Viscosity of Water Interfaces with Hydrophobic Nanopores: Application

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Viscosity of water interfaces with hydrophobic nanopores: application to water flow in carbon nanotubes Mohamed Shaat Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02752 • Publication Date (Web): 16 Oct 2017 Downloaded from http://pubs.acs.org on October 17, 2017

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Viscosity of water interfaces with hydrophobic nanopores: application to water flow in carbon nanotubes M. Shaat a Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. ABSTRACT The nanoconfinement of water results in changes in water properties and nontraditional water flow behaviors. The determination of the interfacial interactions between water and hydrophobic surfaces helps in understanding many of the nontraditional behaviors of nanoconfined water. In this study, an approach for the identification of the viscosity of water interfaces with hydrophobic nanopores as a function of the nanopore diameter and water-solid (nanopore) interactions is proposed. In this approach, water in a hydrophobic nanopore is represented as a double phase water with two distinct viscosities: water interface and water core. First, the slip velocity-to-pressure gradient ratio of water flow in hydrophobic nanopores is obtained via molecular dynamics (MD) simulations. Then, the water interface viscosity is determined via a pressure gradient-based bi-layer water flow model. Moreover, the core viscosity and the effective viscosity of water flow in hydrophobic nanopores are derived as functions of the nanopore diameter and water-solid interactions. This approach is utilized to report the interface viscosity, core viscosity, and effective viscosity of water flow in CNTs as functions of the CNT diameter. Moreover, using the proposed approach, the transition from MD to continuum mechanics is revealed where the bulk water properties are recovered for large CNTs. KEYWORDS: water depletion; nanoconfinement; carbon nanotubes; hydrophobic; hydrophilic; water interface.

INTRODUCTION The study of the interfacial interactions between water molecules and hydrophobic nanopores is of a particular interest because it explains many of nontraditional phenomena of micro/nanofluidics and nanoconfined fluid flow systems. These interactions are the main cause of the changes in water

____________________________ * Corresponding author: Electronic mail: [email protected]; [email protected] (M. Shaat).

Tel.: +15756215929.

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properties. Moreover, the characteristics of nanoconfined water flow systems are strictly related to these interactions. Studies have been conducted on the characteristics of water adjacent to hydrophobic surfaces [1-4]. Some observations can be highlighted from these studies. First of all, it was demonstrated that, when water flows near a hydrophobic surface, a depletion layer is formed which is distinguished with an intensive decrease in the water density [1]. The Synchrotron x-ray reflectivity measurements of the interface between water and methyl-terminated octadecylsilane monolayers revealed a depletion layer with 0.2 - 0.4 nm thickness and lower density of 0-40% that of the bulk water [2]. Using direct noninvasive neutron reflectivity measurements, a sigmoidal reduced-density profiles of water in the depletion layer were revealed [5]. Second, molecular dynamics (MD) and experimental models have revealed high flow rates of water flows in hydrophobic nanopores. For instance, when water flows in a Carbon Nanotube (CNT) (i.e. a hydrophobic nanopore), flow rates with 4-5 orders-of-magnitude higher than those of the conventional water flow are observed [6-12]. This has been attributed to the water depletion [1, 10] where near the hydrophobic surface, jumps in the water flow rate profiles can be observed [10]. Third, because of the depletion layer and the water-solid interaction, the water density at the first water layer after the depletion layer sharply increases (i.e. layering of water near the solid surface) [1,5, 8, 10, 13, 14]. It was demonstrated via MD simulations that the water density close to the depletion layer increases with the increase in the water-solid interaction potential [1, 10]. Joseph and Aluru [10] pointed out that the increase in the water density in the first water layer is accompanied with a decrease in the water’s slip velocity near the nanopore’s wall. Finally, it can be observed from the MD simulations that the hydrophobicity depends on the water-solid interaction energy and the water-surface separation. The increase in the water-solid interaction energy is accompanied with a decrease in the system hydrophobicity [1, 13]. In addition, the hydrophobic force decreases with the increase in the water-surface separation [3, 4].

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It follows from the aforementioned observations that, when water flows in a hydrophobic nanopore, (1) the decrease in the nanopore diameter results in a decrease in the water-surface separation, an increase in the hydrophobic force, an increase in the hydrophobicity, an increase in the water depletion, a decrease in the water viscosity near the solid surface, and an increase in the slip velocity. Moreover, (2) the decrease in the water-solid interaction energy increases the system hydrophobicity, decreases the density of the first water layer, increases the water depletion, decreases the water viscosity near the solid surface, and increases the slip velocity. In order to compensate these changes in the hydrophobicity and water properties when change the feature and/or the strength of the water nanoconfinement, the viscosity of water should be determined as a function of the nanopore diameter and water-solid interactions. In contrast to a hydrophobic surface, water particles stick to a hydrophilic surface (i.e. no water depletion). This gives an increase in the water viscosity when it flows near a hydrophilic surface. The viscosity of water interfaces with hydrophilic surfaces were reported in various studies utilizing experimental models. Using AFM techniques, Goertz et al. [15] and Li et al. [16, 17] measured the shear viscosity of water interfaces with hydrophilic surfaces. For nanopores with diameters 0.3-0.7 nm, the viscosity of the water interface was obtained higher than the bulk water viscosity by ~2.6 − 3.3 × 10

factor [15], while, for a nanopore with diameter 0.5 nm, it was obtained 4-orders of magnitude higher than the bulk water viscosity [16,17]. In another study, Goertz et al. [15] measured the viscosity of water interface with an amorphous silica surface (hydrophilic surface) with 6-orders of magnitude higher than the bulk water viscosity. Because of the sharp increase of the water viscosity near the hydrophilic surface, the total effective viscosity was determined much higher than the bulk water viscosity. For instance, using MD simulations, the effective viscosity of water was obtained ~3 times its bulk water counterpart for hydrophilic nanopores [18, 19]. Using an experiment on water flow in 3-4 nm hydrophilic nanopores, Kelly et al. [20] obtained the effective viscosity of water as 1.6 times the bulk water viscosity. Using MD simulations of a 3 ACS Paragon Plus Environment

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hydrophilic nanotube, the effective viscosity of water was obtained ~1.5 times the bulk water viscosity for a nanotube with 1 nm diameter [21, 22]. Although the viscosity of water interfaces with hydrophilic surfaces were reported in various studies, to the author’s knowledge, the viscosity of water interfaces with hydrophobic surfaces has not been reported before. The reason for this is the formation of the water depletion layer at the interface; thus, the experiment and MD simulations breakdown at this region. Therefore, the existing studies were limited for determining the total effective viscosity of water confined between hydrophobic surfaces by assuming arbitrary constant values for the viscosity of the water interface. These values of the viscosity of the water interface were arbitrary assumed to fit the results of the MD simulations. For example, by assuming a constant viscosity for the water interface of ~0.655 mPa. s, Thomas and McGaughey [23] determined the effective viscosity of water in CNTs (i.e. this effective viscosity was determined as the weighted-average of the viscosities of the water interface and water core) decreasing with the decrease in the CNT diameter. Chen et al. [8] and Xu et al. [24] performed MD simulations for water in CNTs where the water was exposed to arbitrary average flow rates. Then, by relating the effective viscosity of the water to the flow rate using the conventional Hagen-Poiseuille model of fluid mechanics, they determined the viscosity with 2-3 orders of magnitude lower than bulk water viscosity. However, in fact, the viscosity of nanoconfined water should be related to the flow rate-to-pressure gradient ratio. This makes the determined viscosities in Chen et al. [8] and Xu et al. [24] came much lower than the predicted ones in other studies [25, 26, 27]. Using Eyring theory, Ye et al. [26] and Zhang et al. [27] determined the decrease in the effective viscosity of water with the decrease in the CNT diameter. The effective viscosity was estimated from obtained water velocity and density profiles by MD simulations considering a water depleted layer of thickness ~0.6 . In another study, Brabu and Sathian [12] used Eyring theory and MD simulations of water in CNTs assuming a depletion layer of ~3.5 . The results of the latter studies

came different because of the different considered densities and structures of water inside the CNT. Bonthuis and Netz [28] used MD simulations to determine the viscosity profiles for water in hydrophobic


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and hydrophilic nanopores. They assumed a depletion layer with 0.15 nm thickness and 0.067 mPa.s viscosity. They stated that, beyond the water depletion, water recovers the bulk water viscosity. Wu et al. [25] related the viscosity of the water interface to the contact angle (wettability) assuming a constant value for the viscosity of water’s interface with nanopores of diameters larger than 1.4 nm. Incorporating a continuum mechanics model, however, resolves the trouble and can be effectively used to report the viscosity of water interfaces with hydrophobic surfaces. In this study, an approach for the identification of the viscosity of water interfaces with hydrophobic nanopores as a function of the nanopore diameter and water-solid interactions is proposed. In the context of this approach, because of the water depletion, water in a hydrophobic nanopore is represented as a double phase water with two distinct viscosities. The viscosities of these two phases are determined as functions of the nanopore diameter and the water-solid interaction parameters. These reported viscosities are needed to understand the nontraditional physical phenomena of nanoconfined water, e.g. water harvesting [29, 30, 31], slip conditions [23, 25, 32], and high flow rates [10, 23, 25, 32]. Utilizing the proposed approach, for the first time, the viscosity of water interfaces with CNTs with various sizes are reported.

VISCOSITY OF WATER INTERFACES WITH HYDROPHOBIC NANOPORES In this study, the pressure gradient-based bi-layer water flow model shown in Fig. 1 is considered to represent water flow in hydrophobic nanopores. A water core with viscosity is surrounded by a

water interfacial-shell with viscosity. The thickness of the water interfacial-shell (i.e. water interface with the hydrophobic nanopores) is considered = 1.1224 i.e. is the Lennard-Jones (LJ)

potential parameter of the water-solid interaction. This thickness, , equals the water-solid separation at which the water-solid attraction force is zero. By solving the steady-state Poiseuille flow of bi-layer fluid flow in pores, the slip velocity, , is obtained in the form: =

" − ̅" ! $ 16 5


(1)

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where

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denotes the pressure gradient, is the nanopore diameter, and ̅ = − 2 is the diameter of the

water core. In the context of the proposed approach, the slip velocity-to-pressure gradient ratio ( ⁄ ) (VPR) is first determined as a function of the nanopore diameter and water-solid interactions (see Fig. 2 and Methods). For this purpose, MD simulations are utilized. Then, according to equation (1), the viscosity of

FIG. 1: A schematic of water flow in a single-walled CNT where two-phases of water can be recognized. The blue solid line represents a schematic of the radial variation of water density (&) inside the CNT. Because the CNT is a hydrophobic solid, a water depletion layer is formed near the CNT wall. The water interface with the CNT is distinguished with a reduced density and viscosity. Moreover, the water core possesses higher density and viscosity than the bulk water (i.e. &' is the bulk water density). the water interface with the hydrophobic nanopore, , is obtained as a function of the nanopore diameter and water-solid interactions, as follows: =

" − ̅" 16 × VPR


(2)

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Accounting for the nanoconfinement size and water-solid interactions, the viscosity of the water core, , is determined in the form (more details about the derivation of the water core viscosity are given in Methods): = ' !1 +

10+ 0 1" ,/ −/ − 0.252$ /2 /2 3̅+

(3)

where ' denotes the bulk water viscosity. + and denote the LJ water-solid potential parameters while + and are the LJ potential parameters of water.

The effective viscosity of the bi-layer water can be determined as the weighted average of the viscosities of the water interface (equation (2)) and the water core (equation (3)), as follows [23, 25, 32]: 3 = !

" − ̅" ̅" + $ ! $ " "

(4)

Equation (4) presents a closed form for the effective viscosity of water flows in hydrophobic nanopores as a function of the nanopore diameter and water-solid interaction parameters. It is should be noted that, unlike many studies in which the viscosity of the water interface was arbitrary assumed [23, 25, 28], is determined as a function of the nanopore diameter and water-solid interactions. VISCOSITIES OF WATER FLOW IN CNTs To demonstrate the effectiveness of the proposed approach for identifying water viscosities (i.e. , , and 3 ), equations (2)-(4) are utilized to report the interface viscosity, core viscosity, and effective viscosity of water flow in CNTs. LJ potential parameters for water particles are defined as + = 0.6502 45/67 and = 0.3169 and for water-CNT as + = 1.4454 45/67 and =

0.3122 [33]. First, depending on the MD simulations available in the literature [10, 23, 34], the slip velocity-to-pressure gradient ratio, VPR, for water flow in CNTs is depicted as a function of the CNT

diameter in Fig. 2 (see Methods for more details). Second, utilizing the obtained VPR, the viscosity of the water-CNT interface, , is determined via equation (2) and plotted in Fig. 3(a). Third, according to equation (3), the water core viscosity of water flow in CNTs is depicted as a function of the CNT 7 ACS Paragon Plus Environment

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diameter in Fig. 3(b). Fourth, according to equation (4), the effective viscosity of water flow in CNTs is plotted in Fig. (4) as a function of the CNT diameter. To validate the proposed approach and equations (2)-(4), the effective viscosity as obtained using equations (2)-(4) is compared with the effective viscosity as obtained by MD simulations and experiments [6, 23, 25, 26, 27, 36, 37]. Finally, these viscosities are numerically reported in Table I for water flow in CNTs with different diameters and chirality.

FIG. 2: The slip velocity-to-pressure gradient ratio (VPR) of water flow in single-walled CNTs as a function of the CNT diameter. For CNTs with diameters of ~2 − 6.5 , the VPR increases with the increase in the CNT diameter (see Methods for more details). For CNTs with dimeters larger than 6.5 nm, the VPR decreases with the increase in the CNT diameter. The inset represents the slip velocity, , of water flow in CNTs as a function of the CNT diameter for different applied pressure gradients, . Fig. 2 shows the slip velocity-to-pressure gradient ratio, VPR, as a function of the CNT diameter. The VPR is obtained to fit the MD simulations available in the literature [10, 23, 34]. The slip velocities of water flow in CNTs have been reported in various studies using MD simulations [10, 23, 31, 32, 34]. For instance, Kannam et al. [34] reported the increase in water’s slip velocity with the increase in the CNT diameter from 1.62 nm to 6.5 nm. Thomas and McGaughey [23] determined the increase in the 8 ACS Paragon Plus Environment

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volumetric flow rate with the increase in the CNT diameter from 1.66 nm to 5 nm. Moreover, Thomas and McGaughey [32] and Qin et al. [31] pointed out that the average flow rate may increase or decrease with the increase in the CNT diameter from 0.83 nm to 1.66 nm. It follows from the aforementioned results that the slip velocity may increase or decrease with the increase in the CNT diameter from 0.83 up to 6.5 . Reasons for these different trends are the dependency of the water flow rates on, both, the water structure, e.g. OH bond orientation and hydrogen bonding in the depletion layer [10], as well as the CNT size. However, as a matter of fact, for large CNT diameters, water structure has a negligible contribution on the water flow rate. Thus, at these large CNT diameters, the hydrophobicity of the water flow in CNTs decreases (and hence the slip velocity decreases) with the increase in the CNT diameter. By considering the previous facts, in Fig. 2, the VPR is obtained decaying as the CNT diameter increases larger than 6.5 . When compared to the experiment [6, 36, 37], an excellent match is obtained as shown in Fig. 2 for CNTs with diameters larger than 6.5 nm. It follows from Fig. 2 that water flows in CNTs with a specific slip velocity-to-pressure gradient ratio (VPR) that depends on the CNT diameter.

(a)


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(b) FIG. 3: (a) The normalized viscosity of the water-CNT interface as a function of the CNT diameter. (b) The normalized viscosity of the water core as a function of the CNT diameter. Fig. 3(a) shows the obtained viscosity of the water-CNT interface, , as a function of the CNT diameter. Indeed, the viscosity of the water-CNT interface is differently considered in various studies. The viscosity of the water interface region is considered independent from the confinement size (i.e. nanopore diameter) in some studies. For example, Thomas and McGaughey [23] assumed the viscosity of the interface region is ~0.655 mPa. s. Myers [35] assumed the interface possesses a constant viscosity of

1.8% of the viscosity of bulk water. Wu et al. [25] assumed a constant water interface viscosity for CNTs with diameters larger than 1.4 nm. However, for hydrophobic surfaces, it has been pointed out that the viscosity of the water interface should be considered decreasing with the decrease in the confinement (nanopore) size [8, 18, 28]. The obtained viscosity of the water-CNT interface fulfills the latter requirement where it decreases with the decrease in the CNT diameter, as shown in Fig. 3(a). Moreover, the obtained water-CNT interface viscosity fits the previously derived observation that the decrease in the nanopore (CNT) diameter is accompanied with an increase in the system hydrophobicity and the water depletion. Indeed, the contribution of water-CNT interaction increases with the decrease in the CNT diameter. This leads to an increase in the system hydrophobicity, and hence more water particles are depleted at the interface region. In contrast, for pores with large diameters (macro-scale sizes), the role of 10 ACS Paragon Plus Environment

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water-CNT interactions demolishes and the water depletion process does not take place (or at least the size of the water depletion layer, if exist, is much negligible). Multiple phases of water are revealed when water flows in CNTs [14]. Because of the water depletion layer, the viscosity of the water-CNT interface is lower than the bulk water viscosity, i.e. ' , as shown in Fig. 3(a). In addition, because water molecules in the water core region are exposed to attraction forces, the viscosity of the water core is higher than the bulk water viscosity. This explains the reductions in the slopes of the velocity profiles revealed via MD simulations [10, 23, 32]. Fig. 3(b) shows the water core viscosity of water flow in CNTs as a function of the CNT diameter. As presented in the figure, the decrease in the CNT diameter is accompanied with an increase in the water core viscosity. This can be attributed to the increase in the water-CNT interaction potential with the decrease in the CNT diameter. In fact, the first water layer after the water depletion layer is subjected to the highest interaction force which decays for the water layers closer to the CNT center. Moreover, the density of the water has a radial distribution. Therefore, the water core viscosity radially varies. However, to a great extent, it is acceptable considering the average of the water core viscosity, as presented in equation (3). The effective viscosity of the water flow in CNTs is plotted as a function of the CNT diameter in Fig. 4. As shown in the figure, the decrease in the CNT diameter is accompanied with a decrease in the effective viscosity of the confined water. According to the proposed approach, the decreased effective viscosity of water in CNTs is mainly attributed to the decrease in the viscosity of the water-CNT interface. Thus, the decrease in the viscosity of the water-CNT interface is much higher than the increase in the viscosity of water core.


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FIG. 4: The normalized effective viscosity of water flow in CNTs as a function of the CNT diameter.

Inspecting Fig. 4, it is clear that the obtained effective viscosity of water perfectly agrees with the results of the MD simulations [23, 26, 27]. In addition, the obtained effective viscosity matches the experimental and MD-based observations in the literature [6,36,37]. For instance, using MD simulations, Chen et al. [8], Thomas and McGaughey [23], and Babu and Sathian [12] demonstrated that the effective viscosity of water flows in CNTs decreases with the decrease in the CNT diameter. Moreover, the reductions in the effective viscosity of water with the decrease in the CNT diameter are revealed utilizing Eyring-MD method in [26, 27]. The reported viscosity in Fig. 4 demonstrates the effectiveness of the proposed approach to identify the viscosities of water confined in hydrophobic nanopores. Table I shows the water-CNT interface and water core viscosities along with the effective water viscosity for different CNTs (10,10) to (72, 72). It is clear that the increase in the CNT diameter is accompanied with an increase in the water-CNT interface viscosity, a decrease in the water core viscosity, and an increase in the effective water viscosity. When water flows in (12,12) CNT, because of the waterCNT interactions, the effective water viscosity is ~53% its corresponding value of bulk water. However, 12 ACS Paragon Plus Environment

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when water flows in (72,72) CNT, the effective water viscosity increases to ~92% of the bulk water viscosity. This demonstrates the fact that the water properties are significantly altered and influenced by water interactions with the nanopore confining it. For large sized-pores, such interactions, although they are exist, barely contribute to the water properties. Table I: The viscosity of the water-CNT interface , the viscosity of the water core , and the effective water viscosity for different single-walled CNTs. Chirality

CNT diameter, (nm)

/' (10; )

/'

3 /'

(10,10)

1.356

11

1.8616

0.4353

(12,12)

1.628

14

1.6247

0.5279

(16,16)

2.17

19

1.3984

0.6420

(20,20)

2.713

23

1.2914

0.7113

(24,24)

3.255

25

1.2296

0.7581

(28,28)

3.798

26

1.1894

0.7981

(32,32)

4.340

27

1.1612

0.8172

(36,36)

4.880

28

1.1404

0.8372

(44,44)

5.968

31

1.1114

0.8663

(48,48)

6.510

34

1.1010

0.8774

(56,56)

7.595

42

1.0851

0.8948

(64,64)

8.680

59

1.0735

0.9081

(72,72)

9.765

90

1.0647

0.9186

CONCLUSIONS An approach has been proposed for the determination of the water properties when flows in hydrophobic nanopores. These water properties have been determined as functions of the nanoconfinement size and water-solid interactions. This approach has been utilized to report the interface viscosity, core viscosity, and the effective viscosity of water flow in CNTs as functions of the CNT diameter. The decrease in the CNT diameter is accompanied with a decrease in the water-CNT interface viscosity and an increase in the water core viscosity. Moreover, a decrease in the effective viscosity of


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water flow in CNTs is obtained with the decrease in the CNT diameter. Using the proposed approach, the transition from MD to continuum mechanics is revealed where, for large CNTs, the obtained water properties recover the bulk water properties. Moreover, the reported water properties in this work can be utilized to propose a continuum mechanics-based model for water flow in nanopores where the slipconditions can be skipped and the original no-slip conditions can be directly employed. METHODS Water core viscosity: Because of the water-solid interactions, the viscosity of the water core increases by a factor that can be considered as the ratio of water-solid interactions to the water-water interactions. Therefore, in this study, the water core viscosity is determined as follows:

where

〉 is considered. The viscosity of the water core, , is, therefore, determined in the form shown in equation (3). The second term in the right hand side of equation (3) presents the correction of the viscosity of bulk water for the nanopore diameter and the water-solid interactions. Slip velocity-to-pressure gradient ratio (VPR): In Fig. 2, the slip velocity-to-pressure gradient ratio (VPR) of water flow in single-walled CNTs is plotted as a function of the CNT diameter. From various MD simulations [10,23,34], the VPR is obtained increasing with the increase in the CNT diameter up to 6.5 nm. However, for large CNTs, the VPR should decrease with the increase in the CNT diameter. Thus, at large diameters, the VPR recovers the one of the conventional Hagen-Poiseuille model (i.e. DEF' = " ⁄16' ). To fulfill this trend, the VPR function is determined with an increasing trend up to ~6.5 followed by a decreasing trend up to ~35 with the increase in the CNT diameter. Thus, at a CNT diameter of ~35 , the classical DEF' is recovered. Consequently, VPR is derived in the following form: DEF = DEF' G

1 − =̅⁄? M 1 − |I@ + J" + K + L| "

where 14 ACS Paragon Plus Environment

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I = G−0.4044 + 0.0074 ! J = G0.5955 − 0.0308 !

+ $M × 10"@ +

+ $M × 101O +

K = G−2.942 + 0.1875 !

+ $M × 100 +

L = 1.002 − 9.613 × 10;O !

+ $ +

It should be mentioned that LJ parameters are incorporated in the previous equations to account for the effects of the water-CNT interactions on the viscosity of the nanoconfined water.

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TOC Graphic


Viscosity of Water Interfaces with Hydrophobic Nanopores: Application

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