Anomalous Capillary Rise under Nanoconfinement: A View of

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Anomalous Capillary Rise under Nanoconfinement: A View of Molecular Kinetic Theory Dong Feng, Xiangfang Li, Xiangzeng Wang, Jing Li, Tao Zhang, Zheng Sun, Minxia He, Qing Liu, Jiazheng Qin, Song Han, and Jinchuan Hu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01397 • Publication Date (Web): 11 Jun 2018 Downloaded from http://pubs.acs.org on June 15, 2018

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Langmuir

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Anomalous Capillary Rise under Nanoconfinement: A View of Molecular Kinetic

2

Theory

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Dong Feng a, b, Xiangfang Li a, b, Xiangzeng Wang c, Jing Li a, d, Tao Zhang a, Zheng Sun a,

4

Minxia He a, Qing Liu a, Jiazheng Qin a, Song Han a, Jinchuan Hua

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a

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Beijing, Beijing 102249, P.R. China.

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b

8

102249, P.R. China.

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c

Shaanxi Yanchang Petroleum (Group) Corp. Ltd., Xi’an 710075, P.R. China.

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d

Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N1N4, Canada

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* Corresponding author: Dong Feng

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E-mail address: [email protected]

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ABSTRACT

State Key Laboratory of Petroleum Resources and Engineering in China University of Petroleum at

MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), Beijing

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Understanding the capillary filling behaviors in nanopores is crucial for many science and

15

engineering problems. Compared with the classical Bell−Cameron−Lucas−Washburn (BCLW)

16

theory, anomalous coefficient is always observed because of the increasing role of surfaces. Here,

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molecular kinetics approach is adopted to explain the mechanism of anomalous behaviors at the

18

molecular level, a unified model taking account of the confined liquid properties (viscosity and

19

density) and slip boundary condition is proposed to demonstrate the macroscopic consequences, the

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model results are successfully validated against the published literatures. The results show that: (1)

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the effective viscosity induced by the interaction from the pore wall, as a function of wettability and

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the pore dimension (nanoslit height or nanotube diameter), may remarkably slow down the capillary

23

filling process than theoretically predicted. (2) The true slip, where water molecules directly slide on

24

the walls, strongly depends on the wettability and will increase as the contact angle increases. Even

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though in the hydrophilic nanopores, the magnitude may be comparable with the pore dimensions

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and promote the capillary filling compared with the classical BCLW model. (3) Compared with the

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other model, the proposed model can successfully predict the capillary filling both for faster or

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slower capillary filling process; meanwhile, it can capture the underlying physics behind these 1

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behaviors at the molecular level based on the effective viscosity and slippage. (4) The surface effects

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have different influence on the capillary filling in nanoslits and nanotubes, the relative magnitude

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will change with the variation of wettability as well as the pore dimension.

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Key words: effective viscosity; true slip; molecular-kinetic theory; pore geometry; capillary filling

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INTRODUCTION

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Understanding the capillary filling through nanoscale confinements is crucial for solving many

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challenging problems in practical applications, such as energy storage, thermal management, oil

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recovery and the fate of fracturing fluid in shale reservoirs, etc.1-6 It has been widely argued that the

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continuum description and classical hydrodynamics remain valid for simple liquids when the pore

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dimension (nanoslit height or nanotube diameter) is down to nanoscale, suggesting that the capillary

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filling behaviors in nanopores with uniform cross section also obey the expected t1/2-dependence.7,8

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However, compared with the micro/macroscale liquid transport, the anomalous coefficient is always

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observed in the experiments or MD simulations, the confined liquid properties (viscosity and density)

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and slip boundary condition are mainly responsible for these anomalous phenomenon.7-10 In fact, a

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wide range of slip length and confined liquid properties with up to several times of magnitude

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discrepancies have been widely reported,11,12 the mechanism and exact controls of how these effects

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occur and influence the anomalous capillary filling behaviors have still remained poorly

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characterized, and the theoretical research for this purpose is urgently desired.

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Many investigations have shown that properties of near-surface water differ drastically from

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those of bulk water, as a result of the varying structure and dynamics induced by an interaction from

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pore walls.11,13 The structured liquid layers may be insignificant in most macroscopic environments,

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however, they are important in nanoscale fluid filling process. Specifically, these solid-like liquid

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layers influence fluid properties, resulting in higher density and viscosity at the wall-fluid boundary.

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The MD simulation results show that the density and viscosity may be several times larger than that

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of bulk water.13-15 Actually, these parameters play an important role in the flow of fluids, and

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therefore they are also expected to affect the capillary rise of confined water strongly.

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However, consideration of the increased viscosity does not entirely explain capillary filling

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behaviors, enhanced uptake of fluid into capillaries has also been observed both from MD

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simulations and experiments,16,17 this discrepancy is attributed to the occurrence of slip boundary 2


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condition. Depending on the wettability of the solid surface, the MD results show that the magnitude

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of slip lengths can reach a few nanometers on the hydrophilic surface, which are comparable with

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the characteristic dimensions of such small confinements and will play key roles in nanoscale fluid

4

dynamics;7, 14, 15 Botan et al. showed that neglecting the slip effect will result in errors for the overall

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flow of hydrophilic nanopores, which may be 27% and 15% with a diameter of 4.5nm and 8nm.15

6

Actually, the simplest way to account for liquid/solid slip is the so-called ‘true slip’ or ‘partial slip’,

7

which is a ratio of the slip velocity to the shear rate at the pore wall. The true slip occurs at a

8

molecular level, which depicts the actual motion of liquid molecules on the bed of solid surface, the

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magnitude strongly depends on the mobility of adsorbent first-layer molecules.18,19

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Given these challenges, there is a need to investigate how these features will result in

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anomalous capillary rise under nanoconfinement. Meanwhile, both of the liquid viscosity and true

12

slip strongly depend on the liquid mobility; thus, understanding these phenomena at the molecular

13

level is very important for the mechanism of liquid transport through nanoscale confinements. In our

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work, the molecular kinetics approach is adopted to characterize the increased viscosity and true slip

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by further considering the high-density layer phenomenon. The molecular kinetics approach is

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established and developed by Frenkel, Eyring, Tolstoi and Blake based on the description of liquid

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flow as a thermally activated rate process, which is well known and give a good explanation for the

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process of liquid transport at the molecular level.20-24 This paper is organized as follows:(1) firstly,

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effective viscosity is adopted with a weighted average of the viscosities in the interface and bulk-like

20

regions, the viscosity in interface region is modeled based on the concept that the flow energy barrier

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is the sum of liquid-liquid interaction and solid-liquid interaction; (2) the true slip is characterized by

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considering the molecular mobility difference between surface liquid and bulk liquid; (3)

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Bell-Cameron-Lucas-Washburn (BCLW) model is revisited by considering the effects of slippage

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and interface viscosity; (4) comparison of models and our findings are reported. Additionally, some

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basic assumptions are made as follows:

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(1) The capillary filling process is an viscous-dominated Stokes flow;

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(2) Other effects, such as dissolved gas, roughness and nanobubbles are ignored;

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(3) The pore geometry is constant and with uniform cross section;

29

(4) The pore dimension in this paper is larger than 1.6 nm because the continuum theory and 3


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classical hydrodynamics remain valid under such condition.7, 19

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THEORETICAL MODELS

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Modeling for effective viscosity

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4

Compared with the bulk water, the properties of nanoconfined water change drastically as a

5

result of the surface effects. In theory, the properties of confined water should be modeled with a

6

continuous model, leading to the complicated form and hard to be managed analytically;25 “Gibbs

7

dividing” is advantageous in mathematical calculation, which suggests the constant properties in the

8

fixed region and abrupt change when the interaction distance is over a critical value, this description

9

performed well in theoretical calculation and molecular simulation, the effective viscosity is

10

described by a weighted-average of the viscosities in the interface and bulk-like regions 9, 19, 26

µeff = µi

 A (d )  Ai (d ) + µb 1 − i  At (d )  At (d ) 

(1)

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Where µeff, µi, µb are the effective viscosity, interface-region viscosity and bulk viscosity,

12

respectively; Ai, At are the areas of interface region and total cross-section. The area of the interface

13

region can be calculated by a critical thickness, which is known as 0.7nm and has been discussed in

14

the published work.9,19 Meanwhile, this magnitude also approaches to the thickness of high-density

15

layer near the nanopore surface.11, 13-15

16

The liquid viscosity strongly depends on the magnitude of interaction energy between the liquid

17

molecules. Taking the viscous flow as a chemical reaction, Eyring and his coworkers proposed a

18

model to give a description for transport mechanism of fluids based on the statistical mechanical

19

theory of absolute reaction rates, in which the molecular motion has to overcome a potential barrier

20

created by its neighbors, the relationship between the viscosity and activation energy can be

21

described by 27-29

µ=

hp vm

exp( ∆Gb / kbT )

(2)

22

Where µ is the viscosity,ΔG is the activation energy for a molecule to escape from the restriction

23

of neighboring molecules , hp, kb, vm, T are the Planck constant, Boltzmann constant, liquid

24

molecular volume and absolute temperature, respectively.

25

The confined-water energy in interface region has two components: the average surface 4


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energy W due to the solid-liquid interactions, including attractive part and repulsive part at short

2

distance; the interaction energy between the liquid molecules, which is the same as that in bulk water.

3

Based on the Lennard-Jones (LJ) and Young–Dupre equation, the average surface energy W for a

4

molecule in the interface region can be described as30

W = γ LV (1 + cos θ ) / ni 5 6

(3)

Where γLV is the surface force; θ is the contact angle; ni is the number of molecules per unit area in interface region, strongly depending on the liquid density in interface region.

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Similar to the assumption made by Blake and Wu,32,38 the activation energy difference was

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assumed to be equal to the energy difference for the same liquid molecules in the interface and

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bulk-like regions. Therefore, the viscosity ratio between the interface region µi and bulk region µb

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can be expressed as

 γ (1 + cos θ )  µi = exp  LV  µb ni kbT  

(4)

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Due to the short-range force, the density oscillation of fluid molecules near the solid surface is a

12

well-known phenomenon (Figure 1), those oscillations decay and converge to the bulk value over a

13

distance about two water-molecule layers.13-15 Meanwhile, the peaks will decline with the increase of

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hydrophobicity, suggesting there is a relationship between local density and contact angle. The

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oscillatory decay of liquid density may be described by an exponentially decaying cos-function of

16

the form 28,31

ρ s (d ) = ρ b − f (θ ) cos [ 2π (d − δ ) / σ ] exp [ −2( d − δ ) / σ ]

(5)

17

Where ρb is the bulk water density, δ is the void space (ﬂuid density was essentially zero),

18

which is always found in MD simulations because of the strong repulsion between particles at such a

19

short distance from the solid surface, the magnitude is approach to the molecule radius, 0.1~0.2 nm

20

in different simulations, 10, 13~15 0.15nm is adopted in our work; σ is the molecule diameter, d is the

21

distance from the pore wall, f(θ) represents the influence of wettability on the local liquid density.

5


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Pore wall ρw1 ρw2

Bulk water

Density profile

Void space

1 2 3

Figure 1. Liquid molecules close to the pore wall, the oscillatory red curves is the liquid density profile. The

4

When 2(d-δ)/σ=1 and 3, the density of first layer and second layer can be obtained. Following

5

the work of Wang and Zhang,32-33 the liquid density in the interface region can be the average value

6

of first layer and second layer. Thus, we have

thickness of obvious high-density layer is about two water-molecule layers

ρi = ρb +  f (θ ) / e + f (θ ) / e3  / 2

(6)

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Where ρi is the liquid density in interface region. To calculate accurately the density profile, a

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linear relationship f(θ)=7.6-0.045θ is fitted with 17 MD simulation cases (shown in the Supporting

9

Information S1).

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Thus, the density profile and average liquid density in interface region can be given as

ρ s (d ) = ρ b − (7.6 − 0.045θ ) cos [ 2π ( d − δ ) / σ ] exp [ −2( d − δ ) / σ ]

(7-a)

ρi = ρb + (1.6 − 0.0095θ )

(7-b)

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By assuming spherical shape with uniform distribution,28-29 the number of molecules per unit

12

area in interface region (ni) can be expressed as 1/di2, di is the effective diameter controlled by a

13

molecule and can be roughly estimated based on the volumetric method (di3=M/(ρiNA)); ρi is the

14

liquid density in interface region, the parameter (nb and db) in the bulk water can be obtained with

15

the same methods. Then, K is defined to represent the ratio of the number of molecules per unit area

16

in interface region and bulk phase, the magnitude can be calculated as

K = [ ( ρb + 1.6 − 0.0095θ ) / ρb ] 17

2/3

Thus, eq 4 can be rewritten as

6


(8)

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Langmuir

 γ (1 + cos θ )  µi = exp  LV  µb  Knb kbT  1

(9)

Modeling for slip length

2

Flow boundary condition (slip/no slip) is another key issue for the nanoscale liquid transport.

3

Usually, the simplest way to account for liquid/solid slip is the so-called ‘true slip’ or ‘partial slip’,

4

which is a ratio of the slip velocity to the shear rate at the solid surface.16 The true slip occurs at a

5

molecular level, which depicts the actual motion of liquid molecules on the bed of solid surface, the

6

magnitude strongly depends on the mobility of the adsorbent first-liquid-layer molecules. By this

7

physically motivated argument, molecular kinetics approach proposed by Tolstoi and Blake is

8

suitable for describing the slippage behavior by considering the difference mobility between surface

9

and bulk liquid molecules.22

10

The molecular mobility, v, is proportional to exp(−W1/kbT), in which W1 is equal to the energy

11

of the hole by vacuuming the molecule from its initial equilibrium position to a new one, this energy

12

is equal to its “surface” energy.22 Blake et al made the assumption that this energy can be treated as

13

AγLV, where A is the effective surface area and can be expressed as πσ2 for a spherical molecule with

14

diameter σ.22,35 For the first-liquid-layer molecules, it is given by the loss of the solid–liquid

15

interfacial area and the gain in solid–vapor and liquid–vapor interfacial areas, which can be

16

expressed as αA(γSV - γSL) + (1- a)AγLV, where γSV and γSL are the solid−vapor and solid−liquid

17

interface tensions, and a is the solid area fraction of the microcavity at the wall. Then, combining the

18

Young’s law (γSV - γSL= γLV cosθ), we have34, 35

νi = exp[α Aγ LV (1 − cos θ ) / kbT ] νb

(10)

19

From the aspect of fluid mechanics, the mobility of molecules is the average velocity when a

20

force of unit magnitude is applied. For the shear rate of dv/dz induced by shear stress τ，the velocity

21

of a molecule would be r(dv/dz), where r is the center-to-center distance, which is equal to the

22

effective diameter controlled by a molecule. Then, the mobility of bulk molecules and surface

23

molecules can be expressed as 35

νb =

1 dv 1 dv ( ) b ,ν i = ( )i dbτ dz d iτ dz 7


(11)

Langmuir

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The (dv/dz)b is the velocity gradient of bulk molecules with nonslip boundary condition, (dv/dz)i

2

is velocity gradient of first layer molecules with slip boundary condition. According to Tolstoi,22 we

3

have

l + di ν i db ( dv / dz )i = = K ( st ) db ν b di ( dv / dz )b 4

(12)

Rearranging eq 11 and inserting eq 12, the magnitude of true slip can be expressed as

{

απσ 2γ LV (1− cos θ )/ kbT  

lst = K −1/2 db exp 

}

−1

(13)

5

Eq 13 describes the true slip based on the liquid mobility at molecular level, the mode shows

6

that the magnitude of true slip strongly depends on the wettability as well as the higher-density layer

7

phenomena. As shown in Figure 2, the model is in agreement with the majority of study cases from

8

the literatures, suggesting the model is reliable to capture the slip boundary condition. Meanwhile,

9

several points are noteworthy, the slip behavior will also be influenced by the shear stress and can be

10

divided into three regimes: (1) No-slip regime, the boundary slip would not occur unless the critical

11

shear stress is reached. (2) Navier slip regime, the shear stress is higher than critical shear stress but

12

not very high, the slip length is a constant. (3) Shear-dependent slip regime; the slip length increases

13

with the shear rate.34 Actually, the capillary rise is usually a spontaneous process, the shear rate is not

14

very high and the slip boundary condition belongs to the first two regions, this has been validated

15

with the molecular simulation.14,36 8

Cu-Ref.(14) Au-Ref.(14) 6 Slip length (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Ref.(37) LVF-Ref.(38) 4

ED-Ref.(38) Fitting curves

2

0 0

16 17 18 19 20 21

20

40 60 Contact angle (θ)

80

100

Figure 2. Comparison of the model results and simulated data for true slip length, the relevant parameters are in Table 1. (1) MD simulations to investigate the slip length as a function of wall-fluid interaction on copper (Cu) surface and aurum (Au) surface.14 (2) Nonequilibrium molecular dynamics simulations about slip length versus the liquid/solid interaction energy.37 (3) Lateral viscous forces (LVF) with atomic force microscopy (AFM) and energy dissipation (ED) are used to measure the slippage.38 8


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1 2 3

Langmuir

Modeling for capillary filling Assuming the laminar flow and roughly constant curvature, the viscously dominated dynamics during the capillary filling with pore dimension larger than 1.6 nm can be described as39,40

Pc = ∆P =

µeff k

(14)

v(t )l (t )

4

Where Pc is the driving force (capillary force), △P is the viscous losses, which can be

5

described by Poiseuille’s equation, l(t) is the capillary filling length, v(t) is the average velocity, k is

6

the permeability which depending on the shape of the cross-section. Meanwhile, the effective

7

viscosity µeff is adopted in nanopores rather than bulk viscosity. lst

γ µeff

H

Pc

W

8 9 10

Figure 3. Illustration of capillary filling in nanoslits (a rectangular cross-section channel with an extremely small

11

When average velocity v(t) can be expressed as dl(t)/dt, the imbibition model can be derived

12

ratio of channel height H to the width W) and nanotubes with uniform cross section

from eq (14)：

l (t ) = 2 Pc k / ueff t1/2 13 14

(15)

According to the Young-Laplace equation, the curvature is also influenced by the shape of the cross-section, capillary force in nanoslits and nanotubes (Figure 3) can be respectively written as41 2γ cos θ 1 1 + ) ≈ LV H W H 4γ cos θ Pc 2 ( D ) = LV D

Pc1 ( H ) = 2γ LV cos θ (

9


(16-a) (16-b)

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1

Where Pc1 (H) is the capillary force of nanoslits, Pc2 (D) is the capillary force of nanotubes, H

2

and W is the height and width of nanoslits (a rectangular cross-section channel with an extremely

3

small ratio of channel height H to the width W), D is the diameter of nanotubes.

4

For rectangular geometry (W>>H) or circle cross section, k can be given as follows39 k1 ( H ) =

( H + 2l st ) 2 ( D + 2l st ) 2 , k2 ( D ) = 12 32

(17)

5

Where k1(H) is permeability of nanoslits, k2(D) is permeability of nanotubes.

6

Taking eqs 16~17 into eq 15, the capillary filling length in nanopores by further considering

7

effective viscosity and slip boundary condition can be expressed as 1/2

1/2

 3µ b   γ H cos θ  0.5 H ( + 2lst )   LV l1 (t ) =   t 3µ b  H µeff ( H ) 3  144   1444  42444 3 4244443

(18-a)

BCLW equation

K1 1/ 2

1/2

 µb   γ D cos θ  0.5 ( D + 8lst )   LV l2 (t ) =   t 4 µb  D µeff ( D )  144  3 1444  42444 424444 3

(18-b)

BCLW equation

K2

8

Where l1(t) and l2(t) are the capillary filling length in nanoslits and nanotubes.

9

Eq 18a and eq 18b demonstrate that the filling kinetics also obey the expected t1/2-dependence

10

at the nanoscale, which have been validated by many experiments and simulations.7,8 Furthermore,

11

compared with the classical BCLW model, anomalous coefficient K1 (nanoslits) and K2 (nanotubes)

12

can be used to explain the significant deviation between experimental results and classical BCLW

13

expectations. It can be inferred from the equation that the anomalous coefficient is not only related

14

to the pore dimension, but also determined by the confined water properties (µeff(H) and µeff(D)) and

15

ﬂow boundary conditions (lst). The equations demonstrate that the anomalous coefficient will

16

decrease and eventually approach to unity as the pore dimension increases. In addition, the effective

17

viscosity and slippage, as a results of varying structure and dynamics near the nanopore walls, can

18

capture the underlying physics mechanism that how the surface effects influence the capillary filling

19

under nanoconfinement. It should also be noted that the anomalous coefficient is practical only when

20

the spontaneous imbibition occurs, implying that the pore surface is hydrophilic and contact angle is

21

smaller than 90°.

10


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RESULTS AND DISCUSSION

2

Model validation

3 4 5

In order to check the model reliability, the model results are compared with the experimental results or MD simulation data in published literatures. Modeling parameters are listed in Table 1. Table 1. Basic modeling parameters in the calculation Parameter

Symbol

Unit

Value

Temperature

T

K

298

Bulk water density

ρb

kg/m3

1000

Water molar mass

M

kg/mol

0.018

Water bulk viscosity

µ0

Pa·s

0.001

Surface tension

γ LV

N/m

0.0727

Water diameter

σ

10-9m

0.28

Contact angle

θ

°

0~90

Molar volume of water

Vm

m3/mol

18 × 10−6

Avogadro constant

NA

mol-1

6.02×1023

Planck constant

hp

J·S

6.63×10-34

Solid area fraction of the microcavity (fitting constant)

a

Dimensionless

0.58

6 7

Validation of interface region viscosity

8

Figure 4 demonstrates the comparison between proposed model and MD simulations as well as

9

the other model. The results show that our model is reliable to capture the variation trend of

10

interface-region viscosity as a function of contact angle. In strongly hydrophilic nanopores (

17

Wu model > present model ≈ LWR-MKT model > LWR-SF model in the descending order (Figure

18

7a). As the solid surface approaches neutral wetting (θ=78.3°), the order of l2(t) becomes present

19

model >Wu model>BCLW model > LWR-MKT model≈ LWR-SF model. Among these models,

20

LWR-MKT curves are usually obtained with a fitting parameter λ. In Figure 7a, λ is 0.2nm and

21

beyond the values in published literatures (0.3~1nm); in Figure 7b, the curve is obtained withλ

22

=0.3nm (minimum in published literatures). Actually, the form of LWR-MKT model demonstrates

23

that its results cannot exceed that of the BCLW model (β=0 is the BCLW model); thus, this model

24

cannot fit well with this published data as well as describe the faster imbibition compared with the 16


Page 17 of 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Langmuir

classical theory.

2

The phenomena and discrepancies in Figure 7 can be explained from the following points of

3

view: (1) In strong hydrophilic nanopores, the strong solid-liquid interaction will induce remarkably

4

increased viscosity and insignificant slippage, resulting in slower imbibition rate; as the

5

hydrophobicity enhances, the effective viscosity will decrease while the slippage increases, the faster

6

capillary filling occurs when the effect of slippage exceed that of effective viscosity. (2) Compared

7

with the BCLW model, the LWR-self-layering model always shows slower capillary filling, the

8

discrepancies will enlarge with the increase of hydrophobicity because this model only considers the

9

increased friction and ignores the true slip, which is significant for a larger contact angle. Besides,

10

LWR-self-layering model is established based on the concept of thin wetting film ahead of the main

11

meniscus, however, some experimental studies and MD simulations have validated that the precursor

12

film phenomenon only occurred on the strongly hydrophilic surface,56,57 there is a doubt that whether

13

this model can be used to predict the capillary filling for a large contact angle. (3) Similarly,

14

LWR-MKT model also ignores the slippage and always demonstrates the slower imbibition, larger

15

discrepancy is found when the slip length is in the superior place for a larger contact angle. Besides,

16

although the model may perform well in strong hydrophilic nanopores, the parameter λ is very

17

difficult to be determined in theory and the reasonability of fitted value is worthy to be discussed. (4)

18

The model proposed by Wu is also based on the concept of true slip and effective viscosity,

19

comparing the Figure 7a and Figure 7b, the Wu model can reflect the influence of increased friction

20

and slippage, but it is hard to match well with the experimental or simulated results. This is mainly

21

because Wu’s work focus on the liquid transport in nanopores with low energy surface, such as CNT,

22

BNNT, polycarbonate, while the capillary rise mainly occurs in nanopores with higher-energy solid

23

surface (silica, glass, mica, clay and mental etc.),11, 19, 27 the detailed mechanism has been explained

24

before.

17


Langmuir 4.5

30

(a)

Simulated data

3.5

Wu model

3.0

l2(t) (nm)

20

LW model

LWR-MKT model LWR-SF model

15

(b)

Simulated data Our model LW model Wu model LWR-MKT model LWR-SF model

4.0

Our model

25

l2(t) (nm)

2.5 2.0 1.5

10

1.0 5

0.5 0.0

0 0

1 2 3 4 5 6

1

2

3

4 Time (ns)

5

6

0.0

7

0.1

0.2

0.3 0.4 Time (ns)

0.5

0.6

0.7

Figure 7. Comparison of the simulated results and proposed model as well as the previous model (a) equilibrium contact angle is 25°, pore diameter is 5nm,54 LWR-MKT model is fitting with λ≈0.2nm, beyond the published possible values (0.3~1nm); (b) contact angle is 78.3°, pore diameter is 2nm,15 LWR-MKT model is fitting with λ≈0.3nm (the minimum value in published literatures).

Anomalous coefficient K1 and K2 analysis

7

Figure 8 demonstrates the anomalous coefficient K1 and K2 with varying pore dimension as well

8

as wettability. As shown in Figure 8a and Figure 8b, the anomalous coefficient will decrease as the

9

pore dimensions increase, the variation is more sensitive in small nanopores. Figure 8c and Figure

10

8d demonstrate that the capillary filling speed may be faster or slower than classical theoretical

11

prediction, the results strongly depend on the wettability of solid surface. In the strong hydrophilic

12

nanopores, the increased viscosity plays a dominant role on the capillary filling behaviors. With an

13

increasing hydrophobicity, the effective flow resistance decreases while the effect of true slip

14

becomes significant. Besides, the enhancement of water flow flux in nanopores with contact angle

15

larger than 70° has been modelled by Wu et al,19 Vo et al. simulated the higher capillary filling rate

16

in 2nm nanopores when the contact angle is larger than 62.5°.14 These reported works provide

17

powerful evidence for our conclusion and emphasize the important of slippage on the capillary

18

filling for a large contact angle. 2

(a)

1.6

θ=0° °

θ=20° °

θ=60° °

θ=80° °

2

θ=40° °

1.2

θ=0° °

θ=20° °

θ=60° °

θ=80° °

θ=40° °

1.2

K2 0.8

0.8

0.4

0.4

0

0 0

19

(b)

1.6

K1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 27

10

20

30

40

50

0

10

20

30

Nanotube diameter D (nm)

Nanoslit height H (nm)

18


40

50

Page 19 of 27 2

2

(c)

1.6

H=5nm

H=10nm

H=50nm

H=100nm

H=30nm

D=5nm

D=10nm

D=50nm

D=100nm

D=30nm

1.2

1.2

K2 0.8

0.8

0.4

0.4 0

0 0

1

(d)

1.6

K1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

10

20

30

40

50

60

70

80

0

90

10

20

30

40

50

60

70

80

90

Contact angle (θ)

Contact angle (θ)

2 3 4 5

Figure 8. Variations of anomalous coefficient K1 (nanoslits) and K2 (nanotubes) versus pore dimension and contact

6

The diversity between nanoslits and nanotubes

angle. (a) anomalous coefficient K1 (nanoslits) versus pore dimension; (2) anomalous coefficient K2 (nanotubes) versus pore dimension; (3) anomalous coefficient K1 (nanoslits) versus contact angle; (4) anomalous coefficient K2 (nanotubes) versus contact angle. The circle represents that this point is out of the range of our work.

7

Traditionally, many capillary-driven models for porous media are derived based on the

8

Hagen-Poiseuille flow in cylindrical capillaries with uniform cross-section.72,73 However, the pores

9

with slit-shape cross section are also common from the SEM or AFM images, understanding the

10

deviation between nanoslits and nanotubes is helpful for the imbibition evaluation in theory,

11

especially for further considering the surface effects at such small dimension. Figure 9 demonstrates

12

how the single factor (effective viscosity and slippage) and their coupled effect influence the ratio of

13

K1 and K2. Figure 9(a~b) shows the effect of effective viscosity: at such condition, the magnitude of

14

K1 and K2 is smaller than unity, the figures show that K1 is larger than K2 and the magnitude of K1/K2

15

will decrease as the contact angle increases. Those information indicates that the increased viscosity

16

has more remarkable influence on the capillary filling of nanotubes. the details can be explained as

17

follows: (1) for the same sized pore, the portion of interface region in nanotubes is larger than

18

nanoslits, resulting in higher flow resistance and slower capillary filling behaviors; more specifically,

19

the discrepancy of proportion will become the greatest when the pore approaches 4nm (Figure 9a);

20

(2) the effective viscosity in interface region will decrease as the contact angle increases, which will

21

lower the value of K1/K2.

22

On the contrary, the effect of true slip will accelerate the capillary filling, resulting that the

23

anomalous coefficient will be larger than unity, Figure 9(c~d) shows that: (1) the true slip has more

24

significant effect on the capillary filling of nanotubes; (2) the larger diversity between nanoslits and

25

nanotubes occurs when the pore is small or the contact angle is large. Finally, Figure 9(e~f) shows 19


Langmuir

1

the coupled effect of increased viscosity and true slip. Although any of the single effect has more

2

significant influence on the nanotubes, it is hard to estimate which one is more affected by the

3

surface effects because the increased viscosity slows down the capillary filling while the true slip

4

plays a positive role. However, combined Figure 8(c~d) and Figure 9(e~f), one features can be

5

concluded that the relative strength of surface effects on the capillary filling in nanoslits and

6

nanotubes will change with the variation of wettability as well as the pore dimension. 1.25 (a)

1.2

θ=0° °

θ=20° °

θ=40° °

θ=60° °

1.3

H=D=5nm H=D=30nm H=D=100nm

(b)

H=D=10nm H=D=50nm

1.2

θ=80° °

K1/K2

K1/K2

1.15 1.1

1.05 1

1.1

1

0.95 0.9

0.9 0

10

20

30

40

0

50

10

20

7 1.02

1.04

(c)

0.96 θ=20° °

0.92

θ=40° °

θ=60° °

0.9

θ=80° °

50

60

70

80

90

H=D=10nm H=D=50nm

K1/K2

1 0.98

K1/K2

0.98 θ=0° °

40

H=D=5nm H=D=30nm H=D=100nm

(d)

1.02

1

0.94

30

Contact angle (θ)

Pore dimension (nm)

0.96 0.94 0.92 0.9

0.88

0.88

0.86 0

10

20

30

40

0

50

10

20

8 1.25 (e)

1.2 1.15

θ=0° °

θ=20° °

θ=40° °

θ=60° °

30

40

50

60

70

80

90

Contact angle (θ)

Pore dimension (nm) 1.3

(f)

1.2

θ=80° °

1.1

K1/K2

K1/K2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 27

1.05 1

H=D=5nm

H=D=10nm

H=D=30nm

H=D=50nm

H=D=100nm

1.1

1

0.95 0.9

0.9 0

9 10 11 12 13 14

10

20

30

40

0

50

10

20

30

40

50

60

70

80

90

Contact angle (θ)

Pore dimension (nm)

Figure 9. Variations of K1/K2 versus pore dimension and contact angle by considering single factor or coupled effects. (a~b) The increased vicosity: K1/K2 versus pore dimension; K1/K2 versus contact angle; (c~d) The true slip: K1/K2 versus pore dimension; K1/K2 versus contact angle; (e-f) The coupled effect: K1/K2 versus pore dimension; K1/K2 versus contact angle. The circle represents that this point is out of the range of our work.

Our study demonstrates that the filling kinetics in ideal nanoslits and nanotubes also obey the 20


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Langmuir

1

expected t1/2-dependence with a anomalous coefficient, which is influenced by the pore properties

2

(dimension and wettability). For the natural nanoporous media, such as quartz and shale, the overall

3

imbibition characteristic is the sum of single pore with varying pore dimensions and geometries,

4

nanoslits and nanotubes with uniform cross-section are the easy geometries for us to conduct

5

researches in theory.58-60 Actually, the geometry of nanopores is not that of a single circular or slit

6

shape;61-62 in other scenarios, Berthier found that capillary filling into pores with arbitrary (but

7

uniform) cross section also obeys the BCLW law with a modified prefactor;63 besides, some

8

geometries may alter the observed scaling considerably, the shape variations along the flow direction

9

will modify the ‘diffusive’ t1/2-dependence. At short time, the scaling exponent of capillary-driven

10

invasion is same as described of BCLW law; but for a long time, different responses occur, the

11

scaling exponent is the function of the shape of gap along the flow direction.64-66 Under such

12

circumstances, the coupled effect of shape variations and surface effects would result in more

13

complicated imbibition characteristic for nanopores and nanoporous media.

14

Limitation and prospect

15

In this work, molecular kinetics approach is adopted to characterize the increased viscosity

16

and true slip by further considering the high-density layer phenomenon, which helps us capture the

17

underlying physics behind the anomalous capillary rise in nanopores at the molecular level.

18

Furthermore, our model can characterize capillary filling behaviors of many cases with varying

19

wettability and pore dimensions, which are significant in some theoretical studies and industrial

20

applications. However, capillary filling is a complicated process, some other factors may also have

21

remarkable impacts in various ways: (1) inertia force plays an important role at the initial regime of

22

capillary filling, where the capillary force is balanced only by the inertial drag and characterized by a

23

constant velocity and a plug flow profile;67 (2) the wall-roughness will reduce the effective hydraulic

24

radius, a similar effect to the increased viscosity; besides, the roughness may also change the

25

wettability of solid surface, which indirectly affects the capillary filling process;68 (3) dynamic

26

contact angle, indicating the wettability is always varying during the capillary filling process, which

27

has great impact on the driving force, effective viscosity and slippage.69 Our model does not

28

consider those effects, which may be the major reasons for the deviations in Figure. 4~6. In addition

29

to those, the properties of water (ionic type and concentration) are also the important factors for the 21


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

capillary filling behaviors.7,42 Therefore, how to overcome the shortcoming and make a more

2

accurate prediction on the anomalous capillary filling are worth to be further studied.

3

CONCLUSIONS

4

In our work, molecular kinetics approach is adopted to explain the mechanism of anomalous

5

behaviors at the molecular level, and a unified model is proposed to demonstrate the macroscopic

6

consequences. The present model takes account of the confined liquid properties (viscosity and

7

density) and slip boundary condition, the results are successfully validated against the published

8

literatures. Additionally, the comparison and analysis of previous models and proposed model are

9

conducted, the influence of wettability and pore dimension, the discrepancy between nanoslits and

10

nanotubes are also discussed in detail. The following conclusions can be drawn:

11

(1) The effective viscosity induced by the interaction from walls, as a function of wettability as

12

well as the pore dimension, may be several-folds larger than that of bulk water and will decrease

13

with an increasing of contact angle and pore dimension.

14

(2) The true slip, where water molecules directly slide on walls, strongly depends on the

15

wettability and will increase as the contact angle increases. Even though in the hydrophilic

16

nanopores, the magnitude may be comparable with the pore dimensions and promote the capillary

17

filling compared with the classical BCLW model.

18

(3) Considering the coupled effects of effective viscosity and true slip, the model can

19

successfully predict the capillary filling both for faster or slower condition. In the strong hydrophilic

20

nanopores, the increased flow resistance plays a dominant role, with an increasing hydrophobicity,

21

the effective flow resistance decreases while the effect of true slip becomes significant.

22

(4) Each of the effective viscosity and true slip has more remarkable influence on the nanotubes

23

rather than nanoslits; however, the opposite role will make the coupled effect more complicated. The

24

relative strength of surface effects on the capillary filling in nanoslits and nanotubes will change

25

with the variation of wettability as well as the pore dimension.

26

ASSOCIATED CONTENT

27

Supporting Information

28

Derivation of the relationship between the density profile and wettability, the detailed analysis

29

about anomalous coefficient, the possible value of jump length λ and the number of adsorption sites 22


Page 22 of 27

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Langmuir

1

per surface area n1 in other published work. This material is available free of charge via the Internet

2

at http://pubs.acs.org.

3

AUTHOR INFORMATION

4

Corresponding Author

5

*E-mail: [email protected]

6

Notes

7

The authors declare no competing financial interest.

8

ACKNOWLEDGMENTS

9

We acknowledge the National Science and Technology Major Projects of China (2017ZX05039

10

and 2016ZX05042), and the National Natural Science Foundation Projects of China (51504269 and

11

51490654) to provide research funding.

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

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Table of Contents Graphic 2 D=5nm

lst

D=10nm

anomalous coefficient K2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

D=30nm

1.5

µb

µeff

γ

O θ

1

1/2

 γ D cos θ  0.5 l2 (t ) = K 2  LV  t 4 µb 144  3 42444

0.5

BCLW equation

0 0

9 10 11

10

20

30

40

50

60

70

80

90

Contact angle (θ)

Capillary filling process in nanopores: (A) schematic diagram (B) anomalous coefficient

Brief summary:

12

The molecular kinetics approach is adopted to explain the surface effects at the molecular level;

13

then, a unified model based on the effective viscosity and slippage is proposed to characterize the

14

anomalous imbibition behaviors of nanoconfined water. The influence of wettability and pore

15

dimension, the discrepancy between nanotubes and nanoslits are discussed in detail.

16

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Anomalous Capillary Rise under Nanoconfinement: A View of

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