Capillary Phenomena in Assemblies of Parallel Cylinders

Capillary Phenomena in Assemblies of Parallel Cylinders I. Capillary Rise between Two Cylinders H. M. PRINCEN Lever Brothers Company, Research and Development Division, Edgewater, New Jersey 07020 Received October 14, 1968 The capillary rise of liquid in the gap between two closely spaced parallel cylinders has been calculated as a function of distance of separation and contact angle. The solutions are valid only when the height of the meniscus is much greater than the radius of the cylinders. With the same restriction, the capillary rise between a cylinder and a flat plate has been evaluated. INTRODUCTION Capillary phenomena associated with the presence of limited amounts of liquid in fibrous media are of importance in a number of fields, such as wetting of and wicking in textiles and paper (1-3). Especially in woven textiles the fibers are more or less parallel inside a yarn, whereas in certain materials the individual fibers are close to cylindrical, e.g., in most synthetic textiles. We have recently embarked on a study of these phenomena, using the model of perfectly parallel and cylindrical rigid rods. Although this model does not reflect completely the detailed structure of these fibrous materials, it can hopefully lead to a better understanding of the behavior of liquids in more realistic systems. The treatment will be limited to equilibrium conditions. In this first part the relatively simple problem of capillary rise between two vertical rods will be investigated. To our knowledge this system has not yet been studied in detail. In future reports we intend to consider the equilibrium configurations of liquid between two horizontal rods; the number of cylinders considered will also be increased. THEORY Capillary Rise between Two Identical Rods. When two vertical cylinders are placed in a liquid, capillary rise of the liquid will occur in the space between the cylinders, provided the contact angle 0, measured through the liquid, is smaller than 90 (Fig. la). When 0 > 90, capillary depression takes place and the configuration will be a mirror image of Fig. la. The meniscus is saddle-shaped and will reach an equilibrium height z~ above the horizontal liquid surface. The radius of the cylinders is r, and the distance between the cylinders is 2d. It is assumed that the contact angle 0 is the same all along the line of three-phase contact. The problem will be solved only for systems in which z2 >> r. [1] In this case the dimensions of the meniscus are negligible compared to z2, and the liquid surface will be essentially vertical from just above the bulk surface to just below the meniscus. Therefore, in this region one of the radii of curvature of the surface is infinite, and the other is given by the radius R of the circular arcs in the horizontal cross section. Figure lb shows such a cross section for the case of complete wetting, which will be treated first. The angle between the line connecting the centers of the cylinders and the radius pointing to the liquid/solid/gas boundary Journal of Colloid and Interface Science, Vol. 30, No. 1, May 1969 69

70 PRINCEN i I (c),'k R/ \ r/ Z 2d (b) ' (0) L_.I }JIG. 1. (a) Capillary rise of liquid between two vertical cylinders. (b) Horizontal cross section of (a) at level z (0 = 0 ). (c) Horizontal cross section of (a) iust below meniscus. is a. The radius R is related to a according to or R r-~-d - r [2] COS G R_l+d/r 1. [3] COS As the capillary pressure across the interface must equal the hydrostatic pressure, R is determined by the height z in the liquid column through ~/n = zpg, [4] where ~, is the surface tension, p is the density of the liquid, and g is the acceleration due to gravity. Equation [4] can be written in dimensionless form where z 1 r r cr 2 ~ [5] c = pg/~. [6] Hence R is simply inversely proportional to z. When the cylinders touch (d/r = 0) this relationship will hold whatever the height above the plane liquid surface and the capillary rise is "infinite." When d/r is Journal of Colloid and Interface Science, ol. 30, No. 1, May 1969 finite, however, the surfaces form a saddleshaped meniscus at a certain height z~. At this height a = a2 and R = R2 (Fig. lc), so that from Eqs. [3] and [5] and R~ _ l + d/r 1 [7] cos a2 Z2 i 7~ r cr 2 R2" Is] To obtain the relationship between the capillary rise z2/r (or r/r2) and d/r, we use the principle that the weight of the liquid column directly below the shadowed area in Fig. lc must be supported by the upward force due to surface tension in the meniscus region. Knowledge of the detailed shape of the saddle-shaped meniscus is not required for this purpose. The area is given by A = 2r~[(1 + R2/r) ~ sin a2 cos ~2 -- as -- (~/2 -- ~)(R2/r)21. Therefore, the weight of the liquid column is [9] F~ = z~pga. [10] The force due to surface tension consists of two terms.

CAPILLARY PHENOMENA IN ASSEMBLIES OF PARALLEL CYLINDERS 71 /^\ / \ R211 \\ Fro. 2. Horizontal cross section just below meniscus between two incompletely wetted cylinders. 1) An upward force resulting from the contact of the liquid and the cylinders along AC and BD. This force equals F2~ = 47ra2. [11] 2) A downward force due to the free vertical liquid surfaces along AB and CD which tend to pull down the liquid column, leading to ~ F22 = -4,(~/2 -- ~2)R2. [121 Substituting for z2 and A in Eq. [10] according to Eqs. [8] and [9], equating F1 and F2~ A- F~2, and rearranging, one obtains (R~2)2 (sin 0/2 cos 0/2 -- 0/2-}- 2) -t-2 (-~)(sin 0/2 COS 12-0/2)[13] -f- sin 0/2 COS 0/2 -- 0/2 = 0. From this quadratic equation R2/r can be calculated as a function of 0/2, and the corresponding value of d/r can be obtained from Eq. [7]. We followed a somewhat different path, however. Instead of 12, we selected d/r as the independent variable. For the purpose of clarification, the reality of F22 can be appreciated also from a fictitious experiment in which two vertical completely nonwetted cylinders of radius R2 are placed in contact with the two cylinders at AB and CD in Fig. lc. As the shape of the liquid surfaces along AB and CD is not affected by this (they would still be vertical), the meniscus shape and height should be unchanged. In calculating the capillary rise in the newly created closed capillary, the necessity of including F2~ becomes obvious. This requires the evaluation of 0/2 and R2/r by a method of successive approximation. An electronic computer was used for this purpose. For incompletely wetted cylinders the same reasoning applies (Fig. 2). In this more general case of finite contact angle 0 the equivalents for Eqs. [7], [9], [11], and [12] are R2 _ 1 + d/r -- cos 12. r cos (0 + 0/2) ' A = 2r212~sina2cos(O+a2) and -- 0/2 -}- sin c~2 cos c~2 (0-~ 0/2) F21 = 4~/ra2 cos 0; )A [7a] [9a] [lla] F22 = -4~(7r/2-0 - 12)R2. [12a] As Eqs. [8] and [10] still apply one obtains by the same procedure (R2/r)2[Tr/2 -- (0 -? 12) -4- sin (0 -t- 12) cos (0 A- 12)] -4-2(R2/r)[sin 12 cos (0 + 12) -- ~2 cos 0] A- sin 12 cos 12 -- 12 = 0. [13a] Equations [7a] and [13a] are then solved for various values of d/r and 0. Capillary Rise between a Rod and a Plate. Above, we have concentrated on the system of two rods of equal diameter and equal contact angles. The same analysis can be applied to two vertical rods of different diameter and/or different wettabilities. As an example of such systems we have considered only the capillary rise between a completely wettable rod and plate. Figure 3 shows a horizontal cross section just below the meniscus in this system. The distance between the cylinder and the plate is 2d. The equivalent equations for Eqs. [7], Journal of Colloid and Interface Ncience, Vol. 30, No. 1, May 1969

72 PI~INCEN ~\\\\\\\\\\\\\\~\\" ~ Fro. 3. Cross section just below the meniscus between a cylinder and a plate (0 = 0 ). [9], [11], and [12] are R2 _ 1 q- 2d/r -- cos a2 [75] r 1 + cos a~ A = r2{[(1 -t- d/r) 2 - (R2/r) 2] tan a2 - ~ - (~ - ~)(R2/r)q; F21 = 2yr[a2 ~- (1 + R2/r) sin a2]; [9b1 [llb] F=2 = -2yr(Tr - a2)r2/r. [125] This, finally, leads to the equivalent for Eq. [13]: (R=/r)2(sin a~ cos a2 - a2 ~- ~) + 2(R2/r)(sin a= cos a2 - a~) [135] -t- sin a2 cos a~ - a~ = 0; which again has to be solved in conjunction with Eq. [7b]. Force versus Free Energy Approach. In the above analysis the problem was solved on the basis of a balance of gravitational and surface tension forces. An alternative approach is the minimization of the free energy of the system. It will be shown that the latter approach leads to the same result by considering the free energy change in the system when the meniscus is displaced over an infinitesimal distance dz from its equilibrium position. The resultant change in gravitational energy is df1 = &Apt dz, [14] where A is defined as in Eq. [9@ In this process the liquid column, the cross section of which is shown in Fig. 2, is extended by an element dz, resulting in the wetting of an area (AC + BD) dz of the originally nomvetted cylinders and in the Journal of Colloid and Interface Science, Vol. 30, No. 1, May 1969 creation of an area (AB -f- CD) dz of free liquid surface. The corresponding free energy change is given by df2 = [(AC + BD)(y~L - "Ysv) + (AB + CD)yLv] dz, [15] where ~/sl and y~r are the solid/liquid and solid/vapor interfacial tensions, and TLv(=~') is the liquid/vapor surface tension. According to Young's equation ~'sv -- ~'~L = ~ cos 0. [16] Hence, Eq. [15] can be written in the form df2 = [(AB + CD) -- (AC + BD) cos 0]~, dz. [171 As the total change in free energy df1 + df2 must be zero at equilibrium, one finds &Apt + [(AB + CD) -- (AC q- BD) cos O]y = O. [18] When z2, A, AB, CD, AC, and BD are expressed in terms of R2 and a2, this leads to the same equilibrium condition as found previously, i.e., Eq. [13a]. RESULTS AND DISCUSSION OF CAPILLARY RISE Table I gives the results for the rod-rod problem. The angle a2 and r/r2 are tabulated as a function of d/r and for various contact angles. As r/r2 varies from infinity to zero, the table does not permit direct, accurate interpolation of this quantity. The parameter d/r~ = (d/r). (r/r2) is more suitable for this purpose, as it varies only between cos 0 (at d/r = 0) and zero. The former limit arises from the fact that for very small values of d/r the capillary rise becomes identical to that between two flat plates at a distance 2d. Values of d/r2 are included for 0 = 0 only. They are readily calculated for the other contact angles as well. In Fig. 4 we have plotted d/r2 versus d/r for five different contact angles and also for the rod-and-plate system (0 = 0 ). For certain d/r, the table gives r/r2 (or d/r2) and, therefore, the capillary rise z2/r through Eq. [8]. The result is meaningful only if &/r, thus calculated, is much greater

CAPILLARY PHENOMENA IN ASSEMBLIES OF PARALLEL CYLINDERS 73 TABLE I PARAMETERS CHARACTERIZING THE CAPILLARY RISE BETWEEN TWO IDENTICAL VERTICAL RODS d/r 0 = 0 0 = 15 O= 30 0 = 45 O= 60 O= 75" a~ r/r2 d/r, a2 r/r2 a~ r/r2 vee r/r~ a2 r/r~ a~ ] r/g2 0.001 0.81 910.2 0.9102 0.002 1.31 442.1 0.8841 0.004 2.13 213.0 0.8522 0.006 2.84 138.2 0.8294 0.008 3.49 101.4 0.8108 0.010 4,09 79.50 10.7950 0.015 5,49 50.84 0.7626 0.020 6.78 36.80 0.7361 0.025 7.99 28.53 :0.7134 0.030 9,15 23.10!0.6931 0.035 10.27 19.28 0.6748 0.040 11.36 16.45!0.6580 0.045 12.42 14.27 0.6424 0.050 13.46 12.55 0.6277 0.06 15.48 10.01 0.6006 I 0.07 17 44 8.228 0.5759 0.08 19.341 6.915 0.5532 0.09 21.21 5.911 0.5320 0.10 23.03 5.121 0.5121 0.12 26.59 3.960 0.4753 0.14 30.05 3.155 0.4417 0.16 33.41 2.566 0.4106 0.18 36.7C 2.120 0.3815 0.20 39.92 1.771 0.3542 0.22 43.07 1.492 0.3283 0.24 46.16 1,265 0.3037 0.26 49.20 1.077 0.2801 0.28 52.17 0,9199 0.2576 0.30 55.09 0,7863 0.2359 0.34 60.78 0,5730 0.1948 0.38 66.27 0,4116 0.1564 0.42 71.58 0.2863 0.1202 0.46 76.70 0.1871 0.0861 0.50 81.64 0.1073 0.0536 0.54 86.43 0.0422 0.0228 0.81 874.6 0.85 1.31 424.5 1.36 2.11 204.2 2.19 2.80 132.3 2.89 3.43 96.89 3.53 4.01 75.92 4.12 5.35 48.43 5.47 6.58 34.99 6.70 7.72i 27.07 7.84 8.81 21.88 8.92 9.86 i 18.23 9.96 10.87 15.53 10.96 11.85 13.45 11.92 12.81 11.81 [2.86 14.66! 9.387 14.67 16.45 7.692 16.41 18.17 6.445 18.08 19.85 5.492 19.71 21.49 4.743 21.2 24.66 3.644 24.33 27.73 2.883 27.26i 30.69 2.328 30.08 33.57 1.908 32.82 36.38 1.581 35.48 39.12 1.319 38.08 41.80 1.107 40.61 44.43 0.9317 43.1C 47.00 0.7849 45.53 49.53 0.6607 47.91 54.44 0.4630 52.55 59.19 0.3140 57.05 63.78 0.1989 68.23 0.1083 72.55 0.0356 774.1 0.92 616.9 1.04 415.9 1.31 188.0 374.2 1.47 295.9 1.66 196.2 2.09 83.91 179.0 2.35 140.0 2.65 90.57 3.34 35.45 115.5 3.10 89.49 3.56 56.78 4.41 20.50 84.25 3.78 64.80 4.25 40.40 5.38 13.47 65.79 4.40 50.26 4.96 30.82 6.28 9.472 41.65 5.83 31.33 6.55 18.49 8.35 4.526 29.90 7.11 22.17 7.99 12.61 10.25 2.307 23.00 8.31 16.83 9.33 9.232 12.04 1.100 18.48 9.44 13.35 10.60 7.059 13.76 0.3687 15.31 10.52 10.92 11.81 5.560 12.97 11.56 9.139 12.97 4.470 11.18 12.57 7.778 14.10 3.649 9.762 13.54 6.709 15.19 7.679 15.41 5.145 17.29 6.228 17.21 4.063 19.32 5.163 18.93 3.276 21.27 4.353 20.61 2.682 23.17 3.718 22.23 2.220 25.02 2.792 25.36i 1.553 28.62 2.154 28.37 1.101 1.692 31.27 0.7778 1,345 34.08 0.5386 1.076 36.82 0.3563 0.8625 39.491 0.2141 0.6902 42.11 0.1011 0.5489 44.68 0.0102 0.4314 0.3326 0.1771 0.0616 3.010 2. 091 1.468 1. 025 0. 6973 0.4478 0.0994!! d R2 1.0 0.8~ pl~gfe 0.6 0.4 02 0 0.2 0.4 d/r 0.6 i i [ 0.8 1.0 FIG. 4. d/r2 as a function of d/r for two cylinders (0 = 0, 30, 45, 60, and 75 ). Upper curve refers to a cylinder-and-plate system for 0 = 0% Journal of Colloid and Interface Science, Vol. 30, No. 1, MZay 1969

74 PRINCEN /// ~// 0.8 FIG. 5. Limiting case when the separation equals 2(d/r).... than unity, as this is the underlying assumption in the whole above treatment. This may b e illustrated by means of the practical example of the rise of water between two completely wetted glass rods at d/r = 0.1. From the table one finds r/r2 = 5.121, so that z2/r = 5.121/cr 2. For this result to be reliable, cr ~ must be much smaller than 5.121. As c ~ 14 cm -2 for water, r 2 must be much smaller than about 0.36 cm 2, or r << 0.6 cm. It is seen in Table I that for each value of there is an upper limit of d/r above which no values are given. This results from the observation that as can never exceed 90-0 0.6 (Z2)o 0.4 d/1 0.2 o 5 0 6 9 FIG. 6. Effect of the contact angle 0 on the capillary rise between two cylinders for various values of d/r. Extreme right-hand curve gives this relationship for a narrow closed capillary of cylindrical cross section. TABLE II PARAMETERS CHARACTERIZING THE CAPILLARY RISE BETWEEN A ]:~OD AND A PLATE (0 = 0 ) d/r a2 r/r2 d/r2 dlr a~ rlr~ dlr2 0.001 1.01 928.4 0.9284 0.24 47.45 2.085 0,5005 0,002 1.62 454.5 0.9089 0.28 52.55 1.689 0.4731 0.004 2.62 221.0 0.8840 0.32 57.27 1.401 0.4484 0.006 3.47 144.4 0.8665 0.36 61.67 1.184 0,4262 0.008 4.25 106.5 0.8522 0.40 65.78 1.015 0,4059 0.010 4.97 84.02 0.8402 0.45 70.54 0.8505 0,3828 0.015 6.62 54.37 0.8155 0.50 74.94 0.7240 0.3620 0.020 8.12 39.78 0.7957 0.55 79.01 0.6236 0.3430 0.025 9.52 31.15 0.7787 0.60 82.78 0.5427 0,3256 0.030 10.84 25.46 0.7638 0.70 89.56 0.4213 0.2949 0.035 12.11 21.44 0.7503 0.80 95.46 0.3357 0.2686 0,040 13.33 18.45 0.7380 0.90 100.65 0.2731 0.2458 0,045 14.51 16.15 0.7266 1.00 105.24 0.2259 0.2259 0,050 15.65 14.32 0.7160 1.2 112.99 0.1608 0.1929 0,060 17.85 11.61 0.6966 1.4 119.28 0.1191 0.1668 0.070 19.94 9.703 0.6792 1.6 124.47 0.0911 0.1457 0,080 21.95 8.291 0.6633 1.8 128.84 0.0713 0.1284 0,090 23.89 7,207 0.6486 2.0 132.56 0.0570 0.1140 0.10 25.76 6.349 0.6349 2.5 139.84 0.0349 0.0871 0.12 29.34 5.083 0.6100 3.0 145.16 0.0229 0.0688 0.14 32.72 4,198 0.5877 3.5 149.22 0.0159 0.0557 0.16 35.93 3.546 0.5674 4.0 152.42 0.0115 0.0460 0,18 39.00 3.049 0.5488 5.0 157.15 0.0066 0.0329 0.20 41.94 2.658 0.5316 Journal of Colloid and Interface Science, VoI. 30, No. 1, May 1969

CAPILLARY PHENOMENA IN ASSEMBLIES OF PARALLEL CYLINDERS 75 (see Fig. 5), in which case R2 = ~ and z2/r = 0. As the capillary rise is zero, the net force due to surface tension (F~I + F2~) should then also be zero. In other words or (AC) cos 0 = (AB) a2 cos 0 = 1 A- d/r - cos a2, which leads to (d/r)... = 0r/2 -- 0) cos 0 A- sin 0 -- 1. [19] For complete wetting (d/r)... = 7r/2-1 = 0.5708. Although for d/r > (d/r)... some capillary rise does take place, it will never obey the condition z2 >> r, so that for these large separations the above treatment breaks down, whatever the value of r. It must be mentioned that the effect of 0 on z2 differs from that in a narrow closed capillary of circular cross section, where a simple cos 0 term suffices. In the present system z2 decreases more rapidly with increasing 0. This is shown clearly in Fig. 6, where the ratio (z2)o/(z~)o=o is plotted as a function of 0 for several values of d/r, on the assumption that the capillary eorlstant of the liquid (c) remains constant. When d/r approaches zero, the ratio does indeed tend to a value of cos 0, but the larger the value of d/r, the more rapidly does z2 decrease with increasing 0. In Table II the results are presented for a rod-and-plate combination. The table is used the same way, i.e., z2/r is calculated from r/r2 through Eq. [8]. However, in this case there is no finite value for (d/r)... It is also noted that for equal d/r the capillary rise z~ is higher in the rod-plate system than in the rod-rod system. CONCLUDING REMARKS In principle, capillary rise between two rods or between a rod and a plate can be used to measure surface and interracial tensions. Compared to the conventional capillary rise method, it has the disadvantage that the rise is considerably smaller than that observed in a cylindrical capillary of diameter 2d. Also, the rods experience an inward pull owing to capillary forces in the liquid column, so that the rods must be sufficiently rigid to ensure constant separation. On the other hand, the meniscus between two rods will always be in equilibrimn with the bulk surface, which cannot be said for the isolated meniscus in a cylindrical capillary. ACKNOWLEDGMENT I thank Dr. E. D. Goddard for stimulating discussions and the Lever Brothers Company for permission to publish this paper. REFERENCES 1. SCHWARTZ, A. M., AND MINOR, F. W., d r. Colloid Sci. 14,572 (1959). 2. MINOR, F. W., SCItWAI~TZ, A. M., WULKOW, E. A.,.~ND BUCKLES, L. C., Textile Res. dr. 29,943 (1959). 3. MINOR, F. W., SCH\V,kl~TZ, A. M., BUCKLES, L. C., AND WULKOW, E. A., Am. Dyestuff Reptr. 49,419 (1960). Journal of Colloid and ffnterface Science, Vol. 30, No. i, May 1969