Capillary 56, rise

The height of the capillary rise for a hydrophilic surface, or the depth of the capillary depression for a hydrophobic surface, can be found by 

An important application of Young s equation and a procedure for measuring the contact angle at the same time, is the rise of a liquid in a capillary tube. If a capillary is lowered into a liquid, the liquid often rises in the capillary until a certain height is reached (Fig. 7.3 top left). For a capillary with a circular cross-section of radius rc, this height is given by 

g is the acceleration of free fall and p is the density of the liquid. 

This is certainly not enough to bring water from the soil to the top of a tree. In fact, other effects contribute significantly. For details see Ref. [228], 

To derive Eq. (7.13) we consider the change of Gibbs free energy upon an infinitesimal rise of the liquid dh. This is simple because the shape of the liquid-vapor interface does not change. The change in Gibbs free energy is  

Not only does a capillary rise occur but, depending on the interfacial tensions, liquid can also be expelled from a capillary. A liquid rises for partially wetted surfaces (0 90°). If the liquid does not wet the inner surface of the capillary and the contact angle is higher than 90° the liquid is pressed out of the capillary. For this reason it is very difficult to get water into polymeric capillaries, as long as the capillary is hydrophobic. This can achieved only by applying an external pressure. 

The equilibrium value of h will be that which minimizes E or 

Thus differentiating Equation (3.14) with respect to h and substituting into Equation (3.15) yields upon rearrangement 

The quantity (ysL — ysv) is the work of immersion, because it quantifies the energy change when a unit area of soUd/vapor interface is replaced by solid/liquid interface. 

In this case ysv may still be determined from Equation (3.16), but the value of h will be negative. 

Adamson and Cast [6] have discussed corrections to the capillary-rise technique when the meniscus is not spherical as well as the experimental issues associated with the measurements. 

---

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

Fig. II-6. Capillary rise (capillary much magnified in relation to dish).

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

While Eq. 11-14 is exact, its use to determine surface tension from capillary rise experiments is not convenient. More commonly, one measures the height, h, to the bottom of the meniscus. 

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

The capillary rise method is generally considered to be the most accurate means to measure 7, partly because the theory has been worked out with considerable exactitude and partly because the experimental variables can be closely controlled. This is to some extent a historical accident, and other methods now rival or surpass the capillary rise one in value. 

Perhaps the best discussions of the experimental aspects of the capillary rise method are still those given by Richards and Carver [20] and Harkins and Brown [21]. For the most accurate work, it is necessary that the liquid wet the wall of the capillary so that there be no uncertainty as to the contact angle. Because of its transparency and because it is wet by most liquids, a glass capillary is most commonly used. The glass must be very clean, and even so it is wise to use a receding meniscus. The capillary must be accurately vertical, of accurately known and uniform radius, and should not deviate from circularity in cross section by more than a few percent. 

As is evident firom the theory of the method, h must be the height of rise above a surface for which AP is zero, that is, a flat liquid surface. In practice, then, h is measured relative to the surface of the liquid in a wide outer tube or dish, as illustrated in Fig. n-6, and it is important to realize that there may not be an appreciable capillary rise in relatively wide tubes. Thus, for water, the rise is 0.04 mm in a tube 1.6 cm in radius, although it is only 0.0009 mm in one of 2.7-cm radius. 

The general attributes of the capillary rise method may be summarized as follows. It is considered to be one of the best and most accurate absolute methods, good to a few hundredths of a percent in precision. On the other hand, for practical reasons, a zero contact angle is required, and fairly large volumes of solution are needed. With glass capillaries, there are limitations as to the alkalinity of the solution. For variations in the capillary rise method, see Refs. 11, 12, and 22-26. 

If APmax is expressed in terms of the corresponding height of a column of the liquid, that is, APmax = Ap gh, then the relationship becomes identical to that for the simple capillary rise situation as given by Eq. II-10. 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

The table is used in much the same manner as are Eqs. 11-19 and 11-20 in the case of capillary rise. As a first approximation, one assumes the simple Eq. II-10 to apply, that is, that X=r, this gives (he first approximation ai to the capillary constant. From this, one obtains r/ai and reads the corresponding value of X/r from Table II-2. From the derivation of X(X = a /h), a second approximation a to the capillary constant is obtained, and so on. Some mote recent calculations have been made by Johnson and Lane [28]. 

Refs. 34 and 35 for a more up-to-date discussion.) This they verified experimentally by determining drop weights for water and for benzene, using tips of various radii. Knowing the values of 7 from capillary rise measurements, and thence the respective values of a, / could be determined in each case. The resulting variation of / with r/V / has been fitted to a smoothing function to allow tabulation at close intervals [36]. 

It should be noted that here, as with capillary rise, there is an adsorbed film of vapor (see Section X-6D) with which the meniscus merges smoothly. The meniscus is not hanging from the plate but rather fiom a liquidlike film [53]. The correction for the weight of such film should be negligible, however. 

Calculate to 1% accuracy the capillary rise for water at 20°C in a 1.2-cm-diameter capillary. 

Derive the equation for the capillary rise between parallel plates, including the correction term for meniscus weight. Assume zero contact angle, a cylindrical meniscus, and neglect end effects. 

Derive, from simple considerations, the capillary rise between two parallel plates of infinite length inclined at an angle of d to each other, and meeting at the liquid surface, as illustrated in Fig. 11-23. Assume zero contact angle and a circular cross section for the meniscus. Remember that the area of the liquid surface changes with its position. 

The following values for the surface tension of a 10 Af solution of sodium oleate at 25°C are reported by various authors (a) by the capillary rise method, y - 43 mN/m (b) by the drop weight method, 7 = 50 mN/m and (c) by the sessile drop method, 7 = 40 mN/m. Explain how these discrepancies might arise. Which value should be the most reliable and why ... 

A liquid of density 2.0 g/cm forms a meniscus of shape corresponding to /3 = 80 in a metal capillary tube with which the contact angle is 30°. The capillary rise is 0.063 cm. Calculate the surface tension of the liquid and the radius of the capillary, using Table II-l. 

Bianco and Marmur [143] have developed a means to measure the surface elasticity of soap bubbles. Their results are well modeled by the von Szyszkowski equation (Eq. III-57) and Eq. Ill-118. They find that the elasticity increases with the size of the bubble for small bubbles but that it may go through a maximum for larger bubbles. Li and Neumann [144] have shown the effects of surface elasticity on wetting and capillary rise phenomena, with important implications for measurement of surface tension. 

The capillary rise on a Wilhelmy plate (Section II-6C) is a nice means to obtain contact angles by measurement of the height, h, of the meniscus on a partially immersed plate (see Fig. 11-14) [111, 112]. Neumann has automated this technique to replace manual measurement of h with digital image analysis to obtain an accuracy of 0.06° (and a repeatability to 95%, in practice, of 0.01°) [108]. The contact angle is obtained directly from the height through... 

For some types of wetting more than just the contact angle is involved in the basic mechanism of the action. This is true in the laying of dust and the wetting of a fabric since in these situations the liquid is required to penetrate between dust particles or between the fibers of the fabric. TTie phenomenon is related to that of capillary rise, where the driving force is the pressure difference across the curved surface of the meniscus. The relevant equation is then Eq. X-36,... 

The Washburn model is consistent with recent studies by Rye and co-workers of liquid flow in V-shaped grooves [49] however, the experiments are unable to distinguish between this and more sophisticated models. Equation XIII-8 is also used in studies of wicking. Wicking is the measurement of the rate of capillary rise in a porous medium to determine the average pore radius [50], surface area [51] or contact angle [52]. 

The expression for the capillary rise in Exercise 5.29 assumes that the tube is vertical. How will the expression be modified when the tube is held at an angle 6 (theta) to the vertical ... 

Alternatively the capillary rise can be measured using a bulk acoustic wave sensor [97Che]. (Data obtained with these methods are labelled BAW). 

Laplace s pressure produces the capillary rise inside a small tube. We propose to examine now the consequences of the solid deformation in capillary flow. 

The wetting ability of the anode electrode was evaluated as the contact angle measured by the capillary rise method. The value of fractal dimension of anode electrode of MCFC was calculated by use of the nitrogen adsorption (fractal FHH equation) and the mercury porosimetry. 

Here, A is the contacting surface area of anode electrode facing with electrolyte and P is the porosity of anode electrode. The average effective radius of pore,, could be calculated from the results of the capillary rise method using ethanol, which shows a contact angle of 0° with the anode electrode. And then, the contact angle 0 could be acquired as the slope from the plot of m versus... 

A liquid-solid contact angle away from 90° induces the formation of a meniscus on the free surface of the liquid in a vertical tube (the solid phase). In the nonwetting case, the meniscus concaves upwards to the air. The upwards meniscus is the result of a downward surface tension at the liquid-tube interface, causing a capillary depression. In the wetting case, the meniscus has a concave-downward configuration. The downwards meniscus is the result of an upward surface tension at the liquid-tube interface, causing a capillary rise. 

Surface tension is independent of tube size. However, the extent of capillary rise or depression by surface tension is dependent on tube size. This can be seen from Equation 18.1 in Section 18.4.6.1. For example, in the case of a capillary rise, the greater the tension, the higher the water rises above the free-water surface. For the same amount of water, the smaller the tube is, the higher the water rises. 

---