Capillary Displacement of Viscous Liquids - Langmuir (ACS

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Capillary displacement of viscous liquids Peter L.L. Walls, Grégoire Dequidt, and James C. Bird Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b00351 • Publication Date (Web): 14 Mar 2016 Downloaded from http://pubs.acs.org on March 20, 2016

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Capillary displacement of viscous liquids Peter L. L. Walls,† Grégoire Dequidt,†,‡ and James C. Bird∗,† Department of Mechanical Engineering, Boston University, Boston, MA 02215, and École Polytechnique, Route de Saclay, 91128 Palaiseau, France E-mail: [email protected]

Abstract When a capillary tube is brought into contact with a wetting liquid, surface tension forces overcome gravity and the liquid spontaneously rises into the tube until an equilibrium height is reached. The early viscous dynamics of the rise typically follow the well-known Lucas-Washburn law, which is independent of gravity and neglects the displaced fluid. Here we explore the early viscous dynamics when the properties of displaced fluid are significant. Using a combination of experiments and theory, we show how the characteristic behavior of the Lucas-Washburn law is modified when the viscosity of the displaced fluid is comparable to or exceeds the wetting fluid. Additionally, we find that the effects of gravity reshape the dynamics of the capillary rise, not only in the late viscous regime, but also in the early viscous regime.

Introduction The spontaneous imbibition of liquid into a tube has been studied for centuries with some of the earliest documented experiments carried out by Hooke and Boyle in the mid-1600s. 1,2 ∗

To whom correspondence should be addressed Department of Mechanical Engineering, Boston University, Boston, MA 02215 ‡ École Polytechnique, Route de Saclay, 91128 Palaiseau, France †

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Classical understanding of capillary rise dynamics is often credited to Lucas and Washburn who, 250 years later, noted that the advancement of the meniscus in time z(t) is diffusive (z ∝ t1/2 ). 3,4 This proportionality is commonly referred to as the Lucas-Washburn law. Continued interest in the study of capillary rise derives from its importance in applications such as hydrology, 5–7 paper-based deposition 8 and microfluidics, 9–11 and oil recovery. 12–14 Our paper examines the capillary rise dynamics when a liquid is being displaced.

Figure 1: The spontaneous displacement of air by silicone oil. (a) When a capillary tube of length L and radius a is brought into contact with a wetting fluid, the imbibing fluid (µ1 ) overcomes viscous and gravitational forces displacing the original fluid (µ2 ). (b) The rise of the wetting fluid follows the classical Lucas-Washburn law when the viscosity of the displaced fluid is negligible (a = 0.2 mm) . In the early stages of rise, increases in viscous stresses cause the meniscus to decelerate. In the late stages of rise, gravitational forces become significant, leading to an equilibrium height h. The capillary rise phenomenon is illustrated in Fig. 1. Here we have inserted a glass tube of length L and radius a into a bath of silicone oil with known density ρ1 , viscosity µ1 , and surface tension γ. By using a relatively viscous silicone oil (µ1 = 97 mPa s), the dynamics of the rise can be seen with the unaided eye and captured with a standard camera. Provided that the imbibing liquid wets the inner surface of the tube, surface tension forces 2 ACS Paragon Plus Environment

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will draw in oil, displacing the air (ρ2 , µ2 ). As liquid continues to enter the tube, the constant capillary forces driving the motion must overcome an increasing viscous resistance, resulting in a decelerating flow. Eventually, gravitational forces become significant, leading to a rise that asymptotes to the familiar equilibrium height h =

2γ cos θ , ∆ρga

where ∆ρ = ρ1 − ρ2 , θ is the

equilibrium contact angle, and g is the acceleration due to gravity. 15 Less clear are the rise dynamics that occur for two immiscible liquids. Relatively few authors consider the role of the displaced phase during spontaneous imbibition, 16,17 and even fewer have explored the displacement of a liquid in the presence of gravity because of the associated experimental challenges. 18,19 Intuitively, displacing a liquid may alter the rise dynamics in several ways, dependent on its viscosity. If the viscosity of the displaced liquid is equal to the imbibing liquid, one might anticipate an initially constant rise speed as any increase in the viscous resistance from the incoming liquid is exactly offset by the decrease in resistance from displacing the original liquid. Similarly, if the viscosity of the displaced liquid is greater than the imbibing liquid, one might anticipate an acceleration in the rise speed as any increase in the viscous resistance from the incoming liquid is small compared to the decrease in resistance from displacing the original liquid. 20 These predictions have been experimentally tested in both a capillary tube and the more complex case of porous media. 18,19 It was found that the meniscus does advance with a constant speed for equal viscosities in a horizontal capillary tube, as anticipated. Similarly, acceleration of the meniscus was observed when displacing a more viscous liquid in both horizontal and vertical setups. 18 However, there have also been documented cases in which matched viscosities in a vertical tube have resulted in diffusive-like rise dynamics, suggesting that the displaced fluid had little to no effect on the overall dynamics. 19 Relying on the published literature, the effects of the displaced liquid and gravity on the rise dynamics are unclear. Here, we untangle these seemingly conflicting observations in a generalized manner.


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Experimental Section We perform a series of experiments in which we vary the properties of both the imbibing and draining fluids (Table 1). Each experiment where a liquid displaces another liquid is carried out in a transparent acrylic container (H × W × D in mm: 280 × 150 × 45). A 70 mm high column of the displacing liquid (water or a water-glycerol mixture) sits below a 200 mm high column of the displaced silicone oil. A borosilicate capillary tube is first fully submerged in the 200 mm deep column of silicone oil (Supporting Information). Next, the tube is slowly brought into contact with the wetting liquid, spontaneous imbibition occurs and is captured with a SLR camera. The radius and length of the capillary tube varies between a = 0.2 mm to a = 3.0 mm and L = 30 mm to L = 115 mm, respectively. The density of the glycerolwater mixture (µ1 = 97 mPa s) is determined from a standard table of glycerol properties based on the weight percentage of glycerol (≈ 85%). 21 We measure the fluid-pair surface tension using the pendant drop method 22 and the liquid viscosity with a SV-10 vibrating plate viscometer (A & D Company, Japan). 23 The contact angle θ is extracted from the equilibrium height h, recognizing that cos θ =

∆ρgah . 2γ

To eliminate the influence of charge

effects and surface inconsistencies noted by previous researchers, we pre-wet the capillary tubes with a thin layer of glycerol. 4,24 Table 1: Measured value of the densities ρ1 , ρ2 , surface tension γ, contact angle θ, and dynamic viscosities µ1 , µ2 for each of the fluid combinations used in our experiments.


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Results and Discussion A series of images showing the rise profile when the viscosity of the displaced liquid equals and exceeds that of the imbibing liquid (µ1 /µ2 = 1 and µ1 /µ2 = 10−2 , respectively) are shown in Figs. 2a and b. We see that the rise profile in the vertically oriented tube is non-linear when µ1 = µ2 (Fig. 2a). The profile also appears to deviate from the Lucas-Washburn law in the early stages of rise. Contrary to earlier findings and predictions, further increasing the viscosity of the displaced liquid does not result in an acceleration as the more viscous fluid is expelled, but instead the rise appears linear in time (Fig. 2b).

Figure 2: Series of images showing the displacement of viscous silicone oil by water and a water-glycerol mixture. (a) When the viscosity of the liquids match (µ1 /µ2 = 1), the rising interface profile deviates from the Lucas-Washburn law (a = 0.9 mm). (b) When the viscosity of the displaced liquid is larger than the imbibing liquid (µ1 /µ2 = 10−2 ), the rising interface initially follows a linear trend. (a = 1.2 mm) To compare the dynamics of each case, we plot the individual experiments from Figs. 1 and 2 on a log-log plot and extract the scaling of the rise height in time (z ∝ tn ). Here, we have rescaled the measured rise heights (Fig. 3 inset) by the equilibrium height h and 5 ACS Paragon Plus Environment

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8µc h , ∆ρga2

the time by the visco-gravitational time

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where µc is the larger of the two viscosities

(Fig. 3). 24 As anticipated, the capillary rise of oil displacing air (µ1 ≫ µ2 ,

) is accurately

described by the Lucas-Washburn law (n = 21 ) at early times. At the later stages of rise, the dynamics transition from scaling as z ∝ t1/2 or “early viscous” to z ∝ (1 − e−t ) or “late viscous” dynamics as gravitational forces become significant. By contrast, when the rising oil displaces an equally viscous liquid (µ1 = µ2 ,

), the behavior deviates from the Lucas-

Washburn law. Finally, when the viscosity of the imbibing fluid is significantly less than the displaced fluid (µ1 ≪ µ2 ,

), the rise of the meniscus in the “early viscous” regime scales

linearly with time (n = 1). We see no evidence of acceleration when displacing a more viscous liquid in a vertical tube. To rationalize this counterintuitive result, we turn to theory. 100

10-1

120 100 80

10-2

60 40 20 0 0

10-3 -2 10

50

10-1

100

100

150

200

101

250

102

Figure 3: Compilation of individual capillary displacement experiments from Figs. 1 and 2. Here the rise in time (inset) for each experiment is rescaled by the equilibrium height h and a characteristic timescale 8µc h/∆ρga2 , where µc is the larger of the two viscosities. When µ1 /µ2 ≫ 1, the initial rise, denoted as early viscous, follows the anticipated z ∝ t1/2 relationship. Whereas when µ1 /µ2 ≪ 1, the early viscous regime dynamics are linear z ∝ t. The force balance on the rising meniscus, extended to include the displaced fluid, can be expressed as


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[ ∆ρ

] ρ2 2γ cos θ 8 L¨ z + z z¨ + z˙ 2 = − 2 [µ1 z + µ2 (L − z)] z˙ − ∆ρgz, ∆ρ a a

(1)

where we have assumed a constant contact angle θ and a poiseuille velocity profile throughout the tube (see Supporting Information for derivation). 24 We again rescale all lengths by the equilibrium height h =

2γ cos θ ∆ρga

and times by the characteristic time τ =

8µ2 h , ∆ρga2

where we

now choose µc = µ2 . Furthermore, whenever the characteristic acceleration of the system is sufficiently small, i.e.

h ρ2 L [ τ 2 g ∆ρ h

+ 1] ≪ 1, the inertial terms are negligible. In our setup,

typical values of the dimensionless ratio

h τ 2g

are ≈ 10−5 − 10−7 ≪ 1, leaving a simplified

governing equation [

L 0=1− + h

(

) ] µ1 z τ z −1 z˙ − µ2 h h h

(2)

that can be solved analytically. With the initial condition z(t = 0) = 0, we find that ) ( ) ( ( µ1 L z) t µ1 z + 1− − ln 1 − = . 1− µ2 h µ2 h h τ

(3)

It is noteworthy that Eq. (3) contains both a linear and logarithmic contribution with prefactors that depend solely on the physical parameters in the setup. In this form, the governing equation has two degeneracies corresponding to when each of these prefactors equals zero. In one limiting case ( µµ21 = 1), the rise dynamics are described by an exponential 2

decay z = h[1 − exp(− ∆ρga t)]. In the other limiting case ( µµ21 + Lh = 1), the rise dynamics are 8µ2 L described by a linear ascent z =

∆ρga2 t. 8(µ2 −µ1 )

Returning to the experimental results in Fig. 2,

we find that the displacement profiles correspond to these two degenerate cases. When the viscosities are equal (Fig. 2a), we observe the exponential decay predicted by our model. Similarly, when µ1 /µ2 ≪ 1 and L/h ≈ 1 (Fig. 2b), we observe the predicted linear rise, thus rationalizing our seemingly counterintuitive results. There are two other natural limits to consider: µ1 ≫ µ2 and µ1 ≪ µ2 when L ≈ O(h)—the case in our experiments. When the incoming liquid dominates (µ1 ≫ µ2 ), Eq. (3) simplifies


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to the conventional form of the Lucas-Washburn equation z = ( aγ2µcos1 θ t)1/2 , after expanding the logarithmic term in the early stages of rise (z ≪ h) . By contrast, when the displaced liquid dominates (µ1 ≪ µ2 ), Eq. (3) simplifies to a linear ascent, z =

1 a γ cos θ t, 4 L µ2

in the early

stages of rise (z ≪ h). Indeed, as demonstrated in Fig. 3 the “early viscous” dynamics are accurately described by these two power laws. If the assumptions used to develop our model for displacing a viscous liquid are appropriate, the velocity in the early stages (v0 =

1 a γ cos θ ) 4 L µ2

should be constant and decrease with

increasing tube length. Indeed, the time series in Fig. 4a supports this prediction when water (µ1 = 1 mPa s) displaces oil (µ2 = 97 mPa s). The converse, when µ1 ≫ µ2 , is also verified in Fig. 4b. Here, we see the rise velocity is unaffected by changes in tube length, when the oil (µ1 = 97 mPa s) displaces air (µ2 = 0.02 mPa s), as expected. To further test our prediction, we perform a series of experiments in which the length and radius of the capillary tube are independently varied. Plotting each test case (Fig. 4c inset) in terms of collapses our data reasonably well to a line with slope

1 4

µ2 v0 γ cos θ

and a/L

(Fig. 4c). Since it has been shown

that the dynamic contact angle tends to increase with meniscus motion, it is plausible that the data falls below our prediction for this reason. 25,26 In all of our experiments with µ1 ≪ µ2 , the velocity is initially constant and decreases only in the moments before coming to rest at h. In fact, we have seen no evidence of acceleration, as was reported 60 years ago by Eley and Pepper for vertically oriented imbibition. 18 However, their results are completely consistent with our model. To recover their observations, we consider the case when z ≤ L ≪ h, which allows us to expand the logarithmic term in Eq. (3). After multiplying all terms by ( Lh )2 we arrive at ] [ 1 z µ1 ( z )2 aγ cos θ − = t. 1− L 2 µ2 L 4µ2 L2

(4)

This limit is reached in one of two ways: either the tube is oriented horizontally or the system is much smaller than the equilibrium rise height if oriented vertically, the latter


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4 4 3.5 3 2.5 2 1.5 1 0.5 0

3.5 3 2.5 2

0

20

40

80

60

100

120

1.5 1 0.5 0 0

0.02

0.04

0.06

0.08

0.1

Figure 4: Comparison of rise speeds for several different lengths L and radii a of capillary tube. (a) A time series of images illustrating that the constant speed of early rise depends on the tube length L when the viscosity of the displaced liquid is more viscous than the displacing liquid (µ2 ≫ µ1 , a = 3.0 mm). (b) By contrast, the speed of rise is independent of the tube length L when the viscosity of the displaced liquid is negligible (µ2 ≪ µ1 , a = 0.2 mm). (c) For µ1 ≪ µ2 , constant early velocities from each experiment (inset) collapse to a line when plotted in terms of a dimenionless velocity µ2 v0 /γ cos θ and aspect ratio a/L.


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being the case for Eley and Pepper’s experiments. Quick inspection of Eq. (4), reveals the profiles that one might anticipate from viscous resistance arguments. Specifically, if µ1 ≫ µ2 , we recover the Lucas-Washburn relation z = ( aγ2µcos1 θ t)1/2 ; whereas if µ1 = µ2 , the profile becomes linear (z = is parabolic

z L

=

aγ cos θ t 4µ2 L2

+

aγ cos θ t) 4µ2 L

1 z2 . 2 L2

for all times. Finally if µ1 ≪ µ2 , the entire rise profile

Indeed, this result illustrates how a short bead pack with

micron scale pores (z, L ≪ h), such as that used in Eley and Pepper’s experiments, would manifest in a noticeable acceleration. It is noteworthy that the early rise velocities in Fig. 4c (µ1 ≪ µ2 ) are identical to the velocities of a horizontal tube when µ1 = µ2 . For this linear ascent to occur, gravity must be playing a significant role in shaping the profile of the “early viscous” regime. Although gravity is not explicitly contained in the expression for the early velocity v0 , it is clear upon returning to Eq. (3) that gravity, quantified by L/h, plays an analogous role to µ1 /µ2 in determining the influence of the logarithmic term.

Conclusions Through a combination of experimentation and modeling, we uncover how viscosity and gravity reshape the dynamics of spontaneous displacement in the viscous regime. Our results are relevant whenever a viscous liquid is spontaneously displaced in the presence of gravity. Particular applications include: carbon sequestration, especially local capillary trapping, 27 spontaneous imbibition oil recovery, 13 and the migration of pollutants into groundwater. 28 Although our experiments focus on capillary rise, our results are equally valid for capillary descent—a term we use to denote capillarity opposing buoyancy. For example, in the case of pollution migration into groundwater, to displace the groundwater, the less dense hydrocarbons must overcome gravity in the form of buoyancy while moving downward. 29 It is known that variations in porosity can modifiy the Lucas-Washburn scaling; 30,31 our experiments and model indicate that changes in scaling may instead result from viscous and gravitational effects, even in the earliest stages of rise.


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Supporting Information Supporting Information Available: A dimensional analysis, the derivation of the governing equation, and details of the pre-wetting procedure for the capillary tubes. The supporting information is available free of charge on http://pubs.acs.org/ at DOI:

Acknowledgement The authors acknowledge financial support from Schlumberger-Doll Research. We also thank T. S. Ramakrishnan for encouraging us to pursue this problem.

References (1) Hooke, R. An Attempt for the Explanation of the Phenomena Observable in an Experiment Published by the Honourable Robert Boyle. Printed by J.H. for Sam. Thomson, London 1661, (2) Boyle, R. New Experiments Made and Communicated by the Honourable Robert Boyle Esquire; about the Superficial Figures of Fluids, Especially of Liquors Contiguous to Other Liquors. Phil. Trans. 1676, 11, 775–787. (3) Lucas, R. Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten. Kolloidn Z. 1918, 23, 15–22. (4) Washburn, E. W. The Dynamics of Capillary Flow. Phys. Rev. 1921, 17, 273–283. (5) Bell, J. M.; Cameron, F. K. The Flow of Liquids Through Capillary Spaces. J. Phys. Chem. 1906, 10, 658–674. (6) Richards, L. A. Capillary Conduction Of Liquids Through Porous Mediums. J. Appl. Phys. 1931, 1, 318–333.


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