Capillary filling under nanoconfinement: The relationship between effective viscosity and water-wall interactions

International Journal of Heat and Mass Transfer 118 (2018) 900–910

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Capillary filling under nanoconfinement: The relationship between effective viscosity and water-wall interactions Dong Feng a,⇑, Xiangfang Li a, Xiangzeng Wang b, Jing Li c, Xu Zhang a a

MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), Beijing 102249, PR China Shaanxi Yanchang Petroleum (Group) Corp. Ltd., Xi’an 710075, PR China c Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N1N4, Canada b

a r t i c l e

i n f o

Article history: Received 5 August 2017 Received in revised form 29 September 2017 Accepted 11 November 2017

Keywords: Effective viscosity Nanoscale Water-wall interactions Pore shape Capillary filling

a b s t r a c t Understanding the capillary filling in nanopores is crucial for a broad range of science and engineering problems. Because of the dominant importance of surface effects at the nanoscale, the properties of confined water in nanopores must substantially differ from that of bulk water. Here, a novel model for effective viscosity in hydrophilic nanopores was proposed by further considering the water-wall interactions and density layering phenomena to modify the capillary filling at the nanoscale, every parameter in the proposed model has clear physical meaning. The presented model is successfully validated against existing experimental data collected from published literatures. The theoretical analysis and comparison denote that pore size, pore shape and nanomaterial properties have significant effects on the effective viscosity: (1) due to the stronger water-wall interactions, the higher the energy surface is, the larger the effective viscosity will be; (2) the effective viscosity in nanocapillaries is larger than that in nanochannels at the same pore size; (3) the effective viscosity of confined water in hydrophilic nanopores can exhibit a dramatic increase, the value could be even three times or one order magnitude greater than that of bulk water for silica or clay nanocapillaries within 10 nm. Meanwhile, compared with the traditional imbibition model in cylindrical capillaries, the coupled effects of pore shape and water-wall interactions would make the capillary filling characteristic more complex, which is one of the reasons for the deviation between real imbibition condition and traditional value for nanomaterial. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction With the rapid development of nanotechnologies over the recent decades, understanding the capillary filling at the nanoscale has been critical in many industries, including the design of nanofluidic, oil recovery, water desalination as well as the multiphase flow in unconventional reservoirs [1–3]. When the scale is down to nanometer, many investigations have demonstrated the smaller quantity than theoretical classical hydrodynamics, these physical phenomena are further augmented by the large surface to volume ratio, indicating higher apparent viscosity values than the known bulk viscosity values [4,5]. However, how the surface effects contribute to the variety of apparent viscosity has remained poorly characterized in theory, the novel accurate researches are strongly desired for revealing the capillary filling characteristic at the nanoscale.

⇑ Corresponding author. E-mail address: [email protected] (D. Feng). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.049 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

In general, viscosity represents the internal friction among molecules when they are in motion acted by some external forces, which is mainly determined by the intermolecular interaction at a constant temperature [6]. However, the confined water in hydrophilic nanopores usually shows drastically change of properties compared with those of bulk water, as a result of interactions from nanopores walls [7–10]. MD simulations results indicate that water density near hydrophilic surfaces may be two to five folds than that of bulk water, and the thickness of high-density layer is about 0.6–0.8 nm (two molecular layers) [7,8]. On the other hand, some simulations suggest one order magnitude increase in apparent viscosity near the hydrophilic surface, such change in confined water properties could obviously increase the filling resistance [9,10]. Moreover, the slower capillary filling rate than expected was also directly observed in the experiments, Haneveld et al. [11] and Persson et al. [12] measured the capillary filling of water in quartz capillaries with height less than 20 nm, the results showed the imbibition length could be reduced by 0.5–2.6 times, the deviation increased with the decreasing of pore size. Actually, several possible mechanisms have been proposed to explain those

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D. Feng et al. / International Journal of Heat and Mass Transfer 118 (2018) 900–910

Nomenclature

l0 NA hp R V T DG0 DGsl

l1 qs Cls

ql Mw

qw qw1 qw2 qbulk A1 A2

the viscosity of bulk water, mPa s the Avogadro constant, mol
1; the Planck constant, J s the gas constant, J/(kmol) molar volume of liquid, L/ mol; temperature, K viscous interactions between the liquid molecules, J interactions between liquid molecules and the solid surface, J the viscosity considering water-wall interactions, mPa s number density of molecules in solid, /m3 van der Waals interaction constant for solid-liquid interaction, J.m6 number density of liquid, /m3 the molar mass, kg/mol average density of confined water, kg/m3 average density of first layer, kg/ m3 average density of second layer, kg/ m3 density of bulk water, kg/ m3 area percentage of first layer, dimensionless area percentage of second layer, dimensionless

simulated results and experimental phenomenon, such as electroviscous effect and liquid-wall interaction [13–16]. In some work, long-range electrostatic forces were employed to explain the increased resistance in hydrophilic nanopores through the Debye-layer correction [13,14]. However, some recent researches demonstrated that this effect known as electroviscous effect will work when the pore size is about 1000 nm, it is insignificant and can be reasonably neglected in silica nanochannels with a height of less than 200 nm [15,16]. Moreover, change in the properties of confined water induced by the water-wall interaction can be another widely accepted way to explain the slow capillary filling process, furthermore, some typical models were proposed to build the correlations between effective viscosity and pore size, which are summarized in Table 1 [17–20]. Those models can be divided into two types: (1) the effective viscosity is in the inverse proportional relationship or exponential form to the pore size, the model possesses the empirical parameter with unknown physical meanings; (2) based on the concept of weighted average, Tomas et al. [19] and Wu et al. [20] divided the confined water into two parts, namely interface region and bulk-like region, the interface region has fixed thickness about 0.7 nm and constant viscosity. However, it is worth noting that the viscosity in the interfacial area should be a function of water-wall distance as well as the properties of solid surface rather than a constant. Actually, the research results by Cao et al. [21] and You et al. [22] suggested that the influence of waterwall interaction on fluid properties depended on the nature of solid

AH Ass All Dr d l(D) l(H) Pc Pc1(D) Pc2 (H)

c r H W h

s f

e1 e2

the Hamaker constant for solid-liquid interactions, J the Hamaker constant of solid surface, J the Hamaker constant of liquid, J the water radius, nm the pore size, nm effective viscosity in nanocapillaries, mPa s effective viscosity in nanochannels, mPas effective viscosity in nanochannels, MPa the capillary force of nanocapillary, MPa the capillary force of nanochannel, MPa the surface tension, N/m the radius of nanocapillary, nm; the height and width of nanochannel, nm the height and width of nanochannel, nm the contact angle, ° the tortuosity, dimensionless the ratio of height and width, dimensionless the ratio of effective viscosity in nanotubes and nanochannels, dimensionless the the ratio of real imbibition and traditional one, dimensionless

surface, it would decrease rapidly with the increase of the fluidwall distance and the interaction regions were only for several nanometers. Thus, those models hardly reveal the change of confined-water viscosity in a physical way, the micromechanisms and representative models are still not sufficiently understood in the published work. In addition to those, pore shape is another important effect for the evaluation of capillary filling characteristics of nanoporous media. However, many traditional and recent capillary filling models are established on the assumptions of Hagen–Poiseuille flow in cylindrical capillaries, actually, the microstructure and connectivity of natural porous nanomaterial are usually tortuous and noncircular [23–25]. Compared with the traditional model, the coupled effect of pore shape and water-wall interaction would make the capillary filling characteristic more complex. More importantly, the continuum theory is the basics of Navier-Stokes equations and Hagen
Poiseuille equations, it is still valid in nanopores with diameters larger than 2 nm, as is discussed in detail in the review by Bocquet et al. [5]. So, 2 nm is the minimum value of pore size in our study. In this paper, we propose a model to characterize the effective viscosity in hydrophilic nanopores by further considering the water-wall interactions; then, the influence of pore scale, pore shape and surface properties on effective viscosity is analyzed; finally, the coupled effect of pore shape and water-wall interactions on capillary filling is studied to explain the complex phe-

Table 1 Some models between effective viscosity and pore size. Model n

l ¼ l0 þ u=d

pffiffiffiffiffiffiffiffiffiffiffi

l ¼ l0 þ bð As Aw
Aw Þ=d h

l ¼ li AAti ðdÞ þ l0 1
AAti ðdÞ ðdÞ ðdÞ

i

Description

Limitation

Ref.

n is the exponent for decay rate of interaction; u is the interaction coefficient As and Aw are the Hamaker constants for solid and liquid; b is the viscosity increase coefficient li, l0 are the viscosity in interaction area and bulk water; Ai(d), At(d)are the areas of interaction region and total crosssectional

Empirical constants n and u have unknown physical meanings Considering the water-wall interaction with an empirical constants b

Li et al. [17]

The critical thickness for interaction area is always 0.7 nm; the li is also a constant

Tomas et al. [19] Wu et al. [20]

Wang et al. [18]

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nomenon in science and engineering. This work is meaningful and helpful for understanding the flowing regularity and capillary filling characteristic at the nanoscale.

molecules and solid surface DGsl. Thus, the viscosity equation for considering the solid-liquid interaction can be expressed as [28,29]:

2. Model establishment

l1 ¼

2.1. Molecular theory of the viscosity Compared with the gas, the molecular structure of liquid is more complicated, the density of liquids is 1000 times larger while the spacing between molecules in liquids is 10 times smaller than that in gas [6]. The intermolecular forces of liquid are much stronger and the motion of liquid is largely confined by the adjacent neighbors. As showed in Fig. 1, for a pure liquid at rest, although the molecules are constantly in thermal motion, it always located around the equilibrium position. However, if the molecule wanted to move to the next vacant site, additional free energy is needed to overcome a potential barrier (DG0) created by its neighbors. Based on the theoretical analysis, molecular theory of the viscosity (Eyring viscosity) is established by combining the Boltzmann statistical mechanical theory of absolute reaction rates. This theory does give qualitative description of the molecular transport mechanisms, and the calculated values show great consistency with the experimental results [26,27]. The Eyring viscosity equation can be expressed as (detailed in Appendix A)

l¼

 2 d NA hp expðDG0 =RTÞ a V

ð1Þ

Where NA, hp, R, V, T are the Avogadro constant, Planck constant, gas constant, molar volume of liquid and temperature, respectively; a is the distance between vacant site, d is the distance between adjacent planes. 2.2. Viscosity model for considering the water-wall interactions Eq. (1) demonstrates that the viscosity is closely related to the liquid properties (such as liquid molar volume V, energy barrier DG0 and absolute temperature T). It also reveals that the viscosity is positive correlation with energy barrier, meaning that the higher viscosity is corresponding to the stronger intermolecular interaction. However, when the scale is down to nanometer, the properties of confined liquid differ drastically from those of bulk liquid, as a result of the varying dynamic of the confined liquid induced by remarkable liquid-wall interaction. As illustrated in Fig. 1, for the water molecule near the pore wall, the total energy barrier should be the sum of contributions from viscous interactions between the liquid molecules DG0 and interactions between liquid

Layer C

Vacant site Layer B Layer A

energy barrier

a

 2 d N A hp expððDG0 þ DGsl Þ=RTÞ: a V

ð2Þ

WhereDGsl is the additional free energy to escape from the confinement of nanopores wall, it is equivalent to the interaction potential of the solid surface, l1 is the viscosity for considering the interactions between water and pore surface. Furthermore, Eq. (2) can be given as the following form to describe the relation between l1 and l0:

l1 ¼

 2 d N A hp expðDG0 =RTÞ  expðDGsl =RTÞ a V

¼ l0 expðDGsl =RTÞ

ð3Þ

Therefore, the key point in the model is the interaction potential of the solid surface. As shown in Fig. 2, the interaction potential between solid surface and a molecule (particle) can be gives as [29]:

DGsl ¼

pqs C ls 6h

ð4Þ

3

Where qs is number density of molecules in solid, Cls is the van der Waals interaction constant between liquid and solid molecules, J.m6; h is distance between solid surface and molecules. Furthermore, another interaction constant AH is usually used to characterize the Van der Waals interactions between materials, which is called the Hamaker constant and can be expressed as [29]:

AH ¼ p2 C ls qs ql

ð5Þ

Where ql is the number density of liquid molecules and usually can be calculated with term NAqbulk/Mw [30]. However, it is worth noting that the layering of fluid molecules near solid surfaces is a wellknown phenomenon, the density of water layers near the pore walls is much higher than that of bulk water and its impact on the water number density is significant in nanopores [31,32]. Considering this phenomenon, a corrected model is proposed to calculate the number density of water molecules in nanopores

ql ¼

N A qw Mw

ð6Þ

Where Mw is the molar mass, 0.018 kg/mol; qw is the average density of confined water by further considering the density layering phenomena of liquid in hydrophilic nanopores. According to our knowledge, Molecular Dynamics (MD) simulation is the common method to determine the density profile, showing that the thickness of high-density layer is about 0.6–0.8 nm (two molecular layers) [7,8,31,32]. In each layer, the density is nonuniform and hard to be described by a continuous function, a simplified density profile proposed by Li et al. [33] is used to determine the average density of high-density layer (detailed in Appendix B). Then, the average density of confined water can be obtained with the following equation:

G0

Gsl

Fig. 1. Illustration of an escape of fluid in the process of flow, where DG0 is viscous interactions between the liquid molecules; DGsl is the additional interactions between the liquid molecules and the solid surface.

h

Fig. 2. The illustration of interaction between solid surface and liquid molecule, h is the distance between solid surface and molecules.

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qw ¼ qw1  A1 þ qw2  A2 þ qbulk  ð1
A1
A2 Þ

ð7Þ

Where qw1 is the average density of first layer, qw2 is the average density of second layer, qbulk is the density of bulk water, A1 is the area percentage of first layer, A2 is the area percentage of second layer, the detailed calculation is in Appendix B. Substituting Eqs. (5)–(7) into Eq. (4). The DGsl can be derived as:

DGsl ¼

AH M w

ð8Þ

3

6ph qw

Furthermore, the Hamaker constant for solid-liquid interaction can be calculated as follows [29]:

AH ¼

pffiffiffiffiffiffipffiffiffiffiffi Ass All

ð9Þ

Where AH is the Hamaker constant for solid-liquid interaction, J; Ass is the Hamaker constant of solid surface, J; All is the Hamaker constant of liquid, J. Combining the Eq. (3) and Eqs. (8), (9), the viscosity as a function of spatial position and the surface properties is obtained

l1 ¼ l exp

pffiffiffiffiffiffipffiffiffiffiffi  3 Ass All MW =6ph qw RT

ð10Þ

2.3. Modeling for effective viscosity It is now clear that the viscosity of water near the hydrophilic surface is spatially varying, as a result of the interaction force exerted by the nanopores walls. However, the calculated or measured viscosity in actual condition is the cross sectional averaged effective viscosity. To obtain the effective viscosity of confined water in hydrophilic nanopores, a weighted-average of the viscosities is usually adopted [1,19,20]. It is worth noting that there is a process called steric hindrance, which is caused by the overlap of molecular clouds and prevents the molecule from approaching the wall too closely, the distance of closest approach is equal to its radius (Fig. 3), for water molecules, the radius ranges from 0.125 nm to 0.2 nm in the published literatures [34,35]. Actually, the water molecule is nonspherical, the equivalent effective radius in spherical state is approximate 0.14–0.15 nm, this value is also used in MD simulation [9,10,36]. Thus, the effective viscosity in hydrophilic nanopores can be calculated as follows:

R d =2

lðdÞ ¼

0

l1 dA

A

ð11Þ

Where d⁄ = d
2Dr; d is the pore size, nm; Dr is the water radius, 0.15 nm. In detail, the effective viscosity varying shape can be calculated as follows:

Nanocapillary:

R D=2
Dr

lðDÞ ¼

0


2prl exp

pffiffiffiffiffiffipffiffiffiffiffi  Ass All MW =6pr 3 qw RT dr A

ð12-aÞ

Nanochannel:

R H=2
Dr

lðHÞ ¼

0


wl exp

pffiffiffiffiffiffipffiffiffiffiffi  3 Ass All M W =6ph qw RT dh A

ð12-bÞ

Where l(D) is the effective viscosity in nanocapillaries, l(H) is the effective viscosity in nanochannels. Eqs. (10-12a, 12b) provides a novel analytical model to calculate the effective viscosity in hydrophilic nanopores. It can be inferred from the equation that, at the constant temperature K, the viscosity is related not only to the nature of liquid (Hamaker constant of liquid All, molar mass Mw, average density of confined water qw ), but also to the properties of solid surface (Hamaker constant of liquid Ass). In contrast to the previous model by considering liquid-wall interactions as an empirical parameter (Table 1), every parameter in this model has clear physical meaning, which reveals the more fundamental mechanisms affecting the viscosity of confined water in hydrophilic nanopores. Most importantly, the proposed model is simple and can be used to estimate the effective viscosity of confined water in hydrophilic nanopores of different material, which overcomes the complex MD simulations with computational limitations and time-consuming experiments with high costs [1,9,10]. 2.4. Modeling for capillary filling at the nanoscale Capillary filling in nanopores plays an important role in science and engineering, such as design of nanofluidic devices, fabrication of nanostructured materials, the transport of hydraulic fracturing fluid in shale reservoir [2,3]. Many experiments and MD simulations had validated the continuum framework of hydrodynamics and the macroscopic Young-Laplace equation are still valid down to the nanometer scale. Assuming the laminar flow of incompressible fluid under the external force, Washburn derived an equation for capillary filling based on Poiseuille flow, which gives the length and velocity of capillary filling as a function of time [29]:

 LðtÞ ¼

v ðtÞ ¼

2KP c lðdÞ

1=2 t 1=2

 1=2 dL KPc ¼ t
1=2 dt 2lðdÞ

ð13-aÞ

ð13-bÞ

Where Pc represents the driving force, it is equal to the capillary force; K is the permeability, l(d) is the effective viscosity, it is equal to the l(D) for nanocapillaries or l(H) for nanochannels.

Fig. 3. The interaction between pore wall and water in nanocapillary and nanochannel.

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H

L(t)

W center H/2

(b) Nanochannels (W>>H)

(a) nanocapillary

Fig. 4. Illustration of the nanocapillaries and nanochannels during capillary filling process.

Table 2 Basic modeling parameters in the calculation. Parameter

Symbol

Unit

Value (nanocapillary)

Value (nanochannel)

Reference value

Gas constant Temperature Water liquid Water molar mass Water bulk viscosity Surface tension Hamaker constant (glass) Hamaker constant (mica) Hamaker constant (clay) Hamaker constant (water) Hamaker constant (solid-water) Water radius Height of nanochannel Diameter of nanocapillary Contact angle Porosity Tortuosity

R T

J/(k mol) K kg/m3 kg/mol Pa s N/m 10
20 J 10
20 J 10
20 J 10
20 J 10
20 J 10
9 m 10
9 m 10
9 m Dimensionless Dimensionless Dimensionless

8.314 293 1000 0.018 0.001 0.0727 6.5 10 12.5 3.7 5.0 0.15 – 6.8 0 0.31 ± 0.02 3.2 ± 0.4

8.314 293 1000 0.018 0.001 0.0727 6.5 10 12.5 3.7 5.0 0.15 5,11,23,47 – 0 1 1

8.314 293 1000 0.018 0.001 0.0727 6.3–6.8 [38,39] 10 [39] 12.5–15 [38,39] 3.7 [38] 4.83–5.02 0.125  0.20 [34,35] – – [36,37] [36,37] [36,37]

q Mw

l0 c Ass Ass Ass All AH 4r H D h U

s

Comparing with the straight capillary tubes in L-W imbibition standard model, the shapes of nanopores in nanomaterial are generally multitudinous, the classical type is the circular cross section and slit shape (Fig. 4, the former is universal in siliceous nanomaterial while the latter is generally observed in shale reservoirs. According to the Young-Laplace equation, the force for nanocapillary and nanochannel can be respectively written as [29]:

Pc1 ðDÞ ¼

4c cos h D

ð14-aÞ 

Pc2 ðHÞ ¼ 2c cos h

1 1 þ H W

 ð14-bÞ

Where Pc1 (D) is the capillary force of nanocapillary, Pc2 (H) is the capillary force of nanochannel, c is the surface tension, r is the radius of nanocapillary, H and W are the height and width of nanochannel, respectively; h is the contact angle. For capillary tubes or rectangular geometry, K can be given as follows [26]:

D2 K 1 ðDÞ ¼ 32s

ð15-aÞ

"  # 1 X H2 1 192 1 1
f tanh ip K 2 ðHÞ ¼ 2 p5 2f 12s i¼1;3;5 i

ð15-bÞ

Where K1(D) is permeability of nanocapillary, K2(H) is permeability of nanochannel, s is the tortuosity, f is the ratio of height and width, f = H/W; when H/W << 1 or H/W >> 1, it can be considered as the nanochannels. Thus, submitting Eq. (12a, 14a, 15a) into Eq. (13a) or Eq. (12b, 14b, 15b) into Eq. (13b), the capillary filling characteristic (the length and velocity) in nanopores can be analyzed. 3. Model validation In order to check the model reliability, the calculated results with the model are compared with the published experimental data at Refs. [36,37], both of the experiments are conducted on the clean and ‘‘thirsty” vycor glass. The calculating parameters are listed in Table 2. 3.1. Validation of capillary filling for nanocapillary The experimental data used to validate the model are obtained within Ref. [36]. Prior to water filling, the porous vycor glass was preferentially cleaned followed by drying at 200 °C. Meanwhile, the structure parameters (such as tortuosity s, porosity U, contact angle h, pore shape, mean pore radius D, etc.) were measured by combining the water adsorption and N2 adsorption analysis, the detailed information has been presented in Table 2. As illustrated

D. Feng et al. / International Journal of Heat and Mass Transfer 118 (2018) 900–910

filling resistance and decrease the effective height of channels. Although the measured results with Atomic Force Microscope (AFM) demonstrated the mean surface roughness only has a value of 0.229 nm on silica surface [40,41], such small values can still have noticeable impact on the capillary filling process, especially for the pores less than 10 nm.

Proposed model Ignore water-wall interaction

m(t)/cross-section area

mg/mm2)

16

12

Experiment data

8 4. Results and discussion 4.1. The effective viscosity in nanopores

4

0

0

4

8 t(hours)

12

16

Fig. 5. Comparison of analytic solutions and experimental results as well as the ideal condition (without pore wall interaction) for nanocapillary. Capillary rise of water in hydrophilic silica pores with 3.4 nm mean radii at T = 298 K (Gruener, et al. 2009) [36], the calculated effective viscosity is 2.72 mPas.

in Fig. 5, the simulation results for water filling in hydrophilic nanocapillary are in good agreement with the experimental results in published literatures, demonstrating the presented model is reliable for analyzing the imbibition characteristics of nanopores with a circular cross section. Fig. 5 also shows that the realistic imbibition weight (height) in hydrophilic nanocapillary is apparently lower than the classical imbibition theory, this phenomenon further emphasizes that the influence of interaction between water and walls is nonnegligible during the imbibition process.

3.2. Validation of water filling for nanochannel The experimental data used to validate the model is obtained within Ref. [37]. The material is the silicon-based rectangular pore, the pore height is 5, 11, 23 and 47 nm while the width is 20 lm, it met the conditions that w >> h and can be regarded as nanochannels. Similarly, the nanochannels were also cleaned and dried to remove the impurity. Fig. 6 shows that although there are some smaller deviations between the analytical solutions and experimental results, simulation solutions calculated by the proposed model are still roughly consistent with the published experimental data. Meanwhile, a possible reasonable interpretation for those deviations may be the surface roughness, which can enhance the

5 H=5nm(Exp) H=11nm(Exp)

4

H=23nm(Exp)

L(mm)

905

3

H=47nm(Exp) H=5nm(Model)

2

H=11nm(Model)

1

H=23nm(Model) H=47nm(Model)

0

0

10

20

30 t(s)

40

50

4.1.1. Effective viscosity versus pore size Fig. 7 describes the analytic effective viscosity calculated by the presented model within 200 nm based on the Eqs. (12-a) and (12-b), the basic calculating parameters are shown in Table 2. The figure shows that proposed model is able to simulate accurately the majority of experimental results both for nanocapillaries and nanochannels (all the experimental materials are glass or silica) [12,36,37,42–46]. Meanwhile, Fig. 7 also demonstrates the comparison between our proposed model and the previous models, the largest deviations were observed in terms of the Tomas & Wu model [19,20], which suggests underestimating the effect of water-wall interaction. Furthermore, although the empirical model proposed by Li et al. [17] and Wang et al. [18] has a relatively good fitness in nanochannel, they still failed to perfectly match the experimental data in nanocapillary. More importantly, the proposed model takes the water-wall interaction and density layering phenomena into account, it successfully captures the physical mechanisms behind the dramatic change of viscosity in hydrophilic nanopores. In particular, calculated results exhibit a dramatic change in viscosity for nanopores with the pore size smaller than 10 nm, the strong water-wall interactions are able to enhance the effective viscosity up to more than 3 times than that of bulk water. If we assume deviation degree of 5% as the critical influence value, the corresponding size for nanocapillaries and nanochannels are 306 nm and 150 nm, respectively. This difference means that the nanopore shape is another influence factor for the effective viscosity of confined water in hydrophilic nanopores, the influencing mechanism will be discussed below. Although the proposed model matches well with the experimental data, the deviations always exist. Some possible explanations, such as surface roughness, formation of bubbles and dynamic contact angle effect, are worth to be discussed [40,41,45–48]. Owing to the size effect, the surface roughness may further increase the flow resistance, the influence could be estimated by correcting effective height of channels with a small distance DH, the measured results by Ouyang et al. [41] have a value of 0.229 nm, however, this value should not be a constant but strongly depend on the materials and fabrication technology [46]. Another influence factor is the formation and entrapment of nanobubbles, which is related to the dissolved gas or contaminant, this effect happens at the advancing liquid meniscus and will cause an immediate decrease in the filling speed [44]. Besides, although the measured static contact angle is always zero for silica surface, dynamic contact angle effect is sometimes observed during the filling process, the MD simulations by Stroberg et al. [48] suggest larger pores possess higher dynamic contact angle. As the contact angle increases, the slip effect will emerge, which may result in larger deviations of effective viscosity [49].

60

Fig. 6. Comparison of analytic solutions and experimental results in nanochannel. Capillary filling experiments are conducted at T = 295 K (Haneveld, et al. [37]), the calculated effective viscosity is 2.37 mPas, 1.66 mPas, 1.33 mPas, 1.16 mPas, respectively.

4.1.2. Effective viscosity versus the properties of pore wall Different porous nanomaterials have different surface properties, therefore, the effective viscosity will vary with the materials because of the distinction of solid-liquid interaction induced by pore wall. Based on the proposed model, the hydrophilic materials

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3.5

(a) nanocapillary

3.0

Ref [42]

3.0

2.5

Proposed model

2.5

Tomas & Wu model 2.0

Li model Wang model

1.5

Ref [12] Ref [37] Ref [43] Ref [44] Ref [45] Ref [46] Proposed model Tomas & Wu model Li model Wang model

(b) nanochannel

Ref [36]

µ(H) (mpa·s)

µ(D) (mpa·s)

3.5

2.0 1.5 1.0

1.0

0.5

0.5 0

50

100

150

0

200

50

diameter(nm)

100

150

200

channel height(nm)

Fig. 7. Comparison of the experimental results and proposed model as well as the previous models. In Tomas & Wu model, the viscosity in interaction area (li) is calculated with the equation (li/l0=-0.018h+3.25) proposed by Wu et al. by further assuming the contact angle is 0°; the empirical parameters in Li model and Wang model are obtained by fitting all the experimental data without considering the pore shape.

4.1.3. Effective viscosity versus the pore shape Furthermore, we analyze the influence of pore shape on the change of effective viscosity, e1 is assumed to represent the ratio of effective viscosity in nanocapillaries and nanochannels with the same pore scale, which is defined as e1 = l(D)/l(H). Fig. 9 demonstrates the relationship between e1 and pore size for different materials (quartz, mica, clay), it shows that the value of e1 is always over 1, which means the effective viscosity in a circular cross section is always larger than that in slit shape, as a result of larger weighted proportion of interaction regions in nanocapillaries. Compared with the different nanomaterials, the value of e1 is clay > mica > quartz in the descending order, which suggests that the influence of pore shape is positively related to the interactions between water and pore wall. Fig. 9 also demonstrates that as pore size increases, the e1 firstly increases and then gradually decreases, the critical pore size that maximizes the e1 value is in the range of 4 nm to 6 nm. This phenomenon can be explained from the following point of view: (1) when the pore size is smaller than the critical value, all water molecules located within the interaction regions of pore wall; meanwhile, for the same wall-water distance, larger weighted proportion in nanocapillaries would make the lower regression rate of effective viscosity; (2) when the pore size is larger than the critical value, there is a state expected to behave like bulk water without being affected by the walls, the bigger discrepancy between interaction regions and bulk water induces higher regression rate of effective viscosity in nanocapillaries; (3) the clay nanopores have the largest critical size and the value approaches to

(such as quartz, mica, clay) widely used in science and engineering are chosen to conduct a sensitivity analysis for the effect of material nature. The values of Hamaker constant for quartz, mica and clay have been listed in Table 2. As shown in Fig. 8, with the same pore size, the effective viscosity for different nanomaterials is clay > mica > quartz in the descending order both for nanocapillaries and nanochannels, which emphasizes that the higher the surface energy is, the higher effective viscosity will be. In addition, many published works have also validated this conclusion, the effective viscosity in the order of mica > glass > graphite are obtained by combining the MD simulation and high-resolution atomic force microscope (AFM) measurements [31], the research results of Raviv et al. [50] indicated that the effective viscosity may be one or two orders of magnitude greater than those of bulk water. In particular, the analytic solutions by the presented model show that the increase of viscosity is more 10 times larger in clay nanopores with the diameters smaller than 10 nm, which is identical with the results from MD simulation by Botan et al. [9] and Haria et al. [10]. This dramatic change in viscosity can also give a reasonable interpretation for the industrial phenomenon, such as the imbibition characteristic of hydraulic fracturing fluid in shale reservoir, in which the clay is abundant and the pore size ranges from 2 to 300 nm. During the imbibition process, the fluids distribution detected by magnetic resonance imaging showed that it is much harder for water to enter the smaller pores than we usually thought, the increase of effective viscosity plays an important role on this phenomenon [2,3].

100

100

(a) nanocapillary

(b) nanochannel

Quartz

Quartz Mica

Clay

µ(H) (mpa·s)

µ(D) (mpa·s)

Mica

10

1

Clay 10

1 0

50

100

diameter(nm)

150

200

0

50

100

150

200

channel height(nm)

Fig. 8. The relationship between effective viscosity and pore scale for quartz, mica and clay, the relevant parameters are in Table 2.

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and nanochannels, the position of the advancing meniscus (imbibition length) is also linearly proportional to the square root of time t1/2 (solid line in Fig. 10(a), (c)), the filling speed decreases with the decreasing capillary diameter or channel height (solid line in Fig. 10(b), (d)). However, there is deviation compared with the macroscopic expectations based on the bulk properties, especially for the nanopores with the pore size smaller than 10 nm. Furthermore, the discrepancy induced by size effect will decrease with the increasing of pore scale and may be negligible at 200 nm. This conclusion is consistent with experimental results and molecular simulation results in the published work [5,42]. In fact, the microstructure of nanoporous media is very complex, the pores are usually tortuous and noncircular. However, many traditional and recently presented imbibition models are derived based on the Hagen-Poiseuille equation in cylindrical capillaries. Therefore, there is a deviation for quantifying capillary filling into these materials. In this paper, we further analyze the coupled effect of pore shape and water-wall interactions on capillary filling, e2 is assumed to represent the ratio of imbibition amount between real condition and traditional one, which is defined as e2 = L/L(ideal), e2 = 1 represents the real imbibition amount is equal to the traditional value, e2 > 1(<1) indicates the real condition is more (less) than the traditional value. As shown in Fig. 11, the results for nanocapillaries suggest that the value of e2 will gradually increase and finally approaches to the asymptotic line (e2 = 1), which indicates the influence of water-wall interaction on capillary filling decreases with the increase of pore size, the deviation is only 3.6% when the pore size is 200 nm. Nevertheless, capillary filling characteristic for nanochannels is more complex comparing with the traditional cylindrical one. In the smaller nanopores, the value of e2 is smaller than unity because the role of water-wall interactions is superior to the pore shape. As the pore size increases, the role reverses and the value of e2 will

2

Quartz Mica Clay

1.6 1.4

1

(dimensionless)

1.8

1.2 1 0

100

200

300

400

500

pore scale (nm) Fig. 9. The relation curves between e1 value and pore size for quartz, mica and clay, the relevant parameters are in Table 2.

6 nm, which suggests that the interaction distance of pore wall is only for a few nanometers and will be greater for the higher energy solid surface. In fact, many experiments or MD simulations have validated this mechanism, the interface region measured with atomic force microscope by Ortiz-Young et al. [51] is 0.5–1.5 nm, the result of molecular dynamics simulation is 3.3  3.8 nm for the activation energy profiles [52], those values are consistent with the results in our work. 4.2. Capillary filling characteristic in nanopores Based on the Eqs. (13-a) and (13-b), the comparison between proposed model and ideal classical model without considering the pore-wall interactions is shown in Fig. 10. For the nanocapillaries 7

Model (10nm) Ideal (10nm) Model (50nm) Ideal (50nm) Model (200nm) Ideal(200nm)

5 4

2 1 0

2

4

6

t1/2 7

0.8 0.4

(c)

10

(d)

1 00

Model (10nm) Ideal (10nm)

1.6

2 1 4

6

t1/2 (s1/2)

8

10

Model (50nm) Ideal (50nm)

1.2

Model (200nm) Ideal (200nm)

0.8 0.4 0

2

1

t (s)

3

0

Model (200nm) Ideal (200nm)

0

10

4

0

Ideal (50nm)

1.2

(s1/2)

L(Imbibition rate (mm/s)

Imbibition Length (mm)

5

8

Model (50nm)

2

Model (10nm) Ideal (10nm) Model (50nm) Ideal (50nm) Model (200nm) Ideal (200nm)

6

Model (10nm) Ideal (10nm)

3

0

( b)

1.6 Imbibition rate (mm/s)

Imbibition Length (mm)

6

2

( a)

1

10

100

t (s)

Fig. 10. Capillary filling characteristic in nanopores (Fig. 10(a), (c): imbibition length as a function of time for nanocapillary and nanochannel; Fig. 10(b), (d): imbibition rate as a function of time for nanocapillary and nanochannel; solid line is the analytical solution of presented model while dotted line is in ideal condition).

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D. Feng et al. / International Journal of Heat and Mass Transfer 118 (2018) 900–910

for silica or clay nanocapillaries with the pore size smaller than 10 nm, which emphasizes the important role of water-wall interactions on the capillary filling. Furthermore, compared with the traditional imbibition model in cylindrical capillaries, the pore shape and water-wall interactions would make the capillary filling characteristic more complex, which is one of the reason for the deviation between real imbibition value and traditional condition for nanomaterial.

1.4 1.2

2

(dimensionless)

1 0.8 circle

0.6

slit

0.4

Conflict of interest

0.2

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.

0

0

50

100 pore scale (nm)

150

200

Fig. 11. The relationship between e2 value and pore size for nanocapillary and nanochannel.

be higher than unity. When pore size is at 24 nm, the coupled effect of pore shape and water-wall interactions will make the real imbibition amount equal to the traditional one. For the realistic nanoporous media, such as quartz and shale, the overall imbibition characteristic is the sum of single pores with varying pore size and varying pore shape, the research results in our work indicate that if the nanomaterial is richer of micropores, the realistic imbibition amount will be lower than traditional one; if the pore shape is more complex or the proportion of nanochannels is higher, the realistic imbibition value will be higher than traditional one. In this work, the mechanism of dramatic increase in water viscosity has been analyzed and a simple and effective model is proposed to characterize the effective viscosity in hydrophilic nanopores by further considering the water-wall interactions and density layering phenomena, every parameter in the model has clear physical meanings; meanwhile, the coupled effects of pore shape and water-wall interaction on capillary filling are also revealed. However, our study focuses on the capillary filling of strong hydrophilic surface, in which the hydrophobic repulsive force may be ignored compared with the hydrophilic one. However, with the increase of hydrophobicity (contact angle), the hydrophobic force may be comparable or exceed the hydrophilic force, resulting in the change of effective viscosity and imbibition characteristics [20,52,53], this question has still not been completely answered and worth to be further studied.

5. Conclusions In this paper, a model for effective viscosity in hydrophilic nanopores is proposed by further considering the water-wall interactions and density layering phenomena, the analytical solutions are validated with the available experimental data and also successful to characterize the capillary filling at the nanoscale, the model can reveal the more physical mechanisms of water behavior in nanopores and every parameter in the proposed model has clear physical meaning. Meanwhile, the sensitivity analysis indicates that pore size, pore shape and the properties of nanomaterial surface have significant influences on the effective viscosity. At the same pore size, the flow resistance in nanopores with higher surface energy (Hamaker constant) is larger due to the stronger water-wall interactions; meanwhile, the effective viscosity in nanocapillaries is larger than that in nanochannels; more importantly, the effective viscosity of confined water in hydrophilic nanopores can exhibit a dramatic increase, the value could be even three times or one order magnitude greater than that of bulk water

Acknowledgments We acknowledge the National Science and Technology Major Projects of China (2017ZX05039005 and 2016ZX05042, 2017ZX05009-003), and the National Natural Science Foundation Projects of China (51504269 and 51490654) to provide research funding. Appendix A. Eyring viscosity equations As illustrated in Fig. 1, the molecules escape from the initial position and move in each of the coordinate directions in jumps of length a at a frequency m, which could be expressed by the absolute reaction rate theory [6,26,27]:

m¼

jT h

expð
DG0 =RTÞ

ðA-1Þ

Where m is the frequency of molecule motion, j and h is the Boltzmann and Planck constants, T is the temperature, R is the gas constant,DG0 is the energy barrier for restricting molecular motion. When the fluid is under the applied shear stress sxy, the frequency for molecule motion has changed due to the work done by shear force, which could be approximated as asxyV/2d. This force helps the forward motion of the molecule by providing additional energy to overcome the energy barrier, but hinders the motion of the molecule in the reverse direction. The energy barrier under shear stress sxy can be expressed as follows:


DG1 ¼
DG0 

  a sxy V  d 2

ðA-2Þ

Where V is the volume of a mole of liquid,DG1 is the energy barrier under shear stress. Thus, the frequency of molecular motion under shear stress can be rewritten as:

m ¼

  asxy V expð
DG0 =RTÞ exp  h 2dRT

jT

ðA-3Þ

Then the net velocity of flow can be derived as:

Dtx ¼ txA
txB ¼ aðmþ
m
Þ

ðA-4Þ

Furthermore, the velocity profile can be considered to be linear between layer A and layer B under the condition of small distance, this gives




dtx Dtx a  ¼ ðmþ
m
Þ d dy b

ðA-5Þ

Combining the Eqs. (4)–(6), the velocity gradient can be expressed as:




  dtx Dtx a jT asxy V expð
DG0 =RTÞ  2 sinh  ¼  d h 2dRT dy b

ðA-6Þ

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D. Feng et al. / International Journal of Heat and Mass Transfer 118 (2018) 900–910

l¼

 2 d NA hp expðDG0 =RTÞ a V

ðA-7Þ

Appendix B. The average density of first/ second water layer Molecular Dynamics (MD) simulation is the common method to determine the density profile, the results usually show that the thickness of high-density layer in hydrophilic nanopores is about 0.6–0.8 nm (two molecular layers) [7,8,31,32]. In our work, the Molecular Dynamics (MD) simulations in silica nanopores by Wan et al. [54] is used to determine the average density of water confined inside hydrophilic nanopores. As shown in Fig. B1, the density profile is simplified to a calculable shape based on the method proposed by Li et al. [33]. The average density of first water layer can be given as:

qw1 ¼

S1 þ S2 ðqw1
qbulk Þ  hmon =2 þ qbulk  hmon ¼ hmon hmon

ðB-1Þ

Where S1 and S2 are the area in Fig. B1; hmon is the thickness of monolayer water layer, 0.3 nm; qbulk is the density of bulk water, 1 g/cm3. qw1 is the density peak of first water layer, 1.5 g/cm3. Similarly, the average density of second adsorbed layer can be described as

qw2 ¼

S3 þ S4 ðqw2
qbulk Þ  hmon =2 þ qbulk  hmon ¼ hmon hmon

ðB-2Þ

Where S3 and S4 are the area in Fig. B1; qw2 is the density peak of second water layer, 1.1 g/cm3. The calculated results show that the average density of first water layer and second water layer are 1.25 g/cm3 and 1.05 g/ cm3, respectively. The results are reasonable and in the range of physically admissible value (0.95–1.40 g/cm3) [55]. Thus, the average density of confined water can be obtained with the following equation:

qw ¼ qw1  A1 þ qw2  A2 þ qbulk  ð1
A1
A2 Þ

ðB-3Þ

Where qw1 is the average density of first layer, qw2 is the average density of second layer, qbulk is the density of bulk water, A1 and A2 are the area percentage of first layer and second layer, respectively. They can be calculated as follows:

1.6 w1=1.5g/cm

1.4

Density (g/cm3)

1.2

w2=1.1g/cm

S2

1

3

3

3 bulk=1.0g/cm

S4

0.8 0.6

S3

S1

0.4 0.2 0

0

0.3

0.6

0.9 Z (nm)

1.2

1.5

1.8

Fig. B1. Density profile of water near the pore wall (Z = 0 is the pore wall, the thickness of monolayer water is 0.3 nm).

1.14 Nanochannel

1.12

Average density (g/cm3)

Furthermore, the term asxyV/2dRT << 1, the sinh (asxyV/2dRT)  asxyV/2dRT is obtained with the Taylor simplification. Thus, according to the Newton’s law of viscosity, the viscosity of the fluid can be obtained [6,26,27]:

Nanocapillary

1.1 1.08 1.06 1.04 1.02 1

0

20

40

60

80

100

Pore size (nm) Fig. B2. Average density of confined water in nanaochannel and nanocapillary.

Nanochannel:

A1 ¼ A2 ¼

2hmon H

ðB-4Þ

Nanocapillary:

A1 ¼

ðD=2Þ2
ðD=2
hmon Þ

ðB-5Þ

ðD=2Þ2 2

A2 ¼

2

ðD=2
hmon Þ
ðD=2
2hmon Þ 2

ðD=2Þ

2

ðB-6Þ

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