The Meniscus on the Outside of a Circular Cylinder: from Microscopic to Macroscopic Scales
aa r X i v : . [ c ond - m a t . s o f t ] A ug The Meniscus on the Outside of a Circular Cylinder: From Microscopic toMacroscopic Scales
Yanfei Tang ( 唐 雁 飞 ) and Shengfeng Cheng ( 程 胜 峰 ) ∗ Department of Physics, Center for Soft Matter and Biological Physics, and Macromolecules Innovation Institute,Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA (Dated: August 30, 2018)We systematically study the meniscus on the outside of a small circular cylinder vertically im-mersed in a liquid bath in a cylindrical container that is coaxial with the cylinder. The cylinderhas a radius R much smaller than the capillary length, κ − , and the container radius, L , is variedfrom a small value comparable to R to ∞ . In the limit of L ≪ κ − , we analytically solve thegeneral Young-Laplace equation governing the meniscus profile and show that the meniscus height,∆ h , scales approximately with R ln( L/R ). In the opposite limit where L ≫ κ − , ∆ h becomes inde-pendent of L and scales with R ln( κ − /R ). We implement a numerical scheme to solve the generalYoung-Laplace equation for an arbitrary L and demonstrate the crossover of the meniscus profile be-tween these two limits. The crossover region has been determined to be roughly 0 . κ − . L . κ − .An approximate analytical expression has been found for ∆ h , enabling its accurate prediction atany values of L that ranges from microscopic to macroscopic scales. I. INTRODUCTION
A liquid meniscus as a manifestation of capillary actionis ubiquitous in nature and our daily life. For example, itsformation and motion play critical roles in water uptakein plants [1]. Capillary adhesion due to the formationof menisci between solid surfaces makes wet hair to sticktogether and allows kids to build sandcastles [2]. Menisciare also involved in many technologies and industrial pro-cesses [3] such as meniscus lithography [4], dip-pen nano-lithography [5], dip-coating (Langmuir-Blodgett) assem-bly of nanomaterials [6–8], meniscus-mediated surface as-sembly of particles [9], meniscus-assisted solution print-ing [10], etc.A meniscus system frequently discussed in the litera-ture is the one formed on the outside of a circular cylin-der that is vertically immersed in a liquid bath. Oneapplication of this geometry is the fabrication of fiberprobes by chemical etching [11]. A cylinder with ra-dius at the nanometer scale has also been attached tothe tip of an atomic force microscope to perform nano-/micro-Whilhemy and related liquid property measure-ments [12]. The shape of the meniscus is governed bythe Young-Laplace equation [13]. Extensive studies havebeen reported for the scenario where the liquid bath isunbound and the lateral span of the liquid-vapor inter-face is much larger than the capillary length of the liquid[14–19]. Different methods have been applied in thesestudies, including numerical integration [15, 16] and an-alytical approaches such as matched asymptotic expan-sions [17–19] and hodograph transformations for cylin-ders with complex shapes [19]. An approximate formulahas been derived for the meniscus height, which dependson the radius of the cylinder and the contact angle of theliquid on the cylinder surface [14, 17]. The meniscus ex-erts a force that either drags the cylinder into or expels ∗ [email protected] it from the liquid depending on if the contact angle isacute or obtuse. A recent study of the meniscus rise on ananofiber showed that the force on the nanofiber highlydepends on the lateral size of the liquid-vapor interfaceif this size is smaller than the capillary length [20].In this paper we consider a geometry as sketched inFig. 1 where a small circular cylinder vertically penetrat-ing a liquid bath that is confined in a cylindrical con-tainer. With the cylinder and the container being coax-ial, the system has axisymmetry that enables certain an-alytical treatments. By fixing the contact angle on thesurface of the container to be π/
2, we have a meniscusthat systematically transits from being laterally confinedto unbound, when the size of the container is increased.For such a system, the meniscus profile is governed by thegeneral Young-Laplace equation that was first studied byBashforth and Adams more than a century ago [21]. Thisequation has been discussed in various systems includingliquid in a tube [22], sessile and pendant droplets [23, 24]and a capillary bridge between two spheres [25]. rz z(r)O L ! R vaporliquid " h ! FIG. 1. A rising meniscus on the outside of a circular cylindervertically immersed in a liquid bath confined in a cylindricalcontainer that is coaxial with the cylinder.
In the limit where the size of the cylindrical containeris much smaller than the capillary length, the gravita-tional term in the Young-Laplace equation can be ne-glected and the equation becomes analytically solvable.Solutions have been reported for various capillary bridgesbetween solid surfaces [26–28] and tested with moleculardynamics simulations [29, 30]. We have obtained a so-lution for the meniscus in Fig. 1 based on elliptic inte-grals when the lateral size of the meniscus is small andfound that the meniscus height depends on the containersize logarithmically. We further numerically solve thefull Young-Laplace equation for an arbitrary containersize and find that the meniscus height approaches an up-per limit found in some early work when the lateral spanof the interface is much larger than the capillary length[14, 16, 17]. Finally, we find an approximate expression ofthe meniscus height on the cylinder that is applicable toany lateral size of the liquid-vapor interface. This workis the basis of a related work on the wetting behavior ofparticles at a liquid-vapor interface [31], where the the-oretical results presented here are applied to study thedetachment of a spherical particle from a liquid bath.
II. THEORETICAL CONSIDERATIONSA. General Equation of the Meniscus Shape
The geometry of the system considered in this paperis sketched in Fig. 1. A circular cylinder with radius R is immersed in a liquid bath confined in a cylindricalwall with radius L > R . The cylinder and the wall arecoaxial and the system is thus axisymmetric. The shapeof the meniscus in this ring-shaped tube is determinedby the surface tension of the liquid, the contact angleson the two surfaces, and possibly gravity. Our interestis to examine the crossover from the case where L − R is small to the case where the cylinder is immersed ina liquid bath with an infinite lateral span. Since in thelatter limit the liquid-vapor interface is flat at locationsfar away from the cylinder, we will set the contact angleon the wall to be π/
2. Then a meniscus will rise (depress)on the outside of the cylinder if the contact angle on itssurface, θ , is smaller (larger) than π/
2. The case where θ = π/ θ < π/ θ > π/ z ′′ (1 + z ′ ) / + z ′ r (1 + z ′ ) / = ∆ pγ + ∆ ρgzγ , (1)where z ( r ) is the meniscus height at distance r from thecentral axis of the cylinder, z ′ ≡ d z d r , z ′′ ≡ d z d r , ∆ p is the pressure jump from the vapor to the liquid phaseat r = L and z = 0, γ is the surface tension of theliquid, ∆ ρ ≡ ρ l − ρ v is the difference of the liquid andvapor densities, and g is the gravitational constant. Abrief derivation of this equation is provided in A. In thefollowing discussion, we use a water-air liquid interfaceat 25 ◦ C as an example, for which γ ≈ .
072 N / m and∆ ρ ≈ kg / m .To facilitate discussion, we define 2 ˜ H ≡ ∆ pγ and κ ≡ ∆ ρgγ , i.e., κ − = q γ ∆ ρg is the so-called capillary length,which is a characteristic length scale of the problem. Forwater at 25 ◦ C, κ − ≈ . x ≡ κr , y ≡ κz . (2)The result is the following nonlinear differential equation y ′′ (1 + y ′ ) / + y ′ x (1 + y ′ ) / = 2 ˜ Hκ + y , (3)with boundary conditions y ′ = − cot θ at x = κR , (4a) y ′ = 0 at x = κL and y = 0 . (4b)As pointed out in Ref. [22], Eq. (3) is invariant underthe transformation y → − y , θ → π − θ , and ˜ H → − ˜ H ,indicating the symmetry between a rising and a depress-ing meniscus. This second-order nonlinear differentialequation can be rewritten in terms of the local tilt an-gle of the liquid-vapor interface, φ , as defined in Fig. 1.Since y ′ ≡ d y d x = d z d r = tan φ , Eq. (3) then becomesd sin φ d x + sin φx = − Hκ − y . (5)Eq. (5) and d y d x = tan φ can be further rewritten into apair of coupled first-order nonlinear differential equationsin terms of x ( φ ) and y ( φ ),d x d φ = − ( 2 ˜ Hκ + y + sin φx ) − cos φ , (6a)d y d φ = − ( 2 ˜ Hκ + y + sin φx ) − sin φ . (6b)with boundary conditions φ = φ at x = κR , (7a) φ = φ at x = κL and y = 0 , (7b)where φ = θ + π/ φ = π for the system sketchedin Fig. 1. Here θ is the contact angle on the wall and isfixed at π/ φ = π − θ for0 ≤ θ ≤ π .In a general case, Eq. (6) can be numerically solvedby the shooting method [32]. For the case where contactangle θ is close to π/
2, a zero-order solution is providedin B. For a general contact angle θ , analytical solutionsof the meniscus can be found when L ≪ κ − , where theterms on the right sides of Eqs (1), (3), and (6) due togravity are negligible [Sec. II B]. In the opposite limitwhere L ≫ κ − , the ∆ p term is negligible and an ap-proximate solution of the capillary rise on the outsideof a small cylinder with R ≪ κ − was found before byJames using the method of asymptotic matching expan-sions [Sec. II C]. Below we discuss these limits and numer-ical solutions of Eq. (6) for R ≪ κ − and an arbitrary L (which is of course larger than R ). The results naturallyshow the crossover from one limit ( R ≪ L ≪ κ − ) to theother ( R ≪ κ − ≪ L ). B. Analytical Solution in the L ≪ κ − Limit
When the radius of the cylindrical wall is small, i.e., L ≪ κ − , the Bond number gL ∆ ρ/γ ≪
1. As a result,the gravity’s effect can be ignored and Eq. (1) reduces to z ′′ (1 + z ′ ) / + z ′ r (1 + z ′ ) / = 2 ˜ H , (8)with ˜ H being the local mean curvature of the liquid-vaporinterface. This equation has been solved analytically be-fore for a capillary bridge between a sphere and a flatsurface [27, 33]. Here we use the same strategy to solveit for the meniscus in a ring-shaped container as depictedin Fig. 1.It is convenient to introduce reduced variables X = r/R, Y = z/R and a parameter u = sin φ . Eq. (8) isthen simplified as − H = d u d X + uX , (9)where H is the dimensionless mean curvature defined as H ≡ R ˜ H . The boundary conditions are φ = φ at X = 1 , (10a) φ = φ at X = l and Y = 0 , (10b)where φ = θ + π/ φ = π , and l = L/R is the scaledradius of the cylindrical container. The solution for Eq.(9) is u = c HX − HX . (11)The boundary condition in Eq. (10b) yields c = 4 H l .The other boundary condition in Eq. (10a) can then beused to determine the dimensionless mean curvature as H = sin φ l − . (12)From Eq. (11) and d Y / d X = tan φ , we obtain theanalytic solution of the meniscus profile, X ( φ ) = 12 H ( − sin φ + q sin φ + c ) , (13) Y ( φ ) = 12 H Z φφ ( − sin t + sin t p sin t + c ) d t . (14) The solution for Y ( φ ) in Eq. (14) can be written in termsof elliptic integrals, Y ( φ ) = 12 H (cos φ − cos φ ) + √ c H h E ( φ, j ) − E ( φ , j ) − F ( φ, j ) + F ( φ , j ) i , (15)where j ≡ − c , E ( φ, j ) = R φ p − j sin t d t is theincomplete elliptic integral of the second kind, and F ( φ, j ) = R φ √ − j sin t d t is the incomplete elliptic in-tegral of the first kind. The meniscus rise can be easilycomputed as ∆ h = RY ( φ ), or explicitly,∆ h = R H (1 − sin θ ) + R √ c H h F ( π/ − θ , j ) − E ( π/ − θ , j ) i . (16)Some examples of the meniscus profile are shown in Fig. 2for L/R = 5 and θ = 0 ◦ , 30 ◦ , and 60 ◦ , respectively. r/R z / R θ = 0 ∘ θ = 30 ∘ θ = ∘0 ∘ FIG. 2. Meniscus profiles from the analytic solution inEqs. (13) and (14) for
L/R = 5 and θ = 0 ◦ (blue solidline), 30 ◦ (green dashed line), and 60 ◦ (red dash-dotted line). The analytical prediction in Eq. (16) actually indicatesthat ∆ h ∼ R ln( L/R ) when κ − ≫ L > R . To seethis scaling behavior transparently, we examine the limitwhere κ − ≫ L ≫ R , i.e., the cylinder is much smallerthan the cylindrical container and both are much smallerthan the capillary length. In this limit we can take l ≫ j → −∞ , and approximate the elliptic integrals inEq. (16) by series expansions. The mathematical deriva-tion is provided in C. The final result on the meniscusheight is ∆ h = R cos θ h ln 2 LR (1 + sin θ ) − i . (17)A more intuitive way to see the logarithmic behavioris to note that in the limit of l ≫
1, the dimensionlessmean curvature H approaches zero and Eq. (1) can berewritten as [13, 20], r (1 + r ′ ) / = R cos θ , (18)where r ′ ≡ d r d z . The solution of this equation is known asa catenary curve [13]. The meniscus is thus a catenoidwith its generatrix given by z ( r ) = R cos θ ln h L + ( L − R cos θ ) / r + ( r − R cos θ ) / i . (19)The meniscus height can be computed as ∆ h = z ( R ) andan approximate expression is∆ h = R cos θ ln h LR (1 + sin θ ) i , (20)where the condition L/R ≫ ≥ cos θ is used. Inboth Eqs. (17) and (20) the scaling dependence of ∆ h on R ln( L/R ) is obvious. However, the expression inEq. (17) for ∆ h is smaller than Eq. (20) by ( R cos θ ) / π/
2. However, Eq. (20) is based on a catenarycurve, for which the contact angle at the wall is close tobut not exactly π/ C. Approximate Solution in the L ≫ κ − Limit
In the literature, the meniscus on the outside of a cir-cular cylinder vertically penetrating a liquid bath wasmostly investigated for the case where the lateral span ofthe liquid bath is much larger than the capillary length[14–19], i.e., L ≫ κ − . In this limit, ˜ H → y ′′ (1 + y ′ ) / + y ′ x (1 + y ′ ) / = y . (21)The boundary condition Eq. (4a) remains the same butEq. (4b) is replaced by y ′ = 0 at x → ∞ and y = 0 . (22)Eq. (21) has been studied with methods of numericalintegration [15, 16] and matched asymptotic expansions[17, 18]. The meniscus height is approximately given bythe Derjaguin-James formula [14, 17],∆ h = R cos θ h ln 4 κ − R (1 + sin θ ) − E i , (23)where E = 0 . ... is the Euler-Mascheroni constant.Eq. (23) is expected to predict the meniscus height ac-curately when the radius of the cylinder is much smallerthan the capillary length that is in turn much smallerthan the lateral span of the liquid bath, namely R ≪ κ − ≪ L . A comparison between the Derjaguin-Jamesformula and numerical results has been fully discussedin Ref. [17] for L → ∞ . This comparison is revisitedin Fig. 3. Practically, for water at room temperature itis legitimate to use the Derjaguin-James formula to esti-mate the meniscus height on a cylinder when its radiusis less than about 0 . −5 −4 −3 −2 −1 R (mm) Δ h / R θ = 0 ∘ θ = 30 ∘ θ = 60 ∘ −2 −1 R (mm) ∘ rr o r FIG. 3. Comparison of the meniscus height (∆ h ) between theDerjaguin-James formula (Eq. (23), solid lines) and numeri-cal results (symbols) using Huh-Scriven’s integration scheme[16] as a function of the radius of the cylinder, R , for differentcontact angles: θ = 0 ◦ (blue line and (cid:13) ), 30 ◦ (orange lineand (cid:3) ), and 60 ◦ (red line and ⋄ ). The lateral span of the liq-uid bath is treated as infinite by using Eq. (22) as a boundarycondition. Inset: the relative deviation of the numerical re-sults on ∆ h from the prediction based on the Derjaguin-Jamesformula is plotted against R . III. NUMERICAL RESULTS AND DISCUSSION
As discussed in Sec. II A, the general Young-Laplaceequation [Eq. (3)] can only be solved numerically. Werewrite Eq. (3) into a pair of coupled firs-order differentialequations [Eq. (6)] and adopt the shooting method toobtain their numerical solutions for a given R that ismuch smaller than κ − and an arbitrary L that variesfrom 2 R to a value much larger than κ − .Figure 4 shows numerical solutions of the meniscusheight on a circular cylinder immersed vertically in wa-ter when L is varied. Cylinders with radii R from 100nm to 10 µ m and contact angles θ from 0 to 60 ◦ areused as examples. The data show the following trends.When L is smaller than 1 mm, i.e., L/R < for R = 10 µ m, L/R < for R = 1 µ m, and L/R < for R = 100 nm, the meniscus height ∆ h is well predicted byEq. (16), which is derived with gravity ignored. In thislimit, ∆ h grows with L logarithmically. In the otherlimit where L is larger than 10 mm, i.e., L/R > for R = 10 µ m, L/R > for R = 1 µ m, and L/R > for R = 100 nm, the meniscus height fits to the Derjaguin-James formula in Eq. (23), which is derived assuming R ≪ κ − and L → ∞ . For L with an intermediatevalue between 1 mm and 10 mm, the numerical data onthe meniscus rise show clearly the crossover between thelogarithmic regime [Eq. (16)] and the saturation regimedescribed by the Derjaguin-James formula. The latterthus provides an upper bound of the meniscus rise on theoutside of a circular cylinder with a radius much smallerthan the capillary length. L/R Δ h / R (a) L/R (b) L/R (c)
FIG. 4. The meniscus height, ∆ h , for different contact angles on the surface of the cylinder: (a) θ = 0 ◦ , (b) 30 ◦ , and (c) 60 ◦ as a function of the lateral span of the liquid bath, L . The solid line is the analytical expression of ∆ h for L ≪ κ − [Eq. (16)].The horizontal dashed lines are the predictions of the Derjaguin-James formula for L ≫ κ − [Eq. (23)]. The vertical horizontallines indicate where L = κ − . The symbols are numerical solutions of Eq. (6) for an arbitrary L using the shooting method.Data are for cylinders with different radii: R = 100 nm (red (cid:13) ), 1 µ m (orange (cid:3) ), and 10 µ m (blue ⋄ ). Both ∆ h and L arenormalized by R . The results in Fig. 4 indicates that for a cylinder with R ≪ κ − , the Young-Laplace equation without gravityas shown in Eq. (8) can be used to describe the meniscuson the outside of the cylinder when L . . κ − , whilethe liquid bath can be considered as unbounded and theDerjaguin-James formula applies when L & κ − . Therange 0 . κ − . L . κ − is the crossover region in whichthe full Young-Laplace equation [Eqs. (1), (3), (5), or (6)]needs to be employed. This conclusion seems to hold forother liquids with different capillary lengths. For exam-ple, we have solved Eq. (6) numerically for a hexadecane-water mixture at 25 ◦ C, for which κ − = 4 .
824 mm, andfound roughly the same crossover zone.An interesting finding is that the intersection betweenthe solid line from Eq. (16) and the corresponding dashedline from the Derjaguin-James formula in Eq. (23) oc-curs at L ≈ . κ − for all the systems consideredhere. The relationship can be understood if we com-pare Eq. (17), which is an approximate form of Eq. (16)on the meniscus height in the limit of κ − ≫ L ≫ R ,to the Derjaguin-James formula in Eq. (23). At L =2 e / − E κ − ≈ . κ − , the two predictions are equal.This estimate is in perfect alignment with the discoverythat at L ≈ . κ − , the meniscus height from Eq. (16)matches that predicted by the Derjaguin-James formula.In a related work, we find that L ≈ . κ − is also thesaturation length of the lateral span of a liquid-vapor in-terface when discussing how the effective spring constantexperienced by a detaching particle depends on the lat-eral size of the interface [31]. Note that Eq. (16) holdsfor κ − ≫ L > R and is thus more general than Eq. (17),which requires κ − ≫ L ≫ R . Our numerical results in-dicate that Eq. (16) provides a good estimate of ∆ h for L up to about 0 . κ − .Based on this observation and the finding that thecrossover zone, 0 . κ − . L . κ − , is relative small,we propose that for a cylinder with radius R ≪ κ − andvertically immersed in a liquid bath with lateral spandesignated as L , the meniscus height on the outside of the cylinder can be computed using Eq. (16) with theparameter l given as follows, l = ( L/R if L ≤ . κ − ,1 . κ − /R if L > . κ − . (24)Note that the parameter l , in addition to θ and R , en-ters in the computation of the parameters H , c , and j in Eq. (16). For L ≤ . κ − , the meniscus height∆ h depends on L logarithmically while it saturates tothe upper bound expressed in the Derjaguin-James for-mula when L > . κ − . Our numerical data indicatethat Eq. (16) with l from Eq. (24) is quite accurate forthe meniscus height. Even within the crossover region0 . κ − . L . κ − , the relative deviation of the actualmeniscus height from the prediction based on Eqs. (16)and (24) is less than 5%, as shown in Fig. D1 in D.By carefully examining the relative error of usingEqs. (16) and (24) to compute the meniscus height ∆ h and how the error depends on L , R , and θ [see D for de-tail], we arrive at an approximate analytical expressionof ∆ h for an arbitrary L that reads∆ h = ∆ h (elliptic) × { − m ( κL )[ κR (1 + sin θ )] . } , (25)where ∆ h (elliptic) is the meniscus height from Eq. (16)based on elliptic integrals with the parameter l given inEq. (24) and m ( κL ) is a universal function given as fol-lows, m ( x ) = ( .
085 exp (cid:2) ( x − . . / . (cid:3) if x ≤ .
85 ,0 .
085 exp [(1 . − x ) / . x > .
85 .(26)Note that m ( κL ) is independent of the contact angle, θ ,and the cylinder radius, R . The dependence of ∆ h on R and θ enters through ∆ h (elliptic) and the κR (1 +sin θ ) . term in Eq. (25).In Fig. 5, the analytical result on the meniscus height∆ h in Eq. (25) is compared with numerical solutions of L/R Δ h / R = 0 ∘ = 30 ∘
10 μm,Δθ = 60 ∘ FIG. 5. The numerical solutions of the meniscus height (sym-bols) at various combinations of R and θ as a function of L are compared to the analytical expression in Eq. (25). the full Young-Laplace equation rewritten as Eq. (6). Avery good agreement has been found between the two,indicating that Eq. (25) can be used to accurately predictthe meniscus height on the outside of a circular cylinderwith R ≪ κ − for a meniscus with an arbitrary lateralspan, including the crossover zone 0 . κ − . L . κ − .However, Eq. (25) is a result obtained by comparing theanalytical expression of the meniscus height in the limitof κL ≪ κL .It remains an open question if the universal expressionof ∆ h in Eq. (25) for arbitrary L , R (as long as it isless than κ − ), and θ can be derived with an analyticalapproach.The results presented in Figs. 4 and 5 are for L chang-ing from 2 R to a value much larger than κ − and for R changing from 100 nm to 10 µ m, i.e., R ≪ κ − . Thereare several limits that are of interest but not explored indetail in this paper. In one, if R is made much larger than κ − , then the system depicted in Fig. 1 can be regardedas a meniscus between two flat walls (or even be reducedto a meniscus on one flat wall if L − R ≫ κ − ) [34].There is a crossover from the R ≪ κ − limit, the focus ofthis paper, and the R ≫ κ − limit. In the crossover, R is comparable to κ − and the numerical procedureof dealing with the Young-Laplace equation rewritten asEq. (6) can be applied. In the limit of R being reduced tonanometer scales, the line-tension effect associated withthe large curvature ( R − ) of the contact line on the sur-face of the cylinder may become important [35]. In thecase where L − R is small enough, factors including dis-joining pressure will kick in [29]. If L − R is furtherreduced such that the molecular nature of a liquid hasto be taken into account, the continuum theory of capil-larity may break down [29]. These limits are intriguingdirections for future studies. IV. CONCLUSIONS
The problem of a small circular cylinder immersed ina liquid bath has been studied for many years. The focuswas mainly on the limit where the liquid bath is muchlarger than the capillary length (i.e., L ≫ κ − ≫ R )[14–19] or on the case where gravity is negligible and theliquid-vapor interface can be described as a catenary (i.e., κ − ≫ L ≫ R ) [13, 20]. In this paper, we provide a com-prehensive discussion of the meniscus on the outside of acircular cylinder with R ≪ κ − vertically positioned in aliquid bath with lateral span L ranging from microscopicto macroscopic scales. We obtain an analytical solutionof the meniscus profile based on elliptic integrals when κ − ≫ L > R and the solution reduces to a catenarywhen κ − ≫ L ≫ R . In these solutions, the menis-cus height ∆ h ∼ R ln( L/R ). Our numerical solutions ofthe full Young-Laplace equation for an arbitrary L indi-cate that ∆ h indeed scales with R ln( L/R ) up to about L . . κ − . In the opposite limit where L & κ − , themeniscus height agrees well with the prediction of theDerjaguin-James formula and scales with R ln( κ − /R ).The range 0 . κ − . L . κ − is the crossover regionwhere the actual value of ∆ h deviates from the predictionof either the analytical solution based on elliptic integralsor the Derjaguin-James formula.Our analyses reveal a universal behavior that theanalytical solution [Eq. (16)], which predicts ∆ h ∼ R ln( L/R ), always reaches the upper bound set by theDerjaguin-James formula at L ≈ . κ − . Therefore,the analytical solution with its parameter l = L/R when L ≤ . κ − and capped at l = 1 . κ − /R when L > . κ − can be used to estimate the meniscus height∆ h . The relative deviation of the actual value of ∆ h de-termined via numerical solutions from this estimate isfound to be only noticeable in the crossover region butstill less than 5%. We further find that the relative er-rors at different R and contact angles at the surface ofthe cylinder, if properly scaled, as a function of κL allcollapse to a master curve. With a fitting function tothis master curve, we obtain an analytical expression[Eq. (25)] that can be used for accurate prediction of∆ h for the whole range of L from microscopic to macro-scopic scales including the crossover zone. Although inthis paper we only consider cases with the contact angleon the wall being fixed at π/
2, the theoretical analysesand numerical treatments of the general Young-Laplaceequation can also be extended to more general cases withother contact angles at the wall surface.
ACKNOWLEDGEMENTS
Acknowledgement is made to the Donors of the Amer-ican Chemical Society Petroleum Research Fund (PRF
Appendix A: Derivation of Young-Laplace Equation
The profile of a meniscus is governed by Eq. (1), whichhas been discussed extensively for the geometry of sessileand pendant drops. Here we provide a simple derivationof this equation. The energy of a liquid bath bound bya cylindrical container and a meniscus on the outsideof a cylinder at the center of the container (Fig. 1) isa sum of surface energy and gravitational terms, G = γS + ∆ pV + U g , where γ is the surface tension of theliquid, S is the surface area of the liquid-vapor interface,∆ p is a Lagrange multiplier, V is the volume of the liquidbath which is fixed, and U g is the potential energy of theliquid in the gravitational field. The meniscus profile canbe found by minimizing G , which can be written in termsof the surface profile z ( r ), G = 2 πγ Z LR r p z ′ d r + 2 π ∆ p Z LR rz d r + π ∆ ρg Z LR rz d r , (A1)We seek the surface profile that will make the energyfunction G = R f ( z, z ′ , r ) d r stationary, i.e., δG = 0. Theresulting Euler-Lagrange equation isdd r ∂f∂z ′ − ∂f∂z = 0 . (A2)After some algebra, we obtain the following equation, γ h z ′′ (1 + z ′ ) / + z ′ r (1 + z ′ ) / i = ∆ p + ∆ ρgz , (A3)where the left hand side comes from the surface energyand the right hand side originates from the volume of theliquid bath being fixed and the gravitational potentialenergy, respectively. This equation is Eq. (1) in the maintext. Appendix B: Solution of Zero-order
If the contact angle θ on the cylinder in Fig. 1 is closeto π/
2, the resulting liquid-vapor interface is almost flatsince the contact angle on the wall surface is fixed at π/
2. In this case z ′ = tan φ ≪ H + κ z = 1 r dd r h rz ′ (1 + z ′ ) / i ≈ r dd r h rz ′ (1 + O ( z ′ )) i , (B1)with the following boundary conditions, φ = φ at r = R , (B2a) φ = φ at r = L and z = 0 , (B2b)where φ = θ + π/ φ = π . The solution of Eq. (B1)which satisfies the boundary condition Eq. (B2b) is, z = 2 ˜ Hκ h K ( κr ) K ( κL ) − i , (B3) and the angle φ is given bytan φ = − Hκ K ( κr ) K ( κL ) , (B4)where K and K are modified Bessel functions of secondkind of order zero and one, respectively. The undeter-mined constant ˜ H can be found using the other boundarycondition Eq. (B2a) and the result is˜ H = − κ φ K ( κL ) K ( κR ) . (B5) FIG. D1. (a) The relative error δh defined in Eq. (D1) as afunction of κL for various combinations of R and θ . (b) Datain (a) are collapsed onto a master curve when δh × [ κR (1 +sin θ )] − . is plotted against κL ; the blue dashed line is thefit in Eq. (26). In both (a) and (b) the gray zone indicatesthe crossover region 0 . κ − . L . κ − . Appendix C: Expansion of Elliptic Integrals
Here we derive the series expansions of incomplete el-liptic integrals F ( φ, j ) and E ( φ, j ) in the limit of j →−∞ . To facilitate the discussion it is helpful to intro-duce a small parameter ǫ > j = − ǫ ; the limit j → −∞ thus corresponds to ǫ →
0. Below we usethe incomplete elliptic integral of second kind, E ( φ, j ), as an example. A similar expansion can be performedfor F ( φ, j ). E ( φ, j ) = Z φ q − j sin t d t ( t sin t )= 1 ǫ h Z √ ǫ p ǫ + t d t √ − t | {z } t ǫt + Z sin φ √ ǫ p ǫ + t d t √ − t | {z } t /t i = 1 ǫ h ǫ Z / √ ǫ p t d t √ − ǫ t + Z / √ ǫ / sin φ p ǫ t d tt √ t − i = 1 ǫ (1 − cos φ ) + ǫ ( − ln ǫ −
12 ln 1 + cos φ sin φ ) + O ( ǫ ) . (C1)In this derivation we have employed the following ex-pansion √ − ǫ t = 1 + ǫ t + O ( ǫ t ) and √ ǫ t =1 + ǫ t + O ( ǫ t ), and assumed that sin φ > √ ǫ . Theexpansion of the incomplete elliptic integral of first kind, F ( φ, j ), can be obtained similarly and the result is F ( φ, j ) = ǫ ( − ln ǫ + 2 ln 2 − ln 1 + cos φ sin φ ) + O ( ǫ ) . (C2)By substituting Eq. (C1) and Eq. (C2) into Eq. (16), wearrive at∆ h = R cos θ h ln 2 LR (1 + sin θ ) − i , (C3)which is Eq. (17) in the main text. Here the relations H = sin φ l − ≈ cos θ l and ǫ = p − j − = √ c are used. Appendix D: Relative Error of Eq. (16) onPredicting ∆ h In order to obtain an even more accurate expressionof the meniscus height that applies to R ≪ κ − and anarbitrary L , we denote the meniscus height predicted inEq. (16) using elliptic integrals with the parameter l givenin Eq. (24) as ∆ h (elliptic). The full numerical solution ofEq. (6) for an arbitrary L is denoted as ∆ h (actual). Therelative error of using Eq. (16) to predict the meniscusheight is thus given by δh = ∆ h (elliptic) − ∆ h (actual)∆ h (elliptic) . (D1)In Fig. D1(a), δh is shown as a function of L that isnormalized by κ − for several combinations of the cylin-der radius, R , and the contact angle on its surface, θ .As expected, the peak value of the relative error occursat κL = 1 .
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