J. Chem. Soc., Faraday Trans. 1, 1984,80, 2375-2393 Meniscus Curvatures in Capillaries of Uniform Cross-section B Y GEOFFREY MASON*? AND NORMAN R. MORROW New Mexico Petroleum Recovery Research Center, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801, U.S.A. Received 27th June, 1983 Menisci contained in capillaries of uniform cross-section can be broadly classed according to whether wedge-like liquid structures exist, as in triangular-section tubes, or do not exist, as in circular-section tubes. In tubes which form wedge menisci the liquid in the wedge adopts a form so that a section through the liquid surface is the arc of a circle. The volume of liquid per unit length of the wedge is constant along the tube. A non-wedging meniscus, however, is locally bounded by its tube and has a curvature inversely proportional to the hydraulic radius of the tube.Mayer and Stowe (J. Colloid Interface Sci., 1965,20,893) proposed an approximate method of determining the mean surface curvature of menisci in sphere packs. It was later applied independently by Princen (J. Colloid Interface Sci., 1969,30,60) to estimating capillary rise in spaces between parallel rods. The method, which incorporates the presence of wedges, is shown to be exact for determining mean surface curvatures in systems where the meniscus is undistorted by gravity. Experimental confirmation of the theoretical predictions to within 1.5% was obtained from measurements of capillary rise of a perfectly wetting liquid in tubes formed either by a rod and a square corner or by two rods and a plate.The conditions of pore geometry and contact angles which give rise to wedge menisci are discussed and illustrated by examples which include menisci in tubes of polygonal section. A porous material partially saturated with liquid has properties that are dominated by the capillary behaviour of the liquid in the pore space. Processes such as desorption of capillary-condensed gases, mercury-intrusion porosimetry and even the water- proofing of fabrics depend upon liquid menisci in pores. An important factor determining capillary behaviour in pore spaces is the pressure differences across the interface, or meniscus, which separates two phases. This pressure difference, usually called the capillary pressure, is proportional to the interfacial tension between the two phases comprising the interface and, for a given pore shape and wetting condition, varies inversely with pore size.A fundamental property of the meniscus in a pore is the maximum mean surface curvature or displacement, which corresponds to the situation when a meniscus passes through a pore. This curvature is often normalized with respect to some characteristic length of the system such as particle radius. In general pores in porous materials have irregular shape, and the meniscus curvatures are not known for such cases. Even for relatively simple pore shapes, such as those defined by spheres, there has been no exact analysis of meniscus curvatures. Several approximate solutions have been evolved in the past. These were mostly for packings formed by equal spheres.We examine here a method put forward by Mayer and Stowel for pores formed by spheres and subsequently by P r i n ~ e n ~ - ~ for spaces given by parallel rods. t Present address: Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU. 23752376 MENISCUS CURVATURES IN CAPILLARIES The basis for computations made by Mayer and Stowel and Prin~en,~-~ referred to here as the MS-P method, are identical. However, Mayer and Stowel assumed that their analysis was exact for interfaces passing through converging-diverging pores formed by spheres. While it provides a useful approximation if the contact angle is zero, it can be expected that errors in the curvatures computed by Mayer and Stowe for spheres will increase significantly with increasing contact angle.P r i n ~ e n ~ - ~ applied the same analysis to spaces of uniform cross-section given by rods in various arrays. However, the power and exactness of the method, as applied by P r i n ~ e n ~ - ~ to systems formed by rods, was partially obscured by restrictions related to deviations from constant curvature that arise from treatment of systems in terms of capillary rise against gravity. In a past investigation5 remarkable agreement was found between the predictions of the MS-P theory and experimental results. The tube geometry of those experiments involved the space enclosed by two equal parallel contacting rods in contact with a plate. Liquids having different contact angles with the solid were used.This present paper reports a rigorous derivation of the theory for systems of constant curvature together with experiments using configurations having very precisely controlled geometry. Two basic configurations were investigated: the space between a rod and a square corner and that between two rods and a plate. A perfectly wetting liquid was used. The theoretical analysis has been extended to incorporate the effect of contact angle. As an example of how changing the contact angle can alter the basic meniscus configuration, the theory is given for the variation with contact angle of the meniscus curvature in polygonal-sectioned tubes. BACKGROUND TO THEORY In general, liquid menisci in pores of complex configuration have complicated shapes.The application of the MS-P theory requires that the form of the basic meniscus in the pore be given, and for this a terminology is needed. In describing the basic form we will use the terms wedge menisci and terminal menisci. Wedge menisci exist in wedge-like spaces such a those formed by two rods in contact as shown in fig. 1 (a), by two plates contacting at an angle as in fig. 1 (b), or by two rods in contact with a plate as in fig. 1 (c). In a mathematical sense they are infinitely long, with the wedge having constant volume per unit length of the capillary. The terminal part of the meniscus, assuming for the present that the tube is perfectly wetted and ony one phase generates wedges, is the region that spans and fills the tubular space. For menisci between two contacting rods and a plate, fig.1 (c), there is a terminal meniscus which merges into three infinite wedge menisci. For two rods which are close together, but not in contact, the terminal meniscus is a saddle-shaped surface between the rods. It is easier to envisage these meniscus shapes as closed surfaces. Consider, for example, a drop of liquid between two separated rods. The drop will be elongated and there will be a terminal meniscus at each end and two wedge menisci along the centre of the drop. As another example, consider the space between two contacting rods and a plate to be totally filled with liquid and imagine a bubble in the interior space. If the bubble is much longer than its diameter it will have a terminal meniscus at each end and three wedge menisci along the central portion of the bubble. In the absence of gravity these menisci are surfaces of constant mean curvature. Note that in each case the curvature of the wedge menisci, and hence total curvature, is set by the terminal menisci.The essence of the MP-P method is the equating of the curvature of the wedge menisci to the curvature of the terminal meniscus. Together with a virtual work (orG. MASON AND N. R. MORROW 2377 A t 1 I I I I I L B ! t C t 3 Fig. 1. (a) Wedge-meniscus formation between two contacting rods. (b) Wedge-meniscus formation in the angle between two flat plates. (c) Wedge menisci and the terminal meniscus in the space between two contacting rods and a plate. force-balance) equation, this enables the curvature of the terminal meniscus to be determined.In physical terms the method centres on a particular feature of the meniscus behaviour. The curvature of a terminal meniscus is determined in part by the local boundary conditions (position of tube walls, contact angle) and in part by the wedge menisci. However, the curvature of the wedge menisci must equal that of the terminal meniscus, so by varying the radius of curvature of the wedge menisci the meniscus seeks out a position where, if possible, the two curvatures are equal. All this is general. One may, however, choose a particular position to apply the curvature conditions in the analysis. At sufficient distance (in practice, this is not far in terms of number of tube radii) from the terminal meniscus, the profile of the wedge meniscus becomes a circular arc of radius r.Its radius of curvature in the plane of the cross- section is r and its other radius of curvature is infinite. Having a position where the section through the wedge meniscus is simply a circular arc is not just convenient but is crucial to the method. As mentioned before, the systems for which the MS-P method was created obscure the simple elegance of the method. Mayer and Stowel used the analysis for pore geometries involving spheres, whereas their analysis was really for rods. P r i n ~ e n ~ - ~ was concerned with capillary rise, and this inevitably involves distortion of the menisci by gravity and wedges of changing curvature. As a consequence he introduced the condition that the meniscus dimensions must be negligible compared with the height of capillary rise.In the absence of gravity the MS-P method is exact for tubes of uniform cross-section. The principles of the application of the MS-P method are straightforward, but the details of application depend on whether or not a wedge-shaped interface exists. For example, wedge menisci never occur in the circular-section tubes but always occur in tubes made up from three cylinders in contact, irrespective of the contact angle. However, for the configuration of a rod inside a tube, there are no wedge menisci when the rod is centrally placed in the tube; when the rod touches the tube, wedge menisci2378 MENISCUS CURVATURES IN CAPILLARIES I I l l I 1 0 1 ' I I ' 1 I " I Fig. 2. Diagram showing a rod-in-a-tube configuration. When the rod touches the tube wall (a) two wedge menisci are formed.When the rod is centrally spaced (b) no wedge menisci are formed. This illustrates the need to identify the configuration of the meniscus and decide whether wedge menisci are formed. exist (fig. 2). These systems have zero contact angle but variable geometry, and the wedge menisci may appear and disappear as the geometry is changed. We shall see later that in systems with fixed geometry the presence of wedge menisci may depend on wetting properties. The method adopted for deciding on the existence, or not, of the wedge menisci is to calculate the meniscus curvature for all possibilities and then assume that the meniscus will adopt the one with the lowest curvature (minimum surface area).Both Mayer and Stowel and P r i n ~ e n ~ - ~ used this method without stating it as a principle. It is not always a reliable method, as cases are possible where metastable menisci may exist which are of physical significance. Such menisci may need to overcome an energy barrier before they adopt a configuration with a lower curvature. THEORY The aim of the theory is to calculate the curvature of the terminal meniscus. For a capillary which does not form wedges, the Gauss equation of capillarity6 relating area, perimeter, contact angle and meniscus curvature can be applied directly. Con- sider a small displacement, dx, of the terminal meniscus. In the absence of wedges, the projected area of the terminal meniscus is equal to the area of cross-section of the capillary, A'.If the capillary has a perimeter P', equating the two virtual works of displacement gives (1) where pc is the capillary pressure, 0 is the interfacial tension and 8 is the contact angle. Capillary pressure is related to surface curvature, C, by pc A' dx = a P cos 8 dx Hence pc = ac. C = P cos 8lA'G. MASON AND N. R. MORROW 2319 A Fig. 3. Cross-section of a kite-shaped pore. This section is useful for illustrating the MS-P analysis as, with a perfectly wetting liquid, it contains a single-wedge meniscus in the corner. which is simply the inverse of the hydraulic radius of the capillary, a result examined by Ca~man,~ who found that for wetting liquids the equation fitted the data of Schultze8v9 well for almost circular capillary tubes but was less adequate for other tubes. Carman’ noted that this was because ‘where capillary walls form a sharp angle, the edge of the meniscus shows a sharp local rise to a considerable height above the bottom of the meniscus’.7 Thus it was established that eqn (3) applies well to what we term non-wedging systems.The Gauss equation can also be applied to wedging systems, but now the small displacement has to allow for the effect of the wedges. For example, the area term is no longer simply the full cross-sectional area of the capillary because the wedges do not move as the meniscus is displaced. This is easily seen in an example. Consider the meniscus formed in the kite-shaped tube of fig. 3. The terminal meniscus spans the tube, and some way from the terminal meniscus a section across the tube will reveal a wedge meniscus in the corner.Let the mean meniscus curvature be C so the capillary pressure pc is aC, where CJ is the interfacial tension. Let the area of ABDE be A and the perimeter be considered in parts : Ps being the solid perimeter (AB + DE + EA) and PL being the liquid perimeter (BD). Let the contact angle with which the liquid meets the solid wall be 8 and consider a small displacement of the meniscus, dx. The virtual work balance gives (4) pc A dx = O( Ps cos 8 + PL) dx. As before, pc is related to r by Pc = a/r and so A/r = pS cos e+ pL. If we define P as p = p s c o ~ ~ + p L (7) and normalise with respect to R we obtain AIRP = r/R. (8) However, Ps, PL and A all independently depend upon r in a simple geometrical manner.It is thus possible to solve the resulting simultaneous equations numerically or graphically to find r. For some systems, the polygonal tube for example (see later), the equations can be solved analytically. For the kite pore of fig. 3, the area-to-perimeter ratio can be readily calculated for various values of r/R and 8 = 0. A graph of y = A/RP against r / R can then be plotted. The intersection of the line y = r / R with2380 MENISCUS CURVATURES IN CAPILLARIES 0.7 0.6 0.5 5 0.4 rj q 0.3 L 5 0.2 0. I 0 / 7 8 = 0 I-3R-l 0 0.5 I .o normalised radius, r/R Fig. 4. Examples of the graphical solution of the meniscus curvature for the kite-shaped pore of fig. 3. The intersection of the two lines gives the solution. The solution is always at the maximum value of A/RP.y = A/RP gives the value of r / R , which is the solution to the equations. This graphical solution is illustrated in fig. 4. Note particularly that the intersection value of A/RP is also the maximum possible value of A/RP (or r). Thus, as might be expected at equilibrium, the meniscus has minimum curvature (maximum radius of curvature) for the particular boundary conditions. This is always true, irrespective of the tube section. ROD-CORNER (8 = 0") Pore shapes involving at least one flat side are useful for experimental work because, if the flat side is transparent, the meniscus can be observed directly and measured. The general analysis will now be applied to a capillary formed by a rod in a right-angled corner (as in fig. 5) and a fluid making a zero contact angle with the solid.There are three wedge menisci, two where the rod touches the plate and the third in the right-angled corner between the two plates. A section through the meniscus some distance above the terminal meniscus will contain these wedge menisci as arcs of circles. The MS-P equation for r, the radius of curvature of the wedge menisci, is, following eqn (8), (9) area (ABCDEF) length (AB + CD + EF) + length (BC + DE + FA) r = where ABCDEF refer to fig. 5. The lengths and areas in eqn (4) are given by arcs of circles or straight lines, and straightforward geometry gives the following equations for the perimeter and area in terms of the angle a (the radius of the rods is R): P/2 = R- R sin a - r( 1 + sin a) + Rn(45 -a)/ 180 + ~ ( 2 2 5 - a)/180 (10) A/2 = R2/2 - R2z45/360 - R( R sin a + r sin a ) ( 1 - cos a) +$r2 cos a sin a + rzn( 180 - a)/360 + R2na/360 -$R2 cos a sin a - r2/2 + r2n45/360.(1 1 )G. MASON AND N. R. MORROW 238 1 Fig. 5. Cross-section of the rod-in-corner space and section through the three wedge menisci. The sections through these wedge menisci are arcs of circles of the same radius, r. 0.18 - - ROD & PLATE SYSTEM 0.14 - CORNER SYSTEM 9 = o - 0.00 0.04 0.08 0.12 0. I6 0.20 radius, r Fig. 6. Examples of the graphical solution for r / R for a perfectly wetting liquid in the rod-in-a-corner configuration and also in the two-rods-and-plate configuration. The solution is where y = r / R (the 45" line) cuts the relevant A / P line. This occurs at the maximum in A/RP. Geometry gives the relation between r and a : and we also have R(1 -cosa) 1 +cos a A / P - r = 0.r = Solving for r by a numerical method gives a = 35.122", r / R = 0.1008 and R / r = 9.985. The graphical solution for the rod-and-corner system and also for the rods-and-plate system is shown in fig. 6. As expected, the solution lies at the maximum of A / P in2382 MENISCUS CURVATURES IN CAPILLARZES INLET 4 A t (3 Q 0 1 'I A B C D E F aD 0 G3 0 I 2286 m m ES COVER PLATE SECTION A-A I+ E E (D t I VENT - Fig. 7. Diagram of the apparatus when viewed from the front. The rod pairs were in the channels labelled A-F. The precision-bore tubes were in the large central channel. Later, channel A was enlarged to accommodate 9.525 mm diameter rods. Rod pairs of such size do not show the full capillary rise because of distortion of the meniscus by gravity.both cases. Previous work5 had shown that the meniscus curvature calculated by this method for the similar geometry of two rods and a plate and zero contact angle was R / r = 6.970. EXPERIMENTAL The experiment principally involved measuring the height of capillary rise in capillaries of constant cross-section made up of rods and plates. There were two configurations, a rod in a corner and two rods and a plate. The apparatus was a compromise between using small and large rods. Small rods maximise the height of capillary rise but leave the geometry affected by the inevitable dimensional errors. Large rods minimise the dimensional errors but introduce error because the menisci are distorted by gravity. We used a series of rods of different diameters creating an apparatus in the optimum range and for which the two sources of error could be quantified.The general form of the apparatus is shown in fig. 7. The main part was an aluminium-alloy slab in which were milled a series of channels, each of precise depth and width, and a series of smaller connecting channels which carried the liquid. Each channel was machined to takeG . MASON AND N. R. MORROW 2383 a pair of precision-ground steel rods side by side. A transparent window covered and sealed the front face of the slab. The channels were dimensioned so that the rod pairs were kept in contact by the sides of the channels and kept in contact with the Lucite cover plate by the bottom of the channels. The aluminium slab also contained a compartment in which was placed a group of five precision-bore glass capillary tubes.The diameter of these tubes was determined by partly filling them with mercury and measuring both the length and weight of the mercury thread. The rod pairs and capillary tubes all had access to liquid via various channels and holes. The liquid reservoir, a 250 cm3 polytetrafluorethylene (PTFE) beaker, was connected to the slab by a length of PTFE tubing. The liquid chosen for the experiment was iso-octane, which perfectly wetted the steel rods, the aluminium slab, the Lucite window and the glass capillary tubes. By raising and lowering the beaker reservoir, the level of iso-octane in the cell could be increased or decreased. The apparatus had quite a slow response to level changes, typically taking 10min or so to reach equilibrium.This was due to the quite large (relative to the rod-plate capillaries) volume of liquid in the neighbouring region of the capillary tubes combined with the resistance to flow of the 1/8 in. 0.d. PTFE tubing and the very small liquid head. In operation, after reaching equilibrium the levels of all of the liquid menisci, both in the test capillaries and the cylindrical glass capillary tubes, were measured with a cathetometer reading to 0.01 mm. The rod diameters had been measured with a micrometer. The apparatus had two finer points. There were two rod channels which were nominally of the same dimensions at opposite ends of the slab. This gave a check on the manufacturing errors of the channels (which turned out to be smaller than expected) and also on the accuracy of horizontal travel of the cathetometer.Two capillary tubes of the same size were included for similar reasons. Before assembly the machined parts were all carefully cleaned to remove traces of coolant and swarf. Experiments were run at ambient temperature, the comparative interpretation of the measurements (see later) being such that closer control was not necessary. It was expected that almost all of the experimental error would be produced by the dimensional tolerances of manufacture. For a given configuration, the position of the meniscus was measured in both the series of calibrated glass capillary tubes and in the constructed capillaries. The levels in the calibrated glass tubes could be extrapolated to give the level of the reservoir liquid, and from this the capillary rise in the test capillaries could be determined.For rise up the glass tubes, the position of the meniscus is given by (14) where hT is the level of the tube meniscus, the suffix T being used to indicate the capillary tube, h, is the reservoir level, o is the surface tension of the iso-octane, p its density, g the acceleration due to gravity and RT the radius of the tubes. A plot of h, against 1/R, is thus a straight line through h, with gradient, g,, of 2a/pg. 20 hT = ho+- PgRT CR0 For the rods and plate we have hR = ho +- pgRR where C, is the normalised curvature of the rod-and-plate system. A plot of hR, the meniscus height, against l/RR, where RR is the radius of the rods, is thus a straight line of intercept h, and gradient CRd/pg (= gR).For the rods in the corners we have - c', d h, = h,+-. pgRR C, is the normalised curvature of the meniscus in the corner. This is also a straight line of intercept h,, but with gradient g, (= C, o/pg). It can be seen that with reference to the gradient of the tube line 2g, C, = -. and also g T2384 MENISCUS CURVATURES IN CAPILLARIES The attraction of this method is that a/pg is a constant which cancels. This makes the method relatively insensitive to temperature changes. Furthermore, the change in curvature with height within the region of the terminal meniscus caused by gravity is also largely compensated by this comparative method. An experimental run consisted of measuring the meniscus height for every meniscus in the apparatus, three menisci for each of the six pairs of rods and one each for the five capillary tubes.The bottoms of the menisci were used when measuring the meniscus height. The initial menisci were recorded again to confirm equilibrium and that there were no leaks. Runs were repeated for several levels in the apparatus, both as a check on the method and an estimate of the scatter produced by the inevitable dimensional deviations. The rods had diameters of CQ. 1/4, 1/5, 1/6, 1/7 and 1/8 in., their actual sizes being 6.243, 5.062, 4.173, 3.645 and 3.175 mm, respectively. The capillary tubes had diameters of 1.631, 1.290, 0.695 and 0.459 mm and showed, as planned, capillary rises in approximately the same range as the constructed capillaries.RESULTS The graphs for the tube meniscus height, hT, against l/RT gave excellent straight lines. Extrapolation enabled the height of the reservoir liquid relative to all the other menisci to be determined. Rather than reproduce all of the data separately, the runs for different levels in the apparatus have been condensed onto a single diagram by referring all of the menisci heights to the value of h, determined from the capillary rise in the tubes. The values of (hR - h,) and (h, -ho) plotted against 1 /RR are shown for the four separate runs in fig. 8. The deviations from the theoretical straight lines through the origin are believed to be caused mainly by the deviations of the dimensions of the apparatus from the ideal, these deviations being the main source of error in the determinations. Using each of the gradients of the separate run lines of fig.8, the values of CR and Cc were determined using eqn (1 7) and (1 8). The values obtained were C , = 6.88 & 0.02 and C, = 9.83 0.04. The theoretical values for zero contact angle are C , = 6.970 and C, = 9.985. The agreement is good, but not perfect. One possible source of error is that we have used capillary-rise measurements to test a theory which strictly applies to surfaces of constant curvature. In the analysis of this experiment, beyond using the comparative technique of determining curvatures from capillary rise, no attempt was made to correct for the gravity distortion of any of the menisci. Any severe distortion effects will increase as the size of the rods is increased and should give rise to increasing deviation from a straight line as 1 /RR decreases in a plot of 1 / R R against curvature.However, it can be seen from the results presented in fig. 8 that gravity distortion cannot be severe. In order to identify clearly the gravity effects, the slab was remachined and one of the pair of channels containing 6.243 mm rods was enlarged to accommodate 9.525 mm diameter rods. In the course of carrying out this alteration, the diameter of the capillary tubes was remeasured. Three complete runs were made with the menisci in different positions in the apparatus. The capillary rise of the meniscus in the space between the new rods was only 4 mm, and the data points fell well below the straight line generated by data obtained for the smaller rods.In capillary-rise experiments the menisci cannot satisfy the constant curvature condition of the theory because curvature varies directly with height. Measurement of the height of rise to the base of the meniscus is an experimental convenience. It also corresponds to the minimum curvature. In the present work, for the height change through the terminal meniscus to be unmeasurable with the cathetometer (< 0.01 mm) the height of capillary rise would have to be ca. 1500 mm. Observed heights of rise were in the range of 4-25 mm. Thus in practice the meniscus will alwaysG. MASON AND N. R. MORROW ( 1 /RT)/mm-’ I 2 3 4 5 6 7 1 I I I I -A.O,O I 2385 hR hC TUBES ROD IN CORNE / J ‘A,O 7 0 0. I 0.2 0.3 0.4 0.5 0.6 0.7 (1 /RR )/mm-‘ Fig.8. Graph of the heights of capillary rise against 1/R, for the capillary tubes and against 1 /RE for the rod-in-a-corner and rod-and-plate menisci. The small scatter of the points is mainly caused by dimensional variations in the machining of the cell. No correction has been made for the effects of gravity distortion on the menisci. Levels as follows: A, 57; 0, 76; 0, 95 and 0, 114 mm. have a measurably finite height. What we need to know is the level in the meniscus that corresponds to its average curvature. This can be estimated as follows. Assume first that the wedge menisci are solid wedges and that the cross-section so obtained applies to the complete length of the tube. There will be no change in shape of the terminal meniscus. Now imagine a plane across the terminal meniscus in such a position that the volume of liquid above the plane equals the volume of space below the plane.The position of such a plane could be computed or perhaps measured by some suitable experiment. The level of the plane gives a first-order correction for the 78 FAR 12386 MENISCUS CURVATURES IN CAPILLARIES effects of gravity on the meniscus height. This is the principle of the Rayleigh correction.1° For the case of menisci in circular tubes, where the meniscus can be assumed to be hemispherical, this correction leads to the addition of one-third of the tube radius to the height of the bottom of the meniscus. Using our experimental data, this correction was applied to the heights of rise measured in the capillary tubes.The height of the virtual liquid reservoir was then redetermined by linear regression of these values against 1 / R T . For the non-hemispherical menisci the correction cannot be applied exactly, but it is possible to estimate its size. For each size of each of the rod- plate and rod-corner menisci an equivalent radius was calculated from the height of capillary rise. This equivalent radius was the radius of a cylindrical tube which would give the same capillary rise. One-third of this equivalent radius was then added to the experimental heights of capillary rise. For the 6 mm rod-and-plate configuration this correction amounted to 5% of the height of rise, and for the 9.525 mm diameter rods the correction was 11 % . Plots of the adjusted heights of capillary rise against 1/R, and 1/R, (see fig.9) gave a straight line for all the points. The gradients were slightly shifted, but the ratios of the gradients changed very little. Values of the meniscus curvature so obtained were CR = 6.91 k0.02 and Cc = 9.85 f0.03. These compare almost exactly with the previous experimental determination (6.88 and 9.83, respectively) but still not perfectly with the theoretical values of CR = 6.970 and Cc = 9.985. That the two determinations, one involving a correction for the gravity distortion of the menisci and the other not, give such similar results can be attributed to the comparative method of measurement and the fact that the heights of rise in the tubes and in the rod-and-corner and rod-and-plate configurations were all roughly the same.A possible contribution to the small systematic deviation from theory is that, because of gravity, the wedge menisci in the experimental system are not precisely vertical. However, a rough estimate of the slope of the wedge in the vicinity of the terminal meniscus for the largest rods indicates that this will not reduce experimental results below theoretical values by > 0.5% at most. It is therefore concluded that the small systematic deviation from theory of ca. 1.5 % is due mainly to limitations on the tolerances of the constructed capillaries. For example, the radius of the ‘equivalent tube ’ which corresponds to the rise between the 3.175 mm diameter rod and the corner is ca. 0.32 mm. A deviation of 0.0025 mm (0.0001 in.), a typical unit of precision machining, on this radius of 0.32 mm corresponds to a change of 0.8% in the height of capillary rise. It is unlikely that our machining operations were completed to better than 0.025 mm.The overall effect of tolerance variation will always be to make the capillaries larger than their nominal values. This is consistent with observed curvatures being slightly less than theoretical values. DISCUSSION It has been shown theoretically that for systems of constant curvature the MS-P method is based on an exact analysis. Furthermore, through measurement of capillary rise, an experimental confirmation of the method has been provided. The MS-P method has great versatility and a variety of potential applications. For example, it can be used to calculate meniscus curvatures for various configurations with particular wetting conditions.Indeed, as contact angles can be notoriously irreproducible, using the MS-P method may be more reliable than experimental results for determining change in meniscus curvatures with contact angle. Furthermore, as the method is exact for uniform sections, it gives a means of calculating meniscus curvatures for complex configurations. This could be of value in systems where close control of the solidG. MASON AND N. R. MORROW (l/R,)/mm’’ 0 I 2 3 4 5 6 7 8 9 25 I I I I / I 1 I I I R, I ROD IN I I I I I I I I I 2387 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ( 1 lR Ymm-’ Fig. 9. Graph of the heights of capillary rise plotted against the reciprocal of the tube and rod radii. These results are from the modified apparatus containing the 9.525 mm diameter rods.A correction for gravity distortion of the menisci has been applied to these results. The effect on the final ratio of line gradients is quite small as the changes are mostly self-compensatory. Levels as follows: 0, 181 ; e, 197 and A, 204 mm. surface is required, such as in contact-angle work. A pore geometry can be assembled from a particular material, and the measurement of capillary rise in some form of constructed capillary of constant cross-section gives a means of estimating the contact angle. An example of this would be a capillary formed from two rods and a plate using a material which cannot, for example, be drawn into tubes. In the application of the MS-P method, the most critical part of the theoretical analysis concerns the existence or non-existence of wedges.For the rod-and- corner example, if the contact angle is zero, wedge menisci always exist at the 78-22388 MENISCUS CURVATURES HALF ANGLE MENISCUS v IN CAPILLARIES Fig. 10. Diagram of a corner of a polygonal section tube. Depending on the contact angle, a wedge meniscus may exist in the corners. rod-and-plate contacts and in the corner. When the contact angle is a variable, the wedging behaviour is changed and the MS-P method has to be modified accordingly. This is best shown in a worked example. Two configurations will be considered: a capillary with polygonal cross-section and then a more complicated example, the rod-corner capillary. THE II-AGON TUBE (NON-ZERO CONTACT ANGLE) Princen3 has given solutions for the curvature of menisci in triangular and square cross-section tubes for zero contact angle.These cases are relatively straightforward because they always have wedge menisci. Concusll has discussed menisci in polygonal- sectioned tubes for large contact angles where the meniscus is part of a spherical surface. These also are straightforward as there are never any wedge menisci. In this section we will bring both analyses together. The results will illustrate how menisci in wedging and non-wedging systems are analysed and permit an extension to the rod-corner configuration for non-zero contact angle. They can be compared with the conclusions of Hwang,6 who did not distinguish between wedging and non-wedging systems and assumed that all systems are non-wedging.This explains why Hwang’s6 results differ from Prin~en’s~-~ results. Consider a portion of an n-sided polygonal tube as shown in fig. 10. Let the in-circle radius of the tube be R and the half-angle subtended by a side be 90 -p. Let the radius of the wedge meniscus (if any) be r and the contact angle be 6. The perimeter of the wetted solid, Ps, for the sector subtended by 90-p is given by R rcos6 ps = --- +rsin0 tan/? tan/? and the liquid perimeter, PL, is given by (90-p-6) nr. 180 PL = The total resolved perimeter is P: P=P,cosO+P, (90 - /? - 6) nr. 180 R rcos6G. MASON AND N. R. MORROW The area, A , of the section is given by R2 r2cos28 r2 zr2 2tanj? 2tanp 2 360 A = - - - + - cos 8 sin 8 + - (90 -8- 8). We also have the MS-P equation: or, rearranging A / P - r = 0 A - r P = 0.2389 (22) Substituting for A and P gives a quadratic equation in r/R: (24) cos28 cosOsin8 7t r2 cos8 r 1 -- (90-p-8)) --- -+- 2 360 R2 tan/?R 2tanB=" There are two roots, only one of which is physically realistic; the other probably corresponds to mensici on the outside of the triangle. However, there are no real roots when the (r/R)2 coefficient becomes zero, which happens when 8+/3 = 90. (25) This represents the point where the wedge menisci disappear, and so the condition (26) for wedge menisci to occur is e s 90-8. Wedge menisci always occur for 8 = 0 except when 8 = go", which corresponds to the number of sides of the polygon becoming infinite, i.e. the cylindrical tube. For 90 > 8 > 90-8, the wedge menisci do not exist and the curvature of the meniscus becomes A / P , the hydraulic radius. Thus A / P = R / ~ c o s ~ (27) becomes the solution.This corresponds to the spherical meniscus in the n-agon tube with in-circle radius Rlcos 8, a result pointed out by Concus.ll The meniscus in such a tube simply runs into the corner but does not form an infinite wedge. Concusll published photographs of menisci in such configurations. We thus have the analysis for the n-agon tube. The condition for wedge menisci to form is 8 < (90-8). This is also the condition for wedge-meniscus formation in any corner of half-angle 8, a fact which will be put to use later in the analysis of the meniscus in the rod-and-corner configuration. As an example of the effect of the wedge menisci on the total menicus curvature, fig.11 shows the meniscus curvature in 3-, 4- and co-sided tubes together with the contact angle at which the wedge menisci disappear. For zero contact angle these results duplicate those of Prin~en,~ although he normalised his results with respect to side length as compared to the in-circle radius used here. ROD-IN-A-CORNER (NON-ZERO CONTACT ANGLE) For this configuration (fig. 12) there are always wedge menisci between the rod and contacting plates. However, the corner has a 90" angle and will only have a wedge meniscus for 8 < 45". If 8 > 45" then no wedge meniscus is formed and the terminal meniscus just runs into the corner. We have two cases. (i) 8 < 45" In this case P is given by P = (R- R sin a- r[sin (0+ a)- sin 81 - r(cos 8- sin 8) + Rn(45 - a)/ 180} cos 0 + m(225 - 38 - a)/ 180 (28)2390 MENISCUS CURVATURES IN CAPILLARIES I I I I 4 1 I I I SIDES POLYGONAL TUBES 1 4 - * 12- 0 2 - 0 I0 20 30 40 50 60 70 80 90 contact angle, elo Fig.11. Curvature of the meniscus in the n-agon tube normalised relative to the radius of the in-sphere is given as a function of contact angle. The infinite side number tube is simply a cylinder and the values agree with those for meniscus curvature in cylindrical tubes. When corners exist, the menisci have wedge menisci in them for low values of contact angle, and this reduces the curvature. Fig. 12. Diagram of the section through a meniscus in the space between a rod and a corner for non-zero contact angle. Wedge menisci always exist between the rods and plate. Depending on the contact angle, a wedge meniscus may exist in the right-angled corner.G .MASON AND N. R. MORROW 2391 Table 1. Curvature ( R / r ) of the fluid meniscus in the space between a rod and a corner. The angle 0 is the angle that the fluid makes with the solid surface. In this table a wedge meniscus exists in the right-angled corner 0 35.12 0.1002 9.985 10 34.12 0.1011 9.892 20 32.95 0.1043 9.589 30 31.60 0.1105 9.049 40 30.03 0.1212 8.249 45 29.14 0.1291 7.747 where a and 0 are shown in fig. 12. The area, A , is given by A = R2/2 - R2n45/360 - R[R sin a + r sin (0 + a)] (1 - cos a) + ar2 sin 0 cos 8 +tr2 cos (0+ a) sin (O+ a) + r2n( 180 - 20 - a)/360 + R2na/360 - $R2 sin a cos a - +r2 cos2 8 + 4r2 sin O cos 0 + r2n(45 - O)/360.(29) Geometry gives the relation between r and a: R( 1 - cos a) r = cos O+cos (e+ a) ' There is also the MS-P equation: A / P - r = 0. (8) Eqn (8) and (28)-(30) can be solved numerically to give r / R as a function of 0. Some values are given in table 1. (ii) 45 d elo < 90 have P = {R - R sin a - r[sin (0+ a) -sin B] + Rn(45 - a)/ 1 80) cos O+ rn( 180 - 20- a)/180 A = R2/2 -nR2 45/360 - R[R sin a + r sin (O+a)] (1 -cos a)+gr2 sin Oeos 0 When 0 > 45", no wedge meniscus is formed in the right-angled corner. We now (31) +4r2cos(O+a) sin(O+a)+r2n(180-20-a)/360+R2na/360-~R2 sinacos a (32) R( 1 - cos a) cos 0 + cos (0 + a) r = A / P - r = 0. (8) Again eqn (8) and (30)-(32) can be solved numerically and we obtain r / R as a function of 0. (See table 2 for some typical values.) In principle the method can be applied to systems which are considerably more complex than those discussed so far.Provided the properties of the capillary are constant with respect to cross-section even variation of the wetting properties around the perimeter are permissible. Capillaries do not necessarily have to have a closed2392 MENISCUS CURVATURES IN CAPILLARIES Table 2. Curvature (Rlr) of the fluid meniscus in the space between a rod and a corner. The angle 0 is the angle that the fluid makes with the solid surface. In this table a wedge menicus does not exist in the right-angled corner 45 29.14 0.1291 7.747 50 28.14 0.1393 5.178 60 25.62 0.1706 5.861 70 21.88 0.2328 4.295 80 15.13 0.4119 2.428 perimeter, but if they do not, then attention must be paid to the effect of neighbouring capillaries.Systems where the wedge interface occurs at edges can also be treated, and the presence of the edge merely provides an additional degree of freedom with respect to minimisation of free energy. The difficult part of the application concerns the existence and position of the wedge menisci. An interesting point, and one which made our experiments easier to perform, is that the detailed surface geometry in the wedging corners is unimportant. Systems can have regions of extreme and arbitrary complexity in the corners, and provided these regions are bounded by wedges they do not affect the overall meniscus curvature. As an extreme example of this, the surface curvature of the terminal meniscus would be unchanged if the wedge menisci in wetting systems were replaced by solid material to give a capillary of reduced cross-section.SUMMARY This paper brings together several existing ideas and methods of calculating the curvature of menisci in tubes of arbitrary cross-section. When wedge menisci are not formed, then the hydraulic-radius method can be used. When wedge menisci are formed, the MS-P method must be used, with the main problem now being where the wedge menisci are formed. The MS-P theory is exact for constant curvature menisci. Nevertheless, it may be necessary to calculate the meniscus curvature for several conditions of existence or non-existence of wedge menisci and assume that the actual curvature adopted will be the one with the lowest curvature. The method has been applied here to the rod-in-a-corner configuration and the analysis tested with an experiment in which the capillary rise of a perfectly wetting liquid was measured for a range of rod sizes. The agreement was good but not perfect, most probably because of slight manufacturing errors in the apparatus which always increase the apparent size of the pore space in the cross-section. The effect of changing the contact angle in systems of fixed geometry has been discussed. In these cases there can be a contact angle at which the wedge menisci cease to exist, and this must be reflected by the analysis. This was highlighted by the n-sided polygonal tube which has a particular critical contact angle at which the wedge menisci disappear. The basic method of analysis, together with the criterion for wedging, was applied to the rod-in-corner configuration for liquids having a non-zero contact angle. The MS-P method is widely applicable in principle and can be used to estimate the meniscus curvature for any particular pore geometry with any particular contact angle.G. MASON AND N. R. MORROW 2393 We thank A. R. Romero of the TERA workshop for machining the cell and C . Lawson and Shang-shi Shu for assistance in taking the measurements. This work was jointly supported by the U.S. Department of Energy, contract no. DE-AS 19-80BC103 10, and the New Mexico Energy Research and Development Institute, project no. 2-69-3309. G. M. was on sabbatical leave from Loughborough University of Technology, Leicestershire. ' R. P. Mayer and R. A. Stowe, J . Colloid Interface Sci., 1965, 20, 893. H. M. Princen, J. Colloid Interface Sci., 1969, 30, 60. H. M. Princen, J . Colloid Interface Sci., 1969, 30, 359 H. M. Princen, J. Colloid Interface Sci., 1970, 34, 171. G. Mason, M. D. Nguyen and N . R. Morrow, J. Colloid Interface Sci., 1983, 95, 494. S. Hwang, Z . Phys. Chem. (Neue Folge), 1977 105, 225. ' P. C. Carman, Soil Sci., 1941, 52, 1. K. Schultz, Kolloid Z . , 1925, 36, 65. K. Shultze, Kolloid Z., 1925, 37, 10. P. Concus, Preprints, 48th National Colloid Symposium, Austin, Texas (June 1974), (Am. Chem. SOC., Washington, D.C., 1974), pp. 12-16. lo A. W. Adamson, Physical Chemistry of Surfaces (Interscience, New York, 1960). (PAPER 31 1 1 1 1)