Article pubs.acs.org/JPCC
Test of Universal Scaling Law for Molecular Diffusion of Liquids in Bulk and Nanotube Confinement Pooja Sahu, Sk. Musharaf Ali,* and K. T. Shenoy Chemical Engineering Division, Bhabha Atomic Research Center, HBNI, Mumbai 400085, India S Supporting Information *
ABSTRACT: The task of observing and understanding the relationship between transport and thermodynamic properties of atomic and molecular fluids has been a continuing open problem in condensed matter. The problem becomes more complex and challenging when the fluids are confined in nanoscale dimensions. In order to address this complex problem, a scaling law has been established by linking the molecular diffusivity and excess entropy of molecular liquids and liquid mixtures both in bulk and under nanoconfinement, which is found to be quite universal and also reproduces the earlier scaling law for atomic diffusion [Nature (London) 1996, 381, 137]. The excess entropy, which is central to this universal scaling law, has been estimated using a robust and very fast “two phase thermodynamic” (2PT) method, where density of states (DOS) has been employed, which may also be obtained from power spectrum of diffusing liquids using scattering experiments thus offers an indirect route to determine the molecular diffusivity. The present scaling law would contribute to the deeper understanding of molecular transport in bulk and through CNTs, which might be very supportive for various related fields of liquid filtration, biological applications, and nanotechnologies. fluids. However, until now the reported studies were confined to the hard-sphere (HS) fluids,1 and their generalization to the real liquids still remains a challenge because of the incapability of hard-sphere (HS) fluids to capture the structural diversities. The HS fluids account only for binary collisions without involving long-range interactions and therefore have serious limitations to be applied for real molecular systems. Some of the literature studies report the scaling relation using LennardJones (LJ)-like atomic fluid models.12,13 However, the use of atomic fluid model is inadequate for fluids with orientationdependent pair potentials and intramolecular degrees of freedom as it completely neglects the intramolecular contribution, which is known to play a vital role in structural relaxations. Therefore, the further studies were carried out for liquid silica, water, and other model liquids with isotropic coresoftened potentials.19,20 Remarkable efforts were made by Yan et al.,21 Zielkiewicz et al.,22 and Agarwal et al.23 to relate the water anomalies with excess entropy. After two decades of
U
nderstanding the relationship between transport coefficients and entropy is of major importance in the various fields of crystallization, nucleation, and glass transition.1−3 Therefore, numerous efforts have been devoted to connect the transport properties in the framework of kinetic theories to thermodynamic quantities.4−7 Most of these kinetic theories describe the diffusion mechanism based on Enskog theory,8 where it is hypothesized that the dynamics of liquid phase is controlled by “cage effect” in which each atom is restricted in a cage formed by immediate neighbors due to local density fluctuations.9 Such structural relaxations are strongly coupled with the diffusive motion of atoms, and this line of thinking has been explored to estimate the liquid state dynamics from the static structures.10 The pioneering work in this direction was first carried out by Rosenfeld5 followed by Dzugutov11 through a universal scaling law, which connects the atomic diffusivity to excess entropy. Dzugutov suggested that diffusion is connected with the frequency of local structural relaxations and introduced new reducing parameters for the Rosenfeld scaling relation. Further, Samanta et al.12−14 gave theoretical background of scaling relation using mode coupling theory.15,16 In sequence, Bastea et al.17,18 extended the scaling relation for low density © XXXX American Chemical Society
Received: May 5, 2017 Published: May 10, 2017 A
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
Figure 1. Scaling relation for D* and Sex with (a) Dzugutov equation. (b) Modified relation of Dm* and Sex.
universal scaling law for atomic diffusion, until there is no unifying description of this law for molecular liquids, liquid mixtures and liquids under nanoconfinement. Also, origin of the apparent connection between the dynamic and thermodynamic details of the fluids is not well understood. The literature studies report different modified scaling laws for various complex systems such as liquid metals,7,14 ionic melts, and fluids within nanoporous silica24 as well as zeolite.25−27 However, transferability of these modified scaling laws is yet to be examined considering the type of molecular fluids as well as nanoscale geometries. Apart from these, the impact of hydrogen bonding on scaling relation is still unexplained. Also, to the best of our knowledge, no scaling relation has been proposed for fluids confined in nanotubes. In reality, it is very difficult to estimate the accurate diffusivity of fluids confined in such one-dimensional geometries from the experiments. Therefore, linking transport coefficients with the experimentally easily accessible thermodynamic quantities will be of great scientific and technological importance. Interestingly, fluids confined in nanotube geometries do not obey the continuum theory and undergoes the phase transition from single file structure to ice like dense packed structure.28 The fluids confined within CNT exhibits structural anomalies21,29,30 and the dynamics of these fluid molecules is driven by the caging effect formed by immediate self-neighbors as well as by confining walls.31 Since confined fluids are known to show structural and dynamical anomalies,32,33 it would be very demanding to establish a scaling relation for these systems. Moreover, the scaling relation for nanotube confinement would be of great practical use as they relate to many biological channels such as in aquaporin, proton pumps, and protein cavities in lysozymes.34 Therefore, considering the importance of scaling relation for complex systems, the present study is dedicated to establish the scaling relationship for molecular liquids in bulk and within CNT using molecular dynamic simulations. The self-diffusion coefficient and the excess entropy are related through the following expression:11 D* = A[exp(BSex )]
The above expression is based on two main arguments. First, the transfer of energy and momentum in liquids are dominated by short-range repulsive interactions, which can be approximated by the binary collisions of hard spheres. So, the diffusion coefficient (D) can be represented in dimensionless form D* via D* = DΓ−1σ−2, where σ is the diameter of hard sphere and Γ is the Enskog collision frequency, given by ⎛ πk T ⎞1/2 Γ = 4σ 2g (σ )ρ⎜ B ⎟ ⎝ m ⎠
(2)
Here, m and ρ represent the molecular mass and density of the system, respectively. The quantity σ is determined from position of the first peak in pair correlation function, and g(σ) is the value of pair correlation function at σ. The second idea central to the relationship between structure and the dynamics of liquid systems, i.e., the frequency of local structural relaxations in the liquids is determined by the number of accessible configurations. Linking both arguments, the normalized diffusivity D* become proportional to exp(Sex), where Sex is the excess entropy. The value of A and B was taken as 0.049 and 1, respectively. The value of B = 1 was decided from the ergodic hypothesis. However, this condition is usually not satisfied in MD simulations due to the finite time scale. Samanta et al.12 provided a theoretical derivation for this B factor to be 2/3; therefore, in the present scaling relation the value of B has been used as 2/3. Recently, Hoyt et al.35 concluded that the excess entropy under two body approximation of pair correlation function g(r) (eq 3) works reasonably well for systems with simple potential model but deviates significantly for complex molecular systems. Sex ≈ S2 = −2πρ
∫0
∞
{g (r ) ln g (r ) − [g (r ) − 1]}r 2 dr (3)
Since the present article involves very complex molecular systems, a very fast and accurate algorithm is required to estimate the excess entropy. Therefore, an efficient and robust two phase thermodynamic (2PT) scheme has been adopted to estimate the excess entropy (details of the 2PT method are given in section S.1.1 of the Supporting Information). 2PT method provides very accurate thermodynamic quantities with
(1) B
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
reported for the scaling law.6 In the present study, the scaled diffusivity data was seen to be overestimated for η < 0.45 and underestimated for η > 0.45 for pure fluids. Therefore, to make the diffusivity data collapse onto scaling law, the diffusivity data was rescaled with λ = 0.4 to tackle the overestimation and λ = 7.2 to deal with underestimation. It is noteworthy that the value of λ remains the same for mixture of fluids as well as fluids under confinement, but the value of C has been varied based on the physical state of the fluid. The finding of physical explanations for these constants is very difficult and formidable task; however, considering the practical importance of the relationship, the physical explanation of these constants will be attempted in near future. The results show that the proposed relation works better for methane than methanol and water, which perhaps can be correlated with the strength of hydrogen (H) bonding for the respective fluids. Though methane being non-H bonding fluid with simple molecular structure is the best to follow the scaling relation, water, a strong H bonding fluid also agrees well albeit with minor deviation. In the proposed scaling relation, the structural rearrangement by H bonds was taken care by including the terms g(σ) and ρ* in factor α. Nevertheless, a small deviation was noticed for water at temperatures near the freezing point. This might be due to very strong H bonding of the water near the freezing temperature known as “bulk ice rule”, where each water molecule acts as double donor as well as double acceptor of hydrogen bonds and therefore connects with exact four nearest neighbors, leading to solid-like structure formation. In other words, the proposed relation is more applicable to the liquid phase as compared to the solid phase. One-to-one mapping between Dm* and exp(Sex) for bulk liquids also agrees well when they are severely confined in nanotubes. However, the extent of collapsing is seen to be governed by (i) H bond characteristics of the confined fluid and (ii) nanoscale dimension of the CNT. This might be the reason that though water and methanol-like H bonding fluids show maximum deviation for CNT(7,7), methane as non-H bonding fluid show the maximum deviation for CNT(6,6). Further, as per literature studies,28,39 the smaller CNTs are known to display the peculiar properties of confined fluids due to the extraordinary structure formation inside the tubes and the same has been captured in our simulation studies40−42 as shown in Figure 2. The snapshots show the structural anomalies for fluids confined in narrow CNTs. CNT(6,6) and CNT(7,7) indicate the formation of single file structure and double file structure, respectively. Interestingly, the water molecules show structural transformation from perfectly hydrogen-bonded tetrahedral network within CNT(8,8) to hexagonal network within CNT(10,10), which is quite different from the original tetrahedral structure of water in liquid phase. The physical states of these confined water molecules are shown by the density of state functions (DOS) in Figure 3. Results illustrate the intermediate states of water during confinement in critical sized nanotubes, where they follow ice-like symmetry with liquid-like hydrogen bonding. The structural diversities were also noted with methanol-like one-dimensional hydrogen bonding fluid. However, for the comparable range of nanoconfinement, the dissimilarities in structure of water and methanol were observed due to the difference in their strength of H bonding. Additionally, methane-like nonpolar fluid was observed to flow in separate fluid lines inside CNTs, which was varied from single flow line to six fluid lines from CNT(6,6) to CNT(10,10). In other words, the orientation of fluid molecules
very short (∼20 ps) MD trajectories (simulation details are provided in section S.3 of the Supporting Information). Most importantly, 2PT method36 estimates entropy from the density state functions, which can also be obtained from the experiments.37,38 The present work thus addresses the ability of earlier scaling relationship to describe the connections between dynamic and thermodynamic properties of the real liquids and liquid mixtures in bulk and within CNT confinement. Figure. 1a illustrates the reduced self-diffusivity using Dzugutov scaling law as a function of the excess entropy over the temperature range of 90−160 K for methane and 240−350 K for water, methanol, and water−methanol mixture. For nanotube confinement, the diameter of CNT was varied from 8−48 Å. The details of the simulated CNT systems is given in Table S1 of the Supporting Information. Results in Figure. 1a show that data points follow a linear relationship for scaled diffusivity vs excess entropy Sex, however, differ from Dzugutov relation with a constant factor. This factor was observed to be fluid dependent. Also, the factor was seen to be changed during mixing of two fluids. Moreover, for the same fluid or fluid mixture, the factor was reformed during nanotube confinement. This was supposed to be originated from the rehabilitation of molecular interactions while mixing or CNT confinement of the fluid molecules. Further, from the fitting of these data points, it was found that all the molecular fluids and fluid mixtures follow the Dzugutov scaling relation if the scaled self-diffusivity is modified with the a factor, α, where ⎛ ⎛ g(σ ) ⎞⎞ α = ⎜λ C ⎜ ⎟⎟ ⎝ ⎝ ρ*σ ⎠⎠
Therefore, the modified scaling law can be written as Dm* = αD*Dzugutov
(4)
where constants λ and C are defined as follows: (i) If η < 0.45; λ = 0.4 and C = 1 for pure, C = 1/7 for confinement, C = 1/2 for mixture, and C = (1/7) × (1/ 2) for mixture in confinement. (ii) If η > 0.45; λ = 7.2 and C = 1 for pure, C = 2/7 for confinement, C = 1/10 for mixture, and C = (2/7) × (1/ 10) for mixture in confinement. η is the packing fraction, π given by η = 6 ρ*. It is evident from Figure. 1b that scaling of D* vs Sex is highly improved for all the systems with the proposed relation than the original Dzugutov scaling. The mapping of data points with single master equation indicates the excellent approximation. The proposed equation reasonably captures the scaling relation for bulk as well as CNT confinement with both the single and multicomponent liquid systems. It is worth mentioning that the proposed scaling factor depends on two constants, yet bear a great significance as it covers a wide range of fluids in bulk, CNT confinement, and mixtures only with two constant parameters λ and C. Here, the value of λ has been chosen based on the packing fraction, whereas the value of C has been considered based on the physical state of the fluid. The value of λ has been fixed as 0.4 for η < 0.45 and 7.2 for η > 0.45 with C = 1 for pure fluids such that the scaled diffusivity data collapse onto a single curve. For η < 0.45, the hard sphere fluids maintain liquid-like characteristic, whereas for η > 0.45, the solid like characteristics start appearing. Recently, the upper range of η = 0.45 has been C
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
support the structural anomalies of fluids during CNT confinement. In case of wider nanotubes, the results show bulk like characteristics. Conversely, the presence of plateau region in smaller nanotubes indicates the large degree of immobilization for the confined fluid molecules. Further, it might be noteworthy to mention that the plateau region of Sex cannot be completely related with the diffusive motion of fluid particles as the excess entropy, Sex involves the contribution of both the translational as well as rotational dynamics, whereas diffusivity, however, considers only the translational movements. Besides, the rotational dynamics of the fluid molecules is seen to be greatly affected by confinement in smaller CNTs32 and therefore leads to a considerable change in rotational entropy, Srot, and thus to the excess entropy, Sex. Also, the results explain that Strans is not the single function of diffusion coefficient. In reality, the translational entropy of fluid molecules is driven by (i) diffusive motions and (ii) low frequency oscillations.43 This is the reason that though CNT(10,10) shows slightly lower D for methane than the CNT(12,12), yet it displays higher Strans. Indeed, a clear gap can be noticed between methane columns inside CNT(10,10) compared to CNT(9,9) and CNT(12,12) (see Figure. 2), which contributes to enhanced low frequency oscillations for methane molecules inside CNT(10,10). In other words, for nearly the same D value of methane molecules inside CNT(10,10) and CNT(12,12), the higher Strans in CNT(10,10) is dominated by low frequency oscillations. Furthermore, from the simulation results of water−methanol mixture, it was noticed that both water and methanol deviate marginally at lower temperature compared to the higher one. The deviation for water was observed to be higher than methanol. This is because the lowest simulation temperature, i.e., 240 K is lower than the freezing point of water (273 K), however, higher than the freezing point of methanol (176 K). Therefore, methanol scales better at this temperature than water (when plotted on same scale curve, water shows the deviation of ∼1.0, whereas methanol of ∼0.5). Another reason that methanol follows scaling relation better than water might be due to the comparatively weak H bond characteristics of methanol than water. The results show that scaling relation holds reasonably well for both the components of mixture in bulk as well as inside CNTs. Remarkably, no water molecules were seen to be confined inside CNT(6,6) and CNT(7,7) indicating 100% water rejection from water−methanol mixture using CNT(6,6) and CNT(7,7). However, CNT(8,8)-CNT(12,12) was found to be 85% occupied with methanol, which was further reduced to 60% for CNT(16,16) and CNT(18,18). CNTs with diameter higher than 27 Å show 50:50 composition of water/methanol in the confinement, similar to the fluid composition outside the nanotube. Also, the extent of deviation was observed to be reduced with increase in nanotube diameter. For example, methanol shows the highest deviation of 22% for CNT(7,7), 12% for CNT(8,8)-CNT(9,9), and 5% for CNT(6,6), which was further reduced to less than 3% for CNTs having diameter more than CNT(10,10). In other words, the proposed relationship appears to be extremely good for CNTs with diameter greater than 20 Å. Moreover, the extent of deviation for both the water and methanol in CNT confinement of diameter higher than 27 Å was found to be same as in the bulk. This indicates that the effect of nanotube confinement on scaling relation of fluid mixtures becomes insignificant for nanotube diameter greater than 27 Å (i.e., CNT(20,20)).
Figure 2. Top view of (a) water, (b) methanol, and (c) methane structure inside CNTs. (d) Side view of water structure inside CNTs.
Figure 3. (a) Translational DOS. (b) Rotational DOS for water in CNT confinement.
was reformed during confinement in nanometer-scaled tubes, which leads to small deviation in the scaling relation. More specifically, the proposed equation captures the scaling behavior extremely well for excess entropy below 7. Figure. 4 represents the details of thermodynamic parameters like excess entropy, total entropy components, and diffusion coefficients of the fluid systems inside CNTs. The results D
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
Figure 4. (a) Excess entropy (left Y axis) and total entropy (right Y axis); (b) entropy components: translational entropy (left Y axis) and rotational entropy (right Y axis) of fluids in CNT confinement. Solid lines and filled symbols correspond to left Y axis, whereas open symbols and dashed lines represent the right Y axis of the graph. (c) Diffusion coefficient of fluids inside CNTs. Solid lines and filled symbols represents the single fluid component inside CNTs, whereas open symbols and dashed lines show the fluid mixture inside CNTs.
atoms and ρ* in the scaling factor α. Another important point is the evaluation of excess entropy using density of states, which either could be estimated from very reliable and fast 2PT method using short MD trajectories or from the power spectrum of diffusing liquids using suitable scattering experiments. The results show that the range of Sex that corresponds to the liquid phase domain can be well mapped with modified D*. However, a slight deviation is observed as Sex enters in the range of solid phase. Furthermore, applicability of the relation for CNT confinement manifests the transferability of proposed scaling equation from cage diffusion to molecular hopping-like special diffusion mechanism. In spite of different diffusion mechanism of fluids in bulk, mixed, and CNT confined states, they can be precisely described by the present scaling relation. A more general conclusion is that the molecular diffusion coefficient D* can be successfully mapped with excess entropy scaling for all the molecular liquids and liquid mixtures in bulk state as well as CNT like one-dimensional confined states. D can thus be calculated from the density of state functions, especially for the CNT confinement where direct measurement of diffusion coefficient is very difficult.
In summary, the results demonstrate persuasively that there exists a universal scaling relation between D* and Sex for molecular fluids also. However, D* for molecular fluids differs from Dzugutov D* by the scaling factor α, which depends on structural correlation and the reduced density of molecular systems. Interestingly, the simulation results of liquid argon indicate that α reduces to one for atomic fluids. In other words, the proposed scaling equation represents Dzugutov relation in the limiting conditions where strong intramolecular interactions can be waved off. The equation captures the scaling relation for the diverse set of hydrogen bonding fluids both in bulk as well as mixed state. Moreover, it has been found that the validity of the proposed relation is not limited to the domain of conventional fluids but also can be reasonably used for CNT like one-dimensional confinement. The final conclusion is that the connection between selfdiffusivity and excess entropy is nearly universal for varieties of molecular liquids, liquid mixtures, and liquids under nanoconfinement if the structural effects caused by strong intramolecular and intermolecular interactions are taken care by the terms g(σ), i.e., confinement by first shell of neighbor E
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C
(11) Dzugutov. Universal scaling law for atomic diffusion in condensed matter. Nature 1996, 381, 137. (12) Samanta, A.; Ali, S. M.; Ghosh, S. K. Universal Scaling Laws of Diffusion in a Binary Fluid Mixture. Phys. Rev. Lett. 2001, 87, 245901. (13) Samanta, A.; Ali, S. M.; Ghosh, S. K. New Universal Scaling Laws of Diffusion and Kolmogorov-Sinai Entropy in Simple Liquids. Phys. Rev. Lett. 2004, 92, 145901. (14) Samanta, A.; Ali, S. M.; Ghosh, S. K. Universal scaling laws of diffusion: Application to liquid metals. J. Chem. Phys. 2005, 123, 084505. (15) Ali, S. M.; Samanta, A.; Choudhury, N.; Ghosh, S. K. Mass dependence of shear viscosity in a binary fluid mixture: Mode-coupling theory. Phys. Rev. E 2006, 74, 051201−6. (16) Ali, S. M.; Samanta, A.; Ghosh, S. K. Mode coupling theory of self and cross diffusivity in a binary fluid mixture: application to Lennard-Jones systems. J. Chem. Phys. 2001, 114, 10419. (17) Bastea, S. Transport properties of dense fluid argon. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 031204. (18) Bastea, S. On entropy scaling laws for diffusion. Phys. Rev. Lett. 2004, 93 (19), 199603. (19) Sharma, R.; Chakraborty, S. N.; Chakravarty, C. Entropy, diffusivity, and structural order in liquids with water like anomalies. J. Chem. Phys. 2006, 125, 204501. (20) Chopra, R.; Truskett, T. M.; Errington, J. R. On the Use of Excess Entropy Scaling to Describe the Dynamic Properties of Water. J. Phys. Chem. B 2010, 114, 10558. (21) Yan, Z.; Buldyrev, S. V.; Stanley, H. E. Relation of water anomalies to the excess entropy. Phys. Rev. E 2008, 78, 051201. (22) Zielkiewicz, J. Two-Particle Entropy and Structural Ordering in Liquid Water. J. Phys. Chem. B 2008, 112, 7810. (23) Agarwal, M.; Singh, M.; Sharma, R.; Alam, M. P.; Chakravarty, C. Relationship between Structure, Entropy, and Diffusivity in Water and Water-Like Liquids. J. Phys. Chem. B 2010, 114, 6995. (24) He, P.; Liu, H.; Zhu, J.; Li, Y.; Huang, S.; Wang, P.; Tian, H. Tests of excess entropy scaling laws for diffusion of methane in silica nanopores. Chem. Phys. Lett. 2012, 535, 84. (25) Borah, B. J.; Maiti, P. K.; Chakravarty, C.; Yashonath, S. Transport in nanoporous zeolites: Relationships between sorbate size, entropy, and diffusivity. J. Chem. Phys. 2012, 136, 174510. (26) He, P.; Li, H.; Hou, X. Excess-entropy scaling of dynamics for methane in various nanoporous materials. Chem. Phys. Lett. 2014, 593, 83. (27) Liu, Y.; Fu, J.; Wu, J. Excess-Entropy Scaling for Gas Diffusivity in Nanoporous Materials. Langmuir 2013, 29, 12997. (28) Thomas, J. A.; McGaughey, A. J. H. Water Flow in Carbon Nanotubes: Transition to Subcontinuum Transport. Phys. Rev. Lett. 2009, 102, 184502. (29) Agrawal, K. V.; Shimizu, S.; Drahushuk, L. W.; Kilcoyne, D.; Strano, M. S. Observation of extreme phase transition temperatures of water confined inside isolated carbon nanotubes. Nat. Nanotechnol. 2016, 1748, 3395. (30) Krott, L. B.; Bordin, J. R.; Barbosa, M. C. New Structural Anomaly Induced by Nanoconfinemment. J. Phys. Chem. B 2015, 119, 291−300. (31) Hummer, G.; Rasaiah, J. C.; Noworyta, J. P. Water conduction through the hydrophobic channel of a carbon nanotube. Nature 2001, 414, 188. (32) Sahu, P.; Ali, S. M.; Shenoy, K. T. Thermodynamics of fluid conduction through hydrophobic channel of carbon nanotubes: The exciting force for filling of nanotubes with polar and nonpolar fluids. J. Chem. Phys. 2015, 142, 074501. (33) Sahu, P.; Ali, S. M. The entropic forces and dynamic integrity of single file water in hydrophobic nanotube confinements. J. Chem. Phys. 2015, 143, 184503. (34) Vardharajula, S.; Ali, S. Z.; Tiwari, P. M.; Eroglu, E.; Vig, K.; Dennis, V. A.; Singh, S. R. Functionalized carbon nanotubes: biomedical applications. Int. J. Nanomed. 2012, 7, 5361.
Overall, the present study propose a new scaling relationship that can be applied successfully for all the liquids and liquid mixtures in bulk and CNT confinement to capture the qualitative and quantitative relationship of self-diffusion coefficient and excess entropy. We firmly believe that this relationship would be of great practical value as they relate selfdiffusivities with experimentally more accessible thermodynamic quantity, i.e., excess entropy (Sex) especially for nanoscale confinements. The results would be also helpful for nanopore membranes and biological channels as they show the phase transition anomalies similar to nanotube confinements29 and protein−DNA interface.44 Work in this direction is in progress in our computational laboratory.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b04265. Details on computational protocol; overview of 2PT method for excess entropy; diffusion coefficient: Einstein relation; pair correlation function; MD simulation details; simulation details of CNT systems (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel: +91-22-25591992. ORCID
Sk. Musharaf Ali: 0000-0003-0457-0580 Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Authors are grateful to Dr. Sadhana Mohan, Associate Director ChEG, BARC Mumbai, for her continuous encouragement and support. Authors also acknowledge the Anupam computational facility of Computer Division for providing computer resources meant for advanced modeling and simulation techniques.
■
REFERENCES
(1) Boon, J. P.; Yip, S. Molecular Hydrodynamics; McGraw-Hill: New York, 1980. (2) Kelton, K. F.; Greer, A. L. Nucleation in Condensed Matter: Applications in Materials and Biology. Pergamon Materials Series; Elsevier: Amsterdam, 2010. (3) Adam, G.; Gibbs, J. H. On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids. J. Chem. Phys. 1965, 43, 139. (4) Rosenfeld, Y. A quasi-universal scaling law for atomic transport in simple fluids. J. Phys.: Condens. Matter 1999, 11, 5415. (5) Rosenfeld, Y. Relation between the transport coefficients and the internal entropy of simple systems. Phys. Rev. A: At., Mol., Opt. Phys. 1977, 15 (6), 2545. (6) Mittal, J.; Errington, J. R.; Truskett, T. M. Relationships between Self-Diffusivity, Packing Fraction, and Excess Entropy in Simple Bulk and Confined Fluids. J. Phys. Chem. B 2007, 111, 10054. (7) Jakse, N.; Pasturel, A. Excess Entropy Scaling Law for Diffusivity in Liquid Metals. Sci. Rep. 2016, 6, 20689. (8) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1975. (9) Balucani, U.; Zoppi, M. Dynamics of the liquid state. Instituto di Eletronica Quantistica CNR, Firenze, Italy, 1994. (10) Royall, C. P.; Williams, S. R. The role of local structure in dynamical arrest. Phys. Rep. 2015, 560, 1. F
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry C (35) Hoyt, J. J.; Asta, M.; Sadigh, B. Test of the Universal Scaling Law for the Diffusion Coefficient in Liquid Metals. Phys. Rev. Lett. 2000, 85 (3), 594. (36) Lin, S.-T.; Maiti, P. K.; Goddard, W. A. Two-Phase Thermodynamic Model for Efficient and Accurate Absolute Entropy of Water from Molecular Dynamics Simulations. J. Phys. Chem. B 2010, 114 (24), 8191. (37) Grasselli, Y.; Bossis, G.; Morini, R. Translational and rotational temperatures of a 2D vibrated granular gas in microgravity. Eur. Phys. J. E: Soft Matter Biol. Phys. 2015, 38, 8. (38) Yuan, L.; Teitelbaum, S. W.; Robinson, A.; Mullin, A. S. Dynamics of molecules in extreme rotational states. Proc. Natl. Acad. Sci. U. S. A. 2011, 108 (17), 6872. (39) Koga, K.; Gao, G. T.; Tanaka, H.; Zeng, X. C. Formation of ordered ice nanotubes inside carbon nanotubes. Nature 2001, 412, 802. (40) Zheng, Y.-g.; Ye, H.-f.; Zhang, Z.-q.; Zhang, H.-w. Water diffusion inside carbon nanotubes: mutual effects of surface and confinement. Phys. Chem. Chem. Phys. 2012, 14, 964−971. (41) da Silva, L. B. Structural and dynamical properties of water confined in carbon nanotubes. J. Nanostruct. Chem. 2014, 4, 104. (42) Mashl, R. J.; Joseph, S.; Aluru, N. R.; Jakobsson, E. Anomalously Immobilized Water: A New Water Phase Induced by Confinement in Nanotubes. Nano Lett. 2003, 3 (5), 589−592. (43) Pascal, T. A.; Goddard, W. A.; Jung, Y. Entropy and the driving force for the filling of carbon nanotubes with water. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 11794. (44) Chong, S.; Ham, S. Anomalous Dynamics of Water Confined in Protein-Protein and Protein-DNA Interfaces. J. Phys. Chem. Lett. 2016, 7, 3967−3972.
G
DOI: 10.1021/acs.jpcc.7b04265 J. Phys. Chem. C XXXX, XXX, XXX−XXX
