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Proc Math Phys Eng Sci. 2015 July 8; 471(2179): 20150184.
PMCID: PMC4528663

An elastocapillary model of wood-fibre collapse

Abstract

An elastocapillary model for drying-induced collapse is proposed. We consider a circular elastic membrane with a hole at the centre that is deformed by the capillary pressure of simply and doubly connected menisci. The membrane overlays a cylindrical cavity with rigid walls, trapping a prescribed volume of water. This geometry may be suitable for studying structural failures and stiction in micro-electromechanical systems during wet etching, where capillary surfaces experience catastrophic transitions. The dry state is determined using the dihedral-angle and volume-turning-point stability criteria. Open and collapsed conformations are predicted from the scaled hole radius, cavity aspect ratio, meniscus contact angle with the membrane and cavity walls, and an elastocapillary number measuring the membrane stretching rigidity relative to the water surface tension. For a given scaled hole radius and cavity aspect ratio, there is a critical elastocapillary number above which the system does not collapse upon drying. The critical elastocapillary number is weakly influenced by the contact angle over a wide range of the scaled hole radius, thus indicating a limitation of surface hydrophobization for controlling the dry-state conformation. The model is applied to the drying of wood fibres above the fibre saturation point, determining the conditions leading to collapse.

Keywords: elastocapillary, stability, wood-fibre collapse, drying, shrinkage

1. Introduction

Cellulose fibres are abundant natural resources with desirable characteristics, such as high-strength to weight ratio, corrosion-resistance, biodegradability and biocompatibility [13]. Besides their traditional application in papermaking, they have been used as reinforcements for plastic [4] and cement [5] composites, scaffolds for tissue engineering [6] and food-packaging materials [7]. The properties of fibre-based materials depend on wood-polymer characteristics and fibre conformation in the material microstructure.

Isolating fibres from parent wood by pulping is accompanied by dramatic conformational changes, the extent of which depends on the pulping process and the mechanical, physical and structural properties of fibres. The extremities of heating and chemical treatment in pulping also play an important role. Fibres from chemical pulping are delignified and usually collapsed, whereas fibres from thermomechanical pulping are lignin-coated and mostly uncollapsed, imparting contrasting properties to fibre-based composites [5]. For example, papers from chemical pulps have high tensile strength and smooth surfaces, suiting high-quality printing, while those from mechanical pulps have high compressive strength and rough surfaces, suiting fast, economical printing [2].

Wood-fibre deformation upon drying results from complex interactions between water and wood tissues in pulping [2]. The degree to which water is associated with wood can be quantified by differential scanning calorimetry [8,9]. Here, three types of water are identified [8]: (i) free water, having the same transition temperature as bulk water, (ii) freezing-bound (physisorbed) water, having lower transition temperature than bulk water, and (iii) non-freezing-bound (chemisorbed) water, which cannot be detected from the first-order transition. Molecules of non-freezing-bound water are directly bound to cellulose hydroxyl groups, whereas those of freezing-bound water and free water are trapped in the lumen and cell-wall micropores by capillary condensation. Noting that the heat of desorption required to break water-hydroxyl group bonds for non-freezing-bound water is much larger than the excess enthalpy of vaporization because of meniscus formation for freezing-bound and free water [2], drying stresses are expected to increase significantly at low moisture contents where the remaining water is mostly non-freezing bound.

Defining the fibre saturation point (FSP) as the moisture content where all the remaining water is adsorbed (non-freezing bound) [2] identifies two drying-deformation regimes [10]: (i) collapse above the FSP where fibre macro- and microscopic structure is affected and (ii) shrinkage below the FSP where cell-wall submicroscopic structure is affected. Inconsistent reports in the literature as to whether deformations upon drying begin above or below the FSP motivate further studies on wood-fibre collapse and shrinkage at various drying stages [1113].

Barber & Meylan [14] developed one of the first theoretical models to quantify shrinkage below the FSP. They considered a flat element of the cell wall, comprising only the S2 layer with cellulose microfibrils embedded in an amorphous matrix. This matrix shrinks isotropically with a linear relationship between normal strain and moisture content, while the cell-wall overall shrinkage is anisotropic because the matrix is constrained by crystalline microfibrils. This model was later extended to account for the effect of a cylindrical cell wall [15], a multi-layer cell wall [16], changing lumen shape during drying [12] and moisture-dependent mechanical properties [17]. Key observations from these studies include: (i) drying deformations and changes in the physical and mechanical properties of wood occur below the FSP, (ii) an anisotropic three-dimensional shrinkage is predicted with the radial component lower than the tangential component at small microfibril angles, agreeing with experiments [18,19], and (iii) volumetric shrinkage is proportional to the amount of water removed during drying.

In contrast with the foregoing studies, Hernández & Pontin [13] reviewed experimental evidence in the literature to show that drying deformations can begin at moisture contents well above the FSP. These are attributed to collapsed lumen and micropores resulting from an interplay between water surface tension and cell-wall elastic resistance [10,20]. In fact, non-collapsed fibres can be produced by drying solvent-exchanged pulps where water is substituted with a low-surface tension liquid before drying [2], supporting the latter proposition. Depending on the cell-wall microstructure, mechanical properties and pore size, a spectrum of collapsed, partially collapsed and uncollapsed pores are observed [2,10], implying that the overall volumetric shrinkage in this regime is not linearly related to the volume of removed water. Here, we consider an elastocapillary model of the lumen and cell-wall micropores, rigorously examining the interactions between elastic and capillary forces during drying to provide a better understanding of drying deformation at moisture contents above the FSP.

Elastocapillary phenomena have been extensively studied over the past two decades in several areas, such as capillary-induced wrinkling [21,22], micro-electromechanical systems [2325], capillary wrapping and origami [26], and self-assembly, coalescence and bundling of lamellae [2730]. Mastrangelo & Hsu [23] studied the capillary-driven deformation of beams and plates in micro-machined structures and derived collapse criteria based on the bending stiffness to surface tension ratio. Neglecting the meniscus contribution to the total energy, continuous equilibrium trajectories without stability exchange that connect the initial and collapsed states were identified as leading to collapsed conformations. A two-dimensional model of capillary rise between flexible sheets is given by Kim & Mahadevan [31] based on the linear bending theory of beams. Here, the final conformation is determined by a balance involving gravity, elastic and contact-line forces. This model was further examined by Kwon et al. [32] for systems in which conformations are controlled by the liquid content. Taroni & Vella [33] studied the equilibrium and stability of the same problem. Accounting for the meniscus shape, the Laplace pressure was also incorporated into the equilibrium equation of the beam. Moreover, multiple stable equilibria were found, and final conformations were determined through a dynamic analysis.

In this paper, we study an elastocapillary model of drying-induced collapse for cavities that are characterized by three length scales. The model comprises a circular elastic membrane with a hole at the centre (figure 1). The membrane is anchored above a rigid plate, trapping a prescribed volume of water. We examine membrane deformations caused by a meniscus that forms at the hole as liquid is removed. This furnishes an idealized structure where stability and equilibrium are determined by the elastic–capillary force interactions, providing collapse criteria for lumen and cell-wall micropores at moisture contents above the FSP. Neglecting the contact-line force, we adopt a variational formulation to examine stability and equilibria. Here, the meniscus geometry is exactly treated for the interfacial energies to account for the significant role of the Laplace pressure when collapsed conformations are approached. Using the stability criteria we recently derived based on a spectral analysis of the potential energy [34], the dry-state conformation is determined from the stability of equilibrium branches. The model is also relevant to structural failures and stiction in micro-electromechanical systems during wet etching, where capillary surfaces undergo catastrophic transitions. These transitions, which have not been addressed in previous studies, are modelled as bubble-to-bridge transitions (BBTs) to elucidate their role in the system stability and dry-state conformation.

Figure 1.
Elastocapillary model; (a) schematic showing simply connected meniscus (top), doubly connected meniscus (bottom), transition from simply to doubly connected meniscus (middle), and (b) contact angles. (Online version in colour.)

Previous elastocapillary models in the literature are mostly based on the linear theory of plates, where the bending contribution to the total potential energy is only considered [3133]. By contrast, the present model only considers the stretching energy, accounting for variable and anisotropic axial forces. The latter is more challenging because the equilibrium equations are nonlinear and the in-plane displacement is not neglected. Moreover, the model does not admit buckling, which typically occurs in geometries with closed structures, such as those of wood fibres. Buckling and the bending rigidity are additional resistances to deformation. Hence, the collapse criteria derived in this paper can be regarded as upper bounds, which are suitable for predicting open conformations from geometrical parameters and mechanical properties.

2. Theory

We consider an elastocapillary model shown in figure 1 comprising a circular elastic membrane with a hole at the centre supported on the sidewall of a cylindrical cavity with rigid walls, trapping a volume vl of liquid below the membrane and volume vg air between the bounding surface and membrane. Interfacial surface areas are denoted Γij. The cavity is open to the atmosphere from the top. A meniscus forms at the hole as the liquid (water) is removed, resulting in a difference between the liquid pressure pl and atmospheric pressure pg that causes the membrane to deform. Here, the membrane radius R, hole radius R0 and cylinder height H are the model length scales that control the interplay between elastic and capillary forces. To determine the equilibria at a given vl, we consider an imaginary bounding surface (dashed line in figure 1a) that covers the cavity from the top. The system is completely isolated from the surrounding by the bounding surface and cylinder walls. The meniscus is initially a bubble, which can bridge the gap upon contact with the plate at the bottom of the cylinder, forming a free contact line with the plate. The resulting surface Γsg is assumed to be an ideal solid–gas interface that is free of liquid. Here, θc and θd are the thermodynamic contact angle and dihedral angle that the meniscus forms with the plate and membrane, respectively. Assuming that all the dimensions are small compared with the capillary length, the gravity force is neglected. The membrane and meniscus are assumed axisymmetric in equilibrium and perturbed configurations. Drying dynamics are assumed to be slow (quasi-static), so that the system evolves through a sequence of equilibrium states. We also assume that the inner surfaces of the plate and membrane that are in contact with water produce the same thermodynamic contact angle θc.

We derived a variational principle for the stability and equilibrium of the elastocapillary model shown in figure 1 [34]. Neglecting the bending contribution to the elastic strain energy, the membrane in-plane and out-of-plane displacement profiles are (see appendix A)

w~(r~p)=w~0(1r~p2)
2.1

and

u~(r~p)=w~02(1r~p2)4r~p(3ν)+κ2[(3ν)(1+ν)κ2(1ν2)]r~p[1ν+κ2(1+ν)],
2.2

where r~p, u~ and w~ are the radial position in the referential coordinate rp, in-plane displacement u and out-of-plane displacement w, all scaled with R; moreover, w~0=(Qc/NC)1/3Kw, and

Kw=3[1ν+κ2(1+ν)]κ(1κ2)(1ν2)[7ν+κ2(1+ν)]1/3.
2.3

Here, κ=R00/R, Qc=PR00/γgl and NC=C/γgl are the scaled hole radius, scaled capillary pressure and elastocapillary number where P=plpg and C, ν, γgl, R00 are the membrane axial rigidity, Poisson ratio, air–water surface tension and hole radius in the referential configuration. Equations (2.1) and (2.2) are derived using von Kármán's plate theory, which requires dw/drp~H/R[double less-than sign]1 [34]. Therefore, we restrict our analysis to cases for which H/R[double less-than sign]1, so that the membrane displacements can be accurately approximated by equations (2.1) and (2.2).

To compute the liquid volume vl, meniscus meridian curve and minimum gap between the membrane and plate h, we neglect the in-plane displacement, which is a reasonable approximation when H/R[double less-than sign]1 [34]. Neglecting the in-plane displacement implies that the hole expansion resulting from the membrane deflection is negligible (R0R00). Furthermore, one must ensure that a given membrane profile corresponding to the capillary pressure of a meniscus does not violate any geometrical constraint. Specifically, the sum of the membrane deflection and the gap between the membrane and plate at the hole edge must equal H, and volume bounded by the meniscus and membrane must equal the liquid volume. These constraints are H=hw(R00) and vl+vg=πR2h, which upon substituting equation (2.1) furnish

κΛΠQcNC1/3Kw(1κ2)=0,
2.4

and

34+3(1κ2)8κ2Λ+3(1κ2)8κ3Πv^av^l=0,
2.5

where Λ=h/R00, Π=H/R and va are the slenderness, aspect ratio and the volume of the air between the meniscus and the plane z=h, and v^i=vi/(4πR003/3).

The existence of a continuous and stable solution branch from the fully saturated state (h=H) to the collapsed state (h=0) is a necessary condition for the elastocapillary system in figure 1 to collapse. Akbari et al. [34] demonstrated that this system loses stability during drying at volume turning points,1 provided the meniscus is pinned to the hole edge and Π23/4(1κ2). However, assuming that menisci are pinned to the hole edge implies that the contact line is restrained by an external force. This assumption can be relaxed by permitting the contact line to move if it is energetically favourable to do so, which leads to the necessary condition θd+β0>θc for menisci to be stably pinned to the hole edge [35], provided the meniscus contact angles with the membrane and plate are the same. Here, β0 measures the membrane slope at the hole edge (figure 1b), which is negligible because tanβ0w~(κ)Πκ1. This simplifies the foregoing stability criterion to

θd>θc.
2.6

Note that, unlike θc, equilibrium imposes no restriction on θd, so θd is calculated as a part of the solution. Moreover, when the meniscus is a bridge, equilibrium solutions of the meniscus meridian curve can intersect the boundaries or be self-intersecting [36]. These menisci are clearly non-physical and must be excluded before examining the stability of equilibrium branches.

The foregoing stability criteria only ensure that the membrane can contact the plate, which corresponds to the ‘contact bound’ defined by Mastrangelo & Hsu [23]. However, these are not sufficient for predicting the dry-state conformation. Accounting for adhesion energy between the membrane and plate is also required to determine whether the membrane and plate remain attached when dry [24]. In the remainder of the paper, assuming that the adhesion energy is always strong enough to maintain the membrane and plate attached upon contact, the dry state is identified as collapsed if there exists a continuous solution branch without volume turning points and non-physical menisci from h=H to 0 along which equation (2.6) is satisfied.

We solve nonlinear systems of equations using Newton–Raphson-based predictor–corrector techniques. Solution branches are constructed using Keller's arclength continuation method [37]. Branch continuation begins from a state that either has an analytical solution or can be readily constructed (i.e. cases with low sensitivity to initial guess and problem parameters) and terminates when the solution does not satisfy the geometrical constraints (e.g. negative v^a or Λ).

3. Results and discussion

We determine the dry-state conformation in the parameter space (κ,NC,θc) at fixed ν and Π. Drying from the fully saturated state, the elastocapillary system of figure 1 must undergo three deformation phases to collapse: (i) The initial phase where the meniscus is a bubble (simply connected), which can bridge the membrane and plate when the bubble apex reaches the plate, (ii) the transition phase where the bubble transforms into a bridge at fixed liquid volume v*l, and (iii) the final phase where the meniscus is a bridge (doubly connected) with the slenderness approaching zero as vl decreases further below v*l.

(a) Initial phase

Here, the meniscus is a truncated sphere because gravity is neglected. Accordingly, the scaled bubble volume and capillary pressure are

v^a=2+3cosθdcos3θd4sin3θd
3.1

and

Qc=2sinθd.
3.2

The system state at any v^l does not depend on θc. Solving equations (2.4), (2.5), (3.1) and (3.2), the state is specified for a given v^l, κ and NC, where the dependent variables are (Λ,θd,Qc,v^a). The tangent bubble and rigid membrane, shown in figure 2, are special states in this phase that are determined by Λ=cot(θd/2) and Λlim=cot(θdlim/2)=Π/κ, respectively. Here, the superscripts ‘*’ and ‘lim’ denote variables in the tangent-bubble and rigid-membrane states. The latter corresponds to NC, where the membrane is rigid and does not deform.

Figure 2.
Special states for simply connected menisci; (a) tangent bubble and (b) rigid membrane (NC). (Online version in colour.)

Note that the dependent variables in the tangent-bubble state are (Λ,θd,Qc,v^a,v^l), and the system state is specified by fixing κ and NC. Choosing θ*d as the solution norm, we first determine equilibria at the tangent-bubble state in the space (κ,NC,θd), as shown in figure 3. At fixed κ, there is a critical κ below which the tangent-sphere state has three equilibrium solutions for a given NC. Similarly, at fixed NC, there is a critical NC above which the tangent-sphere state has three equilibrium solutions for a given κ. Here, the stability of the equilibrium branch from the fully-saturated to tangent-bubble state must be examined to determine which equilibrium solution is realized for given (κ,NC). Hereafter, we refer to equilibrium branches parametrized with vl as ‘drying trajectories’, which can be conveniently represented by xl, Qc, θd and Λ. Note that

xl=vl(πR2H)
3.3

is a volume fraction, measuring the liquid content relative to the fully saturated state.

Figure 3.
Equilibria at the tangent-bubble state (figure 2a) with ν=0.3, Π= 0.2. Circles indicate the critical point where ([partial differential]NC/[partial differential]θ*d)κ and (2NC/θd2)κ=0; (a) constant-κ ...

Figure 4 illustrates a drying trajectory corresponding to (κ,NC) for which the tangent-bubble state has one equilibrium solution. The equilibrium branch has no turning point in vl, so no stability exchange occurs from the fully saturated state to the tangent-bubble state. At the tangent-bubble state, θ*d[similar, equals]110°. Note that θd monotonically decreases with decreasing vl. Therefore, when θc>110°, the contact line is detached from the hole edge before the bubble can bridge the membrane and plate according to equation (2.6), implying that the dry-state conformation is open. Figure 5 shows a drying trajectory where the tangent-bubble state has three equilibrium solutions. The equilibria at B and C are past the turning point in vl on an unstable branch [34], and, thus, cannot be practically realized. The equilibrium at A, where θ*d[similar, equals]121°, is on a stable branch; thus, drying from the fully saturated state, the tangent-bubble state at A can be practically realized. Similarly to figure 4, when θc>121°, the contact line is detached from the hole edge before the bubble can bridge the membrane and plate.

Figure 4.
(a) Drying trajectory with ν=0.3, Π=0.2, κ=0.15 and NC=5000. Circle indicates the tangent-bubble state at xl[similar, equals]0.7505. (b) Conformations along the drying trajectory at xl=0.95, 0.9, 0.85, 0.8, 0.78, 0.76 and 0.7505 (downward). ...
Figure 5.
(a) Drying trajectory with ν=0.3, Π=0.2, κ=0.01 and NC=8000. Circles indicate multiple equilibria at the tangent-bubble state. (b) Conformations at the tangent-bubble state corresponding to xl[similar, equals]0.5140, 0.5932 and 0.6410 ...

To determine the region in (κ,NC) in which the tangent-bubble state is reached before stability is lost at a volume turning point, we determine tangent-bubble states that coincide with volume turning points (thin-dashed lines in figure 6). These states are determined by (v^l/θd)κ,NC=0, where (Λ,θd,Qc,v^a,v^l,NC) are specified for a given κ. Moreover, at fixed κ, there is a critical NC, determined by (v^l/θd)κ,NC=0 and (2v^l/θd2)κ,NC=0, at which the two volume turning points on drying trajectories coalesce into an inflection point; above this elastocapillary number, drying trajectories have no volume turning points and no stability exchange occurs along equilibrium branches. These criteria define another state (thick-dashed lines in figure 6) that is specified for a given κ, similarly to the tangent-bubble state at volume turning point. The shaded region in figure 6a, constructed based on the foregoing states, furnishes a range of (κ,NC) for which the tangent-bubble state is reached before the volume turning point. However, one must also ensure that equation (2.6) is satisfied along the drying trajectory from the fully saturated to tangent-bubble state. The shaded regions in figure 6bf are subregions of the shaded region in figure 6a in which the dihedral-angle and turning-point stability criteria are simultaneously satisfied, therefore, providing a range of (κ,NC) in which the tangent-bubble state can be realized.

Figure 6.
Phase diagrams providing upper bounds on the region of collapsed conformation in the parameter space from the locus of volume turning points coinciding with the tangent-bubble state (thin-dashed), volume inflection points (thick-dashed) and the dihedral-angle ...

The tangent-bubble state coincides with volume turning points only for θ*d<π/2. To prove this, we choose θd as an independent variable, so that v^l=v^l(κ,NC,θd) and Λ=Λ(κ,NC,θd) for the equilibria from the fully saturated to tangent-bubble state. Differentiating equation (2.5) with respect to θd furnishes

v^lθdκ,NC=34+3(1κ2)8κ2Λθdκ,NCdv^adθd,
3.4

leading to

v^aΛκ,NC=34+3(1κ2)8κ2>0
3.5

at volume turning points. Because Λ decreases (increases) with increasing v^l when the bubble is smaller (larger) than a hemisphere, it follows that (v^a/Λ)κ,NC<0 ((v^a/Λ)κ,NC>0) when θd>π/2 (θd<π/2). Noting that equation (3.5) holds at all volume turning points, specifically those coinciding with the tangent-bubble state, it follows that θ*d<π/2 along the thin-dashed lines in figure 6.

(b) Transition phase

When the bubble reaches the plate in the tangent-bubble state, it transforms into a doubly connected meniscus by bridging the membrane and plate. The dynamics of such transformations are comparable to those of stability loss in liquid-bridge breakup [38,39], so the time scale of the BBT is very small compared with that of drying. Therefore, this transition is volume preserving (vl=const.), and the state after the transition is determined by Akbari et al. [36]

1+a2+2acosτ0|Qc|=0,
3.6

τ0τ11+acost1+a2+2acostdt+ΛQc=0,
3.7

sign(Qc)1+acosτ0asinτ0tanθd=0,
3.8

sign(Qc)1+acosτ1asinτ1+tanθc=0
3.9

andτ0τ11+a2+2acost(1+acost)dt+43Qc3v^a=0.
3.10

Equation (2.5) with v^l=v^l(κ,NC), and equation (2.4), furnishing the dependent variables (Qc,τ0,τ1,a,θd,Λ,v^a). Here, the superscript ‘**’ denotes variables after the BBT, and the system state is specified for a given κ, NC and θc. Note that τ0 and τ1 are, respectively, the mean-curvature-scaled arclengths at the hole edge and the bridge contact line with the plate, and a is the bridge shape parameter [36,4042].

The catenoidal bridge and rigid membrane are special limiting states in this phase, as shown in figure 7. After the transition, the meniscus mean curvature is zero (Qc→0), and the membrane is undeflected in the catenoid limit. The dependent variables in this state (θdcat,v^acat,NC) are determined by

1s^02+1lns^1+s^12+1s^0+s^02+1Πκ=0
3.11

and

s^1s^12+1s^0s^02+1+(Π/κ)s^02+1(s^02+1)3/28v^a3=0,
3.12

and equation (2.5) with v^l=v^l(κ,NC). Moreover, s^0=cotθdcat, s^1=cotθc [43], Qcatc=0 and Λcat=Π/κ with the superscript ‘cat’ denoting variables in the catenoid limit. This state is specified by fixing κ and θc. Similarly to the initial phase, the rigid-membrane limit (NC) is determined by Λlim=cot(θdlim/2)=Π/κ, where the membrane is not deformed by the capillary pressure.

Figure 7.
Special states for doubly connected menisci; (a) catenoidal bridge (Qc→0) and (b) rigid membrane (NC). (Online version in colour.)

Figure 8a shows the system state after the BBT at fixed θc in the catenoid limit. For a given θc, the catenoidal-bridge state separates (κ,NC) on the left for which Qc<0 from those on the right for which Qc>0 after the transition. This contrasts with simply connected menisci, where Qc is always negative. The membrane is deflected downward (upward) when Qc is negative (positive), so (κ,NC) on the right of the catenoid curves are desirable if collapse is to be prevented. Moreover, no equilibrium solution exists for the catenoid state in figure 8a when θc30. This is because at any given θc there is a slenderness Λmax above which no catenoid exists [43]. Since Λmax0.2 at θc=30°, and noting that Λmax monotonically decreases with decreasing θc [43], κmin=Π/Λmax1 at Π=0.2 when θc30, which is not physically possible. Therefore, hydrophilically modifying the membrane and plate enlarges the undesirable region of (κ,NC), which can span the entire parameter space if θc becomes smaller than a critical value corresponding to a given Π.

Figure 8.
System state after the BBT with ν=0.3 and Π=0.2; the bridge after the transition is (a) a catenoid and (b) at the dihedral-angle stability limit (θd=θc) according to equation (2.6). Dashed line in the ...

Figure 8b shows the dihedral-angle stability limit at fixed θc with θd=θc. Here, NC is a dependent variable, and the state is specified for a given κ and θc. For a fixed θc, there is a κ at which NC at the stability limit, corresponding to the rigid-membrane limit (figure 7b). When θc<90°, (κ,NC) in the complement of the region acba correspond to states in which the bridge after the transition satisfies equation (2.6). However, this behaviour changes when θc>90°. For example, at θc=120°, the curve corresponding to the dihedral-angle stability limit intersects the catenoid curve for the same contact angle at E. Here, (κ,NC) in the region dEfd correspond to states in which the bridge after the transition satisfies equation (2.6) and Qc<0. Interesting to note is the opposite behaviour of the dihedral-angle stability region for hydrophobic and hydrophilic surfaces. For hydrophobic surfaces, the stability region spans an area of the parameter space in which κ and NC are large. As will be discussed in the next section, when the meniscus is a bridge, the elastocapillary system tends to lose stability at a volume turning point before collapse at large NC. By contrast, for hydrophilic surfaces, decreasing θc enlarges the stability region, which spans a wide range of large and small κ and NC. Therefore, hydrophilically modifying the membrane and plate broadens the range of κ and NC in which the dihedral-angle and volume-turning-point stability criteria are simultaneously satisfied during drying, thereby favouring collapsed conformations.

(c) Final phase

Similarly to the initial phase, drying trajectories in this phase, where the meniscus is a bridge, describe how the system evolves with decreasing vl. One must determine whether stability is lost from the state following the foregoing BBT to the collapsed state along drying trajectories. Thus, equilibrium branches, parametrized with vl, must be constructed to identify volume turning points on drying trajectories. This is accomplished by solving equations (2.4), (2.5) and (3.6)–(3.10), providing the dependent variables (Qc,τ0,τ1,a,θd,Λ,v^a) for a given v^l, κ, NC and θc. We first consider the collapsed state, which is identified by Λ=0, furnishing

Qccol=ΠNC[Kw(1κ2)]3,
3.13

cosτ0col=1+tanθc(acol)2(1+tan2θc)11+tan2θc
3.14

andv^lcol=3(1κ2)Π8κ3andxlcol=12(1κ2),
3.15

where acol is a solution of

1+(acol)2+2+2tanθc(acol)2(1+tan2θc)11+tan2θc(Qccol)2=0.
3.16

Note that, in the limit Λ→0, the meniscus is vanishingly small and the meniscus meridian curve is linear. Consequently, τ1col=τ0col, θcold=πθc and v^acol=0, where the superscript ‘col’ denotes variables in the collapsed state. Furthermore, θcoldθc when θcπ/2, implying that, according to equation (2.6), the collapsed state is always unstable for hydrophobic surfaces. Therefore, in the remainder of this section, we focus on hydrophilic surfaces with θc<π/2.

At a fixed κ, there is a critical NC below which equation (2.6) is satisfied along the entire drying trajectory, from the state after the BBT to the collapsed state. This critical NC corresponds to a state, which we refer to as the dihedral-angle turning point limit, where θd=θc at a dihedral-angle turning point where the dependent parameters (Qcdtp,τ0dtp,τ1dtp,adtp,Λdtp,v^adtp,v^ldtp,NC) are determined by (θd/v^l)κ,NC,θc=0, equations (2.4), (2.5) and (3.6)–(3.10). Here, the superscript ‘dtp’ denotes variables in this state, which is specified for a given κ and θc. When the meniscus is a bridge, this state (thin-solid lines in figure 9) provides a boundary in the parameter space, separating (κ,NC) for which equation (2.6) is satisfied along the drying trajectory from those for which equation (2.6) is not satisfied on a segment of the drying trajectory. Overlaying this stability region with those constructed in figures 6 and and88b provides a region in the parameter space in which the volume-turning-point and dihedral-angle stability criteria are satisfied along drying trajectories from the fully saturated to collapsed state, except the volume-turning-point stability criterion for doubly connected menisci. Moreover, one must ensure that all menisci are physically realizable along drying trajectories. The resulting region is, therefore, an upper bound on the region of collapsed conformations in the parameter space.

Figure 9.
Phase diagrams providing upper bounds on the region of collapsed conformation in the parameter space from the locus of volume turning points coinciding with the tangent-bubble state (dotted), volume inflection points when the meniscus is a bubble (thick-solid), ...

Figure 9 illustrates the forgoing region for θc=30° and 60°. The curve corresponding to the dihedral-angle turning point limit itself has a turning point in κ at κ[similar, equals]0.0006 and 0.0062 for θc=30° and 60°, respectively. When the meniscus is a bridge and below these values, equation (2.6) is satisfied along the entire drying trajectory for all NC, and the dry-state conformation is determined only by the volume-turning-point stability criterion. This appears as a sudden jump in the boundary of the shaded areas in figure 9, from the dihedral-angle stability limit at the tangent-bubble state (dash-dotted line) to the dihedral-angle turning point limit (thin-solid line). Moreover, it is clear from figure 9 that, except for κ[double less-than sign]1, the stability region is not significantly influenced by the contact angle, and NC(ν=0.3,Π=0.2)~104 furnishes an upper bound on the critical elastocapillary number, separating open and collapsed conformations the elastocapillary system of figure 1 assumes upon drying.

We illustrate how to examine the volume-turning-point stability criterion from the state after the BBT to the collapsed state by an example. Drying trajectories are shown in volume versus pressure and dihedral angle versus slenderness diagrams in figures 10 and and1111 when ν=0.3, Π=0.2, κ=0.1 and θc=60°. The former is suitable for examining the volume-turning-point stability criterion as stability can be related to the slope of drying trajectories [34], whereas the latter is useful for ascertaining the dihedral-angle stability criterion along the drying trajectories. When κ=0.1, the dihedral-angle stability limit after the BBT has the solutions logNC3.7538 and 6.0523 (figure 8b), and the dihedral-angle turning point occurs at logNC3.4576 (thin-solid line in figure 9b). As shown in figure 10, the state after the BBT and collapsed state lie on disconnected branches at logNC=3.75 and 3.6, and the system loses stability at a volume turning point before collapse. Decreasing the elastocapillary number to logNC=3.55, the two disconnected segments merge, forming a continuous solution branch from the state after the BBT to the collapsed state. However, equation (2.6) is not satisfied on the middle part of the branch (figure 11c), and the contact line is detached from the hole edge before collapse. When the elastocapillary number drops below logNC3.4576, for example, at logNC=3 (figures 10e and and1111e), both stability criteria are satisfied simultaneously, and the system collapses upon drying. Therefore, in this example, the stability, and, thus, dry-state conformation of the system is determined by the dihedral-angle turning point limit.

Figure 10.
Drying trajectories in volume versus pressure diagrams for doubly connected menisci with ν=0.3, Π=0.2, κ=0.1, θc=60° and (a) logNC=3.75, (b) logNC=3.6, (c) logNC=3.55, (d) log ...
Figure 11.
Drying trajectories in dihedral angle versus slenderness diagrams for doubly connected menisci with ν=0.3, Π=0.2, κ=0.1, θc=60° and (a) logNC=3.75, (b) logNC=3.6, (c) logNC=3.55, (d) ...

Figures 10f and and1111f show the drying trajectory when logNC=6.1. Although the corresponding point in the parameter space lies in the dihedral-angle stability region after the BBT (figure 8b), the state after the BBT is unstable because the corresponding point lies on an unstable branch, according to lemmas 2 and 3 of Akbari et al. [34]. The dihedral-angle stability criterion is not satisfied along the entire branch, and the system does not evolve towards the collapsed state. In this case, stability is lost during the BBT.

We conclude this section by demonstrating how the elasticapillary system of figure 1 evolves along drying trajectories that lead to open and collapsed conformations upon drying. Figure 12 illustrate the drying trajectory at κ=0.1, logNC=3.6 and θc=60°. These parameters correspond to a point inside the shaded region in figure 6c and outside the shaded region in figure 9b. Starting from the fully saturated state at A and removing water (decreasing xl), the system follows the dashed line until the bubble reaches the plate at B, corresponding to the tangent-bubble state, where Λ*[similar, equals]0.8097, θ*d[similar, equals]102° and xl0.6970. As predicted in figure 6c, the tangent-bubble state is on a stable branch, between the volume turning point and fully saturated state, where equation (2.6) is satisfied. Then, the system undergoes a volume-preserving BBT (xl=const.) to the state C where Λ**[similar, equals]0.8165 and θ**d[similar, equals]73°, which also satisfies equation (2.6) (figure 8b). According to lemmas 2 and 3 of Akbari et al. [34], this state is on a stable equilibrium branch (figure 10b). Decreasing xl below x*l, the system follows the solid line and reaches the dihedral-angle stability limit at D where Λ[similar, equals]0.7601, θd=60° and xl[similar, equals]0.6825. Beyond this point, the contact line does not remain pinned to the hole edge with decreasing xl, as predicted in figure 9b. Note that the drying trajectory from C to the collapsed state at G is not continuous, and the system loses stability at the volume turning point E before collapse, even if the contact line is maintained pinned to the hole edge by an external force. The dry-state conformation is consequently open.

Figure 12.
System conformations along a drying trajectory when the meniscus is a bubble (dashed) and bridge (solid) with ν=0.3, Π=0.2, κ=0.1, θc=60° and logNC=3.6. (Online version in colour.)

Figure 13 shows the drying trajectory at κ=0.1, logNC=3 and θc=60°. These parameters correspond to a point inside the shaded regions in figures 6c and and99b. Here, the system behaves similarly to the previous case (figure 12) until the point C. According to lemmas 2 and 3 of Akbari et al. [34], the corresponding state is stable because C lies on a stable equilibrium branch (figure 10e). However, the drying trajectory has a turning point in θd at F, and no turning point in vl from C to the collapsed state at G. As predicted in figure 9b, θd>θc along the entire trajectory from the fully saturated to collapsed state, and stability is not lost with respect to the dihedral-angle stability criterion. Moreover, the drying trajectory continuously connects C to G, implying that both stability criteria are satisfied along the entire trajectory from A to G, and the dry-state conformation is collapsed. Note also that in figures 12 and and13,13, all menisci are physically realizable from the state after the BBT at C to the collapsed state at G.

Figure 13.
System conformations along a drying trajectory when the meniscus is a bubble (dashed) and bridge (solid) with ν=0.3, Π=0.2, κ=0.1, θc=60° and logNC=3. (Online version in colour.)

4. Wood-fibre lumen collapse

Here, we apply the elastocapillary model to lumen collapse of wood fibres, providing a quantitative estimate of the critical NC, determining the dry-state conformation. The model has the length scales R0, R and H, which we adopt to characterize the meniscus curvature, deformation length and maximum deflection. Wood-fibre geometry, mechanical properties and ultrastructure have been extensively studied for various species in the literature in the past few decades [2]. Here, we choose Norway spruce as an example with which to estimate the foregoing length scales (figure 14) and approximate the critical NC.

Figure 14.
Schematic representation of wood fibres, highlighting the characteristic length scales.

Cristian Neagu et al. [44] reported the average lumen width W=H+2b≈20 μm for early and late wood, and Yan & Li [45] measured an elastic modulus E≈1.4 GPa and wall thickness b≈1.4 μm for Kraft spruce. Sirviö & Kärenlampi [46] reported the normalized size of bordered pits R0/H≈0.075 with an average longitudinal spacing δ≈10 μm. Taking Rδ/2, we find κ≈0.22, Π≈3.4 and NC≈30 500. However, from figure 9, NC≈3700,2300 are the upper bounds on elastocapillary numbers for which the dry-state conformation is collapsed when Π=0.2 and θc=30°, 60°. Noting that the critical NC decreases with increasing Π, it follows that the capillary pressure of menisci, having length scales of the same order as the pit-hole diameter and lumen width, is smaller by at least an order of magnitude to overcome the cell-wall stretching resistance and induce collapse.

5. Concluding remarks

We have developed an elastocapillary model of drying-induced collapse to predict the dry-state conformation from geometrical and mechanical properties. This model simulates drying-induced structural deformations arising in cavities with three characteristic length scales, providing a low-dimensional model to describe the collapse of lumen or cell-wall micropores at moisture contents above the FSP. The dry-state conformation was determined based on the dihedral-angle and volume-turning-point stability criteria. Fixing the cavity aspect ratio, an upper bound on the critical elastocapillary number, corresponding to a scaled hole radius and contact angle, was provided, above which the system does not collapse upon drying. The critical elastocapillary number has a weak dependance on the contact angle over a wide range of the scaled hole radius, indicating a limitation of surface hydrophobization for controlling the dry-state conformation. Accounting for the BBT on the system stability and dry-state conformation, this model is also relevant to structural failures and stiction in micro-electromechanical systems during wet etching.

Applying the model to lumen collapse based on the structural and mechanical properties of Norway spruce fibres revealed that the capillary pressure of menisci, spanning pit-hole openings and/or the lumen width, is not strong enough to overcome the cell-wall stretching resistance, so is unlikely to cause collapse. However, the capillary-induced collapse of cell-wall micropores is yet to be understood. Granted, this is more challenging because cell-wall microstructural properties, such as pore geometry and pore-size distribution have not been established in the literature. Nevertheless, the nonlinear trend of the overall volumetric shrinkage versus moisture content observed above the FSP [13] as compared with the linear trend below the FSP can serve as a suggestive piece of evidence. This nonlinear trend signifies deformations in the collapse regime and may be attributed to the capillary-induced collapse of micropores in a transition phase of drying when the moisture content approaches the FSP from above. Understanding the transition between the collapse and shrinkage regimes during drying demands reproducible and reliable characterizations of cell-wall microstructural properties to provide an accurate prediction of the dry-state conformation.

We considered an idealized model of drying-induced collapse in wood fibres, capturing the basic elements of the underlying physics. The bending contribution to the total potential energy and membrane profile were neglected, limiting the accuracy for thick-walled and slender cavities. Accounting for the bending energy requires a more complicated test function for the membrane deflection in the bending- and stretching-dominated regimes. However, implementing complicated test functions is computationally expensive and algorithmically challenging. Moreover, our model does not account for geometrical complexities of cavities in wood fibres. Specifically, the lumen and cell-wall micropores have a closed structure, exhibiting a buckling resistance to compressive loads. However, the membrane in the present model can continuously deform from the fully saturated to collapsed state without buckling. The critical elastocapillary number is consequently smaller than predicted in this work. Nevertheless, our results furnish an upper bound on the critical elastocapillary number, and can reasonably estimate the range of parameters for which the dry-state conformation is open. More realistic geometries will surely improve quantitative predictions, but at higher computational cost. We hope that this study will motivate future investigations into the underlying mechanisms of wood–water interactions, to improve understanding of drying deformation in the shrinkage and collapse regimes over the full moisture-content range.

Acknowledgements

The authors gratefully acknowledge financial support from the NSERC Innovative Green Wood Fibre Products Network and the McGill University Faculty of Engineering.

Appendix A

The in-plane and out-of-plane equations of equilibrium for the membrane in the elastocapillary model of figure 1 are [34]

rpNrr+NrrNttP(rp+u)w=0
A 1

and

(Nrrwrp)+P(rp+u)(1+u)=0
A 2

with boundary conditions

Nrr=0atrp=R00
A 3

and

u=0,w=0atrp=R,
A 4

where

Nrr=Cu~+w~22+νu~r~p
A 5

and

Ntt=Cνu~+νw~22+u~r~p.
A 6

As we focus on cases where Π[double less-than sign]1, we have w′,u[double less-than sign]1, and the capillary pressure can be regarded as acting in the z-direction, furnishing

rpNrr+NrrNtt=0
A 7

and

(Nrrwrp)+Prp=0.
A 8

The above equations are nonlinear and do not have tractable closed-form solutions. Note that the nonlinearity can be removed if Nrr is known in equation (A 8). Thus, as a first approximation, w is estimated from the out-of-plane equilibrium by substituting the radial distribution of the axial force from a simpler problem: undeflected circular plates under boundary tension, such that the boundary conditions equations (A 3) and (A 4) are satisfied. The radial and tangential components of the axial force in this problem are [47]

Nrr=Arp+B
A 9

and

Ntt=Arp+B
A 10

with A=TR2R002/(R2R002) and B=TR2/(R2R002), where T is the radial tension at rp=R. Solving equation (A 8) using the radial component of the axial force in equation (A 9) furnishes

w~(r~p)=w~0(1r~p2).
A 11

The radial distribution of the in-plane displacement u~(r~p), given by equation (2.2), is obtained by solving equation (A 7) using equation (A 11).

We use equation (A 11) as a test function to derive a variational approximation for the membrane equilibrium. The total potential energy ΩT comprises the membrane stretching energy

ΩS=2πC2R00Ru2+uw2+2νuurp+νuw2rp+u2rp2+w44rpdrp
A 12

and the work of the capillary pressure

ΩP=2πR00RPwrpdrp.
A 13

Introducing the dimensionless energies Ωi=Ωi/(2πCR2) and the scaling forms

w~(r~p)=w~0w¯(r~p),
A 14

u~(r~p)=w~02u¯(r~p),
A 15

ΩS=w~04Ω¯S
A 16

andΩP=w~0Ω¯P,
A 17

the total potential energy can be written

ΩT=w~04Ω¯S+w~0Ω¯P,
A 18

where

Ω¯S=12κ1u¯2+u¯w¯2+2νu¯u¯r~p+νu¯w¯2r~p+u¯2r~p2+w¯44r~pdr~p
A 19

and

Ω¯P=QcκNCκ1w¯r~pdr~p.
A 20

Substituting u¯ and w¯ from equations (2.1) and (2.2) in equations (A 19) and (A 20) leads to

Ω¯S=(1κ2)3(1ν2)[7ν+κ2(1+ν)]48[1ν+κ2(1+ν)]
A 21

and

Ω¯P=Qc(1κ2)24NCκ.
A 22

The membrane equilibrium is identified by δΩT=(dΩT/dw~0)δw~0=0, furnishing

w~0=Ω¯P4Ω¯S1/3=QcNC1/3Kw.
A 23

Since the test function in equation (A 11) is derived based on the axial-force distribution of undeflected circular plates, the variational approximation must approach the exact solution at vanishingly small deflections. For the same reason, this approximation is expected to be less accurate at large deflections.

Footnotes

1Unless stated otherwise, volume turning point refers to a turning point in vl.

Authors' contributions

A.A. performed all the calculations, interpreted the results and wrote the manuscript. R.J.H. and T.G.V. helped to interpret the results, and edited the manuscript. All authors gave final approval for publication.

Competing interests

The authors have no competing interests.

Funding

NSERC Innovative Green Wood Fibre Products Network (R.J.H). McGill University Faculty of Engineering (McGill Engineering Doctoral Award, A.A.).

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