An empirical experimental observations and MD simulation data-based model for the material properties of confined fluids in nano/Angstrom size tubes

The transport of fluids in nanometer and Angstrom-sized pores has gotten much attention because of its potential uses in nanotechnology, energy storage, and healthcare sectors. Understanding the distinct material properties of fluids in such close confinement is critical for enhancing their performance in various applications. These properties dictate the fluid’s behavior and play a crucial role in determining flow dynamics, transport processes, and, ultimately, the performance of nanoscale devices. Remarkably, many researchers observed that the size of the geometry, such as the diameter of the confining nanotube, exerts a profound and intriguing influence on the material properties of nanoconfined fluids, including on the critical parameters such as density, viscosity, and slip length. Many researchers tried to model these material properties: viscosity η, density ρ, and slip λ using various models with many dependencies on the tube diameter. It is somewhat confusing and tough to decide which model is appropriate and needs to be incorporated in the numerical simulation. In this paper, we tried to propose a simple single equation for each nano confined material property such as for density ρ(D)/ρo=a+b/(D−c)n , viscosity η(D)/ηo=a+b/(D−c)n , and the slip length λ(D) = λ 1 D e −n D + λ o (where a, b, c, n, λ 1, λ o are the free fitting parameters). We model a wealth of previous experimental and MD simulation data from the literature using our proposed model for each material property of nanoconfined fluids at the nanometer and Angstrom scales. Our single proposed equation effectively captures and models all the data, even though many different models have been employed in the existing literature to describe the same material property. Our proposed model exhibits exceptional agreement with multiple independent datasets from the experimental observations and molecular dynamics simulations. Additionally, the model possesses the advantageous properties of continuity and a continuous derivative, so the proposed model is well-suited for integration into numerical simulations. Further, the proposed models also obey the far boundary conditions, i.e., when tube diameter D ⟹ ∞, the material properties tend to the bulk properties of the fluid. Due to the models’ simplicity, smooth, and generic nature, this heuristic model holds promise to apply in simulations to design and optimize nanoscale devices.

( ) ( ) h h = + -, and the slip length λ(D) = λ 1 De − n D + λ o (where a, b, c, n, λ 1 , λ o are the free fitting parameters).We model a wealth of previous experimental and MD simulation data from the literature using our proposed model for each material property of nanoconfined fluids at the nanometer and Angstrom scales.Our single proposed equation effectively captures and models all the data, even though many different models have been employed in the existing literature to describe the same material property.Our proposed model exhibits exceptional agreement with multiple independent datasets from the experimental observa-

Introduction
The study of fluid behavior and transport processes at the nanoscale has garnered significant attention in recent years due to its relevance in various scientific and wide-ranging engineering applications in fields such as nanotechnology, microfluidics, and biophysics [1][2][3][4][5][6][7][8].Nanoconfined fluids, which are fluids confined within nano or ngström-sized channels, exhibit unique and complex properties that differ markedly from their bulk counterparts [9][10][11][12][13][14][15][16].Consequently, characterizing the properties of nanoconfined fluids has become a pivotal area of research.In nanoscale confinements, the high surface-to-volume ratio leads to dominant surface forces governing transport [17][18][19][20].Surface roughness and surface-liquid interactions play a crucial role in defining fluid flow.As dimensions approach molecular scales, discrete molecular motion becomes evident, resulting in distinct transport properties compared to macroscopic systems [21].Among the fundamental properties of nanoconfined fluids, density (ρ), viscosity (η), and slip length (λ) are of paramount importance.These properties dictate the fluid's behavior and play a critical role in determining flow dynamics, transport processes, and, ultimately, the performance of nanoscale devices.Accurate characterization of these properties within confined geometries is essential for designing and optimizing nanofluidic systems and understanding phenomena at the nanoscale [6][7][8]22].Remarkably, the size of the geometry, such as the diameter of the confining tube (as shown in figure 1, where we display a schematic diagram of the flow Q in a nanotube with length L and the cross-section radius R (and diameter D)) or the height of the nanochannel have been observed to exert a profound and intriguing influence on the material properties of confined fluids, including on the critical parameters such as density (ρ), viscosity (η), and slip length (λ) [13][14][15][16].Many researchers have extensively explored and substantiated this notable phenomenon dependent on the diameter D, as evidenced by a substantial body of literature [11,23].For example, The fluid flow conditions within a nanocapillary, such as one with a 5 nm diameter, can markedly differ from those in capillaries with a 0.5 nm diameter.This discrepancy arises due to the existence of hydration layers, extending approximately 0.6 nm from the solid surface.Consequently, the presence of these hydration layers has the potential to alter viscosity and induce localized variations in density, as investigated by Uhlig and Garcia [24].Further, on graphene and graphite-like materials immersed in aqueous solutions the interface of the nanochannel might contain hydrocarbon layers.Those layers might also affect the viscosity and density properties of the nanotubes, specifically the ones with the small diameters, say below 2-3 nm [25].
Over time, numerous functional forms have been proposed and employed to model these material properties within the intricate realm of nanoconfined fluids [11,23,26].According to a previous model by Thomas and McGaughey [10], MD simulation was employed to examine pressure-driven water flow through seven armchair CNTs, each with diameters ranging from 1.66 to 4.99 nm.Subsequently, equilibrium simulation was utilized to predict water viscosity within each tube.The viscosity was predicted based on the axial selfdiffusion coefficient, ζ, using the Einstein relation where k B is the Boltzmann constant and A is the molecular diameter [27,28].The results gathered from these simulations collectively support the idea that even in highly narrow CNTs with a diameter as small as 1.66 nm, the flow of water behaves following the slip-modified Hagen-Poiseuille relation.Interestingly, the actual flow rates they observed were higher than what we would expect based on the traditional no-slip Hagen-Poiseuille relation [29][30][31], where water sticks to the tube walls.However, their theory failed to support the claims made by Majumder et al [15] and Holt [32,33] in their previous studies, where they suggested that water behaves very differently in these nanotubes.In another modelling effort, Ye et al [11] proposed a pressure, temperature and tube diameter dependent model as where a 1 , a 2 , a 3 , a 4 , a 5 are the fitting parameters, η bulk is the bulk viscosity and c 1 , c 2 , c 3 are the parameters that depends on the pressure and temperature.Further, Suk and Aluru [34] studied the water flow in carbon nanotubes using both the experiments and the MD simulation and proposed that the viscosity of water in the nanotube is given by where a, b are the fitting parameters and η bulk is the bulk viscosity at D ⟹ ∞ .Similarly, the density in nanotubes changes depending on on the diameter of the tube.Borg et al [23] developed empirical equations to describe variable materials quantities, including density (ρ), and slip length (λ), in the context of the Hagen-Poiseuille-Washburn (H-P-W) equation.These equations were fitted based on their multiscale simulations and were functions of the carbon nanotube (CNT) diameter (D).For density (ρ), Borg et al [23] proposed a quadratic inverse dependence on the diameter as given by where, ρ o , and ρ 1 are free parameters.On the other hand, when we fitted the density data by Ye et al [11] (as shown in figure 2), we found the density depends on the inverse of the diameter as given by Further, Alexiadis and Kassinos [35] proposed that the density of water in the nanotube is given by where ρ o is the bulk density and the ρ * , and D * are the free parameters.Also, Srivastava et al [36] proposed that the density of water in the nanotube is given by where m is the mass flow of water, L is the length of the nanotube, and a is the ehe Lennard-Jones (LJ) parameter for carbon-oxygen interaction.The slip length in nanoconfined geometries like channels and tubes changes depending on the height of the channel and the diameter of the tube.Borg et al [23] proposed that the slip length (λ) is inversely proportional to the fifth power of the CNT diameter, which is given by where, λ o , and λ 1 are free parameters.In another paper by Kannam et al [37], they employed a slip-modified Poiseuille flow Navier-Stokes solution to derive an expression for slip length (λ) and interfacial friction between water and a carbon nanotube (CNT).The slip length (λ) was expressed as Comparison of the fitted density using our proposed model with the data in the literature by Borg et al [23] (in orange color with right-side triangle symbols), Alexiadis and Kassinos [35] (in purple color with diamond symbols) and Ye et al [11] (in red color with square symbols) , respectively.
where ρ represents the fluid density, m is a slope obtained by plotting slip velocity against the external field in the low field range (linear regime), and η o represents a reference viscosity.In this equation, Kannam et al [37] derived the slip length (λ), which depends on the fluid density (ρ), the slope (m) characterizing slip behavior, the reference viscosity (η 0 ) and the dependence on the inverse of the diameter D of the nanotube.Further, Suk and Aluru [34] studied the water flow in carbon nanotubes using both the experiments and the MD simulation and proposed that the slip length of water in the nanotube is given by where a, b are the fitting parameters and λ ∞ is the bulk slip length at diameter D ⟹ ∞ .Besides these, numerous molecular dynamics (MD) simulation investigations have been conducted, primarily driven by the practicality of simulating these scenarios instead of the formidable challenges associated with conducting nanoscale fluidic experiments.Notably, Kotsalis et al [38] discerned slip lengths of merely 11 nm, 13 nm, and 15 nm for tubes with respective diameters of 2.71 nm, 4.07 nm, and 5.42 nm.Thomas and McGaughey [10] reported a consistently diminishing slip length from 105 nm to 30 nm within the diameter range of 1.66 nm to 4.99 nm.This phenomenon persisted, approaching 30 nm even with larger diameter tubes and a planar graphene surface.
Subsequently, they uncovered non-continuous (pertaining to fluid structure dependency) and nonmonotonic transport behaviors within the diameter range of 0.83 nm to 1.66 nm, characterized by slip lengths falling below 1000 nm.The observed non-monotonic behavior was attributed to fluctuations in the fluid structure.In an endeavor to elucidate the mechanisms underlying rapid transport, Joseph and Aluru [39] identified a slip length of 556 nm for a 2.17 nm diameter tube.Du et al [13] reported a slip length of 260 nm for a 4 nm diameter tube.In recent years, Falk et al [12], Babu and Sathian [40] separately observed decreasing slip lengths with increasing tube diameter, albeit with varying magnitudes.Falk et al [12] recorded slip lengths ranging from 2.6 μm to 120 nm for tubes with diameters spanning from 0.81 nm to 7 nm.In contrast, Babu and Sathian [40] found slip lengths of 3 nm to 0.3 nm for tube diameters of 0.83 nm to 5.42 nm.
These studies have indicated that tube diameter influences critical fluid properties.Also, different researchers have used various models and equations to fit density, viscosity, and slip length, which creates confusion and makes tough to make decisions to which model is appropriate or needs to be incorporated in the numerical simulation.To remove this ambiguity, in this paper, we propose a generic model that can be used to model density, viscosity, and slip length in the nanoconfined tubes.Using our model, we fit various experimental data and MD simulation data from the literature, even though many models have been used to model a given material property for the same data.Our suggested model demonstrates exceptional agreement with multiple independent datasets derived from experimental observations and molecular dynamics simulations.Furthermore, the model possesses the advantages of continuity and a continuous derivative, making it highly suitable for integration into numerical simulations.Moreover, the proposed models also adhere to far boundary conditions, meaning that when the tube diameter D approaches infinity, the material properties tend to converge with the bulk properties of the fluid.Given the model's simplicity, smoothness, and versatility, this heuristic approach holds great potential for utilization in simulations aimed at designing and optimizing nanoscale devices.

Results and discussion: models and comparison of confined fluid properties in nano/ Angstrom size tubes
In this section, we propose a single general model for the density, viscosity, and the slip-length.We also fit literature data on these material properties from the experiments and the MD simulation.In section 2.1, we model the density.In section 2.2, we model the nanoconfined viscosity in the nanotube.In section 2.3, we model the slip length.

Density of nanoconfined fluid
An extensive body of research has emerged in the realm of understanding the density of confined fluids within nano and Angstrom-sized tubes, offering various functional forms to describe this intricate phenomenon.Diverse dependencies on the diameter of the nanotube models have been proposed to represent the density (ρ) of confined fluids, as discussed in the introduction of this paper, each attempting to capture the relationship between density and tube diameter (D).
In order to make the modelling effort simpler and more efficient, we need a consolidated model which can model all experimental or MD simulation data researchers obtained on density.For that, we propose a general diameter dependence power-law model with additional free parameters as given by Here, ρ(D) is the tube's diameter-dependent density of the fluid.The ρ o is the bulk density at D ⟹ ∞ .a, b, c are the free parameter.The power-law index n shows the diameter dependence.We find that as D tends to infinity, the density ratio approaches a constant value a, whereas if a = 1, it approaches a regular bulk density value.We also find that the proposed function is continuous and has a continuous derivative, making it appropriate to use in numerical simulations.Also, as the variable ρ(D) represents the fluid's average density in a capillary of diameter D. In alignment with findings in existing literature [11,23,36], where density data modelling spans from the mesoscopic scale of 5 nm to achieving bulk properties at the macroscopic level within a few hundred micrometers, our proposed model adheres to a similar approach.However, our model adopts a more generalized perspective, making it applicable for any diameter exceeding D 5 nm.Further, the model comprehensively addresses water behavior across various phases within the specified range [41].This encompasses liquid, vapor, and potentially confined or interfacial phases contingent on the specific system under examination.
Further, Borg et al [23] explored carbon nanotubes (CNTs) with diameters ranging from 8 to 40 Å.They utilized these data in the Hagen-Poiseuille-Washburn (H-P-W) equation.On the other hand, Ye et al [11] and Alexiadis and Kassinos [35] investigate the impact of tube size on water density, focusing on single-walled carbon nanotubes (SWCNTs) with diameters spanning from 8 Å(corresponding to (6, 6) SWCNT) to 55 Å(related to (40, 40) SWCNT).Their main objective was to calculate the water density within different SWCNTs.
Using our proposed model, we compare the modelling results with the molecular dynamics simulation predictions performed by Borg et al [23] (shown with orange color right-side triangle symbols), Ye et al [11] (shown with red color square symbols), and Alexiadis and Kassinos [35] (in purple color with diamond symbols), respectively as shown in figure 2. In order to model Borg et al [23]  We find that all MD simulation data have different power-law dependencies on the diameter (see equations (4), (5), and (6)), and our single simpler model can give an excellent fit to all their respective data sets.Therefore, this proposed model offers a broader and more adaptable framework for the density of confined fluids in nano and Angstrom-sized tubes.Additional parameters allow for a more precise and comprehensive characterization of the density profile, accommodating a wider range of experimental conditions and simulations.
Nevertheless, our proposed models refrain from explicitly making inferences about system boundaries.This is because material properties can be influenced by factors such as temperature, nanocoating, hydration layers and various other properties across boundaries [42].Notwithstanding, for uncomplicated nanocapillaries without intricate boundary considerations, the models adeptly predict the data, as illustrated in figure 2. Additionally, depending on the availability of data with specific boundary conditions, a model fit can be attempted and subsequently validated.Moreover, while the model refrains from explicit boundary inferences, it implicitly addresses the interfacial region's influence on material properties.The success in predicting data for uncomplicated nanocapillaries and the flexibility for refinement based on specific boundary conditions showcase the model's robustness and potential for adaptation as more information becomes available.

Viscosity of nanoconfined fluid
Many diameter-dependent nanotube models have been proposed to represent the nanoconfined viscosity (η) of confined fluids, as discussed in the introduction of this paper, each attempting to capture the relationship between viscosity and tube diameter (D).Similarly to the nanoconfined density, to make the modelling effort simpler and more efficient, we need a consolidated model that can model all experimental or MD simulation data researchers obtained on viscosity.For that, we propose a general diameter dependence power-law model for the nanoconfined viscosity with additional free parameters as given by Here, η(D) is the tube's diameter-dependent viscosity of the fluid.The η o is the bulk viscosity at D ⟹ ∞ .a, b, c are the free parameters and capture the specific behavior of the fluid's viscosity in confinement.The power-law index n shows the diameter dependence.We find that as D tends to infinity, the relative viscosity approaches a constant value a, whereas if a = 1, it approaches a regular bulk value of the viscosity.We also find that the proposed function is continuous and has a continuous derivative, making it appropriate to use in numerical simulations [43][44][45][46].[14] in the literature.Also, Suk and Aluru [34] studied the water flow in nanotubes using both the experiments and the MD simulations.
Wu et al [9] proposed a modeling tool for water flow in nanopores with diameters larger than 16 Å, whereas Falk et al [12] studied water viscosity within CNTs, considering diameters ranging from 1 to more than 100 Å. Majumder et al [15] measured the viscosity of water flowing through carbon nanotubes (CNTs) membranes with diameters ranging from 13 to 70 Å.Further, Du et al [13] developed a straightforward method to create super long vertically aligned carbon nanotubes (SLVA-CNT) and epoxy composite membranes with sizes ranging from 13 to 100 Å.They also examined the viscosity of various liquids, including water, hexane, and dodecane, as they passed through these SLVA-CNT membranes.They also validated their results using molecular dynamics (MD) simulations.
In the case of data from the MD simulations, as shown in figure 3, the molecular dynamic (MD) simulations conducted by Thomas and McGaughey [10] to investigate the phenomenon of pressure-driven water flow within a set of seven carbon nanotubes (CNTs).This examination encompassed CNTs characterized by diameters spanning the 16 to 50 Årange.They demonstrated that even within the confines of CNTs exhibiting exceptionally small diameters, such as 1.66 nm, the water flow impeccably adhered to the theoretical framework provided by the slip-modified Hagen-Poiseuille equation.On the other hand, Zhang et al [14] explored the viscosity of water confined within single-walled carbon nanotubes (SWCNTs) ranging in diameter from 8 to 54 Å.They employed an "Eyring-MD" (molecular dynamics) method, which combines Eyring's viscosity theory with molecular dynamics simulations to model viscosity.Similarly, Ye et al [11] investigated how the size of SWCNTs, ranging from 8 Å(associated with (6, 6) SWCNT) to 54 Å(related to (40, 40) SWCNT), affects the viscosity of water within these tubes.Further, Suk and Aluru [34] studied the water flow in carbon nanotubes ranging in diameter from 5 to 24 Å, where they found the relative viscosity to be between 0.4 to 0.75.We find that different researchers obtained different data on water transport through the nanotubes.This shows that it might be tough to model all data for a specific power-law.Therefore, we need a generic consolidated model to fit all data as proposed above in equation (12).To model data on the solid-black line, we use equation (12) [11] and Suk and Aluru [34] proposed three different power-law dependencies viscosity model (see equations (1) and (2), and (3)), we predicted all of their data, using our single generic proposed model for the viscosity.We find all the data from MD simulation as well as from different experiments, which have various power-law dependencies on the diameter; our single simpler model (i.e., equation ( 12)) can give an excellent fit to all respective data sets.Figure 3 was created to show a graphical synthesis of these findings, including both experimental data, MD simulations data, and the fitted proposed model, to show a broader context that the single proposed model is enough to fit a vast array of data from various sources.Notably, our modeling approach was very compatible with the experimental and MD simulation results, demonstrating the adaptability of our technique.This contrasts with the need to use several different models researchers approach for different experimental datasets.This shows the efficacy and versatility of our simple single-fitting method in elucidating various features of fluid dynamics within nanotubes.

Slip length of nanoconfined fluid
Similarly, for the slip length (λ), many diameter-dependent nanotube models have been proposed to model the nanoconfined slip as discussed in this paper, each attempting to capture the relationship between slip and tube diameter (D).For the slip, to make the modelling effort simpler and more efficient, we need a consolidated model which can model all experimental or MD simulation data, researchers obtained on it.For that, we propose a generic nanotube diameter dependence exponential-law model for the nanoconfined slip length with additional free parameters as given by Here, λ(D) is the tube's diameter-dependent slip length of the fluid.The λ o is the slip of the bulk at D ⟹ ∞ .λ 1 is a free parameter and captures the specific behavior of the fluid's slip in confinement.The exponential rate n shows the diameter dependence.As D tends to infinity, the slip length approaches a constant value λ o .We also find that the proposed function is continuous and has a continuous derivative, making it appropriate to use in numerical simulations.Figure 4 shows the comparison of the results by our proposed model (shown with black solid line) and those from 10 different cases, composed of 5 cases from experiments by Du et al [48], Holt [13], Trivedi and Alameh [49], Secchi et al [32], Yang and Zheng [47] and 4 cases from MD simulations by Thomas and McGaughey [37], Borg et al [23], Kannam et al [10], Joseph and Aluru [39] in the literature.Also, Suk and Aluru [34] studied the water flow in nanotubes using both the experiments and the MD simulations.
Various approaches can be found in the existing literature for estimating the slip length in nanoconfinements.These methods utilize either equilibrium or non-equilibrium MD simulations.In equilibrium simulations, the commonly used Green-Kubo relation is applied to ascertain the friction coefficient, with the slip length defined as the ratio of the friction coefficient to shear viscosity [50,51].On the other hand, non-equilibrium simulations involve introducing external perturbations to the simulation domain to induce liquid flows.In prior experiments by Holt [32], it was observed that water exhibited a substantial slip effect within CNTs, with reported slip lengths ranging from 0.8 μm (at D = 17 Å) to 54 μm for varying tube diameters.In figure 4, the slip length values of water in nano/Angstrom nanotubes are compared with multiple MD studies as reported in the literature by Thomas and McGaughey [10], Joseph and Aluru [39].Thomas and McGaughey [10], employed MD simulation to examine pressure-driven water flow through seven CNTs, each with diameters ranging from 16 to 50 Å.They used equilibrium simulation to predict the viscosity of water within each tube.Finally, the slip length within each CNT was predicted using the calculated viscosity, pressure gradient, and velocity profile.In another MD simulation study by Borg et al [23], they observed slip lengths of water within CNTs ranged from 750 nm to 900 nm for tube diameters between 10 and 13 Å.Further, Kannam et al [37] predicted 180-75 nm slip length of water in CNTs using equilibrium MD simulations for tubes of  [37], Borg et al [23], Holt [10], Suk and Aluru [47], Kannam et al [48], Joseph and Aluru [39], Trivedi and Alameh [13], Secchi et al [49], Yang and Zheng [32].
diameter 16.6-65 Å(shown in magenta hexagon symbol).On the other hand, in an experimental study by Secchi et al [48], they observed that slip lengths of water within CNTs ranged from 270 nm to 320 nm for various tube diameters spanning between 75 and 90 Å.They observed a decrease in slip length as the tube diameter increased (as shown with blue diamond symbols).Further Yang and Zheng [49] studied tube diameter span between 8 to 30 Åand found slip length between 300 to 1250 nm with decreasing slip as tube diameter increases and Trivedi and Alameh [47] found slip length of 100 nm at diameter D = 80 Å.Du et al [13] and Joseph and Aluru [39] found the slip length of approximately 300 nm (shown in black asterisk) and 500 nm (shown in red cross symbols) at D = 38 Å, and D = 22 Å, respectively.Further, Suk and Aluru [34] studied the water flow in carbon nanotubes ranging in diameter from 5 to 24 Å, where they found the slip length to be between 75 to 110 nm.
Using equation (13), we model the experimental and MD simulation data as shown in figure 4 with a solid black line.For that, we get λ 1 = 10, 000, n = − 1.9 × 10 9 /m, and λ o = 260 × 10 −9 m, respectively.Further, we model data by Borg et al [23], and Trivedi and Alameh [47] as shown with green color solid line.For that, we get λ 1 = 8, 000, n = − 2.2 × 10 9 /m, and λ o = 100 × 10 −9 m, respectively.We also model data by Kannam et al [37] as shown with blue color solid line.For that, we get λ 1 = 3, 000, n = −2.2× 10 9 /m, and λ o = 80 × 10 −9 m, respectively.Finally, we model data by Suk and Aluru [34] as shown with cyan color solid line.For that, we get λ 1 = 500, n = −2.0× 10 9 /m, and λ o = 45 × 10 −9 m, respectively.We find that despite Kannam et al [37], Borg et al [23], and Suk and Aluru [34] proposed three different power-law dependencies model (see equations (8), ( 9) and ( 10)), we predicted all of their data, using our single generic proposed model for the slip.We find all the data from MD simulation as well as from different experiments, which have various power-law dependencies on the diameter; our single simpler model (i.e., equation ( 13)) can give an excellent fit to all respective data sets.Figure 4 was created to show a graphical synthesis of these findings, including both experimental data, MD simulations data, and the fitted proposed model, in order to show a broader context that the single generic proposed model is enough to fit a vast array of data from various sources.
Significantly, our modeling strategy aligned seamlessly with experimental data and MD simulations, underscoring the flexibility of our approach.This differs from the requirement to employ multiple models adopted by other researchers to analyze various experimental datasets.This illustrates the effectiveness and adaptability of our straightforward single-fitting approach in unraveling multiple aspects of fluid dynamics within nanotubes.

Conclusions
The study of fluid behavior at the nanoscale has emerged as a crucial area of research with significant implications for various scientific and engineering applications.Nanoconfined fluids, confined within nano or ngström-sized channels, exhibit intricate properties distinct from their bulk counterparts.These properties encompass density (ρ), viscosity (η), and slip length (λ), and they govern fluid behavior, impacting flow dynamics, transport processes, and the performance of nanoscale devices.Characterizing these properties within confined geometries is essential for designing nanofluidic systems and understanding nanoscale phenomena.The diameter of the confining tube has been shown to exert a profound influence on the material properties of confined fluids, a phenomenon well-documented in the literature.Various modeling approaches have been proposed earlier, which use many different equations for fitting viscosity, density, and slip length, which is somewhat confusing and tough to make decisions to incorporate in the numerical simulations.
In We have discovered that a single proposed equation effectively captures and models all the data, even though many different models have been employed in the existing literature to describe the same material property.We also find that our proposed model is continuous with continuous derivative, which makes it well-suited for integration into numerical simulations.Further, the proposed models also obey the far boundary conditions, i.e., when tube diameter D ⟹ ∞ , the material properties tend to the bulk properties of the fluid.
However, at the nanoscale, where surface-to-volume ratios are significant, intermolecular interactions dominate, leading to the formation of distinct layers of liquid molecules adjacent to the solid surface.These layers exhibit solid-like behavior due to strong molecular interactions and confinement effects [52,53].The arrangement of molecules within these layers is influenced by factors such as surface chemistry, temperature, and pressure, among others.Consequently, the interfacial solid-like liquid layering profoundly impacts the properties and behavior of the interface, influencing phenomena such as wetting behavior, adhesion, and lubrication.These phenomenon can lead to confinement-induced modifications in material properties.Further, these interactions can promote or hinder molecular transport and also affect the slip-length overall.The properties of solid-liquid interfaces at the nanoscale can be described at a macroscopic, continuum level but are ultimately governed by atomistic details, which are the subject of study through molecular dynamics simulations and high-resolution imaging techniques.Therefore, before delving into future investigations and advaning to these nanoscale fluid properties modelling, it is essential to prioritize a thorough understanding of the interfacial solid-like liquid layering phenomenon.
Nevertheless, unlike conventional approaches that necessitate different functional forms for fitting various experimental and MD simulation data at the same diameter, our model offers a unique advantage.It allows for the fitting of diverse datasets, both experimental and molecular dynamics simulations, at the same diameter using a unified model with generic variables.This eliminates the need to redefine separate models for each dataset, streamlining the fitting process and underscoring the versatility and efficiency of our proposed model.Therefore, it provides a unified and adaptable method for characterizing the intricate interplay between fluid properties and tube diameter.It contributes to our understanding of fluid behavior at the nanoscale and facilitates advancements in nanotechnology, microfluidics, and biophysics.

Figure 1 .
Figure 1.Schematic diagram of the flow Q in a nanotube with length L and the cross-section radius R (and diameter D).In (a), we show the complete view of the nanotube.In (b) we show the side-view of the nanotube.

Figure 3
Figure3shows the comparison of the results by our proposed model (shown with black and green solid lines) and those from 8 different cases, composed of 4 cases from experiments Wu et al[9], Falk et al[12], Du et al[13], Majumder et al[15] and 3 cases from MD simulations Thomas and McGaughey[10], Ye et al[11], Zhang et al[14] in the literature.Also, Suk and Aluru[34] studied the water flow in nanotubes using both the experiments and the MD simulations.Wu et al[9] proposed a modeling tool for water flow in nanopores with diameters larger than 16 Å, whereas Falk et al[12] studied water viscosity within CNTs, considering diameters ranging from 1 to more than 100 Å. Majumder et al[15] measured the viscosity of water flowing through carbon nanotubes (CNTs) membranes with diameters ranging from 13 to 70 Å.Further, Du et al[13] developed a straightforward method to create super long vertically aligned carbon nanotubes (SLVA-CNT) and epoxy composite membranes with sizes ranging from 13 to 100 Å.They also examined the viscosity of various liquids, including water, hexane, and dodecane, as they passed through these SLVA-CNT membranes.They also validated their results using molecular dynamics (MD) simulations.In the case of data from the MD simulations, as shown in figure3, the molecular dynamic (MD) simulations conducted by Thomas and McGaughey[10] to investigate the phenomenon of pressure-driven water flow within a set of seven carbon nanotubes (CNTs).This examination encompassed CNTs characterized by diameters spanning the 16 to 50 Årange.They demonstrated that even within the confines of CNTs exhibiting exceptionally small diameters, such as 1.66 nm, the water flow impeccably adhered to the theoretical framework provided by the slip-modified Hagen-Poiseuille equation.On the other hand, Zhang et al[14] explored the viscosity of water confined within single-walled carbon nanotubes (SWCNTs) ranging in diameter from 8 to 54 Å.They employed an "Eyring-MD" (molecular dynamics) method, which combines Eyring's viscosity theory with molecular dynamics simulations to model viscosity.Similarly, Ye et al[11] investigated how the size of SWCNTs, ranging from 8 Å(associated with (6, 6) SWCNT) to 54 Å(related to (40, 40) SWCNT), affects the viscosity of water within these tubes.Further, Suk and Aluru[34] studied the water flow in carbon nanotubes ranging in diameter from 5 to 24 Å, where they found the relative viscosity to be between 0.4 to 0.75.We find that different researchers obtained different data on water transport through the nanotubes.This shows that it might be tough to model all data for a specific power-law.Therefore, we need a generic consolidated model to fit all data as proposed above in equation(12).To model data on the solid-black line, we use equation(12) and get the fitting parameters as a = 1, b = − 4.21 × 10 −10 m, c = − 1 × 10 −10 m, and n = 1.On the other hand, to model the data on the solid-green line, we get a=0.9, b = − 3.21 × 10 −10 m, c = 1 × 10 −10 m, and n = 1.We find that despite Thomas and McGaughey [10], Ye et al[11] and Suk and Aluru[34] proposed three different power-law dependencies viscosity model (see equations (1) and (2), and (3)), we predicted all of their data, using our single generic proposed model for the viscosity.We find all the data from MD simulation as well as from different experiments, which have various power-law dependencies on the diameter; our single simpler model (i.e., equation (12)) can give an excellent fit to all respective data sets.Figure3was created to show a graphical synthesis of these findings, including both experimental data, MD simulations data, and the fitted proposed model, to show a broader context that the single proposed model is enough to fit a vast array of data from various sources.
Figure3shows the comparison of the results by our proposed model (shown with black and green solid lines) and those from 8 different cases, composed of 4 cases from experiments Wu et al[9], Falk et al[12], Du et al[13], Majumder et al[15] and 3 cases from MD simulations Thomas and McGaughey[10], Ye et al[11], Zhang et al[14] in the literature.Also, Suk and Aluru[34] studied the water flow in nanotubes using both the experiments and the MD simulations.Wu et al[9] proposed a modeling tool for water flow in nanopores with diameters larger than 16 Å, whereas Falk et al[12] studied water viscosity within CNTs, considering diameters ranging from 1 to more than 100 Å. Majumder et al[15] measured the viscosity of water flowing through carbon nanotubes (CNTs) membranes with diameters ranging from 13 to 70 Å.Further, Du et al[13] developed a straightforward method to create super long vertically aligned carbon nanotubes (SLVA-CNT) and epoxy composite membranes with sizes ranging from 13 to 100 Å.They also examined the viscosity of various liquids, including water, hexane, and dodecane, as they passed through these SLVA-CNT membranes.They also validated their results using molecular dynamics (MD) simulations.In the case of data from the MD simulations, as shown in figure3, the molecular dynamic (MD) simulations conducted by Thomas and McGaughey[10] to investigate the phenomenon of pressure-driven water flow within a set of seven carbon nanotubes (CNTs).This examination encompassed CNTs characterized by diameters spanning the 16 to 50 Årange.They demonstrated that even within the confines of CNTs exhibiting exceptionally small diameters, such as 1.66 nm, the water flow impeccably adhered to the theoretical framework provided by the slip-modified Hagen-Poiseuille equation.On the other hand, Zhang et al[14] explored the viscosity of water confined within single-walled carbon nanotubes (SWCNTs) ranging in diameter from 8 to 54 Å.They employed an "Eyring-MD" (molecular dynamics) method, which combines Eyring's viscosity theory with molecular dynamics simulations to model viscosity.Similarly, Ye et al[11] investigated how the size of SWCNTs, ranging from 8 Å(associated with (6, 6) SWCNT) to 54 Å(related to (40, 40) SWCNT), affects the viscosity of water within these tubes.Further, Suk and Aluru[34] studied the water flow in carbon nanotubes ranging in diameter from 5 to 24 Å, where they found the relative viscosity to be between 0.4 to 0.75.We find that different researchers obtained different data on water transport through the nanotubes.This shows that it might be tough to model all data for a specific power-law.Therefore, we need a generic consolidated model to fit all data as proposed above in equation(12).To model data on the solid-black line, we use equation(12) and get the fitting parameters as a = 1, b = − 4.21 × 10 −10 m, c = − 1 × 10 −10 m, and n = 1.On the other hand, to model the data on the solid-green line, we get a=0.9, b = − 3.21 × 10 −10 m, c = 1 × 10 −10 m, and n = 1.We find that despite Thomas and McGaughey [10], Ye et al[11] and Suk and Aluru[34] proposed three different power-law dependencies viscosity model (see equations (1) and (2), and (3)), we predicted all of their data, using our single generic proposed model for the viscosity.We find all the data from MD simulation as well as from different experiments, which have various power-law dependencies on the diameter; our single simpler model (i.e., equation (12)) can give an excellent fit to all respective data sets.Figure3was created to show a graphical synthesis of these findings, including both experimental data, MD simulations data, and the fitted proposed model, to show a broader context that the single proposed model is enough to fit a vast array of data from various sources.
this paper, we tried to propose a simple single equation for each nano confined material property such as for density D a the slip length λ(D) = λ 1 De − n D + λ o (where a, b, c, n, λ 1 , λ o are the free fitting parameters) and model various data sets from the experiments and MD simulations from the literature.