Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

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M19e “Surface Tension” Tasks

1. Determine the surface tension σ of an organic liquid using the anchor-ring method. Use three

different lengths of the anchor ring. Calculate σ according to Lenard’s formula.

2. Determine the surface tension of water with the capillary-rise method. Additional task: Measure the surface tension of a water-ethanol mixture with the droplet method. Literature Physics, P. A. Tipler, 3rd Edition, Vol. 2, Chap. 11-5 Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik, 5.0, 5.1, 5.2 Accessories Torsion spring balance, ‘anchor rings’ (frames) with compensation masses and thread suspension, beaker, microscope with eyepiece micrometer and object micrometer, colour filter, capillary (micropipette), capillary holder, test cell, telescope (cathetometer), syringe, thermometer, test fluids Keywords for preparation - Molecular forces in liquids, surface energy and surface tension - Interface tension, fluids in capillaries, wetting angle - Capillary-rise method, anchor-ring method, other methods to measure surface tension - Temperature dependence of the surface tension - Microscopic length measurements, Snell’s law

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Remarks

In task 1 the anchor-ring method is used with frames of length l = 2 cm, 3 cm and 4 cm as well as a torsion-wire balance for the measurement of the tear-off force, to measure the surface tension of an organic liquid. The surface tension is defined as ratio of the amount of energy dW necessary to

increase a given surface area by dA, i.e. σ =

dW/dA. The surface tension might be related to the force necessary to pull a straight wire from the fluid, see Fig. 1. If the wire frame of length l is pulled from the fluid, the surface tension holds it

back with a force Fσ. This might be related to the

surface tension by calculating the necessary work

dW = Fσ dx as well as the increase in surface area

dA = 2 l dx to obtain

Fig. 1 Anchor-ring method.

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F

lσσ = , (1)

with which the surface tension can be approximately calculated (neglecting the correction after

Lenard). In this case dx is the height of the fluid lamella and Fσ is the tear-off force. Lenard derived a

more accurate equation for the surface tension determined by the anchor-ring method, taking the finite wire diameter and the mass of the fluid lamella into account. This yielded

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2 4

F F g F F gr r g

l l l l lσ σ σ σρ ρπσ ρ

⎛ ⎞ ⎛ ⎞⎛ ⎞= − − + + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠. (2)

l is the length of the wire, r its radius and ρ is the density of the liquid.

Perform three measurements each for the tear-off force Fσ; for the calculation of the surface tension

use the average of the tear-off force. Note that the torsion-spring balance is calibrated in mass units

(mg), i.e. the tear-off force has to be calculated from Fσ = mg (g = 9.81ms-2).

Another method to measure the force Fσ is the ring-method by du Noüy. It is used to measure the

surface/interfacial tension using a standardized ring with a ring/plate tensiometer. The ring is immersed into the liquid and then pulled out. This forms a liquid lamella that is stretched to its maximum. The surface/interfacial tension is calculated from the resulting force with appropriate corrections.

In task 2 the height of the capillary rise h of water in a volume scaled pipette should be measured. From this the surface tension can be calculated. Consider a surface between two fluids. In the absence of normal pressure, this surface will be flat. On the other hand, if the surface is curved, the pressure p1 on the concave side is higher than the pressure p2 on the convex side, since the creation of a curved liquid surface from a plane one causes a surface enlargement and thus the expenditure of work. The pressure difference is related to the surface tension and the principal radii of curvature R1 and R2 and is given by the Young-Laplace equation

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1 21 2

1 1p p

R Rσ

⎛ ⎞− = +⎜ ⎟

⎝ ⎠. (3)

When a cylindrical capillary tube with small radius r is considered, the meniscus between liquid and air can be approximated as spherical. According to Eq. (4) a pressure difference is created at the surface that makes the liquid in the capillary rise until it is equalized by the weight of the liquid

column. We get with R1 = R2 = r/cosθ (θ contact angle, see Fig. 2)

1 2

2 cosp p

rσ θ− = . (4)

Fig. 2 Capillary-rise method.

The pressure difference produces a rise h of the liquid level in the capillary given by

L

2 cos( )g h

rσ θ ρ ρ= − . (5)

In case of strongly wetting liquids (θ = 0°) with high density ρ » ρL compared to the density of air one obtains

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2grhσ ρ= . (6)

For small liquid reservoirs with test cell dimensions a and b and an external radius of the capillary r1 a correction term due to the change of the liquid level in the reservoir has to be taken into account:

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1

1 ( )1

2r a b r

grhab r

πσ ρπ

−⎡ ⎤+ += −⎢ ⎥−⎣ ⎦

. (7)

The capillary radius can be calculated from the given volume-scaled micropipettes with ring mark or using a microscope. In the first case you can measure the distance dV between the end of the pipette and the ring mark and calculate the radius from the given volume Vcap. In the second case an optical microscope is used, see below. Since the surface tension depends on the temperature of the liquid under study, take a note of this parameter in all measurements.

Hints towards the experimental realization

Anchor-ring method

The torsion-spring balance is brought to level with the adjustable screws (1). In the arrested state of the balance – adjusting knob (2) with the white dot to the front – the hook (3) is loaded with the suspension thread and the 3 cm anchor ring, whereas the hook (4) is loaded with the corresponding compensation mass. In order to de-arrest the balance the adjusting knob (2) is turned by 90°, such that the white dot is not any longer visible. The balance-beam indicator (8) should point to zero (indicator (10) in zero position) in the relaxed state of the balance. If this is not the case, carefully turn the adjusting knob (9) at the rear wall, until the balance beam is brought to the zero position.

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Fig. 3 Torsion spring balance. For the measurement a glass bowl containing the liquid under study is brought into a position – by adjustment of the arm (5) or the plate (6) – such that the measurement wire of the anchor ring dips into the liquid. The spring of the balance is tensioned by turning the adjusting knob (7), up to the point at which the liquid lamella rips off the measurement wire. During the tensioning the height of the liquid bowl has to be adjusted in such a way that the balance beam indicator (8) shows zero just before the rip-off of the liquid lamella. Before

rip-off the tension must be increased very carefully. Vibrations and shocks have to be avoided. If the measurement range ("250 mg") of the torsion spring balance is not sufficient, the balance should be arrested and another appropriate compensation mass should be attached to hook (4). Capillary-rise method

As capillaries single-use micro-pipettes with a defined volume are used. These thin-walled capillaries are vertically fixed in the holder and are immersed into the liquid under study. Then the liquid is sucked into the pipette with a syringe. After removing the syringe, within a short time the liquid column settles to an equilibrium height that can be measured either with a length scale or with a telescope (cathetometer). The capillary-rise method should be realized with two different micro-pipettes (with different capillary radii).

Fig. 4 Capillary-rise method.

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Measurement of the capillary radius with a microscope Due to refraction and total reflection, the image appears in the field of view of the microscope as bright and dark areas.

Fig. 5 Microscopy image of a capillary.

Fig. 5 shows a microscopy image of a capillary. The bright area 4 is outside of the capillary. The center of the capillary 1 also appears bright. Above and below the center dark areas 2 are seen. Within these dark areas the rim of the inner capillary cylinder appears as two bright lines 3; these are generated by reflection of light at the inner

cylinder walls. The distance 2s between the bright lines in Fig. 6 is measured with the eyepiece micrometer that has been calibrated beforehand. From this the inner radius of the capillary can be calculated.

Fig. 6 Calculation of the capillary radius.

Using the movable microscope table the size of the outer capillary diameter 2r1 has to be determined; a millimeter scale with nonius allows a reading with sufficient precision. Since the microscope has only a limited depth of sharpness and since the cross-section of the capillaries is not exactly circular, the microscope has to be focused on each peripheral ray individually. The reason for this is the different height of the reflections from the inner wall of the capillary for the two peripheral rays in Fig. 4.

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Data Length of measuring wire for the anchor-ring method

Nominal 20 mm 30 mm 40 mm

Set1 (19,85 ± 0,05) mm (29,93 ± 0,05) mm (40,00 ± 0,05) mm

Set2 (19,82 ± 0,05) mm (29,85 ± 0,05) mm (40,02 ± 0,05) mm

Radius r of the measurement wire r = 0.009 cm Density of the test liquids

V1: ρ = 782,4 kg m-3 V2: ρ = 999 kg m-3 V3: ρ = 801,6 kg m-3

20°C...25°C (0,5%) Dimensions of the glass cuvette

l = (30,6 ± 0,1) mm b = (20,0 ± 0,1) mm

Refractive index of the capillary glass

nD = (1.515 ± 0.005) [Metal interference filter λ=589nm]

Droplet method (additional task) For the determination of the surface tension of a liquid against air (using the droplet method) a defined liquid volume V is dripping out of a capillary with plane polished end face and known radius rK (e.g. stalagmometer, Fig. 5); the number of droplets is counted. The analysis of the data starts from the assumption that a droplet drips out of the capillary, when its weight G TF m g= (8)

is equal (or marginally larger) than the force that acts due to surface tension along the circumference of the droplet-capillary contact-line, i.e. σ K2πF r σ= . (9)

Since the mass of the droplet is given by T Fl Tm Vρ= , one obtains for the force equilibrium in the

moment of separation: Fl T K2πV g rρ σ= . (10)

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Here Flρ denotes the density of the fluid, VT the volume of the droplet, g the acceleration of gravity,

rK the radius of the capillary and σ the surface tension. The volume of one droplet VT can be

calculated from the known volume V of the liquid dripping from the capillary and the number of

droplets n: T /V V n= . In reality the mass of a falling droplet is smaller than expected from Eq. (8).

This is related to kinetic processes during the tear-off of the droplet from the capillary. Therefore an empirical correction factor C is introduced in Eq. (10) that accounts for this problem: Fl T K2πV g r Cρ σ= , (11)

such that the surface tension obtained is given by

Fl T

K2π

g V

r C

ρσ = . (12)

If the correction factor C is unknown, it can be obtained from a calibration measurement. Usually water is used as a liquid for calibration, since its surface tension against air is well known as a

function of temperature ϑ and is given by the phenomenological equation

[ ]2

3 -1H O 72,9 0,155( 18 C) 10 Nmσ ϑ −= − − °

in the temperature range between 15 °C and 60 °C (deviations smaller than 1 %). The stalagmometer used in the experiment consists of a capillary and a reference volume with ring marks, see Fig. 7. It follows from Eq. (12) that the surface tension of a test liquid can be obtained relative to that of water, by counting of the particular number of droplets (zW: number of water droplets, z: number of droplets of the liquid under study):

W

W W

zz

σ ρσ ρ

= . (8)

For this the temperature has to be constant throughout the measurements and the flow characteristics of the two fluids should be similar.

The stalagmometer is clamped to a tripod; a tube is attached to the upper end that can be opened/closed by a hose clip. Using a blowball the liquid under study is sucked into the clean and dry stalagmometer such that the liquid level is above the upper ring mark. Then the hose clip is closed and the blowball removed. Subsequently, the hose clip is carefully opened such that droplets are countably dripping off (about 2 droplets per second). While the liquid meniscus is moving from upper to lower ring mark, the droplets are counted. After finishing the measurement, the stalagmometer is flushed with an ethanol/water mixture and the procedure begins anew.

Fig. 7 (a) Stalagmometer with reference marks (schematically), (b) Stages of forming of a droplet (schematically).