Capillary Rise: Validity of the Dynamic Contact Angle Models

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Capillary Rise: Validity of the Dynamic Contact Angle Models Pingkeng Wu, Alex D. Nikolov, and Darsh T. Wasan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01762 • Publication Date (Web): 19 Jul 2017 Downloaded from http://pubs.acs.org on July 20, 2017

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Capillary Rise: Validity of the Dynamic Contact Angle Models

Pingkeng Wu, Alex D. Nikolov, Darsh T. Wasan

Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL, 60616

Corresponding author. Email: [email protected]

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Abstract: The classical Lucas-Washburn-Rideal (LWR) equation, using the equilibrium contact angle, predicts a faster capillary rise process than experiments in many cases. The major contributor to the faster prediction is believed to be the velocity dependent dynamic contact angle. In this work, we investigated the dynamic contact angle models for their ability to correct the dynamic contact angle effect in the capillary rise process. We conducted capillary rise experiments of various wetting liquids in borosilicate glass capillaries and compared the model predictions with our experimental data. The results show that the LWR equations modified by the molecular kinetic theory and hydrodynamic model provide good predictions on the capillary rise of all the testing liquids with fitting parameters, while the one modified by Joos’ empirical equation works for specific liquids, such as silicone oils. The LWR equation modified by molecular selflayering

model

predicts

well

the

capillary

rise

of

carbon

tetrachloride,

octamethylcyclotetrasiloxane and n-alkanes with the molecular diameter or measured solvation force data. The molecular self-layering model modified LWR equation also has good predictions on the capillary rise of silicone oils covering a wide range of bulk viscosities with the same key parameter W(0), which results from the molecular selflayering. The advantage of the molecular self-layering model over the other models reveals the importance of the layered molecularly thin wetting film ahead of the main meniscus in the energy dissipation associated with dynamic contact angle. The analysis of the capillary rise of silicone oils with a wide range of bulk viscosities provides new insights into the capillary dynamics of polymer melts.

Keywords: Capillary rise; Lucas-Washburn-Rideal equation; Dynamic contact angle;

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Molecular self-layering; Solvation force

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1 Introduction 1.1 Background The fluid flow through porous media driven by the capillarity is a hot research topic and has many important applications in various fields, such as geoscience1, biotechnology2 , lab-on-a-chip3, and enhanced oil recovery4, 5. To simplify the model system, the porous medium is often approximated as capillaries with rectangular6, 7 or cylindrical cross sections, with the latter being the most common choice. The classical model describing the fluid flow dynamics in cylindrical capillaries driven by capillarity is the Lucas-Washburn-Rideal (LWR) equation8, 9, 10: ρπr 2

d dh dh (h ) = 2 πr γ cos θ e − πr 2 ρgh − 8 πµ b h dt dt dt

(1)

where h is the fluid penetration height, ρ is the liquid density, r is the capillary radius, γ is the liquid surface tension, θe is the equilibrium contact angle in the liquid phase,

µb is the

liquid bulk viscosity, and g is the gravitational acceleration constant. The LWR equation has numerous important applications in areas such as inkjet printing11, 12, food processing13, and petroleum engineering14. It can describe the flow dynamics of a liquid penetrating into capillaries and can be used to determine the physical-chemical properties of porous media and liquids, such as the contact angle15, the pore size16, and liquid surface tension17. As important and useful as it is, experimental data show a slower capillary rise process than that predicted by the LWR equation18, 19, 20, 21, 22, 23. Previously, this deviation was attributed to the uncertainties of the capillary radius and possible formation of the gas bubbles in the penetrating liquid22. Recently, analyses have been performed on the

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entrance effect23,

24, 25

, effect of the capillary element aspect ratio26 (capillary length

divided by capillary radius), effect of the displaced gas phase25, and dynamic contact angle18, 19. Among all the possible factors that contribute to the deviation, the dynamic contact angle is believed to be dominant. Therefore, a correction must be made in the LWR equation by replacing the equilibrium contact angle θe with the velocity dependent dynamic contact angle θ d . For an advancing meniscus, there are two contact angles: the macroscopic contact angle θ ma formed by the extrapolated free interface with the solid surface, which can be measured from an optical photograph of the shape of the interface, and the microscopic contact angle θ mi , determined by the intermolecular forces acting at the three-phase contact line on molecular scale. Though the two contact angles are correlated27, here we only discuss the macroscopic contact angle as the capillary force in the LWR equation is directly related to the macroscopic contact angle. The effect of the dynamic contact angle θ d during the capillary rise process can be corrected by replacing

θe with θ d in LWR equation: ρπr 2

d dh dh ( h ) = 2 πr γ cos θ d − πr 2 ρgh − 8 πµ b h dt dt dt

(2)

For small capillaries and viscous liquids, the inertia term in the LWR equation can usually be neglected19, 28, 29. An analysis of the significance of the inertia term in the LWR equation was proposed by Andrukh24, who argued that the inertia term ρπr 2

the LWR equation can be ignored when r < rc , where rc =

β2 γρ cos θ

d dh ( h ) in dt dt

is the critical

capillary radius and β is the frictional coefficient quantifying the energy dissipation at the three-phase contact line. Under these conditions, the dynamic contact angle modified 5


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LWR equation can be simplified as 2 πr γ cos θ d − πr 2 ρgh − 8 πµ b h

dh =0 dt

(3)

with the initial conditions that at t = 0 , h = 0 . Though there are theoretical and empirical studies on the dynamic contact angle30, 31

, here we only discuss the LWR equations modified by the hydrodynamic model27, 32, 33,

molecular kinetic theory34,

35, 36

, Joos’ empirical equation37 , and our newly proposed

molecular self-layering model31.

1.2 The LWR equation modified by the hydrodynamic model (LWRHydrodynamic) The commonly used hydrodynamic model was developed by Cox27, Voinov32, and Dussan33. In the hydrodynamic model, liquid is allowed to slip on the solid surface at a molecular scale to remove the conflict between the moving three-phase contact line (TPCL) and the conventional no-slip boundary condition. The dynamic contact angle results from the viscous bending of the liquid-gas interface on the mesoscale. The dependence of the dynamic contact angle on the capillary number is shown below:

G(θd ) = G(θmi ) + ln( where G (θ ) = ∫

θ

0

L )Ca Lm

(4)

x − sin x cos x dx , θmi is the microscopic contact angle between the solid 2sin x

and the local tangent to the interface; L is the appropriate macroscopic length scale and usually takes the value of the size of the meniscus; Lm is microscopic length scale, which is the cutoff length below which the continuum theory breaks down; Ca = µb v / γ is the capillary number, with v being the contact line velocity. Cox27 analyzed Hoffman’s38 6


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data and suggest that θmi could be reasonably assumed as θe . While θmi and Lm are usually set as constants when applying equation (4) to describe the dynamic contact angle behavior29,

39, 40

, Ramé and Garoff41 showed that θmi and Lm vary with contact line

velocity. The same author also measured the viscous bending of the meniscus close to capillary wall42. However, the velocity dependent θmi and Lm and the viscous bending shown by Ramé and Garoff41, 42 do not significantly affect the capillary rise dynamics. Therefore, we took the assumption that θmi equals θe and Lm is independent of contact line velocity. When θ < 3π / 4 , G (θ ) = ∫

θ

0

x − sin x cos x dx ≈ θ 3 / 9 , and Equation (4) 2sin x

reduces to

θd 3 −θe3 = 9χCa where χ = ln(

(5)

L ). Lm

Brochard-Wyart and De Gennes43 derived a similar equation

θd (θd2 −θe2 ) = 6χCa

using the lubrication approximation for small equilibrium contact angles. Since the driving force for the contact line movement in the model is the pressure gradient in the flow direction, the hydrodynamic model is able to describe the flow dominated by both the capillary regime and the gravity regime. The model reveals that

θ d 3 is proportional to the capillary number, which has been confirmed by experiments38, 40

. However, the key parameter χ is very difficult to estimate or measure, and thus is

usually used as a fitting parameter. Besides, the hydrodynamic model only considers the viscous energy dissipation in the wedge region formed by the solid surface and interface and neglects the liquid-solid interaction at the molecular level, which is significant in 7


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capillary dynamics31, 34, 44. The LWR-Hydrodynamic is the combination of Equation (2) and Equation (5): ρπr 2


θ d 3 − θ e3 = 9 χ Ca

(6)

In the case where r < rc , the inertia term is negligible and the LWR-Hydrodynamic model can be reduced to 2 πr γ cos θ d − πr 2 ρgh − 8 πµ b h

dh =0 dt

θ d 3 − θ e3 = 9 χ Ca

(7)

The combined equations can only be solved numerically. The LWR-Hydrodynamic predicts the capillary rise of diethylene glycol in PMMA capillaries and ethylene glycol in glass capillaries40 quite well with the fitting parameter χ.

1.3 The LWR equation modified by the molecular kinetic theory (LWRMKT) Blake et al.34,

35, 36

proposed the molecular kinetic theory based on Eyring’s

activated-rate theory considering the adsorption and desorption dynamics of liquid molecules on a solid surface near the TPCL. The driving force is the unbalanced surface tension force γ (cos θ e − cos θ d ) , and it is balanced by the energy dissipation during the adsorption-desorption process of the liquid molecules on the solid surface. The linearized equation for the velocity dependent dynamic contact angle is

γ (cos θ e − cos θ d ) = β v where β =

Vm

λ3

exp(

λ 2Wa k BT

(8)

) µb is the frictional coefficient between the liquid and solid

8


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surfaces at the TPCL. Vm is the molecular volume for simple liquids, λ is the distance between the adsorption sites on a solid surface, Wa =(1 + cos θe )γ is the work of adhesion, k B is the Boltzmann constant, and T is the absolute temperature. The molecular kinetic theory can be applied to a wide range of equilibrium contact angles and is in good agreement with various systems36. While the molecular kinetic theory shows the significance of the solid surface properties in dynamic wetting, it overlooks the effect of the shape of the liquid molecules, which is also important, as shown in this work and in the literature45. Additionally, the key parameter λ is very difficult to measure. When the dynamic contact angle is given by Equation (8), we have the LWRMKT: ρπr 2


γ (cos θ e − cos θ d ) = β v

(9)

In the case where r < rc , the inertia term is negligible and Equation (9) can be reduced to 2 πr γ cos θ d − πr 2 ρgh − 8 πµ b h

dh =0 dt

γ (cos θ e − cos θ d ) = β v

(10)

Equation (10) can be solved analytically: t d = −(

with β =

Vm

λ3

exp(

8 µb 2 β 16 µbγ cos θe ρgr + ) log(1 − h) − h 2 2 3 ρgr ρ g r 2γ cos θe ρgr 2

λ 2γ (1 + cos θe ) k BT

(11)

) µb .

The prediction of the LWR-MKT is shown to have a good agreement with the capillary rise data of water, ethanol, and silicone oils in glass capillaries18, 19 the with

9


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fitting parameter β .

1.4 The LWR equation modified by Joos’ empirical equation (LWRJoos) Joos37 measured the dynamic advancing contact angle when a continuous solid strip is drawn into a large liquid pool and obtained the empirical equation

cosθd = cosθe − 2(1+ cosθe )Ca1/ 2

.

(12)

By combining Equation (2) with Equation (12), we generate LWR-Joos: ρπr 2


cosθd = cosθe − 2(1+ cosθe )Ca1/ 2 (13)

In the case where r < rc , the inertia term is negligible and Equation (13) can be reduced to 2 πr γ cos θ d − πr 2 ρgh − 8 πµ b h

dh =0 dt

cosθd = cosθe − 2(1+ cosθe )Ca1/ 2

(14)

The capillary rise data for silicone oils in different capillaries can be welldescribed by LWR-Joos28, 46.

10


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1.5 The LWR equation modified by the molecular self-layering model (LWR-Self-layering)

FIG. 1 Dynamic contact angle considering the LMTWF ahead of the main meniscus.

θ d is the apparent dynamic contact angle between the main meniscus and the LMTWF, which has a thickness of several molecular layers.

Recently, we developed the molecular self-layering model31 to explain the dynamic contact angle based on the existence of the layered molecularly thin wetting film (LMTWF) ahead of the main meniscus due to the self-layering of the liquid molecules close to the solid surface under the confinement of the liquid-solid and liquid-air interfaces (Fig. 1). The energy dissipation associated with the dynamic contact angle is assumed to be the viscous dissipation in the LMTWF.

γ (cosθe − cosθd ) = µ f v where µ f = exp(

(15)

2γ ) µb is the viscosity of the LMTWF; n is the number of molecules nk BT 11


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per unit area in the LMTWF. For spherical molecules, n = 1 / σ 2 , where σ is the effective diameter of the liquid molecules in the LMTWF. For non-spherical molecules, n can be estimated from the measured solvation force with the relation W (0) = nk BT , where W (0) is the entropic solvation energy extrapolated to zero distance from the solid surface. The LWR-Self-layering has the same mathematical form as the LWR-MKT: With liquid inertia term: ρπr 2


γ (cosθe − cosθd ) = µ f v (16)

When the liquid’s inertia term is negligible under the condition r < rc , then td = −(

with µ f = exp(

2µ f ρgr

+

16 µbγ cos θe 8 µb ρgr ) log(1 − h) − h 2 2 3 ρ g r 2γ cos θe ρgr 2

(17)

2γ ) µb . W (0)

The LWR-Self-layering yields good predictions without fitting parameters of the capillary rise of spherical CCl4, disk octamethylcyclotetrasiloxane (OMCTS) and cylindrical n-alkanes31. The above models have experimental support, but they are based on very different mechanisms of the energy dissipation of the dynamic contact angle. A comparison of these models is thus needed for a thorough understanding of the dynamic contact angle. Popescu29 compared the theoretical predictions of LWR-Hydrodynamic, LWR-MKT, and LWR-Joos for the capillary rise of water/glass and silicone oil/glass systems. However, there is no systematic comparison of the dynamic-contact-angle-modified LWR equations based on the experimental data. The general validity of the models and the mechanism of the energy dissipation leading to the dynamic contact angle are still unknown. 12


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In this work, we systematically compared the predictions of the LWR equations modified by the dynamic contact angle models on the capillary rise dynamics of different wetting liquids in borosilicate glass capillaries. The results show that the LWR equation modified by the molecular self-layering model has an advantage over the other models in predicting the capillary rise dynamics without the fitting parameters when the solvation force data is available. For silicone oils, with a wide range of bulk viscosities, the molecular self-layering model explains the capillary rise dynamics with the same parameter W(0). The validity of the molecular self-layering model demonstrates that for wetting liquids, the layered molecularly thin wetting film is essential in understanding the dynamic contact angle.

2. Experimental Section 2.1 Materials The silicone oils (10 cst and 100 cst) and OMCTS (98%) were from Sigma Aldrich. Carbon tetrachloride (99%), n-dodecane (99%), n-tetradecane (99%), nhexadecane (99%), and 1-dodecanol (98%) were from Fisher Scientific. The physical properties of the testing liquids are listed in Table 1. All the chemicals were used without further treatment. The capillaries used were micro syringe borosilicate glass capillaries from Drummonds Scientific Company, Broomall, PA, USA. The capillaries are made by melting, and the capillary surface is optically smooth. We measured the capillary size at different points by optical microscopy instead of relying on the manufacturer’s quoted value. We randomly chose a few capillaries and found no significant differences between

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the measurements at various points. The tolerance of the inner diameter of the glass capillaries is 1%. Before use, all the capillaries were cleaned by immersion in chromic acid overnight; the capillaries were connected to a pump and flushed with 20 mL DI water, and dried by suctioning out the air for 10 minutes. Table 1. Physical properties of the testing liquids47, 48, 49. Surface Tension 25℃ mN/m

Viscosity 25℃ cP

Density 25℃ kg/m3

Silicone Oil (10 cst)a

20.1

9.3

930

Silicone Oil (100 cst)a

20.9

96.0

960

Carbon Tetrachloride

26.4

0.9

1587

OMCTS

19.3

2.6

956

n-dodecane

24.9

1.4

749.5

n-tetradecane

26.1

2.0

756

n-hexadecane

27.0

3.0

773

1-dodecanol

29.4

16.1

830.9

a. From the manufacturer

2.2 Capillary rise measurement The capillary rise measurements were conducted at room temperature (25℃). The movement of the meniscus inside the capillary during the capillary rise process was recorded by a Cannon G15 camera. More details of the experimental set-up for the measurement of the capillary rise was presented in our previous work31. The video was then analyzed frame by frame to obtain the capillary rise height h(t) at time t. The experimental data were averaged using the same approach as that found in Radke’s work50. For each run, h(t) was obtained by best fitting the experimental data with a smooth curve. Curves for duplicate runs were combined using an arithmetic average of the h-values at the same time t. 14


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3. Results and Discussion 3.1 Predictions of the modified LWR equations The measured capillary rise dynamics of the testing liquids listed in Table 1 in borosilicate glass capillaries are presented in Figs.2-5 along with the predictions of the above-mentioned LWR equations modified by the four dynamic contact angle models. Since all the testing liquids have a very small equilibrium contact angle (smaller than 10º) on the borosilicate glass surface, we used cos θ e = 1 in all the analyses. There are no fitting parameters in LWR-Self-layering for predicting the capillary rise of carbon tetrachloride, OMCTS, n-dedocane, n-tetradecane, and n-hexadecane, as the measured solvation force data for these liquids under confinement are available31. For 1-dodecanol and silicone oils, there is a fitting parameter, W (0) , in LWR-Self-layering since the molecules are not spherical and there is no solvation force data reported under confinement on molecular scale. LWR-Hydrodynamic and LWR-MKT contain the fitting parameters χ and λ, respectively. The values of W(0), χ, and λ are listed in Table 2. The maximum capillary and Reynolds numbers for each experiment are also presented in Table 2. The capillary force dominates the capillary rise process in all the experiments, as the maximum capillary number is much smaller than 1. It seems that the Reynolds number in the beginning of the capillary rise is quite large, and the liquid’s inertia plays a role. However, what the Reynolds number reflects is the magnitude of the liquid’s inertia relative to the viscous force within the bulk liquid. In the capillary rise process, the frictional force close to the three-phase contact line is known to be significant and thus the traditional Reynolds number cannot accurately reflect the importance of the liquid’s inertia. The rising velocity of the liquid inside the capillary deceases very fast and so does 15


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the Reynolds number. The relatively large value of the Reynolds number in the beginning is not a suitable criterion for the importance of the liquid’s inertia in the whole capillary rise process. A better criterion is the critical capillary radius , rc = 24

µf 2 γρ cos θe

, below

which the liquid’s inertia can be neglected. Table 2 compares the experimental capillary radii and the critical capillary radii. The experimental capillary radius is larger than the critical capillary radius only for carbon tetrachloride and n-dodecane, and thus the inertia term is considered in all the dynamic-contact-angle-modified LWR equations. For all the other experiments, the capillary radius is smaller than the critical capillary radius and the inertia term is neglected. LWR-MKT and LWR-Self-layering have the same mathematical formula and thus the same predictions for 1-dodecanol. The difference between these two models involves the physics of the key parameters, which we discuss later in this work. Figure 2 shows the predictions of the dynamic-contact-angle-modified LWR equations for the capillary rise of CCl4 and OMCTS. All the modified LWR equations generated similar and good predictions in the early stage of the capillary rise. After around 60% of the capillary rise, the predictions of the different models diverged. LWR-Joos began to deviate from the others with its slower prediction, while LWR-MKT and LWR-Self-layering tended to over-predict the capillary rise. Figure 4 shows that for n-alkanes, Joos’ equation overpredicted the early stage of the capillary rise and under-estimated it close to equilibrium. Figure 4d shows that Joos’ equation does not predict the results for polar 1-dodecanol. It is not surprising that Joos’ equation worked quite well for the two silicone oils shown in Fig.5, as Joos’ empirical equation is based on the data from silicone oil/borosilicate glass/air systems. The accuracy of the predictions of the 4 modified LWR equations were 16


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quantified by the correlation coefficient and the normalized root mean square error listed in Table 3.

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Height (mm)

a. Carbon Tetrachloride

b. OMCTS

Height (mm)

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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FIG. 2 Predictions of the modified LWR equations of the capillary rise of CCl4 and OMCTS in borosilicate glass capillaries with an inner diameter of 0.72 mm. 18


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14

a. n-dodecane 12

Height (mm)

10 LWR-Hydrodynamic LWR-MKT LWR-Self-layering LWR-Joos LWR Experimental Data

8 6 4 2 0 0

15

0.1

0.2 0.3 Time (s)

0.4

0.5

b. n-tetradecane

10 LWR-Hydrodynamic LWR-MKT LWR-Self-layering LWR-Joos LWR Experimental Data

Height (mm)

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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5

0 0

0.2

0.4 0.6 Time (s)

0.8

1 20


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Height (mm)

c. n-hexadecane

d. 1-dodecanol

Height (mm)

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FIG. 3 Predictions of the modified LWR equations for the capillary rise of n-alkanes and 1-dodecanol in borosilicate glass capillaries with an inner diameter of 1.10 mm. 21


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Height (mm)

a. silicone oil 10 cst

b. silicone oil 100 cst

Height (mm)

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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FIG. 4 Predictions of the modified LWR equations for the capillary rise of silicone oils (10 and 100 cst) in borosilicate glass capillaries with an inner diameter of 0.72 mm. Table 2. Maximum capillary and Reynolds numbers under our experimental conditions; The critical and experimental capillary radii; W(0) in the molecular selflayering model, and the fitted values of χ in the hydrodynamic model and λ in the molecular kinetic theory.

χ

Experiment al Capillary Radius mm

Critical Capillary Radius mm

Maximum Capillary Number

Maximum Reynolds Number

0.56±0.01

8.6±0.5

0.36

0.05

0.006

177.5

9.8b

0.48±0.2

7.8±0.6

0.36

0.59

0.014

13.6

n-dodecane

12.5b

0.47±0.02

11.0±1.6

0.55

0.37

0.006

35.8

n-tetradecane

12.5b

0.45±0.01

11.5±1.5

0.55

0.73

0.008

19.2

n-hexadecane

12.5b

0.44±0.02

11.4±1.4

0.55

1.77

0.015

17.3

1-dodecanol

13.1±0.3

0.48±0.01

23.0±3.0

0.55

85.95

0.019

1.0

10.0±0.2

N/A

12.2±2.0

0.36

16.66

0.019

1.4

10.4±0.2

N/A

11.8±1.5

0.36

1419.95

0.032

0.02

W(0) mN/m

λ nm


13.6a

OMCTS

Silicone oil (10 cst) Silicone oil (100 cst)

a. Estimated from the molecular diameter using the relation W (0) =

k BT

σ2

.

b. Estimated from the measured solvation force data31.

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Table 3. Correlation coefficients of the various model predictions with the experimental data and the normalized root mean square error LWRHydrodynamic R2 NRMSE % %

LWR-MKT

LWR-Self-layering

LWR-Joos

R2 %

NRMSE %

R2 %

NRMSE %

R2 %

NRMSE %


99.81

4.8

99.74

5.6

99.74

5.6

99.77

5.2

OMCTS

99.95

1.5

99.95

1.1

99.95

1.1

99.92

3.3

n-dodecane

99.85

1.7

99.62

3.2

99.62

3.2

99.81

1.8

n-tetradecane

99.92

1.2

99.85

2.0

99.85

2.0

99.13

2.8

n-hexadecane

99.59

4.4

99.92

5.6

99.92

5.6

98.77

4.7

1-dodecanol

99.92

1.2

99.84

3.0

99.84

3.0

98.94

9.3

99.91

2.1

N/A

N/A

99.59

3.3

99.70

3.5

99.98

1.7

N/A

N/A

99.98

1.8

99.97

2.0

Silicone oil (10 cst) Silicone oil (100 cst)

3.2 Validity of LWR-MKT In the molecular kinetic theory, the interaction between liquid molecules and the solid surface controls the dynamic contact angle and the key fitting parameter is λ, the distance between the adsorption sites on the solid surface. Since all the experiments were conducted in the same type borosilicate glass capillaries, λ should be the same for all the liquids. However, the fitted values for λ listed in Table 2 show a dependence on the 24


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testing liquids. It seems that there is only a small change in λ for all the testing liquid (less than 0.15 nm). However, in the molecular kinetic theory, the frictional coefficient β is very sensitive to the value of λ. Figure 5 shows the ratio between the frictional

coefficient β and bulk viscosity µb for n-dodecane, n-tetradecane, and n-hexadecane at different λ values. In the region of 0.40 nm to 0.60 nm, the ratio increased dramatically with the rising λ value. For example, the λ value obtained from the experimental data of 1-dodecanol will fail to predict the capillary rise dynamics of n-hexadecane, as shown in Fig.6.

200 180 160

Hexadecane Tetradecane Dodecane

b

140 β/µ

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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120 100 80 60 40 0.4

0.45

0.5

λ(nm)

0.55

0.6

FIG. 5 The ratio β / µb as a function of λ for n-dodecane, n-tetradecane, and nhexadecane on a glass surface.

This inconsistency can be well-explained by the existence of the LMTWF ahead of the meniscus for a spreading wetting liquid, which was confirmed by recent experimental studies51, 52, 53 and molecular simulations54, 55. Since the droplet slides on the 25


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LMTWF, it should be the interaction between the meniscus and LMTWF, rather than the interaction between the liquid and solid surface, that directly controls the macroscopic dynamic contact angle. The formation and fluidity of the LMTWF depend on the properties of the liquid molecules, such as molecular shape. Consequently, the dynamic contact angle will show some dependence on the liquid molecules, as reported in literature45.

14 12 10 Height (mm)

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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8 6

Experimental Data LWR-MKT

4 2 0 0

0.1

0.2

0.3

0.4

0.5 0.6 Time (s)

0.7

0.8

0.9

1

FIG. 6 Prediction of LWR-MKT for the capillary rise of n-hexadecane in a borosilicate glass capillary with an inner diameter of 1.1 mm when λ has the value obtained from 1-dodecanol.

3.3 Validity of LWR-Self-layering In the molecular self-layering model, the meniscus slides on the LMTWF and the

26


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fluidity µ f = exp(

2γ ) µb of the LMTWF dominates the dynamic contact angle. It is n, W (0)

the number of liquid molecules per unit area of the LMTWF, that plays a key role. n can be estimated from the molecular diameter for spherical molecules such as CCl4 and from the measured solvation force for non-spherical molecules, such as OMCTS and nalkanes. The good predictions of LWR-Self-layering for the capillary rise of CCl4, OMCTS, and n-alkanes shown in Figures 2 and 3 prove that the molecular self-layering model is valid for wetting liquids when the LMTWF can be formed ahead of the main meniscus. The molecular self-layering model, for the first time, shows the importance of the liquid molecular shape on the wetting dynamics. The fitted W(0) values for the two silicone oils listed in Table 2 are indistinguishable, with a mean value of 10.2 mN/m and a deviation of less than 2%. In the measurement of the solvation force of the linear n-alkanes confined between two mica surfaces, Christenson56 showed that the solvation force does not vary with the length of the n-alkanes since they have the same repeating unit, CH2. The silicone oils used in this experiment were polymerized siloxanes and chain molecules with the same repeating unit (monomer) [SiO(CH3)2]. The wide range of bulk viscosities is equivalent to the wide range of molecular weights (molecular length) as the viscosity and molecular weight of silicone oils are correlated57. The only difference between the silicone oils was the length of the molecules. Molecular simulations have shown that for chain molecules, the molecular selflayering is mainly determined by the repeating segment58. Deloche59 showed that the ordering dynamics of silicone oil is controlled by the segment diffusion and is independent of the molecular weight. Horn60 also suggested that for silicone oil confined 27


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between two mica surfaces, the short sections of the neighboring polymer chains, rather than the entire molecule, determined the molecular self-layering. Therefore, it was expected that W(0) would have the same value for different silicone oils. To further justify the molecular self-layering model and show that the solvation energy of the silicone oils does not change with the molecular length, the capillary rise data of silicone oils with a higher viscosity (molecular weight/length) reported by Joos28 were analyzed with the average W(0) values of the two silicone oils listed in Table 2.

Table 4. Physical properties of the silicone oils, calculated LMTWF viscosity, the ratio between the LMTWF and bulk viscosity

Surface Tension mN/m

µf 2γ = exp( ) , and χ µb W (0)

Bulk Viscosity Pa·s

Density kg/m3

LMTWF Viscosity Pa·s

Ratio µf µb

χ

Silicone Oil 1

20.5

0.34

970

18.9

55.7

12.0

Silicone Oil 2

20.5

0.485

970

27.0

55.7

12.0

Silicone Oil 3

21.3

12.25

980

797.5

65.1

12.0

Silicone Oil 4

21.3

29.4

980

1913.9

65.1

12.0

Silicone Oil 5

21.3

58.8

980

3827.9

65.1

12.0

The surface tension, density, and viscosity data are from Joos28. Joos28 reported the capillary rise data of silicone oils with viscosities ranging from 0.34 Pa·s to 58.8 Pa·s in borosilicate glass capillaries. The solvation energy arises from the geometrical confinement of the solid surface and thus is sensitive to properties of the solid surface. Since both our and Joos’ experiments were conducted in borosilicate glass 28


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capillaries, the W(0) value obtained from our experiment can be applied to analyze Joos’ data. The LMTWF viscosities for the silicone oils listed in Table 4 were calculated through the relation µ f = exp(

2γ ) µb with W(0)=10.2 mN/m, as estimated from our W (0)

capillary rise data of low viscosity silicone oils. We note that the LWR-Self-layering now contains no fitting parameters in predicting the capillary rise of the five silicone oils tested by Joos.

Table 5. Correlation coefficients of the various model predictions with the experimental data and the normalized root mean square error. LWRHydrodynamic

LWR-Self-layering

LWR-Joos

R2 %

NRMS %

R2 %

NRMSE %

R2 %

NRMSE %

Silicone Oil 1

99.97

0.9

99.92

2.4

99.93

2.0

Silicone Oil 2

99.89

1.8

99.77

4.0

99.89

1.5

Silicone Oil 3

99.97

1.0

99.90

1.8

99.94

1.2

Silicone Oil 4

99.94

1.5

99.90

1.5

99.80

3.1

Silicone Oil 5

99.88

1.4

99.71

3.4

99.86

1.9

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18

a. Silicone Oil 1: µb=0.34 Pa·s

16

Height (mm)

14 LWR-Hydrodynamic LWR-Self-layering LWR-Joos LWR Experimental Data

12 10 8 6 4 2 0 0

50

100

150 Time (s)

200

250

300

18 b. Silicone Oil 2: µb=0.485 Pa·s

16 14 Height (mm)

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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12 10

LWR-Hydrodynamic LWR-Self-layering LWR-Joos LWR Experimental Data

8 6 4 2 0 0

50

100

150 200 Time (s)

250

300

350 30


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18 c. Silicone Oil 3: µb=12.25 Pa·s

16

Height (mm)

14 12


10 8 6 4 2 0 0

18

1000 2000 3000 4000 5000 6000 7000 8000 Time (s)

d. Silicone Oil 4: µb=29.40 Pa·s

16 14 Height (mm)

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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12


10 8 6 4 2 0 0

5000

10000 Time (s)

15000 31


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18 16 14 Height (mm)

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

12

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10 8 6

e. Silicone Oil 5: µb=58.8 Pa·s

4 2 0 0

2000

4000

6000 Time (s)

8000

10000

FIG. 7 Predictions of the modified LWR equations for the capillary rise of silicone oils in borosilicate glass capillaries with an inner diameter of 0.50 mm. Experimental data are from Joos28. W(0) and χ were estimated from our capillary rise data and so there are no fitting parameters in predicting Joos’ data.

Figure 7 shows that with the W(0) value obtained from our capillary rise data of the low bulk viscosity (low molecular weight) silicone oils, the LWR-Self-layering predicts well the capillary rise of silicone oils with a much higher viscosity (much higher molecular weight). Table 5 lists the correlation coefficients and the normalized root mean square errors quantifying the accuracy of the predictions of the 3 modified LWR equations.

32


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This finding agrees with the experimental and theoretical studies on the molecular self-layering phenomena quite well and further shows the validity of the molecular selflayering model for the dynamic contact angle in a wide range of systems. The constant W(0) value also connects the capillary dynamics of silicone oils with different viscosities and advances the understanding of the microscopic interaction between polymer melts and solid surfaces.

3.4 Validity of LWR-Joos The capillary rise of all the silicone oils (Figs. 4 and 7) can also be explained by Joos’ empirical equation. We explored the relation between the molecular self-layering model and Joos’ empirical equation. In the molecular self-layering model, the equation between the dynamic contact angle and capillary number can be rewritten as cos θ e − cos θ d =

µf Ca µb

The ratio between the LMTWF and bulk viscosity

.

(18)

µf for all the silicone oils ranges µb

from 55.7 to 65.1, as shown in Table 4. The maximum capillary number for all the silicone oils during the capillary rise process reported by Joos is 0.008. Figure 8 compares the values for cos θ d estimated from Joos’ empirical equation and the molecular self-layering model based on the ratio between the LMTWF and bulk viscosity. Except at the very beginning of the capillary rise (which does not significantly affect the whole capillary rise process), the maximum difference between the estimated cos θ d from Joos’ empirical equation and the molecular self-layering model is less than 13%. This explains why Joos’ empirical equation predicts

33


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the capillary rise of silicone oils quite well. Figure 8 also shows that Joos’ empirical equation estimates a larger value for cos θ d (smaller θ d ) at a high capillary number and a smaller value of cos θ d (larger θ d ) at a low capillary number, which corresponds well with the over-prediction in the beginning and under-prediction close to the equilibrium of the capillary rise process.

1 0.8 0.6

cosθd

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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Joos

0.4

Molecular Self-layering ratio=55.7 Molecular Self-layering ratio=65.1

0.2 0 0

0.002

0.004

0.006

0.008

Capillary Number FIG. 8 Comparison of the value of cos θ d estimated from Joos’ empirical equation and the molecular self-layering model at the experimental capillary number The ratio between the LMTWF and bulk viscosity of the three n-alkanes has a value between 55 and 75, a similar value as that of the silicone oils. Therefore, Joos’ empirical equation also provides good predictions of the capillary rise of n-alkanes. The ratios between the LMTWF and bulk viscosity for OMCTS (40) and 1-dodecanol (90) are far from those of the silicone oils. Therefore, the ability to make predictions using Joos’ empirical equation worsens---it even fails to make a prediction. 34


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In conclusion, the good predictions from Joos’ empirical equation of the capillary rise of silicone oils and n-alkanes are because the equation happens to show the mathematical relation between the dynamic contact angle and capillary number for specific systems. The molecular self-layering model not only explains the applicability of Joos’ empirical equation in silicone oils and n-alkanes, but also offers a physical understanding of the dependence of the dynamic contact angle on the capillary number.

3.5 Validity of LWR-Hydrodynamic Both Joos’ empirical equation and the molecular self-layering model reveal that the dynamic contact angle depends on the capillary number. The hydrodynamic model also relates the dynamic contact angle with the capillary number by one fitting parameter,

χ = ln(

L ) . While the macroscopic length L can usually take the value of the Lm

geometrical parameter of the meniscus, such as the capillary radius, the microscopic length Lm is very difficult to estimate or measure. Consequently, χ = ln(

L ) was used as Lm

a fitting parameter and the value was obtained from the regression of the experimental data. All the experimental data show that the LWR-Hydrodynamic provides good predictions of the capillary rise. Table 2 shows that the three linear alkanes have the same fitted χ value. Although viscosity of the two silicone oils changes ten times, they still share the same χ value. The predictions of the LWR-Hydrodynamic with χ obtained from low viscosity silicone oils on the capillary rise of silicone oils with higher viscosities reported by Joos are shown in Fig.7. The good predictions for all the silicone oils indicate that χ most likely depends on the molecular width rather than the molecular length. Since the meniscus is sliding on the wetting film, the microscopic slip length should be related 35


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to the film properties such as film thickness. Under confinement in the thin wetting film, silicone oil molecules tend to lie parallel to the solid surface and thus the film thickness is determined by the molecular thickness, but not molecular length. When we assume the macroscopic length scale L is the capillary radius, Lm has the value of 2.2 nm. Horn60 reported that the width of the molecules of silicone oils under confinement is around 0.70 nm. Thus, the microscopic slip length corresponds to around three molecular layers. The relation between the microscopic slip length with the molecular self-layering phenomena is still unrevealed. Though the LWR-Hydrodynamic predicts the capillary rise for all testing liquids quite well, the key fitting parameter still lack physical understanding.

4 Conclusions In this work, we presented the capillary rise data of various wetting liquids in borosilicate glass capillaries and systematically compared the predictions of the LWR equations modified by dynamic contact angle models, including the hydrodynamic model, molecular kinetic theory, molecular self-layering model, and Joos’ empirical equation. The capillary rise of silicone oils with a wide range of bulk viscosities (with a variation of more than 6,000) reported by Joos was also analyzed with the LWR equations modified by the hydrodynamic model and the molecular self-layering model. The conclusions of this study can be summarized as follows: 1. The LWR equation modified by the molecular kinetic theory predicts well the capillary rise of the testing liquids with one fitting parameter λ , the distance between the adsorption sites on the solid surface. However, the experimental data shows there is dependence of λ on the testing liquids, which conflicts with the theory as λ is the 36


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property of the solid surface and should be independent on the testing liquid; 2. The LWR equation modified by the hydrodynamic model works well for all the testing liquids with one fitting parameter χ , which relates to the microscopic slip length. The constant values of χ for n-alkanes and all the silicone oils suggesting that χ is also possibly related to the LMTWF. However, the physics of χ still lacks understanding. 3. There is good agreements between the predictions of the LWR equation modified by Joos’ empirical equation and the capillary rise of specific liquids when the ratio between the LMTWF and bulk viscosity is between 50 to 80, such as silicone oils and n-alkanes. 4. Without any fitting parameters, the LWR equation modified by the molecular self-layering model explains well the capillary rise of CCl4, OMCTS, and linear alkanes with the molecular diameter or the measured solvation force data. The capillary rise dynamics of silicone oils with a wide range of bulk viscosities are also well-explained with the same parameter W (0) , which resulted from liquid molecules self-layering on the solid surface. The molecular self-layering model is the only model that shows the importance of the liquid molecular shape on the capillary dynamics. In conclusion, the present study shows the validity of the LWR equations modified by the hydrodynamic model, molecular kinetic theory, molecular self-layering model, and Joos’ empirical equation in explaining the capillary rise of various wetting liquids. For wetting liquids, the advancing meniscus slides on the LMTWF, which forms ahead of the main meniscus due to the liquid molecular self-layering close to the solid surface. Therefore, the molecular self-layering model has an advantage over the other models by clearly showing the role of the LMTWF on the velocity dependent dynamic 37


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contact angle.

5 Outlook The molecular self-layering model has demonstrated that the dynamic contact angle resulted from the viscous energy dissipation between the advancing meniscus and the LMTWF. However, the detail structure and in-layer structure of the LMTWF requires further study. More experimental and theoretical efforts are needed to identify how the liquid molecules of various shapes self-layer into a layered thin wetting film close to the solid surface to complete the understanding of the dynamic contact angle and the capillary dynamics. This work focuses on the small equilibrium contact angle. For a large contact angle, there might be no wetting film ahead of the meniscus and the lubrication approximation might not be applicable.

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TOC Graphic

Height (mm)

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FIG. 4 Predictions of the modified LWR equations for the capillary rise of silicone oils (100 cst) in borosilicate glass capillaries with an inner diameter of 0.72 mm.

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Capillary Rise: Validity of the Dynamic Contact Angle Models

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