they increase at higher densities. However, this behavior is much more pronounced in the case of τres, where we may observe a significant increase with density at this high density region. From Figure 12, we may see that this behavior for τres of the first solvation shell is observed
In the present study, the MD simulation technique was employed to investigate the local density inhomogeneity effects and related dynamics in sc ethanol. More specifically, the static LDA effects, as well as the effect of the length scale on the local density reorganization and residence dynamics, have been systematically explored. Concerning the static local structural effects, the calculated static LDA and LDE values for sc ethanol have been found to be comparable to those of sc methanol at similar conditions, but significantly lower than those of the strongly hydrogenbonded associating fluid water. This fact indicates the effect of hydrogen-bonding interactions on the creation of LDIs in scfs. Furthermore, the extent of local density fluctuations has been investigated in terms of the local coordination number distribution, and it has been found that the static local density fluctuations are maximized in the density range close to the critical one. The bulk density dependence of local density reorganization dynamics has been studied as a function of the shell cutoff radius, revealing a significant change at length scales higher than the position of the first maximum of the radial pair
Structural Effects and Related Dynamics in sc EtOH distribution function. In general, we have observed that the density dependence of the local density reorganization times τ∆Fl exhibits three different behaviors at three different length scales. First, at very short intermolecular distances (up to 4-5 Å) we may observe a monotonic decrease of these times when the density is increasing. By slightly increasing this length scale up to about 6-7 Å (which in many cases correspond to the first solvation shell boundary for several scfs), possibly due to the appearance of contributions arising from collective effects, this reorganization process starts to change. In the range of low up to intermediate densities, we do not observe a very significant change in the behavior of τ∆Fl, followed by a significant decrease at higher densities. At length scales higher than this cutoff, LDIs affect more significantly the reorganization of the local environment, and this becomes even more apparent at length scales corresponding to the second solvation shell radius. In the low density region, we observe an increasing behavior of τ∆Fl with density exhibiting a maximum in the range of densities where the critical fluctuations are also maximized. At higher densities, we also observe a rapid decrease of τ∆Fl even at this length scale. Additionally, the residence dynamics for the region extending up to the first solvation shell of ethanol was investigated, and the results obtained reveal that the density dependence of local density dynamics exhibit differences in correspondence to local density dynamics, something that becomes also apparent at length scales higher than the position of the first peak of the pdf g(r). Moreover, a change in the mechanism responsible for residence dynamics is observed also at length scales higher than the position of the first maximum of the com pdf. However, we may observe some resemblances in the density dependence of the fast component of τ∆Fl for the first shell of ethanol mainly with the one corresponding to the continuous residence times at short length scales. This fact might possibly be attributed to some contributions of binary collision effects on the dynamics of these local structural effects. However, these contributions seem not to be very important at higher length scales, and collective effects related also with the existence of LDI seem to control the dynamic behavior of the local structure. Acknowledgment. One of the authors (I.S.) acknowledges the postdoctoral financial support of the Department of Physics and Nuclear Engineering (DFEN, http://www.fen.upc.edu) of the Technical University of Catalonia (UPC), Barcelona-Spain. The CPU time allocation on the facilities of the Computer Simulation in Condensed Matter Research Group in DFEN is also gratefully acknowledged. E.G. gratefully acknowledges financial support from the Direccio´ General de Recerca de la
J. Phys. Chem. B, Vol. 113, No. 26, 2009 8897 Generalitat de Catalunya (Grant 2005SGR-00779) and from the Ministerio de Educacio´n y Ciencia of Spain (Grant FIS200612436-CO2-01). References and Notes (1) Kiran, E.; Debenedetti, P. G.; Peters, C. J., Eds. Supercritical Fluids: Fundamentals and Applications; NATO ASI Science Series E Applied Sciences; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000; Vol. 366. (2) Johnston, K. P.; Kim, S.; Coumbes, J.; Penninger, J. M. L., Eds. Supercritical Fluid Science and Technology; American Chemical Society: Washington DC, 1998; Chapter 5 and refs therein. (3) Eckert, C. A.; Knutson, B. L.; Debenedetti, P. G. Nature 1996, 383, 313. (4) Besnard, M.; Tassaing, T.; Danten, Y.; Andanson, J. M.; Soetens, J. C.; Cansell, F.; Loppinet-Serani, A.; Reveron, H.; Aymonier, C. J. Mol. Liq. 2006, 125, 88. (5) Tucker, S. C. Chem. ReV. 1999, 99, 391. (6) Kajimoto, O. Chem. ReV. 1999, 99, 355. (7) Skarmoutsos, I.; Samios, J. J. Phys. Chem. B 2006, 110, 21931. (8) Skarmoutsos, I.; Samios, J. J. Chem. Phys. 2007, 126, 044503. (9) Lalanne, P.; Andanson, J. M.; Soetens, J. C.; Tassaing, T.; Danten, Y.; Besnard, M. J. Phys. Chem. A 2004, 108, 3902. (10) Barlow, S. J.; Bondarenko, G. V.; Gorbaty, Y. E.; Yamaguchi, T.; Poliakoff, M. J. Phys. Chem. A 2002, 106, 10452. (11) Hiejima, Y.; Yao, M. J. Chem. Phys. 2003, 119, 7931. (12) Skarmoutsos, I.; Dellis, D.; Samios, J. J. Chem. Phys. 2007, 126, 224503. (13) Guardia, E.; Marti, J. Phys. ReV. E 2004, 69, 011502. (14) Inomata, H.; Saito, S.; Debenedetti, P. G. Fluid Phase Equilib. 1996, 116, 282. (15) Flanagin, L. W.; Balbuena, P. B.; Johnston, K. P.; Rossky, P. J. J. Phys. Chem. 1995, 99, 5196. (16) Impey, R. M.; Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071. (17) Guardia, E.; Laria, D.; Marti, J. J. Phys. Chem. B 2006, 110, 6332. (18) Berendsen, H. J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Di Nola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (19) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comp. Phys. 1977, 23, 327. (20) Jorgensen, W. L. J. Phys. Chem. 1986, 90, 1276. (21) Padro, J. A.; Saiz, L.; Guardia, E. J. Mol. Struct. 1997, 416, 243. (22) Saiz, L.; Padro, J. A.; Guardia, E. Mol. Phys. 1999, 97, 897. (23) Saiz, L.; Guardia, E.; Padro, J. A. J. Chem. Phys. 2000, 113, 2814. (24) Dellis, D.; Chalaris, M.; Samios, J. J. Phys. Chem B. 2005, 109, 18575. (25) Paschek, D.; Geiger, A. MOSCITO 4.140; University of Dortmund: Dortmund, Germany, 2007. (26) Song, W.; Biswas, R.; Maroncelli, M. J. Phys. Chem. A 2000, 104, 6924. (27) Nugent, S.; Ladanyi, B. M. J. Chem. Phys. 2004, 120, 874. (28) Skarmoutsos, I.; Dellis, D.; Samios, J. J. Phys. Chem B. 2009, 113, 2783. (29) Tucker, S. C.; Maddox, M. W. J. Phys. Chem. B 1999, 102, 2437. (30) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (31) Samios, J.; Dorfmu¨ller, Th. J. Chem. Phys. 1982, 76, 5463. (32) Yamaguchi, T.; Kimura, Y.; Hirota, N. J. Chem. Phys. 1999, 111, 4169.
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