Axisymmetric Liquid Hanging Drops - ACS Publications

Research: Science and Education

W

Axisymmetric Liquid Hanging Drops Erich C. Meister* Laboratory of Physical Chemistry, ETH Honggerberg, CH-8093 Zurich, Switzerland; *[email protected] Tatiana Yu. Latychevskaia Physics Institute, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

A lot of important mass and heat transfer processes occur through or at liquid–solid or liquid–gas interfaces, such as vaporization and condensation, sublimation, wetting, washing, adsorption, diffusion, and heterogeneous reactions. Interfacial or surface tension is one of the key parameters that play a significant role in all of these processes, especially at large ratios of interfacial area to volume. Changing the area, O, of an interface requires mechanical energy dw = σdO, proportional to the change dO in area, with σ being the surface or interfacial tension. Thermodynamics shows that for an isothermal, isobaric, and reversible process dw =

dG

p ,T

=

∂ G ∂ O

dO = σ dO p ,T

(1)

where G is the Gibbs energy. Therefore the need arises to obtain reliable data for systems in a broad range of conditions (e.g., temperature, pressure) and as a consequence a manifold of instruments have been designed to measure interfacial tensions under static or dynamic conditions. These instruments are based on the measurement of forces, pressures, masses, or geometrical quantities such as lengths, volumes, angles, and shapes of interfacional surfaces. Some of these tensiometers have also found application in undergraduate physical chemistry laboratory courses mainly to determine the surface tension of pure liquids, of mixtures, or of the critical micelle concentration (cmc) of detergent solutions. Widely used textbooks (1–3) describe experiments using the capillary-rise and maximum bubble-pressure methods, respectively. A more detailed list of methods is found in refs 4 and 5. Articles in this Journal cover the capillary-rise method (6–9), the bubble-pressure method (6, 10–13), force methods using du Noüy ring or Wilhelmy plate (14–18), goniometry on the sessile drop (19–23), the vibrating-jet method (24), and the drop-mass or volume method (25–27). A novel and interesting treatise of the floating-needle experiment has been recently published by Condon et al. (28) who calculated energy versus immersion depth curves for cylindrical metal needles in water. Although the drop-mass (or volume) method, on which we will focus, is inherently connected to the shapes of hanging drops, to our knowledge no descriptions have been given in this Journal of the geometrical properties of drops attached on a horizontal base and of the rules that determine the growth and finally the detachment of such drops. The underlying physical and mathematical relations have been understood for a century and results of its analysis appeared regularly in dedicated journals; however, they are scarcely known to the nonspecialized audience. Therefore, the aims of this article are to present first the necessary theoretical background leading to the differential equation that shapes every drop. The properties of static hanging drops attached on cywww.JCE.DivCHED.org

•

lindrical capillaries of different sizes will then be determined in a straightforward manner from the results of extensive numerical calculations using dimensionless variables. This allows one to characterize special drops, such as the longest as well as the largest possible static drop and to specify the radii of the capillaries to which they are attached. A correction curve, which is necessary in drop-volume tensiometry, is finally calculated from drop-shape analysis and compared with accepted experimental data. The Shape of a Drop Hanging on a Capillary Attempts to describe mathematically the shape of a hanging drop began parallel to the study of capillary phenomena. The German physicist and mathematician Johann Andreas von Segner introduced in 1751 the idea of a surface tension in liquids (29). To calculate the shape of a liquid drop, von Segner considered the curvature of a meridian section of the drop. (However, the correct mathematical description involves the mean of two curvatures, κ1 = 1兾r1 and κ2 = 1兾r2, see below.) In 1804 Thomas Young, professor at the Royal Institution (London), started to study capillary phenomena experimentally and observed that the contact angle between the surface of a given liquid and a solid is constant. He published his work one year later (30). Independently, Pierre-Simon Laplace, member of the Paris Academy of Sciences and also working on capillarity, performed mathematical calculations and achieved results that were identical to those of Young. Laplace presented his theory as a chapter in his mechanical course in 1805 (31) in the same year. Both, Young and Laplace stated that the pressure at the concave side of a curved liquid surface exceeds the pressure at the convex side by a quantity that is proportional to the mean curvature, H = 0.5[(1兾r1) + (1兾r2)], 1 1 + (2) r1 r2 where σ is the interfacial tension or, in the case of a liquid– gas interface, the surface tension and r1 and r2 are the radii corresponding to the principal curvatures. This well-known equation of Young and Laplace allows one to calculate the shapes of liquids that are formed under the influence of surface tension. To define the radii r1 and r2, the geometry of the hanging drop should be considered. The surface of a drop hanging on a cylindrical capillary is shown in Figure 1. The drop is axisymmetric to the vertical axis Oz, with an apex O that defines the origin of the cartesian coordinate system x, y, and z. (The fact that the drop is a figure of revolution allows one to discuss the shape with respect to only two dimensions rather than three.) The circumference where the drop surface contacts the outer diameter of the wetted capillary is called the basis, B, of the drop.

Vol. 83 No. 1 January 2006

∆p = σ

•

Journal of Chemical Education

117


At a point P on the surface one can create a set of three mutually perpendicular planes crossing at P. One of these planes is tangential to the surface, the others (N1 and N2, respectively, see Figure 1) contain the normal n to the curved surface at point P. There is only one possible tangential plane at point P and N1 is chosen to contain the symmetry axis Oz. The three-dimensional surface at any point can be characterized by r1 and r2 defined by the principal curvatures. Two circles of radii r1 and r2, which lie in the N1 and N2 planes, respectively, can be plotted. The centers of the circles M1 and M2 are located on the normal n to the surface. To find mathematical expressions for r1 and r2, we consider a plane that contains the symmetry axis Oz and cuts the surface of the drop in a meridian m, going from the apex to the basis. At point P, the tangential plane creates the inclination angle tanϑ = dz兾dx with the horizontal z = 0 plane. For an infinitesimal length ds of the meridian ds r1 = dϑ

(3)

The center M2 of the second circle of curvature lies on the

Oz axis, thus, r2 =

x sin ϑ

(4)

Substituting eqs 3 and 4 into eq 2 yields

∆p = σ

dϑ sin ϑ + ds x

(5)

The radii of curvature in the apex of the drop, at z = 0, are r1 = r2 = b. The hydrostatic pressure inside the drop decreases with height z > 0, leading to

2σ − ρ l − ρg g z (6) b

∆ p = ∆ papex − ρ l − ρg g z =

where g is the gravitational acceleration and ρl and ρg are the densities of the liquid and the surrounding gas phases, respectively. (In the case of a liquid–liquid interface ρg is replaced by the density of the surrounding liquid phase.) Combining eq 5 and eq 6 gives

dϑ sin ϑ 2 + = − cz ds x b Oz

with the capillary constant c, a

C

ρ l − ρg g

c = B

m z

ds

dz

ϑ

m

dx M1

M2 r2 r1

For a given liquid–gas system at specified values of temperature and pressure, c is a material constant with the dimension of a reciprocal area. It is advantageous to convert eq 7 into a dimensionless form by scaling one-dimensional quantities (lengths, radii) by c1兾2, two-dimensional quantities (areas) by c and three-dimensional quantities (volumes) by c3兾2, respectively. After introducing the new variables, X = xcc

P N2

Y = yc Z = zc

O

n

1 1 1

2

R1 = r1c

2

R2 = r2 c

2

A = ac

Figure 1. The surface of a pendant drop hanging on the axisymmetric capillary C of radius a in the cartesian coordinate system x, y, z with origin O. The basis B of the drop is at the end of the capillary. For a given point P on the drop surface three mutually orthogonal planes crossing at P can be plotted: the tangential plane and two planes N1 and N2. The plane N1, containing the symmetry axis Oz, cuts the surface of the drop in the meridian m. M1 and M2 are the centers of the circles of curvature with radii r1 and r2, respectively. n is the normal to the surface. ϑ is the inclination angle or the angle between the tangential plane and the horizontal plane z = 0. The contact angle of the drop at the basis of the capillary is θ. The inset shows a small portion of the meridian m to define infinite small increments of the meridian length s and the corresponding coordinates x and y.


•

1

1

2

2

1

2

V = vc

3 1

B = bc

2

2

(9)

O = oc

x

S = sc

N1

118

(8)

σ

θ

N

E

(7)

1

2

which are denoted with capital letters, eq 7 modifies to the dimensionless differential equation dϑ sin ϑ 2 + + Z − = 0 dS X B

(10)

(Note the absence of any material constant in this equation.) Equation 10 can also be rewritten in the cartesian coordinate system. The increment of the curve’s length is dS = d X


•

2

+ dZ

2

1

2

= dX 1 +

www.JCE.DivCHED.org

dZ dX

2

1

2

(11)


Substituting ϑ and S and making use of some trigonometric relations,

dX d sin ϑ and = dS dϑ

cos ϑ =

sin ϑ =

Numerical Integration of the Drop-Shape Equation

dZ dX

dZ = dS

dZ dX

1+

1

2

(12) 2

Equation 10 is now expressed as d 2Z dX2 dZ dX

1+

2

3

2

dZ dX

1 + X 1+

dZ dX

relative simplicity of these differential equations, up to now no analytical solution of the drop-shape equation has been found. Therefore, a numerical integration method is required to solve eq 10.

2

1

2

(13) 2 = 0 B

+ Z −

The drop-shape differential equation in both forms, eqs 10 and 13, can be found in most textbooks treating surface phenomena, see, for instance, refs 34 and 4. The parameter B, which is equal to the dimensionless apex radius of curvature, determines the shape of the drop completely. Despite of the

The numerical calculation of a drop shape that is associated with a particular apex radius of curvature B involves the variables X, Z, S, and ϑ, eq 10, and eq 12. The volume V and the surface area O of the drop between apex and height Z follow by the integration of (14) dV = π X 2 d Z and (15) dO = 2 π X dS respectively. The integration starts at the apex where X = Z = S = V = O = ϑ = 0 with dS兾dϑ = X兾sinϑ = B and goes on along the drop’s meridian m up to the basis B of the drop. Devised by Euler (33), an approach for the numerical integration, also known as the method of “difference equations”, replaces the derivatives by finite differences between two nearby points. The differential dS is accordingly set to a sufficiently small step size ∆S. Discrete values of ϑ, X, Z, V, and O result from the governing eqs 10 and 12 and with 14 and 15: X i +1 = Zi + 1 = ϑi + 1 =

4.0

3.0

Zc(A)

L

2.5

0

M

1.

Z(X) and Zc(A)

3.5

2.0 1.5

1.6

1.0 0.5 0.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

X and A Figure 2. Calculated complete drop shapes Z versus X (dimensionless) resulting from different apex curvatures B (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.7789, 0.9, 1.0, 1.2, 1.275, 1.4, 1.6, 1.8, 2.0, 3.0, 4.0, 5.0). The profile sections from the origin up to the intersection with the critical curve Zc(A), plotted with thick lines, refer to stable hanging drops. Instable parts of the drop profiles are drawn using dotted lines. Zc(A) ends in the largest possible hanging drop (M) with B = 1.275. The point L indicates the longest possible critical drop with B = 0.88.

Table 1. Definition of the Drop-Shape Types Type

Equator

Neck

I

no

no

II

yes

no

III

yes

yes

www.JCE.DivCHED.org

•

X i + ∆ X i = X i + ∆S cos ϑi Z i + ∆ Zi = Zi + ∆ S sin ϑi ϑi + ∆ ϑi = ϑi + ∆S 2 B − Z i − sin ϑi X i

Vi + 1 = Vi + ∆Vi Oi + 1 = Oi + ∆Oi

(16)

= Vi + ∆S π X i 2 sin ϑi = Oi + 2 ∆ S π X i

The procedure is started with X1 = Z1 = ϑ1 = V1 = O1 = 0 and stopped at the second point of inflection of the meridian curve Z(X ). This calculation scheme is readily implemented and gives, even when using a constant step size ∆S in the order of 105B, already quite reliable drop shapes. However, to improve the accuracy over Euler’s method we used a fourth-order Runge–Kutta interpolation method (34). Given an apex radius of curvature B for a particular drop shape a set of narrowly spaced values of ϑ, X, Z, V, and O results and is stored in a file. In this way a library with data from several hundred drop shapes in the range 0.1 ≤ B ≤ 10 has been calculated.1 Close inspection of these data shows an almost perfect agreement with those of Hartland and Hartley (35) and of del Río and Neumann (36). Some drop shapes with different values of the parameter B are shown in Figure 2. Broad and flat drop profiles result from large B values, whereas small B values give nearly spherical profiles. Upon inspecting the calculated drop profiles, one finds that curves with values B > 0.7789 have inclination angles ϑ always less than 90, that is, these drops never have an equator. On the other hand, curves with values B < 0.7789 might have an equator and possibly a neck, depending on the length of the drop. It is convenient for the further analysis to define three drop shape types I–III, according to the scheme in Table 1.


•


119


The Growth of a Hanging Drop

Drop Volume A drop that hangs on a capillary grows by continuously gathering volume from the liquid delivered through the tube. The volume of the drop finally reaches a maximum or critical value, defined by the balance between the forces acting upwards and downwards on the drop. Any further increase of the volume of this critical drop upsets the force balance and the drop will detach from the capillary. When following the volume of the drop, at the very beginning V = 0, the apex radius of curvature is infinity, B = ∞. As the volume increases, B decreases and takes values in the order of the capillary diameter. The shape of any drop, that is, the curve Z versus X from X = 0 to X = A, is determined solely by the “profile parameter” B, and one can use the library of the data produced with the numerical integration technique described above to evaluate the drop volumes V (and further quantities such as the length and contact angle) for the chosen capillary radius A. The resulting function V(B) is characteristic for the capillary in question. Two such curves are shown in Figures 3A and 4A for the values A = 0.65 and

V 2

3

0

20

40

60

80 100 120

1.0

A

1.0

90°

B

0.9

0.9

0.8 0.7

B

c III

r

0.6

0.7

II

r

0.6

0.5

0.5

120

1

C

II

100

D

90°

80

r

c

4

6

2.0

8

0

A

20

40

60

80 2.0

B

-1

r

r 1.0

1.0

-2

c

20

1.5

1.5

r

40

c

c

-3

0 0

1

2

3

-2

-1

V

0

1

0.5

2

X

0.5

80

E

C

60

V Figure 3. Shape analysis of hanging drops attached on a capillary with dimensionless radius A = 0.65 as shown in D and E. A, X, Z, V, and B are given as dimensionless quantities. (A–C): Each point of the curves results from X = A of the respective drop shape. The drop grows along the thick dotted curves, following the direction given by arrows, up to the critical drop (marked with “c”) whose shape is drawn in D. The thin curves beyond the critical point refer to numerical solutions with no physical significance. The shape of the residual drop (“r”) which, according to Lohnstein’s hypothesis, makes the same angle θ at the capillary as the critical drop is also shown in D. The lines intersecting the growth curves give the limits of the drop shape types I–III; thus, the critical drop here is of type III, whereas the residual drop is of type I. E: The growing drop at 10%, 20%, …, 100% of its critical volume.

120

2

0

III

I

60

V 0


•

r

1

D

0

c r

40 20

-1

Z

I

III c

II

Z

B

0.8

I

B

1

B

0

A = 1.30, respectively. In the case of water drops (c = 1.344 × 105 m2 at 20 C), these dimensionless values correspond to capillary diameters 2a of 3.55 mm and 7.09 mm, respectively. To determine the maximum or critical volume Vc of the drop suspending from the given capillary with radius A, we follow the curve V(B ) in the direction of drop growing (see arrows in Figures 3 and 4), until we reach the maximum volume (indicated by the letter “c”). Upon growing, the drop accumulates volume from V = 0 to V = Vc and simultaneously changes its profile (determined by B) from B = ∞ to B = B(Vc) = Bc. Since there are no solutions for V > Vc on the equilibrium curve, a drop with a volume larger than Vc is unstable and will detach from the capillary. Further, after reaching the critical volume at Vc the size of the hanging drop cannot decrease, thus the light dotted branches of the curves represent physically not accessible solutions of the differential equation. A plot of the volume Vc of critical drops as a function of the capillary radius A is given in Figure 5. Vc increases with A in a slight sigmoid manner around a straight line with slope 2π (for a discussion see below) and levels off at a maximum volume of Vc = 18.964 at radius A = 3.22. The curve coincides perfectly with the theoretical data given by Pu and Chen (37). Morgan and Stevenson (38) and, very recently, Gunde et al. (39) have measured the critical volume of drops from several liquids with great care. Some of their data are summarized in Table 2 and plotted in Figure 5. They show an almost perfect agreement between experiment and calculation. The maximum drop-volume tensiometer therefore is a valuable instrument, free of any correction factors. Its experimental realization has been thoroughly discussed by Gunde et al. (39) and by Pu and Chen (37, 40).

-2

c

0

-3 0

2

4

6

8

-2

-1

v

0

1

2

x

E

V Figure 4. Shape analysis of hanging drops attached on a capillary with dimensionless radius A = 1.30. Here, all possible drops, including critical and residual drops, are of type I. For details refer to caption of Figure 3.


•

www.JCE.DivCHED.org


20

Critical Volume Vc

M

15

decreases again, finally ending in the critical value θc where dθ兾dV → ∞. As can be concluded from Figure 6, θc approaches 0 for large and 90 for small capillaries. Tubes with radii larger than A ≈ 0.92 always lead to contact angles between 0 and 90, that is, flat type I drops without neck and equator. Note from the figure that the critical angle is just 0 when A ≈ 3.22, which will be discussed below. The development of the drop shape on a capillary with A < 0.92 is more complex, as can be verified in the existence diagram (Figure 7) that shows how the drop-shape type depends on the capillary size and the relative drop volume. The type I shape of the beginning drop changes to type II with an equator when θ > 90 and finally to type III with an additional neck just before the drop becomes critical. 160

A = 0.2 II A = 0.2

140

Contact Angle / deg

Contact Angle Once the volume Vc of the critical drop and its associated profile parameter Bc are determined, other quantities of this drop can easily be evaluated from its profile. One of these quantities is the critical value θc of the angle θ that the meridian curve takes at the capillary circumference. The relationships between V, B, and θ are plotted in Figures 3 and 4 for two capillaries of different size. They allow an easy and straightforward characterization of the growing and critical drop (and also of the drop remaining at the capillary after separation, see below). The qualitative drop shape (type I, II, or III) is readily determined by inspecting the θ versus V projection and following the curve in the direction of increasing volume: type I drops occur in the first branch where 0 ≤ θ ≤ 90, type II drops at θ > 90, and type III drops in the last branch again at θ < 90 (see Figures 3, 4, and 6). The contact angle θ of the drop surface with the flat end of the capillary depends characteristically on the size of the capillary and changes with the drop volume (Figure 6). In all cases θ increases first, passes through a maximum and

III

120

A = 1.0 I

100 90°

A = 3.0

80

1.0 I

60 40

2.0

20

3.0 10

0 5

0

10

15

20

Volume V Figure 6. Contact angle θ of hanging drops as a function of drop volume, shown for capillary radius A = 0.2, 0.4, ..., 3.2 (see also labels on the curves). The growth curve for a particular radius A ends at the critical point given by the critical values Vc and θc. Some drops suspending from different tubes are shown to illustrate the three shape types I–III (drawn to scale).

5

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Radius A 1.0

Figure 5. Volume Vc of the critical drop as a function of capillary radius A. The squares show experimental values as given in Table 2. M indicates the largest possible hanging drop. The dotted straight line Vc = 2πA gives the critical volume of the “ideal” drop (see text).

III 0.8

II

Liquid

a/mm

Vccalc/ mm 3

Vcexp/ mm3

Ref

Water

3.11

126.1

127.3

38

1.699

61.23

60.88

39

1.203

42.60

42.47

39

Diethylether

3.11

48.58

48.5

38

Clorobenzene

3.11

59.96

59.7

38

Ethanol

1.699

25.86

25.8

39

1.203

17.12

17.02

39

1.704

29.60

29.52

39

Toluene

NOTE: Capillary constants c = g∆ρ/σ at 20 °C are (41, 42): water, 1.344 x 105 m2; diethylether, 4.108 x 105 m2; chlorbenzene, 3.277 x 105 m2; ethanol, 3.456 x 105 m2; and toluene, 2.977 x 105 m2.

www.JCE.DivCHED.org

•

0.6

V / Vc

Table 2. Calculated and Experimental Critical Drop Volumes of Several Liquids against Air Attached on Different Cylindrical Capillaries

0.4

I 0.2

0.0 0.5

0.0

1.0

1.5

Radius A Figure 7. Existence diagram for the three shape types of stable hanging drops, attached on a capillary with radius A. V/Vc is the relative size of the drop with respect to its critical volume. For a particular capillary size drop growth is followed in the diagram in vertical direction from V = 0 to V = Vc.


•


121


Drop Length

3.0

L

1.0

2.0

2.5

M

3. 0

2.0

Height Z

The development of the drop length Z of a growing drop is shown in Figure 8. For all capillaries, the length is a steady increasing function of the drop volume,2 up to the critical drop where dZ兾dV → ∞. Small drops suspending from narrow capillaries elongate steeper than large drops hanging on wide tubes or on a flat surface. Owing to the large slope of the function Z(V ) near the critical drop an exact experimental measurement of the maximum drop length Zc, contrary to the maximum volume, is practically impossible and cannot be used for the determination of interfacial tension. A plot giving the length Zc(A) of critical drops hanging on capillaries of different radii A is included in Figure 2. This curve intersects all drop profiles at their critical coordinates, thus separating the regions of stable and instable drops. Tubes with large radii yield flat critical type I drops with clear intersection at θc well below 90 (see also Figure 6). On the other hand, small critical type III drops are nearly spherical with contact angles θc ≈ 90. Close inspection of the curves shows that the longest possible critical drop hangs on a tube with radius A ≈ 1.68 and forms an apex radius Bc ≈ 0.88; its length and volume are Zc ≈ 2.64 and Vc ≈ 10.30, respectively (see also point L in Figure 8). Table 3 compares this result for different liquids. One should note that the shape of this drop does not depend on the nature of the liquid, however, the absolute lengths and volumes scale with c1兾2 and c3兾2, respectively. An interesting feature is the endpoint M of the critical curve at X = A = 3.22 in Figures 2 and 5 that represents the largest radius a drop can have at its basis when hanging on a flat surface. Increasing the drop shape over this value yields an impossible contact angle θc < 0 in the numerical solution of the drop shape. This drop thus has the largest possible volume and forms θc = 0 with the supporting surface as has been already mentioned when discussing Figures 5 and 6. Therefore, the largest possible thin-walled tube producing a hanging drop has a radius A = 3.22.3 In the case of water at 20 this corresponds to a = 8.78 mm. Again, it has to be

1.5

1.0

0.5

0.0 5

0

10

15

20

Volume V Figure 8. Drop growth, shown as drop height (or length) Z as a function of hanging drop volume V. Growth curves are shown for capillary radii A from 0.2 to 3.2, incremented by 0.2. Each curve ends in the point characterized by the critical values of volume, Vc, and length, Zc, respectively. L is the longest, M the largest possible drop.

pointed out that the shape of this drop is the same for all liquids. Further specifications of this largest drop are collected in Table 4.

Drop Surface The surface of a drop hanging on a particular capillary increases continuously with its volume, as illustrated in Figure 9. Critical drops hanging on narrow capillaries show a considerable relative increase of their surface. On the other hand, the surface area of the largest possible drop (A = 3.22, type I) exceeds the starting value πA2 at V = 0, that is, the

Table 3. Data for the Longest Possible Critical Hanging Drop as Obtained from Drop-Shape Analysis Liquid

Tube Radius

Apex Radius

All

A = 1.68

Bc = 0.88

Zc = 2.64

Vc = 10.30

Oc = 20.52

Water

a = 4.58 mm

bc = 2.40 mm

zc = 7.20 mm

vc = 209 mm3

oc = 153 mm2

Ethanol

2.86 mm

1.50 mm

4.49 mm

51 mm3

59 mm2

5.01 mm

3

74 mm2

Mercury

3.19 mm

1.67 mm

Length

Volume

70 mm

Surface Area

NOTE: The values are given in dimensionless units as well as in metric units for water, ethanol, and mercury. Capillary constants c = g∆ρ/σ at 20 °C are (41, 42): water, 1.344 x 105 m2; ethanol, 3.456 x 105 m2; mercury 2.777 x 105 m2.

Table 4. Data for the Largest Possible Critical Hanging Drop as Obtained from Drop-Shape Analysis Liquid

Tube Radius

Apex Radius

All

A ≥ 3.22

Bc = 1.275

Zc = 2.151

Vc = 18.964

Oc = 40.49

Water

a ≥ 8.78 mm

bc = 3.48 mm

zc = 5.87 mm

vc = 385 mm3

oc = 301 mm2

Ethanol

5.48 mm

2.17 mm

3.66 mm

93 mm3

117 mm2

Mercury

6.11 mm

2.42 mm

Length

Volume

4.08 mm

130 mm

Surface Area

3

146 mm2

NOTE: The values are given in dimensionless units as well as in metric units for water, ethanol, and mercury. Refer to Table 3 for capillary constants.

122


•


•

www.JCE.DivCHED.org


the critical angle θc of the drop of maximum volume just before detachment.4 This criterion can be used to find the drop shape or, more precisely, the apex radius of curvature B, that leads to an angle θ = θc at X = A. Since the numerical relations between the quantities B, V, and θ are already known from our previous drop-shape analysis, B and Vr of the residual drop can now be obtained by the relations B ↔ θ ↔ Vr (see Figures 3B and 4B). Plots of the residual and the maximum drop shapes are shown in Figures 3D and 4D. It is found that the ratio of the residual volume to the falling volume depends on the capillary size. From very narrow capillaries approximately the whole hanging drop actually falls, while up to 40% of the hanging critical drop volume remains at larger capillaries.

40

Surface Area O

3.2 30

3.0 2.8 2.6 2.4 2.2 2.0 1.8

20

10

0 0

5

10

15

20

Volume V

Drop-Volume Method of Surface Tension Determination

Figure 9. Drop growth, shown as surface area O versus volume V of hanging drops. Growth curves are shown for capillary radii A from 0.2 to 3.2, incremented by 0.2.

cross-sectional area of the capillary, by only some 25%. For all critical drops the slope dO兾dV → ∞ (see Figure 9). Detachment of a Hanging Drop If the drop volume below the capillary exceeds the critical maximum value given by Vc, the drop becomes unstable and separates from the capillary. This process is fast and cannot be followed by eyes in the case of low viscosity liquids such as, for example, water and ethanol. However, high-speed photography equipment (45, 46) has revealed fascinating details of the dynamics of separating drops. When taking a closer look at these pictures, one can recognize that the drop does not detach at the end of the capillary, which means that the actually falling volume is not equal to the critical volume Vc of the hanging drop. Instead, a neck is formed that grows longer and narrower and finally separates the falling main part of the drop and the part remaining at the capillary. Still connected to the fluid in the capillary, the drop forms an ellipsoid comparable to a water-filled rubber balloon held at its nozzle (47, 48). After separation, this main drop oscillates during free fall owing to tension forces and momentum conservation. The fate of the neck that has elongated to a long stem depends on the diameter of the capillary and the properties of the liquid itself: It may completely return to the capillary or it may separate from the liquid remaining there and follow the main drop at some distance as one or more violently oscillating satellite droplets (49–51). The remaining liquid at the end of the capillary relaxes in a damped oscillation with amplitude and frequency depending on the viscosity and drop age (52). The Residual Drop: Lohnstein’s Hypothesis After detachment of the falling volume Vf, a residue of volume Vr = Vc − Vf stays attached on the capillary. Up to now, exact constraints that could lead to the shape of this residual drop were not known. On the basis of visual observations Lohnstein (44) suggested in his 1906 article that the meridian curve of the residual drop forms an angle θ with the horizontal end plane of the capillary that matches closely www.JCE.DivCHED.org

•

Drop-volume tensiometers measure either the critical volume vc of the hanging drop or, more frequently, the falling volume vf of the liquid after separation from the capillary. It is therefore necessary to discuss how the surface tension can be determined from these experimental quantities. Mechanical equilibrium for a drop hanging vertically at the end of a capillary with radius a in the state of its maximum or critical volume vc is given by the balance between forces acting upwards and downwards, 2π a σ sin θc + vc ρg g = π a 2∆ p + vc ρ1 g

(17)

Rearranging and substituting ∆p from eq 6 gives the volume vc and the mass mc of the hanging critical drop

vc =

mc =

2π a σ g ρl − ρg

2π a σ ρg g 1− ρl

sin θc −

a azc c + bc 2

sin θc −

a azc c + bc 2

(18)

(19)

A Note on the “Ideal” Drop Early attempts to describe the maximum volume of hanging drops started with a simpler force balance vcid ρ l − ρg g = 2π a σ

(20)

that is, neglecting the excess pressure ∆p arising from the surface curvature and assuming a critical angle of θc = 90. This version of the Tate equation5 includes the buoyancy effect. Proposed by Harkins and Brown (54), drops following this behavior are called ideal drops. Their critical volume is vcid =

2π a σ ρ l − ρg g

=

2πa c

(21)

or, in dimensionless form, simply

V c id = 2 π A

(22)

suggesting a linear dependence of the critical volume on the capillary radius. This behavior is depicted in Figure 5.


•


123


Equation 22 has to be compared with the dimensionless form of the exact eq 18, Vc = 2πA sin θc −

A A Zc + Bc 2

(23)

(Note that with the substitution A = X and cancelled subscripts “c” eq 23 is valid also for subcritical hanging drops.) Vc is equal to the critical volume 2πA of an “ideal” drop only if the bracketed expression in eq 23 becomes unity. Numerical evaluation of (sinθc − A兾Bc + AZc兾2) for different capillary sizes gives approximately 1.00 for A ≈ 1.80 and A ≈ 2.95, as can also be verified in Figure 5 at the intersection of the straight “ideal” drop line with the S-shaped curve for real drops. While the mass mc or the volume vc of the critical drop can be calculated by integration of the Laplace equation as described above, their exact measurement is an experimental challenge (37–39). Commercial drop tensiometers, therefore, make use of the mass mf or volume vf of the drop that has fallen from the capillary. These latter quantities can be measured much easier but, on the other hand, are accessible to computation only when applying the weakly-founded hypothesis of Lohnstein (which turns out to be a fairly good approximation, however). Due to the lack of a theory-based relation between Vc and Vf, Harkins and Brown (54) defined an empirical correction factor φ as the ratio of the falling volume (or mass) to the maximum hanging volume (or mass) of an ideal drop, mf vf Vf Vf φ = = id = = id id 2π A mc vc Vc

(24)

They, and later Wilkinson (55), carefully determined correction factors experimentally for capillaries with different di1.1

ameters. They found excellent agreement of φ for different liquids and capillary materials—an important finding concerning the applicability of the falling-drop method for surface tension measurements. Moreover, no apparent viscosity effect could be measured (40). Experimental data (54, 55) and a recommended numerical fit (56) of φ versus the dimensionless fraction q defined by q =

a vf

1

= 3

A Vf

1

(25)

3

are shown in Figure 10. The curve φ(q), being independent on the drop or capillary material, has a minimum at q ≈ 0.85 where φ ≈ 0.60, that is, only about 60% of the maximum hanging mass detaches from the capillary. The exact characteristics of the function φ(q) is absolutely essential to obtain accurate numeric results for surface tension determinations using the drop-volume or drop-weight method.6 It is an interesting fact that even modern commercial drop-volume tensiometers still make use of φ(q) based on experimental data that have been presented some 80 years ago. Figure 10 also contains results from our numerical drop-shape analysis using Lohnstein’s hypothesis with identical contact angles of critical and residual drops. Although calculated and experimental data deviate in a systematic manner from each other, agreement for q < 1.1 is quite satisfactory. Equations 18 and 24 yield σ =

ρg mf g 1− 2πa φ ρl

=

vf g ρ l − ρg 2π a φ

(26)

a convenient form to evaluate the surface tension from experimental data. The extensive compilation of Earnshaw et al. (56), who, based on the data given by Harkins and Brown (54), tabulate the correction factor φ(q) in steps of ∆q = 0.002, can be used to calculate the falling volume as a function of A now on the basis of experimental data. Figure 11 shows dimensionless critical, falling, and residual volumes,

0.9

Vc

1.2

calc

0.8

2πA

Vcid

1.0

2πA 0.7

M Vf

0.8

2πA

exp

0.6

V 2πA

Correction Factor φ

1.0

0.6

0.4

0.5

Vr 2πA

0.2

0.4

0.0

0.2

0.4

0.6

0.8

q ⴝ

1.0

1.2

1.4

1.6

1.8

a

Vf

0.0

Figure 10. Correction factors φ plotted against the dimensionless ratio q = a/Vf1/3. Experimental data from Harkins and Brown (54) and Wilkinson (55) appear as open symbols. The solid line is a cubic spline fit to the experimental data as recommended by Earnshaw (56). Heavy dots show calculated correction factors on the basis of Lohnstein’s hypothesis.

124


0

0.0

1 3

•

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Radius A Figure 11. Maximum, falling, and residual drop volumes, normalized to the volume 2πA of the “ideal” drop. The curve Vc has been obtained from the drop shapes (see also Figure 5), whereas the curve Vf is a fit to empirical values (56). Vr is given by the difference Vc − Vf.


•

www.JCE.DivCHED.org


normalized to “ideal” volume, as a function of the capillary radius. They clearly show that (i) from narrow capillaries small drops fall almost completely and (ii) at relatively large capillaries (A ≈ 2) nearly “ideal” drops with Vc = 2πA form but detach only partially. Inspection of Figure 11 shows that the curve associated with the falling volume, as fitted by Earnshaw et al. (56), lies erroneously above the critical volume curve at narrow capillaries (A < 0.3) and, as a consequence, negative residual volumes arise in this region. This feature is most probably founded in errors in the quite scattered basic experimental data of Wilkinson (55). Narrow capillaries should therefore not be used in combination with the correction factors provided so far. Note that while all falling-drop tensiometers depend on the empirical curve labeled “Vf” in Figure 11, the maximum volume-drop tensiometer (37, 39) makes use of curve “Vc” that is computable to any degree of accuracy. However, this principal advantage is in common lowered by the experimental difficulty to measure vc. Numerical integration of the drop shape, as obtained by processing the video camera captured critical drop image, may be the method of choice to obtain reliable experimental values of the critical as well as of the residual volume. Since surface tension and gravity effects govern the shape of the static hanging drop the former can be determined in principle by analyzing the measured drop geometry at any stage of formation (59–61). Summary The geometry of drops hanging on a circular capillary can be determined by numerically solving a dimensionless differential equation that is independent on any material properties. A single parameter B, the radius of curvature in the apex of the drop, determines the shape of the drop. Other quantities, such as the volume and the surface of the hanging drop, can be calculated straightforwardly from its geometrical shape. This enables one to follow the change of the height, surface area, and contact angle of drops hanging on a particular capillary as a function of its volume from V = 0 up to a maximum or critical volume at which detachment occurs. The critical volume is calculated as a function of the capillary radius, showing excellent agreement with experimental data from the literature. It is found that very narrow capillaries support nearly spherical critical drops that will separate almost completely from the capillary. Wide tubes support drops with a broad and flat profile and hold back substantial volumes of the drop after exceeding the critical volume. The existence of a largest possible drop hanging from a flat surface is derived and its size parameters are given accordingly. Finally, the application of drop tensiometers for the determination of surface tension is discussed on the basis of the calculated drop shapes and experimental working functions found in the literature. Dedication This article is dedicated to Prof. Urs P. Wild on the occasion of his retirement. Acknowledgment The authors would like to thank Arthur Schweiger for careful proofreading of the manuscript. www.JCE.DivCHED.org

•

W

Supplemental Material

Program code and drop data sets are available in this issue of JCE Online. Notes 1. The program was implemented in the programming language Pascal using a SUN Pascal 4.2 compiler. When running the program on a SUN Ultra 10 workstation under the Solaris 2.5 operating system the calculation of this library took only a few minutes. Program code and drop data sets are available in the Supplemental Material.W 2. This finding seems to be in contradiction to refs 35 and 43, which stated that drops of volume less than Vc but length greater than Zc exist for a given capillary. A misinterpretation of the function Z (V ) for a given capillary size may be responsible for that discrepancy. Each function Z (V ) indeed has a maximum with Z > Zc but this maximum belongs to the instable branch of the drop length curve. Figure 8 shows only the stable branches. 3. The early value A = 2.710 for the maximum drop base radius, given by Lohnstein (44), may possibly be incorrect due to inaccuracies in his integrations. Hartland and Hartley (35) present a curve that is in quantitative agreement with ours. 4. In Lohnstein’s German words (44, p 255): In dieser Hinsicht schien mir nun die Beobachtung zu lehren, daß der am Röhrenrande befindliche Endteil der Meridiankurve des Tropfenmeniskus annähernd die gleiche Neigung gegen den Horizont aufweist wie der Endteil der Meridiankurve des hängenden Tropfens unmittelbar vor dem Abreißen. 5. Equation 20 is the combination of the so-called first and second laws of Thomas Tate who, based on experimental results with water, stated in 1864 (53) a proportionality between the weight of the falling drop and (i) the diameter of the tube and (ii) the weight of the liquid that is raised in this tube owing to capillary action (which is proportional to the surface tension). Careful measurements, however, have shown later a systematic deviation from this “law”. For details of this controversy see, for example, refs 38 and 58. 6. This applies also to the stalagmometer (57), a simple glass apparatus used to determine the surface tension by means of counting the number of drops that are delivered from a given volume and fall from a capillary. The surface tension of the liquid under question is obtained by comparing the drop number of that liquid with that of a reference liquid with known surface tension.

Literature Cited 1. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996. 2. Sime, R. J. Physical Chemistry. Methods, Techniques, and Experiments; Saunders College Publishing: Philadelphia, 1990. 3. Försterling, H.-D.; Kuhn, H. Praxis der Physikalischen Chemie, 2. Aufl.; VCH Verlagsgesellschaft mbH: Weinheim, Germany 1985. 4. Adam, N. K. The Physics and Chemistry of Surfaces; Dover Publications, Inc.: New York, 1968. 5. Alexander, A. E.; Hayter, J. B. Determination of Surface and Interfacial Tension. In Techniques of Chemistry: Physical Methods of Chemistry, Vol. I; Weissberger, A., Rossiter, B. W., Eds.; Wiley-Interscience: New York, 1971. 6. Steinbach, O. F. J. Chem. Educ. 1944, 21, 582.


•


125

Research: Science and Education 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

35.

36. 37.

Kay, M.; McClure, D. W. J. Chem. Educ. 1970, 47, 540. Meyer, E. F.; Wyshel, G. M. J. Chem. Educ. 1986, 63, 996. Munguia, T.; Smith, C. A. J. Chem. Educ. 2001, 78, 343. Haendler, H. M.; McGuire, W. S. J. Chem. Educ. 1937, 14, 591. Hazlehurst, T. H. J. Chem. Educ. 1942, 19, 61. Castro, M. J. L.; Ritacco, H.; Kovensky, J.; Fernández-Cirelli, A. J. Chem. Educ. 2001, 78, 347. Rosenthal, A. J. J. Chem. Educ. 2001, 78, 332. Macy, R. J. Chem. Educ. 1935, 12, 573. Sánchez-Rubio, M.; Gordillo, B.; Rushforth, D. S. J. Chem. Educ. 1983, 60, 70. Sánchez-Rubio, M.; Castellanos-Ortega, J. R.; Puig, J. E. J. Chem. Educ. 1991, 68, 158. Digilov, R. M. J. Chem. Educ. 2002, 79, 353. Dolz, M.; Delegido, J.; Hernández, M.-J.; Pellicer, J. J. Chem. Educ. 2001, 78, 1257. Kabza, K. G.; Cochran, K. J. Chem. Educ. 1997, 74, 322. Kabza, K.; Gestwicki, J. E.; McGrath, J. L. J. Chem. Educ. 2000, 77, 63. Gesser, H. D. J. Chem. Educ. 2000, 77, 58. Dionísio, M.; Sotomayor, J. J. Chem. Educ. 2000, 77, 59. Craig, V. S. J.; Jones, A. C.; Senden, T. J. J. Chem. Educ. 2001, 78, 345. Srisankar, E. V.; Shah, J. P.; Narayan, K. S. J. Chem. Educ. 1987, 64, 378. Ewart, H. A.; Hyde, K. E. J. Chem. Educ. 1992, 69, 814. Worley, J. D. J. Chem. Educ. 1992, 69, 678. Spanoghe, P.; Cocquyt, J.; Van der Meeren, P. J. Chem. Educ. 2001, 78, 338. Condon, F. E.; Condon, F. E., Jr. J. Chem. Educ. 2001, 78, 334. von Segner, J. A. Comment. Soc. Reg. Goetting. 1751. Young, Th. Phil. Trans. Roy. Soc. 1805, 95, 65. Laplace, P. S. Traité de Méchanique Céleste, 2nd ed.; Bachelier: Paris, 1805; Vol 2, p 479. Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons: New York, 1982. Euler, L. Institutiones Calculi Differentialis; Kaiserliche Akademie: St. Petersburg, 1755. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Pascal; Cambdridge University Press: Cambridge, 1989. Hartland, S.; Hartley, R. W. Axisymmetric Fluid-Liquid Interfaces; Elsevier Scientific Publishing Company: Amsterdam, 1976. del Río, O. I.; Neumann, A. W. J. Colloid Interface Sci. 1997, 196, 136. Pu, B.; Chen, D. J. Colloid Interface Sci. 2001, 235, 265.

126


•

38. Morgan, J. L. R.; Stevenson, R. Zeitschr. f. physik. Chemie 1908, 63, 151. 39. Gunde, R.; Kumar, A.; Lehnert-Batar, S.; Mäder, R.; Windhab, E. J. J. Colloid Interface Sci. 2001, 244, 113. 40. Pu, B.; Chen, D. J. Colloid Interface Sci. 2001, 235, 273. 41. CRC Handbook of Chemistry and Physics, 74th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1993. 42. Landolt-Börnstein, Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik, 6. Aufl., II. Band, 3. Teil, Schmelzgleichgewichte und Grenzflächenerscheinungen, Springer-Verlag: Berlin, 1956. 43. Hartland, S.; Srinivasan, P. S. J. Colloid Interface Sci. 1974, 49, 318. 44. Lohnstein, T. Ann. Physik 1906, 20, 237. 45. Hauser, E. A.; Edgerton, H. E.; Holt, B. M.; Cox, J. T., Jr. J. Phys. Chem. Baltimore 1936, 40, 973. 46. Peregrine, D. H.; Shoker, G.; Symon, A. J. Fluid Mech. 1990, 212, 25. 47. Eggers, J.; Dupont, T. F. J. Fluid Mech. 1994, 262, 205. 48. Eggers, J. Phys. Bl. 1997, 53, 431. 49. Boys, C. V. Soap Bubbles. Their Colours and the Forces Which Mold Them; Dover Publications: New York, 1959 (a republication of the 1911 edition). Boys presents a ready-to-made thaumatrope to show the formation and oscillations of drops. 50. Recently published photographic pictures and graphical results from numerical simulations have shown the influence of viscosity to the shape of the stem at the moment of separation: Shi, X. D.; Brenner, M. P.; Nagel, S. R. Science 1994, 265, 219. 51. Zhang, X.; Basaran, O. A. Phys. Fluids 1995, 7, 1184. 52. Zhang, X.; Harris, M. T.; Basaran, O. A. J. Colloid Interface Sci. 1994, 168, 47. 53. Tate, T. Phil. Mag. 1864, 27, 176. 54. Harkins, W. D.; Brown, F. E. J. Am. Chem. Soc. 1919, 41, 499. 55. Wilkinson, M. C. J. Colloid Interface Sci. 1972, 40, 14. 56. Earnshaw, J. C.; Johnson, E. G.; Carroll, B. J.; Doyle, P. J. J. Colloid Interface Sci. 1996, 177, 150. 57. Jaycock, M. J.; Parfitt, G. D. Chemistry of Interfaces; Ellis Horwood Limited: Chichester, United Kingdom, 1981. 58. Lohnstein, T. Z. Phys. Chem. 1907, 64, 686. Lohnstein, T. Z. Phys. Chem. 1913, 84, 410. 59. Cheng, P.; Li, D.; Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Colloids Surfaces 1990, 43, 151. 60. Pallas, N. R.; Harrison, Y. Colloids Surfaces 1990, 43, 169. 61. Wulf, M.; Michel, S.; Grundke, K.; del Rio, O. I.; Kwok, D. Y.; Neumann, A. W. J. Colloid Interface Sci. 1999, 210, 172.


•

www.JCE.DivCHED.org

Axisymmetric Liquid Hanging Drops - ACS Publications

Recommend Documents