MEASUREMENT OF INTERFACIAL TENSION IN FLUID-FLUID …

MEASUREMENT OF INTERFACIAL TENSIONIN FLUID-FLUID SYSTEMS

J. DrelichCh. FangC.L. WhiteMichigan Technological University, Houghton, Michigan

INTRODUCTION

For more than a century, a variety of techniques have been

used to measure interfacial tensions between immisci-

ble fluid phases. A recent monograph by Rusanov and

Prokhorov (1) provides a broad review of the technical

literature on the interfacial tension techniques with de-

tailed discussion of the theoretical bases and instrumenta-

tion. Additional valuable sources of information on the in-

terfacial tension measurement methods include selected

chapters in Refs. 2–5. In this article, we present a very

brief overview of the most common techniques used in

interfacial tension measurements. The reader is encou-

raged to explore Refs. 1–5 and references therein for fur-

ther details.

This article is organized as follows. ‘‘Classical Inter-

facial Tension Measurement Methods’’ reviews the me-

thods that are used in surface chemistry laboratories. A

short comparison of these techniques is presented at the

end of the section. This comparison has been prepared to

guide a selection of the experimental method for mea-

surements of interfacial tension in liquid-fluid systems,

including systems with surfactants, viscous liquids, or mol-

ten metals. Many of the industrial operations involve the

liquid-fluid interfaces, for which the composition is cons-

tantly refreshed and does not reach equilibrium. The im-

portance of such dynamic interfacial tensions is increa-

singly recognized to be essential to the understanding and

control of interfacial processes in multiphase, multicom-

ponent systems. ‘‘Dynamic Interfacial Tension Measure-

ments’’ discusses a freshly created interface. In ‘‘Measure-

ment of Ultralow Interfacial Tension’’ an example of:

when the value of interfacial tension is significantly less

than 1 mN/m is discussed. Ultralow interfacial tensions

are common in the fluid systems of advanced tech-

nologies of liquid-liquid emulsification processes when

effective surfactant solutions are used. Finally, in ‘‘Mic-

rotensiomery,’’ we discuss the methods of interfacial

tension measurements that have been applied (or have

potential to be applied) to microinterfaces of microdrop-

lets. Fundamental research on the interfacial properties of

nanomaterials (materials and particles with microstructur-

al features on the micrometer or nanometer scale) and

droplets of micrometer-sized or nanometer-sized dimen-

sion will be an important challenge in the rapidly deve-

loping field of nanotechnology.

CLASSICAL INTERFACIAL TENSIONMEASUREMENT METHODS

Fig. 1 shows a classification of common interfacial ten-

sion measurement methods, both classical and modern.

Group I represents examples of techniques commonly

used for direct measure of the interfacial tension with a

microbalance. The techniques in group II are those in

which interfacial tension can be determined from direct

measurement of capillary pressure. Analysis of equilib-

rium between capillary and gravity forces is employed in

the techniques of groups III and IV. Group III techniques

rely on the balance between surface tension forces and a

variable volume of liquid, whereas Group IV techniques

fix the volume of a liquid drop and measure the distortion

of that drop under the influence of gravity. Group V in-

cludes techniques where the shapes of fluid drops are dis-

torted by centrifugal forces and are used to measure ultra-

low interfacial tensions.

Group I: Direct MeasurementUsing a Microbalance

Interfacial tension at fluid-fluid interfaces is a reflection

of the excess energy associated with unsaturated inter-

molecular interactions at the interface. This excess energy

tends to drive interfaces to adopt geometries that mini-

mize the interfacial area, and this tendency can be inter-

preted as a physical force per unit length (i.e., a tension)

3152 Encyclopedia of Surface and Colloid Science

Copyright D 2002 by Marcel Dekker, Inc. All rights reserved.

applied in the plane of the interface. The excess energy

per unit area (E/A) is numerically equal to this force per

unit length (F/L), which is numerically equal to the in-

terfacial tension (g).

To directly measure interfacial tensions using a mic-

robalance, a plate, ring, rod, or other probe of simple shape

is brought into contact with the interface. If the probe is

completely wetted by one of the liquids, this liquid will

adhere to the probe and climb as the result of capillary

force, increasing the interfacial area and leading to a force

tending to pull the probe toward the plane of the interface

(Figs. 2 and 3). This restoring force is directly related to

the interfacial tension and can be measured by a micro-

balance. The force (F) acting along the three-phase contact

Fig. 1 Classification of techniques for interfacial tension measurements that are discussed in this article.

Fig. 2 A schematic of the Wilhelmy plate method. Fig. 3 Illustration of the ring method.

M

Measurement of Interfacial Tension in Fluid-Fluid System 3153

line is exactly equal to the weight of the liquid meniscus

standing above the plane of the fluid-fluid interface. This

force, measured by the microbalance, is used to calculate

the interfacial tension:

g ¼ F

p cos yð1Þ

where p is the perimeter of the three-phase contact line and

y is the contact angle measured for the liquid meniscus in

contact with the object surface.

The two principal techniques used for direct measure-

ment of interfacial tension using the microbalance are

Wilhelmy plate and du Nouy ring methods. The Wilhelmy

plate technique is used in both static and detachment

modes, whereas du Nouy ring technique is strictly a de-

tachment technique. In the static measurement, the plate

remains in contact with liquid during the entire cycle of

interfacial tension measurement. If the instrument ope-

rates in the detachment mode, the interfacial tension is

measured by measuring the force required to separate the

ring or plate from contact with the interface.

Wilhelmy plate technique

A vertical thin plate is used in this technique (Fig. 2) (6).

The commercial plates are made of roughened platinum-

iridium alloy or platinum. The metal plate must be cleaned

from organic contaminants by an organic solvent and then

flamed before the experiment. Both roughening and clean-

ing of the plate surface are used to maintain good wetting

of the plate by the test liquid. It should be noted that

materials other than platinum or platinum-iridium alloy,

such as glass, mica, and steel (1, 5) have also been used.

Nevertheless, good wetting of the test liquid to the plate is

always necessary. The use of plates made of material other

than metal is a ‘‘must’’ requirement in the case of certain

liquids such as during the measurements of the interfacial

tension between a heavy nonpolar liquid (i.e., carbon tet-

rachloride) and immisible, but lighter, polar liquid (i.e.,

water). For such systems, the plate should be hydrophobic.

Several polymers, especially fluorinated polymers, can be

used for this purpose. Adsorption and self-assembling of

organic amines on the surface of the platinum plate could

also be a solution to this problem.

In the Wilhelmy plate method, the plate is put in a fixed

position relative to the horizontal surface of the liquid

(Fig. 2). Then, the force (F) vertically acting on the plate

by the liquid meniscus is measured by using a microba-

lance. The force applied to the plate is equal to the weight

of the liquid meniscus uplifted over the horizontal surface.

By measuring this force, the interfacial tension can be

calculated by using Eq. 1 where p = 2(L + t). Modern

instruments use plates of standard dimensions so that

measurements of the plate size and its weight are not re-

quired. Adsorption of organic compounds from the labo-

ratory environment or test solutions can be a major source

of experimental error when measuring surface tensions

using the Wilhelmy plate method.

Du Nouy ring method

In this method, the interfacial tension relates to the force

required to pull a wire ring off the interface (Fig. 3) (7, 8).

As in the case of Wilhelmy plate, the ring is usually made

up of platinum or platinum-iridium alloy of a radius (R)

of 2–3 cm. The radius (r) of the wire ranges from 1/30 to

1/60 of that of the ring (9).

Again, Eq. 1 describes in general the calculation pro-

cedure of the technique. Here, the perimeter ( p) of the

three-phase contact line is equal to twice the circumfe-

rence of the ring: p = 4pR. Because additional volume of

liquid is lifted during the detachment of the ring from the

interface, a correction factor ( f ) is required in Eq. 1 (8):

g ¼ F

p cos yf ð2Þ

The correction factor varies from about 0.75 to 1.05 and

depends on the dimensions of the ring (R, r), its surface

wettability (y), and difference in fluid density (Dr). The

tabulated f values in relation to R/r (for y = 0) can be

found in Ref. 8, and also calculated from the following

approximate equation (10):

f ¼ 0:725þ 9:075 � 10�4F

p3DrgR3� 1:679r

Rþ 0:04534

� �1=2

ð3Þ

The application range of Eq. 3 is: 0.045 � DrgR3/F �7.5. The maximum force is measured by the microbalance

(F ) and corresponds to detachment of the ring from the

interface. The F value is measured experimentally and

then Eq. 3 is used to calculate the correction factor f. The

interfacial tension is then calculated from Eq. 2. The in-

terfacial tension reading made by modern computerized

instrumentation does not require separate calculation of f,

since its calculation is incorporated in the software.

The high-accuracy measurements with the ring method

require that the plane of the ring remain parallel to the

interface. The major error in this technique is caused by

deformation of the ring, which is a very delicate probe and

subject to inadvertent deformation during handling and

cleaning. It is also important that perfect wettability of the

ring surface by the denser fluid be maintained (y = 0). If

perfect wetting is not achieved, additional correction of the

instrument reading is needed. Poor wetting of ring by the

3154 Measurement of Interfacial Tension in Fluid-Fluid System

denser fluid makes the measurement of interfacial tension

impossible to carry out. In the case of special measure-

ments requiring homemade rings, very large rings should

be avoided to avoid the small value of the correction factor

(see Eq. 3). If all of the necessary experimental precau-

tions are observed, this method can guarantee higher ac-

curacy than any other detachment method.

Group II: Measurement of Capillary Pressure

Interfacial tension is defined as the work required to create

a unit area of interface at a constant temperature, pressure,

and chemical potential. Because it is always positive for

interfaces between immiscible phases, interfacial tension

always tends to decrease the area of interface. This ten-

dency gives rise to a pressure difference between fluids on

either side of a curved interface, with the higher pressure

on the concave side of the interface. This pressure dif-

ference results in phenomena such as a capillary rise,

bubble and drop formation, etc. A formula describing the

pressure difference (DP) across the curved interface is

known as the Young-Laplace equation (11, 12):

DP ¼ g1

R1

þ 1

R2

� �ð4Þ

where R1 and R2 are the radii of curvature.

The pressure difference across a curved interface (DP)

can be measured in a number of ways (e.g., using a pres-

sure sensor or observing a capillary rise) and then be used

to calculate g if the radii of curvature are known. The most

common and probably one of the oldest methods in this

group of interfacial tension measurement techniques is a

maximum bubble pressure method that is briefly described

in the next paragraph. Modification of the maximum bub-

ble pressure method based on a continuous measurement

of varying pressure during growing bubble or drop is now

a basic technique in examination of dynamic (not equi-

librated) interfacial tension and is further discussed in

‘‘Dynamic Interfacial Tension Measurements.’’

Maximum bubble pressure

This method is based on measuring the maximum pressure

( p� ) to force a gas bubble out of a capillary into a liquid

(Fig. 4) (13, 14). The measured pressure is the sum of

capillary pressure (DP) caused by the interfacial tension

and the hydrostatic pressure (rAghA) caused by the liquid

column above the orifice of the capillary:

DP ¼ p� � rAghA ð5Þ

This pressure can be expressed as the height (h) of the

column of an imaginary liquid of density (Dr = rA�rB):

h ¼ DP

Drgð6Þ

Sugden (13) derived an expression to relate h with the

Laplace capillary constant a = 2g/(Drg) and the bubble

meniscus:

r

X¼ r

bþ r

a

� � zc

b

� � b2

� �1=2

ð7Þ

where X = a2/h, b = 2b2/a2, zc is the height of the bub-

ble, and b is the curvature radius at the apex (lowest point

of the bubble). Then he tabulated the minimum values of

Fig. 4 Maximum bubble pressure method. (A) A sequence illustrating the shape of bubble at three different stages of bubble growth.

(B) Relationship between pressure inside the bubble and radius of the bubble.

M


X/r as dependent of a given value of r/a within the range

0 < r/a � 1.5. Using this table, the surface tension can be

calculated by following an iteration procedure.

Direct and easier, but a little less accurate, calculation

of the interfacial tension can be done by using the fol-

lowing equation (5):

g ¼ DPr

21 � 2rDrg

3DP� ðrDrgÞ2

6DP2

!ð8Þ

As further discussed in the next section; the maximum

bubble and drop pressure technique or its modifications

have been very useful in studying the dynamic interfacial

tensions. This technique has also been attractive to exa-

mination of surface tension for molten metals (2).

Group III: Analysis of the Balance BetweenCapillary and Gravity Forces

Methods based on analysis of capillary effects, other than

the shape of a drop or meniscus, such as capillary rise and

drop volume or weight, are among the oldest surface ten-

sion measurement methods in use. A variety of modern

instruments, usually fully automated and computerized

(groups I, II, and IV), have replaced these methods in most

laboratories. We provide a short review of two techniques

that still might be attractive to researchers who have limi-

ted access to modern instrumentation.

Capillary rise method

The basis for the capillary rise method is to measure the

height h of the meniscus in a round glass tube having the

known inner radius r, as shown in Fig. 5 (2, 15). For

small-diameter tubes (i.e., r < < h) the shape of the me-

niscus is spherical, and the surface tension can be calcu-

lated by using the following equation:

g ¼ Drghr

2 cos yð9Þ

Since the glass tubes are easy to clean with acids, bases,

and organic solvents, and because many of the liquids

perfectly wet the glass surface, the cosy term in the above

equation will often equal unity.

If the shape of the meniscus is not spherical, Eq. 9

should be replaced with (15):

g ¼ 1

2Drgrh 1 þ r

3h� 0:1288

r2

h2þ 0:1312

r3

h3

� �ð10Þ

The capillary rise method can be one of the most ac-

curate techniques used to make surface tension measure-

ments. Technical problems with the technique are related

to fabrication of a uniform bare capillary tube and precise

determination of its inside diameter. In addition, the ca-

pillary rise method is not very convenient for measuring

the interfacial tension between two liquids.

Drop volume or weight

In this method, the weight or volume of a drop falling from

a capillary with a radius r is measured (Fig. 6) (16, 17).

The weight (W) of the drop falling off the capillary cor-

relates with the interfacial tension through the following

equation (5):

W ¼ VDrg ¼ 2prgfrffiffiffiffiV3

p� �

ð11ÞFig. 5 Illustration of the capillary rise method.

Fig. 6 Schematic illustration of drop volume or weight method.


where V is the drop volume, r is the radius of the capillary,

and f is the correction factor required because only a

portion of the drop volume is released from the capillary

during detachment (2). The correction factor is a function

of r/V1/3, and this correlation was empirically determined

and tabulated by Harkins and Brown (17). It can also be

calculated from the empirical function (5):

frffiffiffiffiV3

p� �

¼ 0:167 þ 0:193rffiffiffiffiV3

p� �

� :0489rffiffiffiffiV3

p� �2

� 0:0496rffiffiffiffiV3

p� �3

ð12Þ

Because of small volume of each drop, many drops need

to be collected for the accurate measurement of drop

weight or volume. In modern instrumentation, the volume

of liquid and the number of droplets released from the

capillary can be determined very precisely and thus the

weight or volume of the individual drop is not difficult to

calculate (18).

Capillaries used in the drop weight or volume tech-

niques are usually made of glass; however, metal capil-

laries are also used on occasion (1). Glass is wetted by

many liquids, is transparent, and is relatively easy to clean.

Capillary tubes specifically fabricated for routine inter-

facial tension measurements are now difficult to purchase

in the U.S. market; however, glass capillaries can be pro-

duced relatively easily in glass workshops.

The measurements of interfacial tension with the drop

weight or volume technique are very simple but, unfortu-

nately, sensitive to vibrations on the other side. Vibrations

of the apparatus can cause premature separation of the drop

from the end of the capillary before the drop reaches the

critical size. In addition, the measurements in multicom-

ponent solutions when adsorption occurs might not reflect

equilibrium saturation of the solutes at the interface.

Group IV: Analysis ofGravity-Distorted Drops

Interfacial tension causes interfaces to behave as elastic

membranes that always tend to compress the liquid. In the

absence of other forces (e.g., in zero gravity), the liquid

surface has a natural tendency to form spherical shapes to

minimize the interfacial area per unit volume of liquid and

thus, to minimize the excess energy of the interface. The

shape of an interface in a gravitational field (Fig. 7) de-

pends on the competition between the capillary and gra-

vitational forces and can be described by the Bashforth-

Adams equation (19):

gsinf

xþ 1

R1

� �¼ 2g

bþ Drgz ð13Þ

Eq. 13 is often expressed in a dimensionless form as:

sinfx=b

þ 1

R1=b¼ 2 þ Drgb2

gz

bð14Þ

where g is the interfacial tension; Dr = �A��B equals

the difference in density of fluids; R1 is the radius of cur-

vature; x is the radius of rotation of point S around the z

axis; f is the angle of R2 vector with the axis of symmetry;

b is the radius of curvature at the apex of the curvature; and

g is the acceleration due to gravity. Fig. 7 shows the de-

tails of drop geometry.

The techniques of curved interface shape analysis are

particularly attractive to researchers because they do not

require advanced instrumentation. The experimental setup

requires a camera with a low-magnification lens to record

the shape of the drop. The interfacial tension can be easily

calculated from the dimensions of the pendant drop, ses-

sile drop, or liquid meniscus taken from the photographic

picture and by using numerical solutions to the above

equations. Modern instruments, however, use image anal-

ysis software whose role is to match the entire drop profile

to the best fit of the theoretical curve (e.g., the Bashforth-

Adams equation) describing the shape of the drop (14).

These advances significantly improved the precision of

the techniques and reduced the time of the measurement,

providing an opportunity for examination of the interface

aging process. Probably the most advanced software, axi-

symmetric drop shape analysis, was introduced by Neu-

mann and co-workers (20). Since advanced instrumen-

tation is not always available to researcher, a brief reviewFig. 7 Definition of dimensions and coordinates describing the

sessile drop.

M


of classical approaches of interfacial tension determination

from the shape of the interface seems to be appropriate.

Pendant drop method

In a simple method, two parameters of the pendant drop

that should be experimentally determined are the equa-

torial diameter D and the diameter d at the distance D

from the top of the drop (Fig. 8) (21, 22). The interfacial

tension is then calculated from the following equation (2,

21, 22):

g ¼ DrgD2

Hð15Þ

The shape dependent parameter (H ) depends on a value of

the ‘‘shape factor’’ S = d/D. Tables including the set of

1/H vs. S values are available in several references (1, 22).

The values of 1/H can also be calculated from the fol-

lowing empirical formula (23):

1

H¼ B4

Saþ B3S3 � B2S2 þ B1S � B0 ð16Þ

where Bi (i = 0, 1, 2, 3, 4) and a are empirical constants

for a certain range of S, which are shown in Table 1.

The pendant drop technique, as other interfacial tension

measurement techniques, requires extreme cleanliness to

obtain good quality and reproducible results. Here, the

needle used for hanging the drop should be well cleaned

and the climbing of the interface over the outer surface of

the needle should be avoided. Needles made of stainless

steel or glass that are relatively easy to clean with acids,

bases, and organic solvents are most often used in surface

chemistry laboratories. It is recommended that needles

with a diameter that is less than 0.5 D be used (21). The

diameter of the needle should not be too small, however,

because this reduces the value of d and, consequently, the

precision of interfacial tension determination.

Sessile drop method

This method is based on the analysis of the profile of the

drop sitting on a solid substrate, as shown in Fig. 7 (24,

25). It is recommended that substrates used in sessile drop

measurements be poorly wetted by the drop, i.e., they

should have a contact angle larger than 90 degrees. In a

simple experimental approach, one first needs to locate

the equator of the drop, and then measure the height from

the top of the drop to its equator (ze). For a very large

sessile drop, an analytical expression for the interfacial

tension is as follows (5):

g ¼ Drgz 2e

2ð17Þ

From a practical point of view, it is often difficult to pre-

cisely locate the equator of the drop and measure ze for

many drops. Although the large drop is almost flat, loc-

ating the top of the drop is sometimes experimentally

difficult. It should be recognized, however, that large

drops are not required if tabulated dependencies of drop

shape parameters, based on the Bashford-Adams analysis

(Eq. 14) are used (19, 25).

Practical Comments

Most of the techniques reviewed in this section have been

commercialized. Table 2 summarizes the accuracy, com-

Fig. 8 Pendant drop.

Table 1 Empirical constants for Eq. 16

Range of S A B4 B3 B2 B1 B0

0.401–0.46 2.56651 0.32720 0 0.97553 0.84059 0.18069

0.46–0.59 2.59725 0.31968 0 0.46898 0.50059 0.13261

0.59–0.68 2.62435 0.31522 0 0.11714 0.15756 0.05285

0.68–0.90 2.64267 0.31345 0 0.09155 0.14701 0.05877

0.90–1.00 2.84636 0.30715 � 0.69116 � 1.08315 � 0.18341 0.20970


mercial availability, and suitability of these techniques for

various types of fluid-fluid systems. The accuracy of most

of these techniques for pure liquid-gas systems is about

0.1 mN/m. The capillary rise technique is capable of

significantly better accuracy than the others in Table 2.

Most of the interfacial tension measurement techniques

listed in Table 2, including the capillary rise method, have

been successfully applied to liquid-liquid systems. Re-

duced accuracy of the detachment du Nouy ring method

is associated with difficulties in calibrating the weight of

the ring immersed in the less dense liquid. The same

problem can be expected in the interfacial tension mea-

surements using the Wilhelmy plate instrument. All tech-

niques in Table 2 yield reduced accuracy when applied to

liquid-liquid interfaces or when one or both of the liquids

is viscous.

The measurements with viscous liquids are always dif-

ficult to carry out due to problems with handling the

liquid, injection of a liquid sample of the required volume

into the instrument, low–velocity liquid flow, and long–

time viscous effects during deformation of the interface.

Two techniques are particularly recommended to examine

the surface tension of viscous liquids: the Wilhelmy plate

(not a detachment option!) and the sessile drop methods.

In both techniques, samples can be equilibrated for se-

veral hours before the measurements are taken.

It is important to recognize that any interfacial ten-

sion measurement can be strongly influenced by inter-

facially active solutes or impurities that are accidentally

introduced into the fluid-fluid system or present on solid

surfaces that act as part of the measurement system (1,

2). Any solid surfaces that make contact with liquids

(e.g., plates, rings, and capillary tubes) must be careful-

ly cleaned prior to making measurements. Furthermore,

some interfacially active contaminants can be introduced

from the skin or breath of laboratory workers. Finally, it is

important to note that interfacial tensions are influenced

by temperature, which should be controlled and reported

for all measurements.

When using techniques that depend on a known wet-

tability of a solid probe by one of the liquids, surface active

solutes (whether present intentionally or as impurities) can

seriously influence the interfacial tension measurements.

Interfacially active solutes in a liquid phase can adsorb not

only on the fluid-fluid interfaces but on the liquid-solid

interfaces as well. This adsorption will affect wettability

of the solid surface (the cosy term in Eqs. 1 and 2) and,

therefore, influence the measured result. In principle, such

effects can be eliminated by employing solid probes of

alternate materials to which the solute does not adsorb, but

plates and rings are not usually available in a wide range

of alternative materials.

Even when adsorption to solid-liquid interfaces is not a

problem, it is important to allow fluid-fluid interfaces to

achieve equilibrium before making a measurement. When

equilibrium is achieved rapidly (within a few seconds),

the drop volume technique may be suitable. If equilibra-

tion requires longer times, sessile drop, Wilhelmy plate,

or other quasi-static techniques may be more appropriate.

This consideration applies to any fluid-fluid system in

which kinetically limited processes (adsorption, viscous

flow, etc.) take place.

Because of the relatively high temperatures involved

and their reactivity with many gases and solids, surface

tension measurements on liquid metals pose special chal-

lenges. The four principal techniques that have been em-

ployed are the maximum bubble pressure method, the

sessile drop method, the drop volume or weight method,

and the pendant drop method (26, 27). It is important that

measurements on liquid metals be carried out in inert

Table 2 Accuracy and suitability of classic techniques used in interfacial tension measurements

Method

Accuracy

½mN=m�

Suitability

for surfactant

solutions

Suitability

for two-liquid

systems

Suitability for

viscous liquids

Suitability for

melted metals

Commercial

availability

Wilhelmy plate �0.1 Limited Good Very good Not recommended Yes

Du Nouy ring �0.1 Limited Reduced accuracy Not recommended Not recommended Yes

Maximum bubble 0.1–0.3 Very good Very good Not recommended Yes Yes

pressure

Capillary rise <<0.1 Very good Very good, Not recommended Not recommended Not

experimentally

difficult

Drop volume 0.1–0.2 Limited Good Not recommended Yes Yes

Pendant drop �0.1 Very good Very good Not recommended Yes Yes

Sessile drop �0.1 Good Very good Very good Yes Not

M


gas environment to avoid reactions with gases and other

phases. Oxygen and other reactive gases are known to exert

strong effects on the surface tension of selected metals,

even when present in parts per million concentration (27).

DYNAMIC INTERFACIALTENSION MEASUREMENTS

In fluid-fluid systems containing interfacially active so-

lutes, a freshly created interface will not generally be in

compositional equilibrium with the two immiscible fluids

it separates. It is only after solute redistribution from one

or both phases (i.e., adsorption) has occurred that this

interface will achieve its equilibrium state. It is sometimes

important to measure the interfacial tension of freshly

created interfaces, and such measurements yield what is

known as ‘‘dynamic surface tension.’’ A detailed review

of experimental techniques, theoretical background, and

literature on the measurements of dynamic interfacial ten-

sions was recently published by Dukhin et al. (3). Another

valuable source of analysis of adsorption at the interface

and dynamic interfacial tension is the book published by

Joos (28).

Table 3 provides a characteristic time range for the

selected interfacial tension measurement techniques. Of

different techniques already discussed in this article, the

capillary rise method is not recommended for dynamic

interfacial tension measurements. The techniques dis-

cussed in the next sections are not very suitable for exa-

mination of dynamic effects at interfaces either.

Interfacial tension changes that occur over time in-

tervals of at least several seconds (and continue over se-

veral minutes, hours, or days) can be studied by using

most of the classical techniques discussed in the previous

section. For example, Fig. 9 shows the results of inter-

facial tension relaxation between bitumen and water of

varying pH value recorded with the Wilhelmy plate ins-

trument (30). In this bitumen-water system, the dynamic

character of the interfacial tension is caused by diffusion

of natural surfactants from the bitumen to interface and

aqueous phase, and surfactant reaction with ions dissolved

in water (31).

As emphasized in the previous section, examination of

the interfacial tension for surfactant solutions using clas-

sical techniques should be carried out with caution. Sur-

factants often adsorb on the solid surfaces of equipment

used in measurements and change the wetting character-

Table 3 Characteristic time range for common interfacial tension measurement techniques

Method Time rangea Comments

Wilhelmy plate >10 s Some of the surfactants might alter the wetting properties

of the plate, causing the change of measurement

conditions (possible source of error)

Du Nouy ring >30 s Same as above

Pendant drop >10 s Strongly surface active chemicals might cause the release

of pending drop before completion of the measurement

Sessile drop >10 s Some of the surfactants might alter the wetting properties

of a solid support substantially changing the shape of

the sessile drop

Drop volume/weight 1 s–20 min Hydrodynamic effects associated with releasing liquid

volume and circulation of liquid inside the drop

sometimes significantly reduce the accuracy of the

interfacial tension measurements

Maximum bubble pressure 1 ms–100 s Difficulties with determination of the real surface age and

problems with hydrodynamic effects in the vicinity

of interface

Growing drop/bubble >10 msb Not available commercially

Oscillating jetc 1–10 ms Not available commercially

Pulsating bubblec 5 ms–0.2 s Not available commercially

aBased on Dukhin et al. (3).bIt is claimed by MacLeod and Radke (29) that the dynamics interfacial tension can be measured for several hours, although Ref. 3 specifies the upper

limit as 600 s.cMethods not discussed in this article.


istics of a solid surface. This effect usually causes expe-

rimental problems in the Wilhelmy plate, du Nouy ring,

and sessile drop techniques (see comments in Table 3).

Short-time interfacial tension and wetting effects play

important roles in high–volume industrial processes such

as froth flotation of particles and droplets, detergency,

foam or froth generation, and stability (3). In these pro-

cesses, dynamic interfacial tensions become more crucial

to the success of the technology than the equilibrium (or

near-equilibrium) interfacial tension. This issue has been

emphasized in the literature (3, 32, 33), but straightforward

links between dynamic interfacial tensions and (e.g., pro-

cess efficiencies) have not yet been well established. This

research area is just evolving, and the continued funda-

mental research will probably establish a better connection

of dynamic interfacial phenomena with practical needs.

Four basic techniques for measurements of the dynamic

interfacial tension at short intervals include the maximum

bubble pressure, growing drop (bubble), oscillating jet,

and pulsating bubble methods (Table 3). Neither the os-

cillating jet technique nor the pulsating bubble technique

is discussed in this article. Bases of these techniques are

provided in Refs. 1–3 and references therein.

The maximum bubble (drop) pressure method and its

modifications have been the most popular techniques used

in research conducted in recent years. Apparatus for ma-

king these measurements are also available commercially.

The maximum bubble pressure technique was briefly des-

cribed in ‘‘Classical Interfacial Tension Measurement Me-

thods.’’ Following is a short description of a modification

of the maximum bubble pressure technique that is suitable

for dynamic surface tension measurements.

Growing Drop (Bubble) Method

Modern instrumentation permits the pressure inside a bub-

ble or drop to be precisely and continuously measured as it

forms and detaches from the end of a capillary (3, 29, 34,

35). The geometry of a drop or bubble can also be mo-

nitored during growth and detachment by using advanced

videographic equipment. This ability to simultaneously

monitor both pressure and geometry (size and shape) of

bubbles or drops as they form allows dynamic interfacial

tensions to be evaluated over a range of growth rates.

Furthermore, the same experimental approach allows mea-

surement of surface tensions using approaches normally

applied to systems in (or near) equilibrium, such as the

drop volume and maximum pressure drop techniques.

Fig. 10 describes an experimental approach used by

MacLeod and Radke (29) for measurement of dynamic

interfacial tension using the growing drop technique. In

Fig. 9 Dynamic interfacial tension measured between bitumen

and water at 60�C using the Wilhelmy plate technique. The pH

values for water at the beginning (t = 0) and at the end of the

measurements (t = 90 min) are listed in the figure legend. (The

results are from Ref. 30.)

Fig. 10 Schematic of the growing drop apparatus used by MacLeod and Radke. (From Ref. 29.)

M


this apparatus, a liquid drop or gas bubble is formed and

released from a capillary by using a precise micropump to

carefully control the growth rate of the drop or bubble. A

pressure transducer is used to simultaneously monitor and

record the internal pressure in the drop or bubble, while

its size and shape are recorded by using a video camera.

These experiments can be carried out for a range of flow

rates ranging from near equilibrium growth (very low

flow rates) to highly nonequilibrium growth conditions

(very rapid bubble or drop growth).

Fig. 11 shows selected results of MacLeod and Radke

(29) for growth of 0.25 mM aqueous decanol droplets for

a range of droplet growth rates between 5 and 100 mm3/

min. The plots of interfacial tension vs. time show ini-

tially increasing, reaching a maximum, and then decrea-

sing as a function of time. The positive slope of the g vs. t

curves for t < 1 results from the depletion of decanol

adsorption due to rapid expansion (stretching) of the

interface when the bubble is small (see Fig. 4A). As the

drop geometry proceeds beyond hemispherical shape

(Fig. 4C), the relative rate of surface area growth decrea-

ses, allowing decanol from the bulk liquid to replenish the

surface adsorption. The decrease in interfacial tension as-

sociated with this increase in adsorption yields the negative

slope observed in the g vs. t curves for t > 1. As expected,

the largest dynamic interfacial tensions, approaching those

for pure water, are observed for droplets formed at the

highest capillary flow rates. Conversely, the lowest values

(approaching the equilibrium interfacial tension for this

solution) were observed for droplets growing at very slow

rates. At drop formation times approaching 100 s, inter-

facial tensions for drops formed at all flow rates approach

the equilibrium value for the 0.25 mM aqueous 1-decanol

solution.

The zero time for each measurement in Fig. 11 cor-

responds to the moment of detachment for the previous

drop. Data points for a given capillary flow rate in Fig. 11

start at the time where the drop reaches a hemispherical

shape (Fig. 4B), which also corresponds to the point at

which the interfacial tension can be evaluated by using the

maximum pressure method. Interfacial tension measure-

ments using the drop-volume method are, of course, de-

termined at the point where the drop detaches from the

capillary and correspond to the last data point in the

corresponding curve for the growing drop method. These

two techniques, therefore, provide ‘‘snapshots’’ of the in-

terfacial tension values in dynamic systems, and they

bracket the family of curves determined by the growing

drop method. The growing drop method provides the very

important advantage of continuously recording the inter-

facial tension throughout the drop formation and allows

the competing kinetic effects of interfacial stretching and

solute transport for adsorption to be explored.

MEASUREMENT OF ULTRALOWINTERFACIAL TENSION

Recovery of petroleum using tertiary oil recovery tech-

nology; cleaning of solid surfaces from dirt, grease, and

oil; formulation of stable emulsions; in situ remediation

of oil-contaminated soil with surfactant solutions; and

other applications often rely on lowering the interfacial

tension between immiscible liquids to ultralow values

(much less than 1 mN/m) using surfactants. The measure-

ments of such low interfacial tensions are extremely dif-

ficult to perform with classical interfacial tension mea-

surement methods reviewed in ‘‘Classical Interfacial

Tension Measurement Methods’’ or the dynamic tech-

niques discussed in the previous section (e.g., see the ac-

curacy values for methods shown in Table 2). For the

measurements of ultralow interfacial tensions, the spin-

ning drop technique has been developed at both laboratory

and commercial scales (36–39). The basis of this tech-

nique is discussed in the following paragraph. An ad-

ditional method designed for measurements of ultralow

interfacial tensions was proposed by Lucassen (40) and

is based on the analysis of the shape of the drop suspended

Fig. 11 Dynamic surface tension of 0.25 mM aqueous decanol

solution droplets growing in air at 23�C. Based on the experi-

mental data presented by MacLeod and Radke. (From Ref. 29.)


in liquid with a density gradient. The need for a strict

control of liquid density limits this latter technique to rela-

tively few applications.

Spinning Drop Technique

This technique relies on the fact that gravitational ac-

celeration has little effect on the shape of a fluid drop

suspended in a liquid, when drop and the liquid are con-

tained in a horizontal tube spun about its longitudinal axis

(Fig. 12) (36, 37). At low rotational velocities (o), the

fluid drop will take on an ellipsoidal shape, but when o is

sufficiently large, it will become cylindrical. Under this

latter condition, the radius (r) of the cylindrical drop is

determined by the interfacial tension, the density diffe-

rence (Dr) between the drop and the surrounding fluid,

and the rotational velocity of the drop. As the result, the

interfacial tension is calculated from the following equa-

tion (38):

g ¼ 1

4r3Dro2 ð18Þ

The spinning drop method has been very successful in

examination of ultralow interfacial tensions down to 10�6

mN/m (39). For example, Fig. 13 shows the interfacial

tension values for octane drops suspended in an aqueous

phase saturated with ethoxylated alcohols (41). As shown

in Fig. 13, the interfacial tension for the octane-water-

CnE4 system varied from about 1 to 10�4 mN/m and de-

pended on both temperature in the system and length of

the hydrocarbon chain in the chemical structure of etho-

xylated alcohol (Ref. 41 provides additional examples).

The minimum in the interfacial tension vs. the temperature

curve coincides with the phase inversion temperature, at

which both hydrophilic and oleophilic natures of surfac-

tant are in balance.

MICROTENSIOMETRY

Criminology, biology, and pharmaceutical processing are

among a number of fields in which material quantities of

interest may be too small to apply conventional tensio-

metric techniques. Furthermore, the developing field of

nanotechnology promises to yield novel structures on a

nanometer scale, the interfaces of which are sure to be the

subject of considerable study. The study of interfaces on

the nanometer scale can be very important because they

may exhibit properties that are very different from their

macroscopic counterparts. For example, knowledge of

partition and adsorption of surfactants and other interfa-

cially active components of two-liquid systems is often

crucial for the control of emulsion formulation and sta-

bility. It has recently been demonstrated that interfacial

tension of microscopic droplets may differ significantly

from the interfacial tension of macroscopic drops for the

same surfactant solutions (42). This difference arises from

dissimilar partitioning of surfactants into two immiscible

liquids and is due to the effect of an enlarged interfacial

area. In this regard, it becomes critical to conduct the study

of interfacial phenomena, including the measurements of

interfacial tension, on a length scale that is characteristic

of the individual elements of the emulsion.

Microtensiometry is the study of interfaces on very

small particles and in finely dispersed systems. The micro-

pipette technique, discussed first, is closely related to tech-

niques depending on pressure differentials across curved

interfaces that have already been presented. Techniques

based on atomic force microscopy (AFM) offer the pos-

sibility of imaging nanometer-sized particles and directly

measuring their interactions. AFM-based techniques are

relatively new and still very much under development.

Fig. 12 Schematic of the rotating drop method.

Fig. 13 The effect of temperature on interfacial tension mea-

sured between octane and aqueous phase saturated with CnE4.

Graph based on the experimental data presented in Ref. 41.

M


Micropipette Technique (43–45)

The micropipette technique was recently developed to

directly measure interfacial tensions of micrometer-sized

droplets. The technique was first used in examination of

vesicles (43, 44) and next liquid droplets (42, 45). In this

technique, the droplet is first captured at the tip of the glass

micropipette and then sucked into the pipette (Fig. 14A).

The interfacial tension is calculated from the minimum

pressure, at which the droplet extends a hemispherical pro-

trusion into the pipette, and by using the Laplace Eq. 4

in the following form (42):

Dp ¼ 2g1

Rp

� 1

Ro

� �ð19Þ

where Rp is the inner radius of the pipette and Ro is the

radius of the exterior segment of the droplet; the dimen-

sions of the pipette’s internal diameter must be smaller

than the diameter of the droplet.

In the conventional technique shown in Fig. 14A, the

large pressure difference required to draw the droplet into

the pipette when the droplet does wet or adhere to the pi-

pette surfaces, is a limitation. To avoid this limitation, a

two-pipette technique, with a separation force applied bet-

ween the pipettes to deform the droplet, has been used as

shown in Fig. 14B (45). In the two-pipette technique, the

separation force between the pipettes must also be mea-

sured, and the interfacial tension is calculated from the

force–drop deformation relation. Table 4 presents the re-

sults of interfacial tension measurements for water–orga-

nic liquid systems determined with this technique.

Atomic Force Microscopy

The application of AFM permits roughness, heterogeneity,

and interaction forces to be studied at submicroscopic

scales that may extend down to molecular sizes. Among

the most popular applications of the AFM are studies of

interactions between substrates and colloidal particles. The

AFM has not been established as a technique to measure

interfacial tensions directly: however, it appears to have

great potential for such measurements at the microscopic

and submicroscopic levels. The technique is currently be-

ing evaluated as a tool for measuring the wetting properties

of colloidal particles (48).

Fig. 14 Illustration of micropipette techniques. (A) Based on analysis of pressure differences required to suck a microdroplet into the

pipette. (B) Based on the examination of the force–drop deformation relation.

Fig. 15 Schematic of atomic force microscopy and its ap-

plication to measurements of surface tension for microdroplets.

Table 4 Results of interfacial tension measurements for the

organic solvent microdroplets in water obtained with the mic-

ropipette technique and their comparison to the similar results

measured for bulk two-liquid systems

System

Interfacial tension

measured for

microdroplets

[mN/m)

Interfacial tension

measured for

macrosystem

[mN/m)

Water-ethyl acetate 6.8 ± 0.6 6.8

Water-chloroform 30.9 ± 1.6 31.6

Water-benzene 33.7 ± 0.7 34.1

Water-toluene 36.4 ± 0.8 36.1

Water-hepto 40.3 ± 1.1 39.6

(From Ref. 45.)


AFM is a scanning probe technique based on measuring

interaction forces between a cantilever tip and a specimen

(Fig. 15). The force measurement is based on measuring

the deflection of the cantilever, which has a known spring

constant. The cantilever deflection is detected by the ref-

lection of a laser beam as shown in the figure. Movement

of the specimen under the cantilever tip (both in the ho-

rizontal plane for scanning, and vertically for force mea-

surements) is controlled very precisely by a piezoelectric

specimen stage (reverse systems with the piezoelectric

stage attached to the cantilever holder are also in use).

Interaction forces as small as 1pN (10�12 N) can be mea-

sured between the probe tip and the specimen.

Fig. 15 shows an approach for measuring the in-

terfacial tension between a probe tip and a microscopic

drop of liquid. The capillarity forces exerted on the tip by

the liquid can be measured as it is inserted into the drop

and as it is withdrawn and detaches from the drop. Cal-

culation of surface tensions from these force-distance

curves will depend on the shape of the probe tip, but the

equations should be similar to those for classical macro-

detachment techniques.

A major challenge in using the AFM technique to mea-

sure interfacial tensions is fabrication of appropriate probe

tips. Cylindrical tubes and spheres offer simple geometries

and may present the best near-term options. Carbon nano-

tubes have recently been used as AFM tips, and may offer

another promising approach (49).

REFERENCES

1. Rusanov, A.I.; Prokhorov, V.A. Interfacial Tensiometry;

Elsevier: Amsterdam, 1996.

2. Adamson, A.W.; Gast, A.P. Physical Chemistry of Surfa-

ces, 6th Ed.; John Wiley & Sons, Inc.: New York, 1997.3. Dukhin, S.S.; Kretzschmar, G.; Miller, R. Dynamics of

Adorption at Liquid Interfaces: Theory, Experiment, Appli-

cation; Elsevier: Amsterdam, 1995.

4. Hiemenz, P.C.; Rajagopalan, R. Principles of Colloid and

Surface Chemistry, 3rd Ed.; Marcel Dekker, Inc.: New

York, 1997.5. Sonntag, H. Koloidy; PWN: Warszawa, 1982.

6. Wilhelmy, L. Ueber die abhangigkeit der capillaritats-

constanten des alkohols von substanz und gestalt des

benetzten fasten korpers. Ann. Phys. Chem. 1863, 4 (29),177–217.

7. Lecomte du Nouy, P. A new apparatus for measuring sur-

face tension. J. Gen. Physiol. 1919, 1, 521–524.8. Harkins, W.D.; Jordan, H.F. A method for determination of

surface and interfacial tension from the maximum pull on a

ring. J. Am. Chem. Soc. 1930, 52, 1751–1772.9. Vold, R.D.; Vold, M.J. Colloid and Interface Chemistry;

Addison-Wesley Publishing Co.: London, 1983.

10. Zuidema, H.H.; Waters, G.W. Ring method for the de-

termination of interfacial tension. Ind. Eng. Chem. Anal.

Ed. 1941, 13, 312–313.11. Young, T. Miscellaneous Works; Peacock, G., Ed.; J. Mur-

ray: London, 1855; Vol. I, 418.

12. de Laplace, P.S. Mechanique Celeste. Supplement to Book

10; 1806.

13. Sugden, S. The determination of surface tension from the

maximum pressure in bubbles. J. Chem. Soc. 1922, 121,

858–866.14. Rehbinder, P.A. Dependence of surface activity and sur-

face tension of solutions upon temperature and concentra-

tion. Z. Phys. Chem. 1924, 111, 447–464.15. Lord Rayleigh, OM.; F.R.S. On the theory of the capil-

lary tube. Proc. R. Soc. London, Ser. A 1916, 92, 184–195.

16. Tate, T. On the magnitude of a drop of liquid formed un-

der different circumstance. Phil. Mag. 1864, 27, 176–180.

17. Harkins, W.D.; Brown, F.E. The determination of surface

tension (free surface energy), and the weight of falling

drops: the surface tension of water and benzene by the

capillary height method. J. Am. Chem. Soc. 1919, 41,499–525.

18. Miller, R.; Hofmann, A.; Schano, K.H.; Halbig, A.; Hart-

mann, R. Measurement of surface and interfacial tensions

with an automatic drop-volume tensiometer. Parfuem. Kos-

met. 1992, 73, 390–397.19. Bashforth, F.; Adams, J.C. An Attempt to test the Theory of

Capillary Action; Cambridge University Press: London,

1883.

20. Lahooti, S.; del Rio, O.I.; Neumann, A.W.; Cheng, P.

Axisymetric Drop Shape Analysis (ADSA). Applied Sur-

face Thermodynamics; Neumann, A.W., Spelt, J.K., Eds.;

Marcel Dekker, Inc.: New York, 1996; 441–507.

21. Andreas, J.M.; Hauser, E.A.; Tucker, W.B. Boundary ten-

sion by pendent drops. J. Phys. Chem. 1938, 42, 1001–1019.

22. Stauffer, C.E. The measurement of surface tension by the

pendent drop technique. J. Phys. Chem. 1965, 69, 1933–1938.

23. Misak, M.D. Equations for determining 1/H versus S

values in computer calculations of interfacial tension by

pendent drop method. J. Colloid Interface Sci. 1968, 27,141–142.

24. Quincke, G. Ueber die capillaritats-constanten des queck-

silbers. Ann. Phys. Chem. 1858, 4 (15), 1–48.25. Padday, J.F. The profiles of axially symmetric menisci.

Trans. R. Soc. London 1971, A269, 265–293.26. Lang, G. Surface Tension of Liquid Elements. Handbook of

Chemistry and Physics, 72nd Ed.; Lide, D.R., Ed.; CRCPress: Boca Raton, 1992; 133–145, Chapter 4.

27. Eustathopoulos, N.; Nicholas, M.G.; Drevet, B. Wettabi-

lity at High Temperatures; Pergamon: Amsterdam, 1999;

Chapter 4.

28. Joos, P. Dynamic Surface Phenomena; VSP: Zeist, The

Netherlands, 1999.

M


29. MacLeod, C.A.; Radke, C.J. A growing drop technique for

measuring dynamic interfacial tension. J. Colloid Interface

Sci. 1993, 160, 435–448.30. Drelich, J. The Role of Wetting Phenomena in the Hot

Water Process for Bitumen Recovery from Tar Sand.

Ph.D. Dissertation, The University of Utah: Salt Lake

City, Utah, USA, 1993.

31. Rudin, J.; Wasan, D.T. Mechanisms for lowering of in-

terfacial tension in alkali/acidic oil systems. Colloids Surf.

1992, 68, 81–94.32. Rosen, M.J. Surfactants and Interfacial Phenomena;

Wiley-Interscience: New York, 1976.

33. Dynamic Properties of Interfaces and Association Struc-

tures; Pollai, V., Shah, D.O., Eds.; ACS Press: Champaign,

Illinois, 1996.

34. Passerone, A.; Liggieri, L.; Rando, N.; Ravera, F.; Ricci,

E. A new experimental method for the measurement of the

interfacial tension between immiscible fluids at zero bond

number. J. Colloid Interface Sci. 1991, 146, 152–162.

35. Horozov, T.; Arnaudov, L. A novel fast technique for mea-

suring dynamic surface and interfacial tension of surfactant

solutions at constant interfacial area. J. Colloid Interface

Sci. 1999, 219, 99–109.36. Vonnegut, B. Rotating bubble method for the determina-

tion of surface and interfacial tensions. Rev. Sci. Instr.

1942, 13, 6–16.37. Princen, H.M.; Zia, I.Y.Z.; Mason, S.G. Measurement of

interfacial tension from the shape of a rotating drop. J.

Colloid Interface Sci. 1967, 23, 99–107.

38. Couper, A.; Newton, R.; Nunn, C. A simple derivation of

Vonnegut’s equation for the determination of interfacial

tension by the spinning drop technique. Colloid Polym.

Sci. 1983, 261, 371–372.

39. Manual of the Spinning Drop Tensiometer Site 04; Kruss

GmbH: Hamburg, Germany, 1995.

40. Lucassen, J. The shape of an oil droplet suspended in an

aqueous solution with density gradient. J. Colloid Interface

Sci. 1979, 70, 355–365.41. Sottmann, T.; Strey, R. Ultralow interfacial tensions in

water-n-alkane-surfactant systems. J. Chem. Phys. 1997,106, 8606–8615.

42. Yeung, A.; Dabros, T.; Masliyah, J. Does equilibrium

interfacial tension depend on method of measurement? J.

Colloid Interface Sci. 1998, 208, 241–247.43. Evans, E.; Skalak, R. Mechanics and Thermodynamics of

Biomembranes; CRC Press, Inc.: Boca Raton, Florida,

1980.

44. Evans, E.; Needham, D. Physical properties of surfactant

bilayer membranes: thermal transitions, elasticity, rigidity,

cohesion, and colloidal interactions. J. Phys. Chem. 1987,91, 4219–4228.

45. Moran, K.; Yeung, A.; Masliyah, J. Measuring interfacial

tensions of micrometer-sized droplets: a novel microme-

chanical technique. Langmuir 1999, 15, 8497–8504.46. Sarid, D. Scanning Force Microscopy; Oxford University

Press: New York, 1991.

47. Wiesendanger, R. Scanning Probe Microscopy and Spec-

troscopy: Methods and Applications; Cambridge Univer-

sity Press: Cambridge, Massachusettes, 1994.

48. Ecke, S.; Preuss, M.; Butt, H.-J. Microsphere tensiometry to

measure advancing and receding contact angles on indi-

vidual particles. J. Adhesion Sci. Technol. 1999, 13, 1181–

1191.49. Dai, H.; Hafner, J.H.; Rinzler, A.G.; Colbert, D.T.; Sma-

ley, R.E. Nanotubes as nanoprobes in scanning probe mic-

roscopy. Nature 1996, 384, 147–151.


MEASUREMENT OF INTERFACIAL TENSION IN FLUID-FLUID …

Documents

Transcript of MEASUREMENT OF INTERFACIAL TENSION IN FLUID-FLUID …