Capillary Force between Flexible Filaments | Langmuir

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Capillary force between flexible filaments Majid Soleimani, Reghan J. Hill, and Theo G.M. van de Ven Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b01351 • Publication Date (Web): 09 Jul 2015 Downloaded from http://pubs.acs.org on July 12, 2015

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Capillary force between flexible filaments Majid Soleimani,† Reghan J. Hill,∗,† and Theo G.M. van de Ven‡ Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 0C5, and Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6 E-mail: [email protected]

Abstract Liquid droplets bridging filaments are ubiquitous in nature and technology. Although the liquid-surface shape and the capillary force and torque have been studied extensively, the effect of filament flexibility is poorly understood. Here, we show that elastic deformation (at large values of the elastocapillary number) can significantly affect the liquid-surface shape and capillary force. The equilibrium state of parallel filaments is calculated using analytical approximations and numerical solutions for the fluid interface. The results compare well, and the numerical solution is then applied to crossing filaments. In the investigated range of parameters, the capillary force increases rapidly when the filaments touch. The force decreases continuously when decreasing the liquid volume for parallel hydrophilic filaments, but produces a maximum for crossed filaments. The liquid volume at the maximum force is reported when changing the filament flexibility, crossing angle, and contact angle. These results may be beneficial in applications where the strength and structure of wet fibrous materials are important, such as in paper formation and when welding flexible components. ∗

To whom correspondence should be addressed Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 0C5 ‡ Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6 †

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Introduction Droplet-on-filament systems have been investigated extensively because of their many applications 1–3 . For example, a network of fibres is used in the fog collection apparatus of the cactus stem 4 , spider silk 5 , and manufactured woven meshes 6 , and is beneficial for controlling evaporation 7,8 . Furthermore, a droplet on a filamentous network can be responsible for the hierarchical helical structures observed in nature 9 , bundles of wet hair 10 , and the water repellency of feathers 11 . Advances in micro- and nanoscale technology bring increasing attention to surface-tensiondriven deformations of elastic solids, i.e., ‘elasto-capillarity’, because the deformations are dominated by surface rather than body forces 12,13 . The liquid-droplet shape between two parallel solid fibres has been studied for decades 14,15 , and was recently used to control evaporation 7 and liquid delivery 7,16 . In an array of parallel flexible fibres, elasto-capillarity furnishes several spreading regimes for the liquid, which can be observed on wet feathers 17 . However, fibres in a random network are rarely parallel or clamped at one end. Therefore, studying the droplet shape between crossed fibres is essential for understanding the capillary impact on fibrous materials when exposed to moisture. Surface tension can attach or detach crossed fibres, which is important in many applications, such as textile wetting and drying 18 , paper drying 19 , filtration and separation of two immiscible liquids 20,21 , electrical resistance and rheology of fibrous materials 22,23 , and nanoparticle assembly 24 . Such fibre deformations can cause surface-adhesion-driven collapse 25 , and they affect the bulk properties of fibrous materials 26 . Virozub et al. 27 numerically investigated the liquid bridges between two identical cylinders at a fixed separation and crossing. With a large separation, the liquid bridge was symmetric, and the filaments were attracted by capillary forces. However, separation reduction at large contact angles furnished stable and apparently unstable solutions with near-zero attractive and repulsive forces, respectively. Bedarkar and Wu 28 studied the capillary torque on the same geometry, using a similar method, explaining the role of capillarity on the 2 ACS Paragon Plus Environment

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alignment of wet rod-like particles and on the distortion of fiber networks in sensor grids and textiles. Claussen 29 proposed an analytical description for a liquid column between two identical filaments with a small finite crossing angle, and compared the results with numerical solutions, characterizing the capillary torque as a power law, which is valid for a very wide range of crossing angles. Sauret et al. 30 investigated the morphologies of a perfectly wetting fluid at the intersection of two fibers with a small crossing angle (a liquid column, a droplet, and a mixed morphology whereby a drop lies at the end of a column). Nevertheless, the effect of the surface-tension-driven deformation of the filaments was not considered. In this work, we study the capillary force and torque produced by a liquid bridge between two elastically-deformed filaments, calculating the equilibrium value via an iterative method. We derive a solution for parallel filaments using an analytical approximation for the liquid bridge between two parallel curved cylinders. This analytical solution is then compared with a numerical solution, which is then applied to crossed filaments. Our primary motivation is to calculate capillary forces between wood fibres in wet paper, which has been shown to account for its strength when containing less than 40% solids 31 . Such forces have been shown to bring the fibres into molecular contact during drying 19 .

Theory Parallel filaments Consider a liquid drop bridging two parallel filaments, as shown in figure 1, with diameters 2r and surface-to-surface distance 2d. At equilibrium, the liquid bridge must satisfy the Laplace condition on its surface, p0 − p = γ

1 1 + R1 R2

,


(1)

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y

π 2

−α −φ

R

α

φ r

O

z d

Figure 1: Left: Schematic of the cross-section of the liquid column between two parallel filaments showing the various geometrical parameters. Right: The three-dimensional visualization of a liquid bridge (left) and a liquid droplet (right) at the intersection of two crossing filaments. and the Dupre-Young condition on its contact line,

γ cos α = γsg − γsl ,

(2)

where R1 and R2 are the principal radii of curvature, p and p0 are the liquid and gas pressures, γsl , γsg , and γ are the solid-liquid, solid-gas, and liquid-gas interfacial tensions, and α is the equilibrium contact angle. These conditions can be derived from the principle of stationary potential energy under constant-volume variations 32 . The Laplace equation is a nonlinear partial differential equation, and is usually solved analytically in two-dimensional geometries, such as axisymmetric or constant cross-sectional interfaces. Princen 14 assumed a constant cross-section for a long liquid column that bridges two parallel horizontal filaments, deriving an analytical solution by balancing the increase in the free energy with the work of the capillary pressure when the liquid volume is increased infinitesimally. As shown in figure 1, the filament separation d = R cos(α + φ) + r cos φ − r,

(3)

where R is the liquid-surface radius of curvature, and φ is the half-opening angle of the solid-liquid interface. The Princen model has been modified for fibres with a small, finite


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crossing angle δ, where d varies slowly along the fibre axes. This modified model agrees with experimental 30 and numerical 29 results; here we extend it for slightly curved, parallel fibres with separation d(x) = R(x) cos[α + φ(x)] + r cos φ(x) − r,

(4)

where x denotes distance along the filament axis, as shown in figure 2.

fi

z

x

δ y Li −mi

L mi

x

δ  Li cos   2

Mi

Figure 2: Left: Filament deformation under the capillary force in the xz-plane and capillary torque in the xy-plane, with simple supports at each end. Right: Top view showing the crossing angle δ and the liquid-bridge length Li . Using the Princen formulation close to the terminal meniscus of the liquid column, we find

14

Re r

2

π 1 2Re 1 − β + sin(2β) + (sin φe cos β − φe cos α) + sin(2φe ) − φe = 0, 2 2 r 2

(5)

where β = α+φe , and Re and φe are the radius of curvature of the liquid surface and opening angle close to the terminal meniscus, respectively. Equations (4) and (5) furnish Re and φe , which together determine the interface terminal cross-sectional profile. The other principal curvature along the filament axis can be neglected, so R is constant along the liquid column, and φ(x) is calculated using Eqn. (4). A prescribed volume then determines the length of the column, and the maximum possible length is achieved when αe reaches its maximum value and the liquid column becomes unstable 14 .


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Crossing filaments For a crossing angle δ < 10◦ , Sauret et al. 30 used the modified Princen model to calculate the length and cross-sectional profile of long liquid columns. However, this failed to predict the liquid-surface shape with small Li /r, where Li is the length of the interface, as shown in figure 2, because the surface curvature along the column axis is not constant. This occurs when either the liquid volume decreases or δ > 10◦ . 30 For a small liquid bridge between parallel or crossing filaments, we use the Surface Evolver software 33 to compute the liquid-surface shape, energy, and capillary torque and force. This minimizes the total energy of a surface with several constraints. The surface is discretized into points and edges that form triangles termed facets, and the constraints are implemented using Lagrange multipliers. The initial geometry is a distorted rectangular prism that is continuously refined after converging at each step to form a uniform dense mesh, giving first-order convergence of the energy with respect to the number of facets. The convergence criterion is when the energy change at each iteration is less than 7 significant figures with more than 60 × 103 facets. Results from Surface Evolver with solid parallel cylinders are compared with the Princen model in figure 3. Note that the numerical results deviate from the theory when Li /d < 6. Furthermore, the Princen model predicts that a liquid column becomes unstable when √ √ d/r > 2 and α = 0, while Surface Evolver provides stable solutions with d/r > 2 when the length of the liquid column is small, but does not converge for long columns. In this paper, we adopt the Princen model when Li /d > 6. Another configuration of a liquid droplet on an array of parallel filaments is a barrelshaped droplet. Princen 14 experimentally showed the existence of these metastable config√ urations when 0.57 < d/r < 2, and Wu et al. 15 numerically showed a transition from a bridge to a droplet when increasing the liquid volume, with a critical volume that depends on the contact angle and filament separation. However, there is no analysis for this transition for crossing filaments, except a qualitative description by Claussen. 29 Therefore, based on 6 ACS Paragon Plus Environment

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Figure 3: Comparison of the analytical Princen model (solid lines) and numerical Surface Evolver (symbols) calculations of a liquid column between two parallel solid filaments showing the length Li , capillary pressure P , and surface energy Ui versus the filament separation d when α = 0, V /r3 = 10 (blue), and 100 (red). the results of Wu et al. 15 for parallel filaments, we limit this analysis to small liquid volumes, small filament separations, and α ≤ 90◦ , focusing on liquid bridges. The capillary force Fi and torque Mi are the surface-energy gradients with respect to separation and rotation, respectively, which we calculate using central finite-differences. The forces and torques were validated using the results of Bedarkar and Wu 28 and Virozub et al. 27

Elastic filaments When a bending moment M is applied to a linearly elastic filament, it satisfies

M = EI

d2 v . dx2

(6)

where EI is the flexural rigidity, v is the deflection, and x is defined in figure 2. To calculate M , the distribution of the capillary force on the filament must be computed. For example, using the modified Princen model for a long liquid column between curved parallel cylinders, the force on a differential element of the filament is

14

nr o dx dFi =2 sin φ(x) + sin[α + φ(x)] , γr R r


(7)

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where R is constant, and φ changes along the interface. The first and second terms on the right-hand side represent the contributions of the capillary pressure and the surface-tension force, respectively. Equation 7 is not valid for small liquid bridges. When the filaments are rotated and deflected, it becomes extremely difficult to compute the force distribution and the exact filament deformation. Figure 4 shows the shape of the solid-liquid interface when rotating crossed filaments. Note that the width of the contact region is almost constant along the filament axis when δ = 0◦ and 90◦ , except when close to the terminal meniscus. However, the contact region does not have uniform width when δ = 30◦ and 60◦ , since the solid-liquid interface deforms around the filament, producing an azimuthal force and an accompanying capillary torque.

Figure 4: Top and accompanying side views when increasing the crossing angle δ = 0◦ , 30◦ , 60◦ , and 90◦ (left to right) with α = 0◦ , V /r3 = 4, d0 /r = 0.5, Ω = 5, 000. To simplify the model, we adopt a linear force distribution in the xy- and xz-planes for the capillary torque and force, respectively, as shown in figure 2. Therefore, the force on a differential element is dFiz = fi dx, γr dFiy 2mi xdx = , γr Li 8 ACS Paragon Plus Environment

(8)

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where Fiz and Fiy are the capillary forces in z- and y-directions, respectively, giving

fi =

Fi , γLi

mi =

6Mi . γL2i

(9)

Although the uniform force distribution significantly reduces the computational cost, the errors are expected to decrease with increasing slenderness of the solid-liquid interface, and when Li is small compared to the filament length, thus limiting the approximation to large values of the elastocapillary number (defined below). The filament is on two simple supports that mimic the connections among fibres in a network, and L is a free length, where a fibre has no contact with other fibres. In this paper, we fix L/r = 15 to remove one degree of freedom and, thus, simplify the model.1 With this load distribution and boundary conditions, the filament deflection is calculated using Eqn. (6). The deflected-filament centerline along the liquid bridge is2 z(x) = b4 x4 + b2 x2 + b0 , (10) 5

3

y(x) = c5 x + c3 x + c1 x, where x, y, and z are normalized with r, and b4 = b2 = b0 = c5 = c3 = c1 =

fi , 24Ω fi Li (Li − 2L), 16Ωr2 fi Li (L3i − 4L2i L + 8L3 ), 4 384Ωr mi r , 60ΩLi mi Li (2Li − 3L), 72ΩLr mi L2i (40L2 − 45Li L + 12L2i ), 2880ΩLr3

1

(11)

Increasing L/r makes the filaments more slender, thus reducing the errors associated with Eqn. (6). However, this causes the model to depart from the wood-fibre networks that motivate this work. 2 A different solution for the filament deflection outside of the liquid bridge is calculated, and the two solutions are matched at the contact line.


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where Ω=

EI γr3

(12)

is the elasto-capillary number. The strain energy of the deflected filament is fi2 L2i m2i L4i Ue 3 2 3 = (5L − 5LL + 2L ) + (35L2 − 54LLi + 21L2i ). i i γr2 490Ωr5 30240LΩr5

(13)

An iterative method is used to compute the filament-liquid equilibrium. First, the values of Fi /(γr) and Mi /(γr2 ) for unstrained filaments are calculated to induce deflection, and then the deflected filaments are used to recalculate the force and torque until the absolute change becomes less than 0.01.

Results and discussion The modified Princen model for the elastic parallel filaments is compared with Surface Evolver computations in figure 5. With d/r = 0.1, the analytical and numerical results are in good agreement, but deviate when d/r = 0.5, either because the increased variation of d along the filaments violates the assumptions, or because the terminal meniscus becomes important. However, the modified Princen model is still a good approximation for d/r = 0.5, achieving less than 15% relative error in Fi /(γr), Li /r, and Ut /(γr2 ). Recall d/r is the initial separation of the unstrained filaments. Analytical results for parallel elastic filaments are shown in figure 6. With the stated elasto-capillary numbers and contact angles, the filaments contact each other at equilibrium when d/r = 0.1, and there is no contact when d/r = 0.5. The volume is increased until the length of the liquid column becomes close to the length of the filaments (L/r = 15). The capillary force continuously increases with volume, only reaching a maximum when d/r = 0.1 and α = 60◦ . Therefore, the capillary force among hydrophilic parallel filaments is unlikely to show a maximum with respect to the liquid volume. The length of the liquid column


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0.4 0.6 0.8 1 V /r 3

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Figure 5: The analytical Princen-model results (solid lines) of the capillary force, liquidcolumn length, and total energy versus the liquid volume for the parallel flexible filaments compared with results from Surface Evolver (symbols). Top: d/r = 0.1, Ω = 5, 000, α = 0◦ (blue), 30◦ (green), and 60◦ (red). Bottom: d/r = 0.6, Ω = 5, 000, α = 0◦ (blue), 30◦ (green), and 45◦ (red).


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increases with volume, as expected, but the total energy can increase or decrease depending on α and Ω. Furthermore, the capillary force and the length of the liquid column decrease when increasing Ω. This decrease is negligible for the attached filaments with d/r = 0.1, but is significant for detached filaments with d/r = 0.5.

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Figure 6: Analytical calculation of the capillary force, length of the liquid column, and total energy versus the liquid volume for parallel flexible filaments. Ω = 5, 000 (solid lines), 10, 000 (dashed lines), and 20, 000 (dash-dotted lines). Left: d/r = 0.1, α = 0◦ (blue), 30◦ (green), and 60◦ (red). Right: d/r = 0.6, α = 0◦ (blue), 30◦ (green), and 45◦ (red). Only the solid lines are shown then the solid, dashed, and dash-dotted lines are very close. The equilibrium capillary force of the crossing filaments when changing the elasto-capillary number is shown in figure 7. The black line shows the force required for the filaments to 12 ACS Paragon Plus Environment

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contact each other, so, filaments touch when the points are above this line. Note that, with d/r = 1.5, the black line is not shown because the forces are very small and there is no contact. From figure 7, the capillary force clearly decreases when increasing the contact angle and crossing angle. With α = 30◦ and d/r = 0.5, there is a rapid decrease in the force with increasing Ω when the filaments detach. This decrease is also observed when α = 60◦ , and with a smaller slope when d/r = 0.1. However, the capillary force reaches a plateau when Ω is large. With α = 90◦ , the force does not vary significantly when changing the crossing angle. Furthermore, it decreases and becomes repulsive when the filaments are very close (see d/r = 0.1). The numerical results for the equilibrium capillary force of the crossing filaments versus the liquid volume are shown in figure 8. Note that, except for d/r = 1.5, the force increases with decreasing volume, and, therefore, in contrast to parallel filaments, reaches a maximum because the capillary force vanishes at V = 0. This agrees with the experiments of de Oliveira et al. 34 for two wet paper sheets held together by capillary forces between (mostly) crossed wood fibres. This force goes through a maximum, and then vanishes as the water volume decreases. The maximum is obvious in figure 8 when α = 90◦ and δ = 45◦ , 90◦ . In these figures, the filaments touch when d/r = 0.1, and there is no contact when d/r = 1.5. When d/r = 0.5, the filaments are in contact above the black line, and there is no contact below the line; consequently, the maximum may exist in both conditions. The liquid volume of the maximum capillary force between the crossing filaments Vm is calculated with various d/r, Ω, α, and φ, and is plotted versus Ω in figure 9. With small Ω, when the filaments are deformed and attached, Vm slightly increases with Ω, but significantly increases when the filaments detach. Furthermore, increasing the filament separation slightly changes Vm if the filaments remain attached, but significantly increases Vm if they are detached. Filament rotation seems to have a weak influence on Vm , while increasing the contact angle decreases Vm . Persson et al. 19 showed that the capillary forces between wood fibres in a wet paper sheet


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Figure 7: Numerical calculation of the capillary force versus the elasto-capillary number for crossed flexible filaments. Top: d/r = 0.1, V /r3 = 1. Middle: d/r = 0.5, V /r3 = 4. Bottom: d/r = 1.5, V /r3 = 30. Solid lines: δ = 30◦ , dashed lines: δ = 60◦ , dash-dotted lines: δ = 90◦ . Blue: α = 0◦ , green: 45◦ , and red: 90◦ . The black lines show the critical force required for filament attachment; in the bottom panel, this line is very close to the vertical axis, and it is not shown.


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Figure 8: Numerical calculation of the capillary force versus the liquid volume for the crossed flexible filaments with Ω = 700. d/r = 0.1 (top), 0.5 (middle), and 1.5 (bottom). δ = 30◦ (solid lines), 60◦ (dashed lines), 90◦ (dash-dotted lines). α = 0◦ (blue), 45◦ (green), and 80◦ (red). The black line in the top-right panel shows the critical force required for filament attachment. Because of numerical instability, no solution could be obtained when d/r = 0.1 and δ = 90◦ , and with large volumes when d/r = 1.5 and δ = 60◦ .


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pull the fibres into closer contact, and conform their surface profiles by breaking and reforming hydrogen bonds. Our calculations may provide a better estimate of the capillary forces between the wood fibres at a specific solid content. The calculations are also important for finding the optimum weld-metal volume for a desired contact between flexible components. 1.5

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Ω

Figure 9: Numerical calculation of the liquid volume of the maximum force versus the elastocapillary number. The filled symbols correspond to attached filaments, and empty symbols are for detached filaments. Top: d/r = 0.1 and bottom: d/r = 0.6. Blue: δ = 45◦ , α = 0◦ . Black: δ = 90◦ , α = 0◦ . Green: δ = 45◦ , α = 45◦ . Red: δ = 90◦ , α = 45◦ .

Conclusions The equilibrium shape of a liquid bridge between two crossing or parallel filaments was calculated iteratively when the filaments are deformed by a capillary force and torque, which are the respective energy gradients with respect to separation and rotation. For parallel 16 ACS Paragon Plus Environment

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filaments, we used an analytical approximation for the liquid-surface shape and computed the equilibrium analytically, and the Surface Evolver software was also used to compute the liquid-surface shape. Numerical and analytical results agreed well. Surface Evolver was also used to compute liquid bridges between crossed filaments, as no analytical solution is available. In the investigated range of parameters, the capillary force decreases when decreasing the liquid volume for parallel filaments, but has a maximum for crossed filaments. The critical volume at which the peak force occurs slightly varies when the filaments are attached, but significantly changes with filament flexibility and separation when detached. This critical volume is a weak function of rotational alignment. A significant decrease in the capillary force with decreasing filament flexibility is predicted when the filaments detach. Finally, the results are consistent with experimental observations of the maximum capillary force between wet paper sheets, and may be useful for analysing the contact of flexible fibres.

Acknowledgement Supported by the NSERC Innovative Green Wood Fibre Network (R.J.H.) and a McGill Engineering Doctoral Award (M.S.).

References (1) Carroll, B. J. The accurate measurement of contact angle, phase contact areas, drop volume, and Laplace excess pressure in drop-on-fiber systems. J. Colloid Interface Sci. 1976, 57, 488–495. (2) Soleimani, M.; Hill, R. J.; van de Ven, T. G. M. Bubbles and Drops on Curved Surfaces. Langmuir 2013, 29, 14168–14177. (3) Fang, J.; Davoudi, M.; Chase, G. G. Drop movement along a fiber axis due to pressure driven air flow in a thin slit. Sep Purif Technol 2015, 140, 77–83.


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Force

Table of Contents Graphic

Force

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Liquid volume

Liquid volume


Capillary Force between Flexible Filaments | Langmuir

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