THE SURFACE TENSION OF THE SURFACE TENSION OF PURE SUBSTANCESPURE SUBSTANCES

INTRODUCTIONINTRODUCTION

IntroductionIntroductionSurface tension is the contractile force

which always exists in the boundary between two phases at equilibrium

Its actually the analysis of the physical phenomena involving surface tension which interests us

Our topics primarily concern onOur topics primarily concern onSurface tension as a forceSurface tension as surface free energySurface tension and the shape of mobile

interfacesSurface tension and capillaritySurface tension and intermolecular forces

Surface Tension As A Force: Surface Tension As A Force: The Wilhelmy PlateThe Wilhelmy PlateThe surface of a liquid appears to be

stretched by the liquid it enclosesExample of this are:

the beading of water drops on certain surfaces; the climbing of most liquids in glass capillaries

The force acts on the surface and operates perpendicular and inward from the boundaries of the surface, tending to decrease the area of the surface

lF2

NotedNoted Equation above defines the units of surface

tension to be those of force per length or dynes per centimeter in the cgs system

The apparatus shown resembles a two-dimensional cylinder/piston arrangement, so its analogous to a two dimensional pressure

A gas in the frictionless, three-dimensional equivalent to the apparatus of the figure would tend to expand spontaneously. For a film however the direction of spontaneous change is contraction

A quantity that is closely related to surface tension is the contact angle , defined as the angle (measured in the liquid) that is formed at the junction of three phases, as shown in figure 6.1b

Although the surface tension is a property of two phases which form the interface, requires that three phases be specified for its characterization

The Wilhelmy Plate

Figure 6.2 The Wilhelmy plate method for measuring . In (a) the base of the plate does not extend below the horizontal liquid surface. In (b) the plate is partially submerged to buoyancy must be considered

a b

Figure 6.2 represent a thin vertical plate suspended at a liquid surface from the arm of tarred balance

The manifestation of surface tension and contact angle in this situation is the entrainment of a meniscus around the perimeter of the suspended plate

Assuming the apparatus is balanced before the liquid surface is raised to the contact position, the imbalance that occurs on contact is due to the weight of the entrained meniscus

Since the meniscus is held up by the tension on the liquid surface, the weight measured by the apparatus can be analyzed to yield a value for

The observed weight of the meniscus w, must equal the upward force provided by the surface

w = 2(l+t) cos is the contact angle, l and t are the length and

thickness of the plate. Because of the difficulties in measuring , the Wilhelmy plate method is most frequently used for system in which = 0 so

w = 2(l + t) Since the thickness of the plate used is generally

negligible compared to their length (t <<< l) equation may approximated:

w = 2l

Surface Tension As Surface Excess Free Energy The work done on the system of figure 6.1 is given by

Work = F dx = 2l dx = dA This supplies a second definition of surface tension, it equals

the work per unit area required to produce new surface If the quantity w’ is defined to be the work done by the

system when its area is changed, then equation becomesw’ = -γdA according to the first law

dE = q - w in which w is the work done by the system and q is the heat absorbed by the system. It relates to Gibbs free energy by following equation :dG = TdS – pdV - wnon-pV + pdV + Vdp – TdS – SdT for a constant temperature, constant pressure and reversible process

dG = -wnon-pV

That is dG equals the maximum non-pressure/volume work derivable from such a process since maximum work is associated with reversible process

We already seen that changes in surface area entail non-pV work, therefore we identify w’ as wnon-pV and write

dG = γdA Even better in view of the stipulations we write

γ = (G/A)T,p This relationship identifies the surface tension as

the increment in Gibbs free energy per unit increment in area

The Laplace Equation

0 surfaceplanar aFor

R surface lcylindrica aFor

2 surface sphericalFor

11

21

21

21

21

pRRR

p

RpRRR

RRp

Our attention will focus on those specific surfaces with most readily allow the experimental determination of

The shape assumed by a meniscus in a cylindrical capillary and the shape assumed by a drop resting on a planar surface (called a sessile drop) are most useful in this regard

Figure 6.4 may be regarded as a portion of the surface of either of these cases

As can be seen the curve represent the profile of a sessile drop; inverted, the solid portion represent the profile of a meniscus

The actual surfaces are generated by rotating these profiles around the axis of symmetry

Because the symmetry of the surface, both values R must be equal at the apex of the drop

The value of the radius of curvature at this location is symbolized b, therefore, at the apex (subscript 0)

b

p 20

Next, let us calculate the pressure at point S. At S the value of p equals the difference between the pressure at S in each of the phases

These may be expressed relative to the pressure at the reference plane through the apex (subscript 0) as follows

In phase A:pA = (pA)0 + Agz

In phase B:pB = (pB)0 + Bgz

Therefore, p at S equals(p)S = pA – pB = (pA)0 – (pB)0 + (A - B)gz

= (p)0 + gz

Where = A - B and we can write it

gzb

p s

2

Notes If A > B, will be positive and the drop will be

oblate in shape since the weight of the fluid tends to flatten the surface

If A < B, a prolate drop is formed since the larger buoyant force leads to a surface with much greater vertical elongation. In this case is negative

A value of zero correspond to a spherical drop and in a gravitational field is expected only when p = 0

Positive values of correspond to a sessile drops of liquid in gaseous environment

Negative values correspond to sessile bubbles extending into a liquid

Notes

The previous statement imply that the drop is resting on a supporting surface

If instead the drop is suspended from a support (called pendant drops or bubbles), g becomes negative, and it is the liquid drop that will have the prolate ( < 0) shape and the gas bubbles the oblate ( >0) shape

Measuring Surface Tension: Sessile DropsMeasuring Surface Tension: Sessile Drops

The Bashfort and Adams tables provide an alternate way of evaluating by observing the profile of a sessile drop of the liquid under investigation

Once known for a particular profile, the Bashfort Adams tables may be used further to evaluate b

For the appropriate value, the value of x/b at = 90o is read from the tables. This gives the maximum radius of the drop in units of b

From the photographic image of the drop, this radius may be measured since the magnification of the photograph is known

Comparing the actual maximum radius with the value of (x/b)90 permits the evaluation of b

The figure can use for example of the procedure described

Theoretically its shown to correspond to a value of 10,0 then b is evaluated as follows

1) The value of (x/b)90 for = 10 is found to be 0,60808 from the tables

2) Assume the radius of the actual drops is 0,500 cm at its widest point

Item (1) and (2) describe the same point; therefore b = 0,500/0,60808 = 0,822 cm

Assuming to be 1,00 g cm-1 and taking g = 980 cm s-

2 gives for

2-

22

cm ergs 3,660,10

}822,0)980(00,1{

gb

Measuring Surface Tension: Capillary Rise A simple relationship between the height of capillary

rise, capillary radius, contact angle and surface tension can derived

2R cos = R2h g (48) Its difficult to obtain reproducible result unless = 0o, so

the equation simplifies to

(49)gRh

2

The cluster constant 2/(g) is defined as the capillary constant and is given the symbol a2;

(50)

ga

22

The apex of the curved surface is identified as the point from which h is measured. As we have seen before, both radii of curvature are equal to b at this point

At the apex of the meniscus, the equilibrium force balance leads to the result

(51)

(52)2

2

abh

ghb

p

Equation (48) is valid only when R = b, that is for a hemispherical meniscus.

In general this is not the case and b is not readily meaured so we have not yet arrived at a practical method of evaluating γ from the height of capillary rise. Again the tables of Bashfort and Adams provide the necessary information

For liquid to make an angle of 0o with the supporting walls, the walls must be tangent to the profile of the surface at its widest point

Accodingly (x/b)90o in the Bashfort and Adams tables must correspond to R/b. since the radius of the capillary is measurable, this information permits the determination of b for a meniscus in which θ = 0

However there is a catch. Use of the Bashfort and Adams tables depends on knowing the shape factor β. It is not feasible to match the profile of a meniscus with theoritical contours, so we must find a way of circumventing the problem

The procedure calls for using successive approximation to evaluate β. Like any iterative procedure, some initial values are fed into a computational loop and recycle until no further change results from additional cycles of calculation

In this instance, initial estimates of a and b (a1 and b1) are combined with Eqs. (46) and (50) to yield a first approximation to β (β1)

The value of (x/b)90o for β1 is read or interpolated from the tables

This value and R are used to generate a second approximation to b (b2). By Eq.(52) a second approximation of a (a2) is also obtained and –starting from a2 and b2 – a second round of calculation is conducted.

The following table shows an example of this procedure

It is sometimes troublesome to find a starting point for these iterative calculations. The following estimates are helpful for the capillary rise problem:

From Eqs (49) and (50) a1 Rh Treating the menicsus as hemisphere b1 R The initial value of table 6.3 assuming R = 0.25 cm and h = 0.40 cm

Measuring Contact Angle

The experimental methods used to evaluate θ are not particularly difficult, but the result obtained may be quite confusing

The situation is best introduced by refering figure right-below which shows a sessile drop on a tilted plane

It is conventional to call the larger value the advancing angle θa and the smaller one the receding angle θr

With the sessile drop, the advancing angle is observed when the drop is emerging from a syringe or pipet at the solid surface

The receding angle is obtained by removing liquid from the drop

Weight of a meniscus in a Wilhelmy plate experiment versus depth of immersion of the plate. In (a) both advancing and receding contact angles are equal. In (b) a > r

Schematic energy diagram for metastable states corresponding to different contact angles

The general requirement for hysteresis is the existence of a large number of metastable states which differ slightly in energy and are separated from each other by small energy barriers

The metastable states are generaly attributed to either the roughness of the solid surface or its chemical heterogeneity, or both.

Cross section of a sessile drop resting on a surface containing a set of concentric grooves. For both profiles, the contact angle is identical microscopically, although macroscopically different

Kelvin Equation

Another result of pressure difference is the effect it has on the free energy of the material possessing the curved surface

Suppose we consider the process of transferring molecules of a liquid from a bulk phase with a vast horizontal surface to a small spherical drop of radius r

Assuming the liquid to be incompressible and the vapor to be ideal, ∆G for the process of increasing the pressure from po to po + ∆p is as follows:

rM

rV

ppRT

ppRT

pppRTG

rVdpVdpVG

L

o

oo

o

LL

pp

pL

o

o

22ln

:equal areG of values two thesem,equilibriuat are vapor and liquidWhen

lnln

: vaporFor the 2.liquid theof memolar volu theisV Where

2

:liquid For the 1.

L

The Kelvin equation enables us to evaluate the actual pressure above a spherical surface and not just the pressure difference across the interface, as was the case with the Laplace equation

Using the surface tension of water at 20oC, 72,8 ergs cm-

2, the ratio p/po is seen to be

r

xrxp

p

o

7

7

1008,1exp2931031,8998,0

8,720,182exp

Or 1,0011; 1,0184; 1,1139; and 2,9404 for drops of radius 10-4, 10-5, 10-6 and 10-7 cm respectively.

Thus for a small drops the vapor pressure may be considerably larger than for flat surfaces

The Kelvin equation may also be applied to the equilibrium solubility of a solid in a liquid

In this case the ratio p/po in equation is replaced by the ratio a/ao where ao is the activity of dissolved solute in equilibrium with flat surface and a is the analogous quantity for a spherical surface

For an ionic compound having the general formula MmXn the activity of a dilute solution is related to the molar solubility A as follows:

oo

nm

SSRTnm

aaRT

rM

nSmSa

lnln2sphere solid afor Therefore

The equation provides a thermodynamically valid way to determine SL, for example the value of SL for the SrSO4-water surface has been found to be 85 ergs cm-2 and for NaCl-alcohol surface to be 171 ergs cm-2 by this method

The increase in solubility of small particles and using it as a means of evaluating SL is fraught with difficulties: The difference in solubility between small particles and

larger one will probably differ by less than 10% Solid particles are not likely to be uniform spheres even

if the sample is carefully fractionated The radius of curvature of sharp points or

protuberances on the particles has a larger effect on the solubility of irregular particles than the equivalent radius of the particles themselves.

The Young Equation Suppose a drop of liquid is placed on a perfectly

smooth solid surface and these phases are allowed to come to equilibrium with the surrounding vapor phase

Viewing the surface tension as forces acting along the perimeter of the drop enables us to write equation which describes the equilibrium force balance

LV cos = SV - SL

First Objections Real solid surfaces may be quite different from the idealized

one in this derivation Real solid surface are apt to be rough and even chemically

heterogeneous If a surface is rough a correction factor r is traditionally

introduced as weighting factor for cos , where r > 1 The factor cos enters equation by projecting LV onto the

solid surface If the solid is rough a larger area will be overshadowed by the

projection than if the surface were smooth Young’s equation becomes

rLV cos = SV - SL A surface may also be chemically heterogeneous. Assuming

for simplicity that the surface is divided into fractions f1 and f2 of chemical types 1 and 2 we may write

LV cos = f1(S1V - S1L) + f2(S2V - S2L) Where f1 + f2 = 1

Second Objection

The issue of whether the surface is in a true state of thermodynamically equilibrium, it may be argued that the liquid surface exerts a force perpendicular to the solid surface, LV sin

On deformable solids a ridge is produced at the perimeter of a drop; on harder solids the stress is not sufficient to cause deformation of the surface

Is it correct to assume that a surface under this stress is thermodynamically the same as the idealized surface which is free from stress?

The stress component is absent only when = 0 in which case the liquid spreads freely over the surface and the concept of the sessile drop becomes meaningless

Notes

We must assume that SV and S may be different Let us consider what occurs when the vapor of a volatile liquid

is added to an evacuated sample of a non volatile solid This closely related to the observation that the interface

between a solution and another phase will differ from the corresponding interface for the pure solvent due to the adsorption of solute from solution

For now we may anticipate a result to note that adsorption always leads to decrease in , therefore:

incorrect)(cos SLSLV o

eSV

SVS o

oS

e difference esignify th to symbol theuse shall we

The equation must be corrected to give

SLeSLV o cos

Figure shows relationship between terms write at the right hand side, it also suggest that the shape of the drop might be quite different in equilibrium and non equilibrium situations depending on the magnitude of e

There are several concepts which will assist us in anticipating the range of e values:

1. Spontaneously occurring processes are characterized by negative values of ∆G

2. Surface tension is the surface excess free energy; therefore the lowering of with adsorption is consistent with the fact that adsorption occurs spontaneously

3. Surfaces which initially posses the higher free energies have the most to gain in terms of decreasing the free energy of their surfaces by adsorption

4. A surface energy value in the neighborhood of 100 ergs cm-2 is generally considered the cutoff value between ‘high energy’ and ‘low energy’ surfaces

ADHESION AND COHESION

Figure illustrates the origin of surface tension at the molecular level

In (a) which applies to a pure liquid, the process consists of producing two new interfaces, each of unit cross section, therefore for the separation process:∆G = 2A = WAA

The quantity WAA is known as the work of cohesion since it equals the work required to pull a column of liquid A apart

It measures the attraction between the molecules of the two portion

G = WAB = final - initial = A + B - AB This quantity is known as the work of adhesion

and measures the attraction between the two different phases

The work of adhesion between a solid and a liquid phase may be define analog:WSL = S + LV - SL

By means of previous equation S may eliminated to givesWSL = SV + e + LV - SL

Finally Young’s equation may be used to eliminate the difference:WSL = LV(1 + cos ) + e

e 0 where the equality holds in the absence of adsorption

High energy surface bind enough adsorbed molecules to make e significant, example of these are metals, metal oxides, metal sulfides and other inorganic salts, silica, and glass

On the other hand e is negligible for a solid which possesses a low energy surface, most of organic compounds, including organic polymers are in this criteria.

The difference between the work of adhesion and the work of cohesion of two substances defines as quantity known as the spreading coefficient of B on A, SB/A:SB/A = WAB – WBB

If WAB > WBB the A-B interaction is sufficiently strong to promote the wetting of A by B (positive spreading). Conversely no wetting occurs if WBB > WAB since the work required to overcome the attraction between two molecules B is not compensated by the attraction between A and B (negative spreading).SB/A = A - B - AB = A – (B + AB)

The Dispersion Component of Surface Tension

1 repulsion2 attraction3 specific Interaction4 resultant

Bulk phaseBulk phase

Interface between two phasesInterface between two phases

A A

A

AA

A

A

AA

A

A

A

Bulk phaseBulk phase

Interface between two phasesInterface between two phases

A B

A

AB

A

A

AA

A

B

B

B

B B

F. Fowkes has proposed that any interfacial tension may be written as the summation of contributions arising from the various types of interactions which would operate in the material under consideration, in general then:

= d + h + m + + i = d + sp

Superscripts refer to dispersion forces (d), hydrogen bonds (h), metallic bonds (m), electron interactions () and ionic interactions (i).

2/12

yield equations twoComparing

2

all themof sum theis interface AB theforming of work Total)(

dB

dAAB

dB

dABA

dB

dAB

dB

dAAAB

dB

dAAA

sAA

dB

dA

s

W

EWork

E

TUGAS KIPER

Gunakan data tabel 6.2 untuk mem-plot profile tetes(drop) dengan = 25. Ukur (dalam cm) jari-jari tetes yang anda gambar pada titik terjauhnya (widest point). Dengan membandingkan nilainya dengan nilai (x/b)90 dari tabel, hitung b (dalam cm) untuk tetes yang anda gambar. Andaikan tetes sesungguhnya memiliki nilai ini (25), jika jari-jari pada widest point 0,25 cm dan = 0,50 g cm-3, berapa untuk antarmuka tetes tsb.