applications of the differential balances to momentum transfer

8/10/2019 Applications of the Differential Balances to Momentum Transfer

1/102

A pp lica t io ns of t h e Dif ferent ia l B alan ces t o

M o m e n t u m T r a n s f e r

Let us now see w hat mu st be don e in order to arrive at detailed d escriptions for

the configurations and motions of adjoining multiple phases within specified

boundaries consistent with the principles we have developed in the preceding

two chapters.

In writing this chapter, we have assumed that you have already been in

troduced to the more standard problems of fluid mechanics involving the flow

of a single phase [42, Chap. 3; 99] and that you wish to learn more about

analyzing multiphase flows when interfacial effects are important.

3 . 1 P h i l o s o p h y

3 .1 .1 S tru c ture o f Pr ob lem

It is relatively easy to outline the issues that must be addressed in order to

describe the configuration and motions of several adjoining phases.

i)

Within each phase,

we must satisfy the differential mass balance, the

differential momentum balance, and an appropriate model for bulk stress-

deformation behav ior such as th e N ewto nian fluid [42, p. 42].

ii) On each dividing surface, we must satisfy the jump mass balance and the

jump momentum balance. Note that, as the result of the discussion in Sect.

2.2.2, we will express the surface stress tensor simply in terms of an interfacial

tension or energy.

iii) At each common line,there are mass and mom entum balances to be obeyed

(Sects. 1.3.8 and 2.1.9).

iv)

Everywhere,

velocity must be finite.

v)

Within each phase,

the stress tensor is bounded, since every second order

tensor in three space is bounded [77, p. 140].


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160 3 Applications of the Differential Balances to Momentum Transfer

vi)

On each dividing surface,

the surface stress tensor is bounded, by the same

argument .

vii) Within each phase in the neighborhood of a phase interface, it is important

to correct for the intermolecular forces from the adjoining phase (see Sect. 2.2).

In the case of a single interface, this may be a direct correction, although

usually it will be done in terms of a surface tension or energy. In the case of

thi n films, we comm only will introd uce a surface tension or energy a ttri bu tab le

to unbounded adjoining phases with a further correction to account for the

fact that at least one of the adjoining phases is not unbounded.

viii) At each dividing surface, the tangential components of velocity are con

tinuous. This is often referred to as the no-slip boundary condition. It is a

consequence of assuming nearly local equilibrium at the phase interface (see

Sect. 4.10.3). Lacking contrary experimental evidence, this condition is ap

plied even when there is a phase change, the normal component of velocity

is not continuous at the interface, and the adjacent phases axe clearly not

in local equilibrium. A careful examination of the no-slip condition has been

given by G oldste in [81, p. 676]. It is know n to fail, when o ne of th e pha ses is

a rarefied gas.

commo n lines on relatively rigid solids

Prob lem s involving comm on

lines on relatively rigid solids require special consideration, because we will

normally decide to ignore the deformation of the solid within the immediate

neighb orhoo d of th e comm on line. We can sum ma rize our discussions in Sects.

2.1.10 and 2.1.13 with these additional comments.

ix ) A common line can not move. As viewed on a macroscale, it appears

to move relative to the solid as a succession of stationary common lines are

formed on a microscale, driven by a negative disjoining pressure in the receding

film (see Sect. 3.3.5). The amount of receding phase left stranded by the

formation of this succession of common lines is too small to be easily detected,

as in the experiments of Dussan V and Davis [16, see also Sect. 1.2.9].

x)

The m ass balance at a comm on line on a rigid solid is satisfied identically,

since all velocities measured relative to the rigid solid go to zero at the common

line.

xi) The mom entum balance at a comm on line on a rigid solid reduces to

Young's equation (Sect. 2.1.10)

7^^^) cos 0 0 + 7^^^^

- 1 ^ ^ ^

= 0 (3.1.1-1)

where 7^'^^ denotes the interfacial tension or energy acting in the

i-j

interface

of Fig. 2.1.10-1 and OQ is the contact angle measured through phase A at the

true common line (rather than the static or dynamic contact angle measured

by experimentalists at some distance from the common line, perhaps with 10

X magnification).


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3.1 Philosophy 161

The preceding discussion assumes that one is willing to solve a complete

problem. In the majority of applications, you will prefer to use a data corre

lation to specify the static or dynamic contact angle at the apparent common

line, the line at which the experimentalist measures the contact angle with

perhaps 10x magnification. If th e ap par ent comm on line is in motion, it mu st

be allowed to slip along the solid surface. Various empiricisms have been sug

gested to account for this pheno me non [83, 85-8 7, 91 , 115, 183-185].

3 . 1 . 2 A p p r o x i m a t i o n s

Although it is relatively easy to outline as we have done in the preceding

section the relationships and conditions that must be satisfied in describing

any multiphase fiow, this is not suHicient, if we are to describe real physical

phenomena. One must decide precisely what problem is to be solved and

what approach is to be taken in seeking a solution. We think in terms of the

approximations that must be introduced. Approximations are always made,

when we describe real physical problems in mathematical terms.

We can distinguish at least four classes of approximations.

a) idealization of physical problem. Usually the physical problem with which

we are concerned is too difficult to describe in detail. One answer is to replace

it with a problem that has most of the important features of our original

problem but that is sufficiently simple for us to analyze. Physical boundaries

are replaced by geometric surfaces. Viscous effects may be neglected in a gas

phase with respect to those in a liquid phase. The motion of boundaries is

simplified. The disturbance caused by a tracer particle floating in an interface

may be ignored. Bulk stress-deformation behavior may be described by the

incompressible Newtonian fluid.

b) limiting case. Even after such an idealization of our original physical prob

lem, it may still be too difficult. We may wish to consider a limiting case in

which one or more of the parameters appearing in our boundary-value prob

lem approach zero. This is equivalent to asking for a solution correct to the

zeroth perturbation in these parameters.

c) integral average. In many cases, our requirements do not demand detailed

solutions. Perhaps some type of integral average is of primary interest.

d) mathematical approximation. Often we will introdu ce a m ath em atic al ap

proximation in seeking a solution to a boundary-value problem. This is inher

ent with numerical solutions.

While we should attempt to hold approximations to a minimum, even

greater effort should be devoted to recognizing those that we are forced to

introduce. For it is in this way that we shall better understand the limitations

of our theories in describing experimental data.


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162 3 Ap plication s of th e Differential Bala nces to M om en tum Transfer

3 .2 On ly In t e r f a c i a l Te n s io n

3 .2 .1 C las se s o f Pr ob lem s

In this section, we will talk about phase interfaces in the absence of deforma

tion. In all cases, there is a frame of reference with respect to which there is

no motion anywhere within the system being considered. Usually this will be

the laboratory frame of reference, but we will see that this is not necessary.

Because of this lack of motion with respect to some preferred frame of

reference, the interface and the adjoining phases axe static, but they should

not necessarily be taken to be at equilibrium. Long after motion can be said

to be absent from a system, its chemical composition may be slowly shifting

as the various species present distribute themselves between the interface and

the adjoining phases. You might expect this point to be made in Chap. 5,

where we explicitly account for mass transfer. But in fact it is the rare ex

periment in which surfactants, perhaps present only in the form of unwanted

contaminants, have been eliminated.

We can distinguish at least two classes of problems.

In the simplest class, we neglect the effect of all external forces. Pressure is

independent of posit ion within each phase and the jump momentum balance

requires

[ P ^ ] =

2H^i

(3.2.1-1)

which implies that the mean curvature H is independent of position on the

interface. These problems are discussed further in Exercise 3.2.1-1 and illus

trated in the sections that follow.

A second and often more difficult class of problems includes those in which

the effect of one or more external forces can not be neglected. In Sect. 3.2.2,

we consider the spinning drop measurement of interfacial tension in which the

drop shape is controlled by the Coriolis force and the effect of gravity is ne

glected. In Exercise 3.2.5-1, we discuss a rotating interface in which the effect

both of gravity and of the Coriolis force play important roles in determining

its shape.

Exercise 3.2.1-1. Axially symm etric interface in absence of external forces [186]

In Exercise

A.5.3-11,

we find that for an axially symmetric surface in cylindrical

coordinates

z = h{r) (3.2.1-2)

the mean curvature H may be described by

H=Yr[rh

+

h'+{h'f][l

+

{h'f]

^ ^ ^ '^'

(3.2.1-3)

-3 /2

2'-dr- \^[i + (ft/)2]V2


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3.2 On ly Interfacial Tension 163

where

h' = ^

(3.2.1-4)

d r

We confine our attention here to a static interface in the absence of gravity, in which

case i f is a con stant . W e assume th at the surface coordina te system has been chosen

such tha t

H >0.

Integrating (3.2.1-3), we find

^ ^ -,

=Hr + Br-^

(3.2.1-5)

[l +

ih'ff'

Notice that a solution of (3.2.1-5) can exist only in an interval of

r

for which

| i f r + S r - ^ | < l (3 .2 .1-6 )

There aie three possibilit ies.

i) If B > 0, prove that solutions exist only if

B < ^ (3.2.1-7)

Prove that there is a solution in the interval (r i ,T-2)

0 < r - i < ^ < r 2 < - i : (3.2.1-8)

where

ri

an d

r2

assumes all values in the indicated ranges as

B

varies from zero to

i t s upper bound and

as r -> n :

h' ^ oo

(3.2.1-9)

as r ^ 1-2 : ft' ^ oo (3 .2.1 -10 )

An example is i l lustra ted in Fig. 3.2.1-1.

H

F i g . 3 . 2 . 1 - 1 . Section of rotationally symmetric interface for B > 0


6/102


ii) If S = 0, prove that the unique solution for (3.2.1-5) is a sphere of radius (in

spherical coordinates)

(3.2.1-11)

iii) If B < 0, prove that there is a solution in the interval (ri,r2)

0 < r-i < oo (3.2.1-12)

4 < ^2 < oo (3.2.1-13)

where ri and r2 increase through all values of their ranges as

B

decreases from 0 to

oo and

as r ^ r-i : h' - o o (3.2.1-14)

as r ^ r2 :

h' ^ oo

(3.2.1-15)

One case is shown in Fig. 3.2.1-2.

r = 0 r i r2

Fig. 3.2.1-2.

Section of rotationally symmetric interface for S < 0

3 .2 .2 Spin ning D ro p Interfac ia l Te ns io m eter [21]

Althoug h not restricted to this l imit, only three experiments have been dem on

strated as being suitable for measuring ultra low interfacial tensions (less than

10 '^

dyne / cm

:

th e spinning drop [187-191], th e microp end ant drop

[192],

and

the sessile drop [193]).

The spinning drop experiment sketched in Fig. 3.2.2-1 is perhaps the sim

plest to construct and to operate. At steady state, the hghter drop phase {A),

the heavier continuous phase

{B),

and the tube all rotate as rigid bodies with

the same angular velocity Q.

Although the analysis for the spinning drop presented by Princen et al.

[188] an d by Ca yias et a l. [190] is correct, it is inconven ient t o use bec ause the ir


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3.2 Only Interfacial Tension 165

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

y y ^ ^ ^ y y ^ ^ ^ y y y ^ ^ y y y ^ ^ y y y ^ ^ y y y ^ ^ ^ y y ^ ^ ^ y y ^ ^ -

Fig. 3.2.2-1. At steady state, the drop phase A, the continuous phase B, and the

tube all rotate as rigid bodies with the same angular velocity Q

results axe given in terms of the radius of a sphere having the same volume

as the drop. An experimentalist normally measures the maximum diameter

(2rniax after correction for refraction) and the length (2/imax) of the drop.

The revised solution that follows avoids this difficulty. It is more than a

simple rearrangement of their result, since the radius of curvature of the drop

at its axis of rotation (a in their notation) is treated in their development as

thought it were known.

Prom the differential momentum balance for each phase (i = A, B)

(3.2.2-1)

5()

i .

where V), is th e modified press ure of ph ase i on the axis of the cylinder. The

(3.2.2-2)

jump momentum balance requires

where we have assumed that the Bond number

AT,

o

(P'

B) _ n(^ )

)9rl

7

< 1

(3.2.2-3)

Here H is the mean curvature of the interface, 7 is the interfacial tension, and

To

is a char acte ristic len gth of th e sys tem t h a t will be defined shortly. If we

describe the configuration of the interface by

z = h{T)

the mean curvature takes the form (Table

B .

1.2-7)

-1 /2

I


8/102


It is convenient to introduce the change of variable [188]

ta.n9 = - (3.2.2-6)

d r

in terms of which (3.2.2-5) becomes

-2H

= - - ^

(r-

sin6) (3.2.2-7)

r dr

Given (3.2.2-1) and (3.2.2-7), we can rearrange (3.2.2-2) as

\ - ^ {r*sin6) = A- r*^ (3.2.2-8)

where we have found it convenient to define

* = IL

~ To

(3.2.2-9)

A=- (p^^ ) - P ^ ^ ) ) To (3.2 .2-10)

and

27

I

(3.2.2-11)

Equation (3.2.2-8) can be integrated once consistent with the boundary con

ditions

(3.2.2-12)

(3.2.2-13)

(3.2.2-14)

at r*

and

at r*

to find

sin^

= 0 :

= r*

' max

=4

61

= 0

: e=

^ * 3

with the requirement

A =

2

' m a x

r* 2

' max

2

n

2

(3.2.2-15)

We can now visualize integrating (3.2.2-6) to learn

h*m^= '

1/2^^*

(3.2.2-16)

Jo ( l - s i n 2 6 l)^


9/102


in which sin

0

is given by (3.2.2-14) and

h*

is defined by analogy with (3.2.2-9).

With the change of variable

g ^ - - (3.2.2-17)

(3.2.2-16) can be written as

rA/2

K.^ =

/ ^

- ^ dq

(3.2.2-18)

91

[{q - qi) [q -

^2)

[q - qs)]

with

qi = ^

(3.2.2-19)

' m a x

r* 2 1 / r * * V^^

92 = ^ + 2

[ - ^ +

^ m a. j (3.2.2-20)

93 = ^ - 2 [f^ +^ m a. j (3.2.2-21)

Let us consider two limiting cases. For a cylindrical drop

sin6l = l (3.2.2-22)

and it follows immediately from (3.2.2-8) and (3.2.2-15) that

< a x = 2 ^ / ' (3.2.2-23)

For a spherical drop whose dimensionless radius is

R*

r*

s i n 6 l = - - (3.2.2-24)

R*

and we conclude from (3.2.2-8) and (3.2.2-15) that

< a x = 0 (3.2.2-25)

This suggests that we confine our attention to the range 0 < r^^x ^ 2^/^

for which

qi>qi> qa (3.2.2-26)

With this restriction, (3.2.2-18) can be integrated to find [194, p. 78].

2

h* =-

' max , .1 /2

[qi -

93)

qiF {o, k) - {qi - qs) E {^o, k)

+

(91 - 93) (t an(t>o)

ya - k^

sin^ 0o

3.2.2-27)


10/102

168 3 Applications of the Differential Balances to Mom entum Transfer

in which

E

{(j)o,

k)= f Vl-k'^sm'^^d^

Jo

Fic^o.k)^

/

Jo

v l - ^ si

sin^

^

3.2.2-28)

3.2.2-29)

(3.2.2-30)

(3.2.2-31)

qi -qs

Following a suggestion of Princen et al.

[188],

the volume of the drop

0o = arcsin

. M-2 , ,^^ /^

A-2q2

k

2 _ q2-q3

becomes

V^

V

+

/?* - 1

3.2.2-32)

We computed rmax/^max = r^^^JKi^ from (3.2.2-27) an d V from

(3.2.2-32) as functions oir'^^^. In these com puta tions , the incom plete elliptic

integrals (3.2.2-28) and (3.2.2-29) were expanded in series [195, p. 300]. The

results are presented in Table 3.2.2-1 and Fig. 3.2.2-2 in a form convenient for

ex pe rim en tal ists : r^g^^ ^^ ^ ^ * ^^ func tions of rmax/Z^max-

' max max

0.4 0.6

1.259

1.250

1.255

0.25

^ m a x / ' ^ m a x

Fig. 3.2.2-2.

rJiax as a function of rmax/^max from (3.2.2-27)

0.35


11/102


Table 3.2.2-1.r^a^ andV*as function of rmax/Vax from (3.2.2-27)

and (3.2.2-32)^

1 '

-msn-v

h

^

h

1.0

0 0

0.3198 1.252 30.6741

0.9997 0.1 0.0042 0.3122 1.253 31.6546

0.9980 0.2 0.0336 0.3038 1.254 32.7883

0.9932 0.3 0.1139 0.2945 1.255 34.1321

0.9840 0.4 0.2725 0.2837 1.256 35.7818

0.9687 0.5 0.5406 0.2708 1.257 37.9195

0.9459 0.6 0.9571 0.2543 1.258 40.9617

0.9140 0.7 1.5745 0.2297 1.259 46.2969

0.8710 0.8 2.4714 0.2262

1.2591

47.1309

0.8148 0.9 3.7782 0.2225 1.2592 48.0733

0.7415 1 0 5.7457 0.2183 1.2593 49.1567

0.6432 1.1 8.9812 0.2136 1.2594 50.4307

0.4928

1 2

15.9687 0.2081

1.2595

51.9769

0.3332 1.25 29.0379 0.2016 1.2596 53.9444

0.3268 1.25 29.8101 0.1932

1.2597

56.6522

In order to use the spinning drop technique to determine interfacial ten

sion, one must measure both r-max and /imaxfo rgiven values of (p^^^ p^ ^^)

and /?. Having rmax/Zimax, one determines r^^x from Table 3.2.2-1 or Fig.

3.2.2-2. When rmax//imax< 0.25, (3.2.2-23) can be used with less than 0.4

error in the interfacial tension. The interfacial tension can then be found by

rearranging the definition for r^ax-

^ ^ 1 / r ^ A UB)_p(A)\^2 (3.2.2-33)

2 \' max/

Princen et al. [188] and Cayias et al. [190] expressed their results as func

tions of a dimensionless shape parameter a, which here becomes

a = (^j^ (3.2.2-34)

While their results can not be readily rearranged in the form given here, the

relation betweenaand rmax/Zimax calculated from (3.2.2-34) and Table 3.2.2-1

agrees with that given by Princen et al. [188].

There are several cautions to be observed in using this technique.

^This table,aminor revision of that presented by Slattery and Chen [21], is

due to Chen

[196],

who gives the computer program as well as a discussion of the

truncation errors.


12/102


1) Prom (3.2.2-3) and (3.2.2-11),

2 V 3 ( p ( B ) _ p ( A ) ) i / 3 g

< 1 (3.2.2-35)

This implies that the angular velocity

O

should be as large as possible. Unfor

tunately, so long as (p^^^ p^^^) i 0, we can approach the rigid body motion

assumed here only asymptotically as

N-QO

^ 0 or as /2> OO. For any finite

value of i7, the axis of the drop or bubble will be slightly displaced from the

axis of the tube, resulting in convective mixing within the two phases

[191].

In addition to negating rigid body motion, these convection patterns have the

potential of creating interfacial tension gradients in the interface leading to a

loss of stability, if the re is a surfac tant pre sen t.

If

fl

is too large, the drop or bubble phase will contact the ends of the

t ube ,

introducing end effects not taken into account in this analysis.

It may be possible to find a compromise by studying 7 as a function of Q

[197].

2) Let us assume that the phases are equilibrated before they are introduced

into the apparatus . As fl begins to increase towards its final value, the drop

lengthens and the area of the drop surface increases. If there is no surfactant

present, the system reaches a steady state rapidly. If there is a surfactant

present, the system approaches a steady state more slowly: as surfactant is

adsorbed in the interface, the interfacial tension falls, the interfacial area of

the drop increases allowing more surfactant to come into the interface,

The problem is obvious, when the phases are not pre-equilibrated [198,

199]. But it could also be a serious problem, if the primary source or surfac

tant is a dilute solution forming the drop phase. If sufficient surfactant were

removed from the drop phase by adsorption in the expanded interface, the

equilibrium compositions of the two phases could be changed.

These may have been the conditions studied by Kovitz

[200].

The drop

phases were dilute solutions of surfactant in octane and the continuous phases

were aqueous solutions of NaCl. After pre-equilibration, the drop phase re

mained the primary source of surfactant. He observed that days or weeks

might be required to achieve an equilibrium interfacial tension with the spin

ning drop experiment compared with minutes or hours in static experiments

such as the sessile drop and pendant drop. Frequently an equilibrium in

terfacial tension could never be measured with the spinning drop interfacial

tensiometer, either because the drop disappeared entirely or because the drop

phase evolved into a shape that was clearly not amenable to data reduction.

(This may have been the result of buoyancy-induced convection becoming

more important as the single-phase region was approached on the phase di

agram

[191].

Lee and Tadros [201] observed interfacial instabilities leading

to disintegration of the interface and the formation of droplets that they


13/102


attributed to interfacial turbulence.) When an equilibrium interfacial tension

was measured, it often differed by an order of magnitude from the values found

with static experiments [meniscal breakoff interfacial tensiometer (Sect. 3.2.3),

du Noiiy ring (Exercise 3.2.3-2), sessile drop (Sect. 3.2.5)]. In contrast, for sin

gle component systems he found excellent agreement between measurements

made with the spinning drop and those made with static experiments.

3) Vib rations shou ld be carefully redu ced, since the y prom ote the developm ent

of instabilities

[191].

4) The spinning tube should be horizontal, in order to prevent migration of

the bubble or drop to one of the ends.

In spite of all of these potential problems, the spinning drop technique has

been used successfully in the measurement of interfacial (and surface) tensions

having a wide range of values.

Wiistneck and Warnheim [202] report excellent agreement between mea

surements made with a du Noiiy ring (Exercise 3.2.3-2) and those made with

a spinning drop for an aqueous sodium dodecylsulphate solution against air

(their Fig. 2). Their surfactant was confined to the continuous phase, elimi

nating any significant change in the equilibrium phase compositions.

In the experiments of Ryden and Albertsson [189] and Torza

[203], signif

icant amounts of surfactant were not present.

3 .2 .3 Menisca l Breakoff Interfac ia l Tens iometer

Figure 3.2.3-1 shows two geometries that may prove to be the basis for a useful

technique for measuring interfacial tension. A circular knife-edge is positioned

so as to just touch the A-B pha se interface. In Fig. 3.2. 3-l(a) , the general level

of th e A-B interface is slowly lowered un til it bre aks away from th e kn ife-edge;

in Fig. 3.2.3-l(b), theA-B interfac e is slowly raise d unt il breakoff

occurs.

From

the measured value

oi H =

-ffmax at

breakoff,

the interfacial tension can be

calculated.

Let us describe the configuration of th e interface in cylindrical co ordina tes

by

r = c{z) (3.2.3-1)

From either the r- or z-component of the jump momentum balance at the

A-B ph ase interface, we have (Table B.1.2-8)

d^c / dc

1

c

dz^

Vd : i + ' d z y

d c V '

(3.2.3-2)


14/102


(a)

gravity

z = 0

(b)

z = -H

z = 0

gravity

Fig. 3.2.3-1.

Meniscal breakoff interfacial tensiometer. (a) When lower phase

pref

erentially wets the knife-edge, the A-B interface is slowly lowered un til breakoff

occurs, (b) When upper phase preferentially wets the knife-edge, the

A-B

interface

is slowly raised until breakoff occurs

Here

po

is th e pressure within phase

A

as the interface is approached in a

region sufficiently iai away from the knife-edge that the curvature of the inter

face is zero;po is the pressure with in pha se B as the interface is approached

in the same region. Equation (3.2.3-2) requires that sufficiently far away from

the knife-edge

at z = 0

Pi^^ Pi^^

(3.2.3-3)

Equation (3.2.3-2) can be more conveniently written in terms of the dimen-

sionless variables

-1/2

(3.2.3-4)


15/102

-1/2

as

{piA)-p{B))g

d^c*

f dc


(3.2.3-5)

z =

dz* dz*

- 1

1 +

d^

dz*

-3/2

(3.2.3-6)

If our objective is to describe the elevated meniscus shown in Fig.

3.2.3-1

(a),

we must solve (3.2.3-6) consistent with the boundary conditions

as z* ^ 0 : c*> oo

at

z* = H* : c* = R*

(3.2.3-7)

(3.2.3-8)

In order to describe the depressed meniscus pictured in Fig.

3.2.3-1

(b),

we

must find a solution for (3.2.3-6) consistent with (3.2.3-7) and

at

z* -H*

R* (3.2.3-9)

Notice that the elevated meniscus problem takes the same form as that for

the depressed meniscus, when

z

is replaced by

z

and

[p^^ p^^^)

by

[p^^^ p^^^)- For this reason, we will confine our atte nti on for th e m om ent

to the elevated meniscus.

Bo th Pa dd ay a nd P it t [204] and K ovitz [205] have considered this p roblem .

Selected members of the family of meniscus profiles as calculated by Kovitz

[205] axe shown in Fig. 3.2.3-2; they are identified by the value of a parameter

Q introduced in his numerical solution. For a selected value of R* (a given

knife-edge diameter and chemical system), the interface can assume a contin

uous sequence of configurations corresponding to increasing the elevation

H*

from 0 to some maximum value -ffmax) beyond which static solutions do not

exist and the interface ruptures.

A sequence of such static configurations for an air-water interface is pre

sented in Fig. 3.2.3-3. As an example. Fig. 3.2.3-4 shows that the profile pic

tured in Fig. 3.2.3-3(f) compares well with Kovitz's numerical computations.

Figure 3.2.3-2 indicates the possibility of two different configurations pass

ing through a given knife-edge denoted by a point (i?*,

h*).

One of these two

configurations always touches the envelope i^âx before contacting the knife-

edge.

Kovitz [205] has never observed such a configuration in his experiments

and believes it to be unstable.

Kovitz [205] provides analytical expressions for various portions of the

envelope curve H^^ = H^^^ (R*) in Fig. 3.2.3-2. For large R*,

H*^.^

= 2-IR*-'-IR*-^ +

0{R*-^

(3.2.3-10)

which may be used with less than 1% error for

R* >

1.9. For 0.2

< R*

i^max?

no

solutions exist

and the

interface

ruptures

D

n = l

where

C i = 0.18080

Ci = 0.29211

C2 = 0.84317

C5 = - 0 . 0 6 2 3 3

C3 = - 0 . 6 9 4 9 4

C s

=

0.00532

For sufBciently small

R*,

R*

=

ê-a+ (1

_

coscj,)-^

iîax = 4 e - ( ^ - ) e - c ^ ( l ^ ) t a n ^

\

1 COS(p J

in which

a = 0 . 5 7 7 2 1 5 7 -

(3.2.3-11)

(3.2.3-12)

(3.2.3-13)

(3.2.3-14)


17/102

3.2 On ly Interfac ial Ten sion 175

F i g . 3 . 2 . 3 - 3 . P ho to gr ap hs by Kov itz [205] of an air -w at er interface for i?* = 1.12.

(a)

H*

= 0.24. Th e knife-edge is below the ai r-w ate r interface, which in tersects

the rod with a f inite contact angle, (b)

H^

= 0.24. Th e air -w ate r interface interse cts

the knife-edge. The contact angle boundary condition is removed, (c)

H*

= 1.20.

(d)

H* =

1.25. (e) if* = 1.30. (f) if* = 1.36. See th e com pari son w ith the or y in

Fig. 3.2.3-4. Breakoff occurred at if* = 1.38

i s t h e E u le r c ons t a n t a n d c o t

(j)

= d c * /dz * . E q ua t io n ( 3 .2 .3 -13 ) c o r r e c t s K o v i t z

[205] Eq . (46) . E qu a t io n (3 .2 .3-13) m us t be so lved s im ul t an eo us ly for va r iou s

values of

0 for

which only one metastable drop configuration exists (G = 3).

H*-G:

-1.05

-0.3-5 0.10

G

I G

=1.90

:0.5

2.35

Fig. 3.2.4-4. Sequence of pendant drops for R* = 1.18 asH* increases [220]

Limiting cases of the inner and outer envelopes in Fig. 3.2.4^2 are derived

by Kovitz

[220],

by Hida and Miura

[221],

and by Hida and Nakanishi

[222].

There are two ways in which the pendant drop or emerging bubble can be

used to measure interfacial tension.

The preferred way of measuring interfacial tension with the pendant drop

is to com pare the experim entally observed drop configuration w ith those pre

dicted theoretically. Tabular numerical solutions of (3.2.4-3) consistent with

(3.2.4-5) and (3.2.4-6) have been presented by a variety of workers. The most

recent tables are those prepared by Hartland and Hartley [223] and by Padday

[224;

these tables are available on microfiches from Kodak Limited, Research

Laboratories, Wealdstone, Harrow, Middlesex, England; see also 212]. Padday

[225,

pp . I l l and 152] summ arizes with comm ents the earlier tables p repared

by Andreas et al .

[226],

by Niederhauser and Bartell [227, p. 114], by Ford-

ham

[228],

by Mills

[229],

and by Stauffer

[230].

In the tables of Hartland and

Hartley

[223],

B

G

(3.2.4-9)

The tables of Padday and Pitt [204, 212] are based upon a rearrangement of

(3.2.4-3)

2+

/3z**

=

2 * *

C

dz**^

+

d^

dz*

+ 1 1 +

dc^

dz*

- 3 / 2

(3.2.4-10)


29/102


where

b

is th e radiu s of curv ature at th e ape x of the drop and

(3.2.4-11)

b

z

0-

b2(^piB) _ piA)> g

(3.2.4-12)

(3.2.4-13)

In arriving at (3.2.4-10), we have noted that at

z** =

0

-2Hb =

c*

2

G

2**

d^c

c

+

z**^ \ dz

dc*

+ 1 1 +

dc^

dz*

-3/2

{piA)_p{B)^g

- 1 / 2

(3.2.4-14)

From (3.2.4-13) and (3.2.4-14), we see th at the sha pe pa ram ete r /3 is negative

for a pendant drop and simply related to G:

0

4

" G 2

(3.2.4-15)

Perhaps the simplest technique is to slowly increase the volume of the drop

until rupture occurs at the intersection of c* = R* with the outer envelope.

Unfortunately, the volume of this drop is not identically equal to the volume

of the drop which breaks away and which would be measured experimentally.

A correc tion m ust b e employed P ad da y [225, p . 131]. Th is is referred t o as

th e drop we ight t echn ique , since experim entally one would mea sure the

weight of the drop which breaks away.

The pendant drop and emerging bubble shown in Fig. 3.2.4-1 may be

thought of as radius-limited, in the sense that the drop configurations must

pass through the knife-edge tip of the capillary. Special cases of this problem

are the contact-angle-limited pendant drop and emerging bubble shown in Fig.

3.2.4-5. Equation (3.2.4-3) must again be solved consistent with boundary

conditions (3.2.4-5) and (3.2.4-6), but boundary condition (3.2.4-7) is now

replaced by a specification of the contact angle at z* = H*. Figure 3.2.4-2

again applies. Given the contact angle between the meniscus and the solid

wall from which the drop is suspended. Fig. 3.2.4-2 can be used to determine

the configuration of the drop for a specified value

H*

G.

Notice that for a

given value of H* G, only one solution is possible. Furthermore, the drop

phase must wet the wall. Multiple solutions are possible as H*G varies (as

the volume of the drop varies).


30/102


(a)

z = H

z H A

gravity

^ +

(b)

gravity

z = -H

Fig. 3.2.4-5. (a) Contact-angle-limited pendant drop in which the density of the

drop is greater than that of the continuous phase, (b) Contact-angle-limited emerg

ing bubble in which the density of the bubble is less than that of the continuous

phase

As in the previous section, the radius of the knife-edge tip of the capil

lary is presumed here to be sufHciently small that the outer solution (outside

the immediate neighborhood of the common line where the effects of mutual

forces such as London-van der Waals forces can be ignored; see Sect. 3.1.1) is

independent of the static contact angle. It is only necessary that the bubble

or drop wet the capillary.

3 .2 .5 Sess i le Drop

Th e captive bu bble a nd sessile drop are shown in Fig.

3.2.5-1.

They are directly

analogous with the contact angle l imited pendant drop and emerging bubble

sketched in Fig. 3.2.4-5. The only difference is that the densities of the two

phases are interchanged. As with the pendant drop and emerging bubble,

interfacial tension can be inferred by comparing their measured profiles with

those expected experimentally.


31/102

3.2 On ly Interfac ial Ten sion 189

(a)

z = H

gravity

(b)

z = -H

gravity

F ig . 3 . 2 .5 - 1 . ( a ) Captive bubble in which densi ty of the drop is less than that of

the continuous phase, (b) Sessile drop in which density of the drop is greater than

that of the continuous phase

Le t us p roceed as we d id in Sec t . 3 .2 .4 to desc r ibe the conf igura t ion of

th e i n t e r f a c e i n c y l ind r i c a l c oo r d ina t e s . Ra th e r t h a n ( 3 .2 .4 - 3 ) , w e ha ve f r om

e i t h e r t h e

r-

o r

z-

c o m p o n e n t o f t h e j u m p m o m e n t u m b a l a n c e a t t h e

A-B

pha se i n t e r f a c e

-G -

* 1

z =

c*

where

G =

(.A) {1

PH -PH

' - (P'

1+

dc^

dz^

-\

- 3 / 2

,(^)-p(^))5^][7(p()-^(^))5]

- 1 / 2

7

1 - 1 / 2

7

(p(B)-piA))g

- 1 - 1 / 2

(3.2.5-1)

(3.2.5-2)

(3.2.5-3)

(3.2.5-4)

I f w e w i sh t o de sc r ibe t he c on f igu r a t i on o f t he c a p t ive bubb le show n in F ig .

3 .2 .5 - l ( a ) , w e m us t so lve ( 3 .2 .5 -1 ) c ons i s t e n t w i th


32/102


at z* = 0 : c* = 0 (3.2.5-5)

0 : ^ ^ 00 (3.2.5-6)

s z

at

z* = H*

c* = 0

dc*

dz*

dc*

dz*

= cot 0('*'**) (3.2.5-7)

in which 0(s*^*) is the static contact angle measured through phase

A.

In order to describe the sessile drop sketched in Fig. 3.2.5-1(b), we must

determine a solution for (3.2.5-1) consistent with (3.2.5-5), (3.2.5-6) and

at z*

-H* :

dc*

dz*

-cot6('*^*)

(3.2.5-8)

The sessile drop problem takes the same form as that for the captive bubble

when z is replaced by z and

[p^^

p^ ^^) by

[p^-^^ p^^^),

which mea ns

that it is not necessary to give separate discussions for these two problems.

In the limit of the large flat captive bubble shown in Fig. 3.2.5-2, (3.2.5-1)

simplifies to

- G - z *

d V

d ^

1+

/dc*Y

-1 - 3 / 2

(3.2.5-9)

If the bubble does not wet the solid surface upon which it rests, then there

z = 0

B

Fig. 3.2.5-2. Large flat captive bubble that does not wet the solid

is an axial position

h

such that

dc*

at z* =

h*

dz

= 0

Since the bubble is relatively flat at its apex, we will require

dc*

dz*

d^c*

as z

0

oo

dz*2

remains finite

(3.2.5-10)

(3.2.5-11)


33/102


This permits us to conclude from (3.2.5-9) that

G = 0 (3.2.5-12)

Finally, we can integrate (3.2.5-9) consistent with (3.2.5-10) and (3.2.5-11) to

find

- 3 / 2

r--/:-(sr

(s)

P^^^)9

(3.2.5-13)

(3.2.5-14)

While this is a simple expression, it is relatively difficult to locate the position

h

of the maximum diameter.

For greater accuracy, it is preferable to consider a smaller bubble and to

compare the experimental and theoretical profiles at several axial positions.

In these cases, we must rely upon the tabular numerical solutions of (3.2.5-1)

consistent with (3.2.5-5) through (3.2.5-7) which are available. The most re

cent tables are those prepare d by Ha rtla nd and H artley [223; see also footnote

a of Sect. 3.2.4] and by Padday^ [224; these tables are available on microfiches

from Kodak Limited, Research Laboratories, Wealdstone, Harrow, Middle

sex, England; see also 212]. Padday [225, pp. 106 and 151] summarizes with

comments the earlier tables prepared by Bashforth and Adams [231] and by

Blaisdell

[232].

The contact angle limited captive bubble and sessile drop are

stable in all configurations as are the analogous contact angle limited pendant

drop and emerging bubble discussed in Sect. 3.2.4.

^These tables were prepared with (3.2.5-1) in the form of (3.2.4-10). Note now

however that

at

z**

= Q:

-2Hb =

= 2

= G

, d_c_

dz**^

H)'"'lh()

- 3 / 2

-1-1/2

(p (S ) -p (A) )3 j

which implies

4

P =

G 2

(3.2.5-15)

(3.2.5-16)

The shape factor /? is positive for the captive bubble-sessile drop but negative for

the pendant drop-emerging bubble.


34/102


Besides their use in determining interfacial tension, the contact angle-

limited captive bub ble and sessile drop ca n also be used to measu re the con tact

angle

[150].

The contact angle limited captive bubble and sessile drop in Fig. 3.2.5-1

may be thought of as special cases of the radius-limited captive bubble and

sessile drop shown in Fig. 3.2.5-3. Equation (3.2.5-1) is again to be solved

consistent with boundary conditions (3.2.5-8) and (3.2.5-9), but boundary

condition (3.2.5-10) is now to be replace by a specification of the bubble

radius

ai z = H.

The radius limited captive bubble is analogous to the radius

limited pendant drop in the sense that not all configurations satisfying these

bo un da ry conditions are stab le [145, 204].

(a)

(b)

z = H

gravity

gravity

z = -H

Fig. 3.2.5-3. (a) Radius limited captive bubble in which density of bubble is less

than that of continuous phase, (b) Radius limited sessile drop in which density of

drop is greater than that of continuous phase

Finally, note that the analysis presented here is for the outer solution

(outside the immediate neighborhood of the common fine, where the effects of

mutual forces such as London-van der Waals forces and electrostatic double-


35/102

3.2 On ly Interfacial Tension 193

laye r forces can be ignored ; see Sec t . 3 .1 .1) . The s ta t ic contac t ang le i s one

t h a t a n e x p e r i m e n t a l i s t m i g h t m e a s u r e w i t h p e r h a p s 1 0 x m a g n i f i c a t io n .

E x e r c i s e 3 . 2 . 5 - 1 .

Rotating menisci

Fi gu re 3.2.5-4 shows an interface formed in

a vertical tube of radius

R

that rota tes with a constant angular veloci ty

O.

T he

contac t ang le measured th rough phase

B

is

0.

z = 0

F i g . 3 . 2 . 5 - 4 .

Interface formed between phases

A

and B in a ver t ica l tube that

rota tes with a constant angular veloci ty

f2

Assume that the configuration of the interface in cylindrical coordinates is de

scribed by z =

h{r).

D e te r mine t ha t

h

can be found by satisfying

1

dr*^ ^ dr* ^ \dr* J ^ \dr* J

- 3 /2

( ^ ( B ) _ ^ ( A )

^

I h*h^-^^r^^^+C

(3.2.5-17)

subject to the boundary condit ions

at r* = 0 :

h* = 0,

^ = 0

'

dr*

at r* = 1

Here

* _ r

' =R'

dh*

dr*

( s i n 0 )

R'

( P ^ ^ ' - P ^ )

(3.2.5-18)

(3.2.5-19)

(3.2.5-20)

Here pg

'

is th e press ure as (2: = 0, r = 0) is app roac hed wit hin pha se

A,

and pg

'

is th e pressure as this sam e point is approach ed w ithin ph ase

B.


36/102

194 3 Ap plication s of th e Differential Balan ces to M om en tum Transfer

This problem was first solved by Wasserman and Slattery [233]. Fur the r numer

ica l resul ts have been presented by Pr incen and Aronson [234], a portion of which

are shown in Fig. 3.2.5-5.

2.4

2.0

1.6

1.2

.8

.4

/i^ 0

.4

.8

1.2

1.6

9 n

1 1 1 1

Q _ ^ ^ . > - < ^

2^^

J\

i Q - ^ / /

3 J / /

1 1 1 1

1 .8 .6 .4 .2 0 .2 .4 .6 .8 1

F i g . 3 . 2 . 5 - 5 .

Computed profiles of rotating menisci from Princen and Aronson [234]

for 6> = 0,

[p^^^ - p^^A gE?h = 1-3389,

and various values of

Q

{R/gf^'^

3 .3 A p p l i c a t i o n s o f O u r E x t e n s i o n of C o n t i n u u m

M e c h a n i c s t o t h e N a n o s c a l e

I n S e c t . 2 . 2 , w e de ve lope d a n e x t e ns ion o f c on t inuum me c ha n ic s t o t he

na n osc a l e ( no t t he mo le c u la r s c a l e ). By na nosc a l e , w e m e a n th e im m e d ia t e

ne i gh bo r ho od o f a ph a se in t e r f a c e . O ur pa r t i c u l a r c onc e r n is w i th t hose p r o b

lem s w he re i t i s no t suff ic ien t to cor re c t for in te rm ole cu la r fo rces f rom an


37/102

3.3 Applications of Our Extension of Continuum Mechanics to the Nanoscale 195

adjacent phase with an interfacial tension or energy as we did in the prior

section.

3 . 3 . 1 S u p e r c r i t i c a l A d s o r p t i o n [ 2 2 ]

Far from a solid-fluid interface on a molecular scale, the density of a gas,

either subcritical or supercritical, approaches the bulk density. Within a few

molecular diameters of the interface, the gas is subjected to intense attractive

intermolecular forces attributable to the solid. The magnitudes of these forces

increase as the distance to the interface decreases. With a subcritical gas or

vapor, condensation occurs. With a supercritical fluid, the density increases

as the interface is approached, becoming similar to that of a liquid. We will

confine our attention here to the case of a supercritical fluid. We explore the

adsorption or densification of thr ee supercritical gases on Gra pho n: argon,

krypton, and methane.

Specovius and Findenegg [235] used a gravimetric method for the deter

mination of surface excess isotherms of supercritical argon and supercritical

methane on Graphon (a graphitized carbon black). Blumel et al. [236] did a

similar study for supercritical krypton on Graphon.

Five analyses of these data starting from a molecular point of view have

been published.

Egorov [237] pre sen ted a microscop ic sta tisti ca l me chan ical the ory of ad

sorption of supercritical fluids. The theory is in excellent agreement with

experimental observations of Specovius and Findenegg [235] and Blumel

et al. [236] for argon and krypton, although the error increases at higher

densit ies as the temperature approaches the cri t ical temperature.

Ra ng araja n et al. [238] developed a mean-field mod el th at superim poses

the fluid-solid potential on a fluid equation of state to predict adsorption

on a flat wall. Their paper shows some predicted adsorption isotherms for

krypton but none for argon or methane.

Sokolowski [239] used the Percus-Y evick e qua tion for th e local den sity of

a gas in contact with a flat solid surface to calculate the adsorption char

acteristics of argon and of methane on graphite. The calculations used the

Boltzmann-averaged potential

[240],

which is calculated from the pa rtic le-

graphite basal-plane potential

[241],

for the gas-surface interaction and

used the Lennard-Jones (12,6) potential for the gas-gas interactions.

Fischer [242] used a mod el for high tem pe rat ur e and high pressure adsorp

tion t h at was the sam e as the one introduced in previous pa pers [243, 244]

for moderate pressures. He assumed that a hard sphere gas was in con

tact with a structureless plane wall and that a Lennard-Jones potential

described the wall-particle interaction.

Aranovich and Donohue [245] used two adjustab le para m ete rs in com par

ing the Ono and Kondo [49] theory to the data reported by Specovius and

Findenegg

[235].


38/102


The first four of these analyses will be discussed in more detail later.

There have been a number of studies of supercritical adsorption on acti

vated carbo ns, both theore tical and experim ental [246-253]. W hile activate d

carbons are of immense practical importance, they are not well-suited to test

theories, because they are porous and the effects of intermolecular forces are

more complex. For this reason, they will not be discussed here.

In Sect. 2.2, we introduced a new extension of continuum mechanics to

the nanoscale. In Sect. 2.2.2 we tested this theory in predicting the surface

tension for the n-alkanes using one adjustable parameter. What follows is the

first test of the theory using no adjustable parameters.

Referring to Fig. 2.2.1 -l(a), let phase A be a static sup ercritical fluid (/ )

and phase B an impermeable crystalline carbon solid (s). Our objective is to

determine the density distribution within the static fluid and therefore the

apparent adsorption of the fluid on the crystalline carbon solid.

This analysis will be done in the context of view (iv) of Sect.

2.2.1,

but ,

because we are interested in only the gas phase, we will have no need for the

jump momentum balance. Start ing with the differential momentum balance

(2.2.1-1),

we will neglect any effects of gravity to write for the fluid

_ V P ( / )

+

b(/'


39/102


The Lennard-Jones (6-12) potential (2.2.0-1) is commonly recommended

for non-polar dilute gases. In this case,

f( /. /) = - V (n(^)'


40/102


Here (dP^-'^^/dc^-''))^ will be ev alua ted using a n em pirical eq uatio ns of stat e for

argon

[257],

for krypton

[258],

an d for m et ha ne [259, 260]; AT is A vog adro 's

number;

c^^' = n'^^'/N

is th e molar density in th e interfacial region. The

solution of this differential equation with the boundary condition

a s 2 > 00 :

c(/) ^ c^f:^-^^) (3.3.1-9)

gives a density distribution.

The surface excess or the apparent adsorption

r ( - )

= r

(c (/) - c(/' '' ^))

Az

(3.3.1-10)

Since

5

is not a known para m eter , it m ust be defined. Prom (3.3.1-4), (3.3.1-6),

and (3.3.1-7), there are two values of

z

at which ^'^f^'^'^'^)

=

0 and c'-^'>

=

^(/,buik) Qj g is as z ^ oo; th e oth er we will define to be

2;

= 8. The graphi te

is assumed to be impermeable to the fluid.

Before comparing (3.3.1-10) with experimental data, let us consider the

sources of experimental error. For graphitized carbon blacks, the uncertainty

in the measured surface area is approximately 10% [235, 254], which trans

lates to 10% uncertainty in the surface excess mass. Specovius and Findenegg

[235] and Blumel et al. [236] found that the maximum relative error in their

buoyancy correction to be 3% and 4% respectively, the values of which could

be expected to increase as the critical point was approached. Other errors

such as base-line drift were rep orted t o be < 1 % . Preclu ding any ran do m or

systematic error, the uncertainty of the results should be less than 15%.

In the comparisons of (3.3.1-10) with experimental data in Figs. 3.3.1-1

through 3.3.1-3, we would like to emphasize that no adjustable parameters

have been used. Equation (3.3.1-10) describes these data within the uncer

tainty range for argon and krypton as well as for methane at 50C.

It is clear that, for both the krypton and for the methane, (3.3.1-10) under

estimates th e dat a, th e error becoming progressively larger as the tem pe ratu re

decreases. There are at least two possible explanations.

As noted earlier, Steele [254, p. 52] expressed some doubt about the as

sumption of additivity of intermolecular forces in using his semi-empirical

extension of the Lennard-Jones potential to the interactions between a crys

talline solid and a gas. He proposed that this problem could be at least par

tially avoided by using a more realistic description of the solid. For this reason,

he described the solid as being discretely rather than continuously distributed

in space. While this proposal has not been previously tested experimentally

to our knowledge, we believe that the excellent agreement between the predic

tions of our theory and the experimental observations for argon does support

the use of pairwise additivity, at least when temperature is sufficiently above


41/102

3.3 Applications of Our Extension of Continuum M echanics to the Nanoscale

7 :

199

(a)

(b)

6

5

4

^ 3

2

1

/

/

f

^

^

^ - ^ '

^ - ^ '

/

1 2 3 4 5

T

6

25C

Fig. 3.3.1-1. r (^mol/m^) for argon on Graphon predicted by (3.3.1-10) (the

solid curves) (a) as a function of

P

(MPa) and (b) as a function of c (mol/L). The

experimental observations are from Specovius and Findenegg

[235].

The dashed and

dashed-double dot curves in (a) represent the computations of Sokolowski [239] for

0C and the computations of Fischer [242] for 50C, respectively. The dashed-dot

curve in (b) indicates the computations of Egorov [237] for 25C

the cri t ical temperature Tc- Notice that for argon Tc = 150.8 K. For kry pto n,

Tc =

209.4 K, a nd we have excellent agreem ent between our prediction s and

th e expe rime ntal observations at 100C, bu t errors increase as th e t em pe rat ur e

decreases, the density increases, and the interference of neighboring of kryp

ton increases. In summary, we believe that pairwise additivity can be used

as suggested by Steele [254, p. 52], so long as the temperature is sufficiently

above the cri t ical temperature.

The poor agreement between our predictions and the experimental obser

vations for methane is due in part to the same failure of pairwise additivity as

th e critical tem pe ra tur e Tc = 190.4 K is appro ached . Bu t the errors are larger


42/102


,

i

*^ ,.

10

(a) r^

(b) r'

^ ^ ^ _

25 C

Fig. 3.3.1-2. r (nmol/m^) for krypton on Graphon predicted by (3.3.1-10) (the

solid curves) (a) as a function of

P

(MPa) and (b) as a function of c (mol/L). The

experimental observations are from Blumel et al.

[236].

The dashed curve in (a)

represents the computations of Rangarajan et al. [238] for0C;the dashed-dot curve

for 50C; the dashed-double dot curve for 100C. The dashed curve in (b) indicates

the computations of Egorov [237] for 25C

than those seen with krypton, which has a higher Tc- We do see the errors

decrease as the temperature is raised, but we don't have measurements at

hight temperatures. However, we aje also concerned that the Lennard-Jones

potential may not be appropriate, particularly in denser fluids, because the

methane molecule is not spherical.

As mentioned in the introduction, there are four analysis based upon sta

tistical mechanics which have been compared with these same data: Egorov

[237],

Rangarajan et al .

[238],

Sokolowski

[239],

and Fischer

[242].

In all cases.


43/102


per

Fig. 3.3.1-3.

r ^

(|i.mol/m^) as a function of

P

(MPa) for methane on Graphon

predicted by (3.3.1-10) (the solid curves). The experimental observations are from

Specovius and Findenegg

[235].

The dashed curve represents the computations of

Sokolowski [239] for0C;the dashed-dot curve for 25C; the dashed-double do t curve

for 50C

we have used their results from their figures without repeating their compu

ta t ions .

Figure 3 .3.1-l(b) com pares the co mp utatio ns of Egorov [237] with (3.3.1-10)

for the argon data of Specovius and Findenegg [235] at 25C; Figure 3.3.1-2(b)

compares the two theories for the krypton data of Blumel et al.

[236].

The

computations of Egorov [237] are excellent, although, as he notes, they begin

to fail at higher densities. Unfortunately, he did not report computations for

higher temperatures.

Figure 3.3.1-2(a) compares the model of Rangarajan et al. [238] with

(3.3.1-10) for the krypton data of Blumel et al.

[236].

Figures 3.3.1-l(a) and

3.3.1-3 compare the Sokolowski [239] model with both (3.3.1-10) and the argon

and methane data of Specovius and Findenegg

[235].

Fischer [242] presented

a plot of his results compared with the data of Specovius and Findenegg [235]

for argon at only one temperature 50C. Figure 3.3.1-l(a) compares his results

with (3.3.1-10). In all cases, (3.3.1-10) was superior.

Our theory does not do as well in describing the supercritical adsorption

of krypton or of methane as the critical point is approached. We do not feel

that this indicates a failure or limitation of the theory, but rather a limitation

in th e way th a t t h e theo ry w as executed . Following Steele [241, 254, 255],

we have assumed the Lennard-Jones potential can be used to describe the

carbon-fluid as well as the fluid-fluid interactions, and we have assumed that

long-range intermolecular forces are pairwise additive.

We believe that the poorer agreement between our predictions and the ex

perimental observations for krypton at lower temperatures is at tr ibutable to

the assumption of pairwise additivity beginning to fail. The critical tempera-


44/102


ture for argon

Tc =

150.8 K; for krypton,

T^ =

209.4 K. Since the assumption

of pairwise additivity would begin to fail

as T

>

Tc

and the fluid became

denser, we could expect to see these effects at higher temperatures with kryp

ton than with argon.

The poor agreement between our predictions and experimental observar

tions for methane is certainly due in large measure to this same failure of the

assumption of pairwise additivity. For methane, Tc = 190.4 K, an d t h e er

ror in our predictions decreases as the temperature is increased. However, we

are also concerned that the Lennard-Jones potential may not be appropriate,

particularly in the denser fluids as Tc is approached, because the methane

molecule is not spherical.

Steele [254, p. 52] clearly recognized that pairwise additivity of intermolec-

ular forces could not be justified in condensed m aterials . He propos ed th at this

problem could be at least partially avoided by using a more realistic descrip

tion of the solid, one in which the crystalline solid was distributed discretely

rather than continuously in space. We believe that the excellent agreement

between the predictions of our theory and the experimental observations for

argon and krypton , particularly at higher tempe ratur es, supp orts his proposal.

We wish to emphasize that the use of pairwise additivity is not an inherent

limitatio n in the use of this theory. For exam ple, in describing the interactio ns

between two condensed media, the limitations of pairwise additivity can be

avoided by describing the two-point interactions using an effective, Lifshitz

t ype ,

Hamaier constant that is both screened and retarded

[170].

3 .3 .2 Stat ic Contact Angle [20]

It has been recognized for a long time that the molecular interactions play

an important role within the immediate neighborhood of a three-phase line of

contac t or comm on line where the meniscal film is very thin. T he configuration

of the fluid-fluid interface in the common line region and static contact angle

depend on the correction for intermolecular forces (see Sect. 2.2).

Rayleigh [261] recognized that, in two dimensions sufHciently far away

from the common line, the fluid-fluid interface approaches a plane inclined

at a angle 0(*^*) to the solid surface. As the common line is approached,

the curvature of the interface is determined by a force balance to form a

different angle 0o with the solid surface. More recently, the configuration of

the interface near the common line has been investigated in the context of

statistical mechanics [262, 263] and by molecular dynamics simulations

[264].

See Dussan V [115] for a further review.

This problem has been discussed theoretically by many investigators [124,

162, 163, 180, 181, 265-270]

Figure 3.3.2-1 shows in two dimensions the three-phase line of contact or

common line formed at equilibrium between phases A, C, and a solid B. The

static contact angle 0(^*^*) might be measured by an experimentalist at some

distance from the common line, using perhaps 10x ma gnification.


45/102


Fig. 3.3.2-1. A thin discontinuous film

C

forms a common line

Our objective is to determine 0(stat)^ which we will distinguish from 0O)

the contact angle at the common line introduced in the context of Young's

equation (2.1.10-3). We will focus on the thin (precursor) film (Sect.

1.2.9)

formed in the neighborhood of the true common line, and we will employ the

correction for intermolecular forces discussed in Sect. 2.2.5.

We will make three assumptions.

i) The solid is rigid, and its surface is smooth and planar. Because the effect

of gravity will be ignored with respect to the correction for intermolecular

forces, its orientation with respect to gravity is arbitrary.

ii) The system is static, and equilibrium has been established. There are no

interfacial tension gradients developed in the system.

iii) In discussing the thin film we will adopt view (v) of Sect.

2.2.1.

The

corresponding correction for intermolecular forces from adjoining phases will

be taken from Sect. 2.2.4.

Referring to Fig. 3.3.2-1 and Table B.1.2-6, our initial objective will be to

determine the configuration of the A-C interface

Z3 = /l (Z i)

assuming that the C B interface is

Z3 = 0

(3.3.2-1)

(3.3.2-2)

Both the z\ and Z3 comp onents of the jum p m om entum balance for the AC

interface reduces to

a t z3 = / i : p ( ^ ) - p ( ^ ) = 7

dzi2

1+

( ^ ) '

- 3 / 2

(3.3.2-3)

Here

p'^^^

and

p^^^

are the pressures in these phases evaluated at the

AC

in

terface. In view of ass um ptio n iii, we conclud e from th e differential m om en tum

balance (2.2.1-1) that


46/102


p ( ^ ) = p ( ^ ) +


47/102


atz^ = Q: h* = 1

dh*

at 2; = 0 : -p- r = ta nOQ

dz*

where

&o

is the conta ct angle at the t ru e com mon Une.

Noting that

d^h*

_ dh* d^h* dz^

dzf ~~d4dzfdh*

_ l_d_ /dh*V dzt

~2dzf \d^J dh*

_ l_d_ /dfe*^^

~

2dh* \dzt^

(3.3.2-12)

(3.3.2-13)

(3.3.2-14)

we may write (3.3.2-11) as

h*^

~ 2dh* \dz*) ^ V d ^ y

-3 /2

(3.3.2-15)

Integrating this once consistent with (3.3.2-12) and (3.3.2-13),we have

-1 /2

( l - F t a n ^ ^ o )

-1 /2

1 +

dh*

dz*

T\h*

- 1

In the limits

a s 2 ;* > 0 0 : / i *

>

CX3

and

dzi

Eq. (3.3.2-16) further reduces to

(1 + tan^ 0o ) ' ^ ' - (1 + tan^ (^t^*))

1/2 ^

B*

or

| c O S 0n |

cos

0(s ta t )

= T-

127r7


48/102


| co s0o |

COS

0(s ta t )

= T-

127r7(5(sc')^

(3.3.2-21)

The angle

OQ

at the true common line should be determined using Young's

equation (2.1.10-3)

7(^^ ) cos 00 =

7 ^^^

-

7^-^^^

After some arrangement, we rearrange (3.3.2-21) as

cos

+

3.3.2-22)

3.3.2-23)

The Hamaker constants

A^^*^^

and yl^^^^ were calculated by Hough an d

W hi t e

[162],

and

j(-^'^)

were measured by Fox and Zisman

[271].

Girifalco

an d G ood [272] (see also A dam son [273, p . 360]) prop ose d th a t 7(^C') could

be estimated as

^ ( B C ) o o ^ f^^{AC)oo _ y ' ^ ( A B ) > y

7

{AB)oo _^_

^ ( ^ C ) o o

2 ( ^ ( A B ) o o ^ ( A C ) o o - j l / 2

3.3.2-24)

It is important to recognize that the n-alkanes are volatile liquids and

that the gas phase is dependent upon the specific n-alkane being considered.

Fox and Zisman [271] observed that there is no difference between contact

angles measured in saturated air and those measured in unsaturated air . The

conclusion is that 7(^^) is independent of the particular n-alkane being

considered. Prom (3.3.2-22) and (3.3.2-24), we conclude that

^{AC)oo^^^Q^^^{AB)oc

iAB)oo _^ ^ ( A C ) o o

2(^{AB)oo^{AC}oo\

1/2-

3.3.2-25)

or

cos 00

-1 + 2

ry{AB)00

3.3.2-26)

As the alkane number n is decreased, there is a limiting value at which spon

taneous wetting occurs, and 0o = 0(^*' '*) = 0 and 7(^^) =

r^(AC)oo_ ^^

.j jg

way, Zisman [274, p. 190] (see also Zisman [160, p. 20]) concluded that

^(AB)oo = i 8 .5n iN /m

3.3.2-27)

Working with different liquids on PTFE, Owens and Wendt [275] came to the

same conclusion.

In their analysis of supercritical adsorption of argon and krypton on im

permeable carbon spheres, Fu et al. [22] chose 6 as the position at which


49/102


attractive and repulsive forces between the carbon and the gas, as described

by a two-point Leonard-Jones potential, were balanced and


50/102

208 3 Ap plicatio ns of th e DifTerential Balan ces to M om en tum Transfer

a n d a pp l i e d a t t h e a p pa r e n t c o m m on l i ne ( a s s e e n w i th 10x m a gn i f i c a t i on ) .

T a b le s 3 .3 .2 - 1 a nd 3 .3 .2 - 2 de mons t r a t e t ha t ( 3 . 3 .2 - 29 ) doe s no t r e p r e se n t t he

e xpe r ime n ta l me a su r e me n t s o f 0 ( ^ *^ * ) w e l l .

S l a t te r y [108 , p . 169] de r iv es Y ou ng 's eq ua t io n as (3 .3 .2-22) , an d app l ie s

i t a t t h e t r ue c o m m on l i ne a s w e do he r e . E qu a t io n ( 3 .3 .2 - 23 ) , w h ic h w a s

d e v e l o p e d u s i n g ( 3 . 3 .2 - 2 2 ), r e p r e s e n t s t h e e x p e r i m e n t a l d a t a i n t h e a b o v e

ta b l e s mu c h be t t e r t h a n do e s ( 3 .3 .2 -29 ) . O ur c onc lu s ion is t h a t ( 3 .3 .2 - 22) i s

t h e p r e f e r r e d f o r m of Y o ung ' s e q ua t ion .

T a b l e 3 . 3 . 2 - 2 . Comparison of ca lcula t ion and exper imenta l da ta of

static contact angle 0^^*^' ' for dispersive liquids on PDMS

liquids

d i iodomethane

bromonapha lene

methylnaphtha lene

ie r t -bu ty l naphtha lene

liq. paraffin

Hexadecane

^{AC)oa

m N / m

50.8

AAA

39.8

33.7

32.4

27.6

^Csc)

A

3.178

2.497

2.378

3.384

3.139

3.044

^(BC') fc

10-2 J

5.57

5.05

4.80

4.76

5.15

4.80

^ ( C C ) ' '

10-2 J

7.18

5.93

5.35

5.23

6.03

5.22

0(cal)'=

Deg.

71.9

66.4

61.3

52.5

50.2

38.6

Q(CB.\)d-

Deg.

66.8

61.0

50.5

50.1

44.5

34.4

@(exp)

Deg.

70.0^

62.0^

52.0^

49.0

40.0^

36.09

Calculated by (3.3.2-28).

' ' P rov ided by Drummond and Chan

[279].

'^ Calcula ted using a common form of Young's equat ion (3.3.2-29) .

C alc ula ted by (3.3.2-23 ).

^ Measured by Chau dhury and W hi tes ides [276].

' ' Measured by Baler and Meyer [277].

9 Measured by Bowers and Zisman

[278].

T hi s i s pe rh ap s th e fi rs t t e s t o f th e form of Yo un g 's equ a t io n (3 .3 .2-22)

de r ive d by S l a t t e r y [ 108 , p . 169 ] f or t h e t r ue c om m on l i ne . C om m on ly Y o ung ' s

e q u a t i o n i s a p p l i e d a t t h e a p p a r e n t c o m m o n l in e w i t h 0 o ( a s o b s e r v e d a t t h e

t r u e c om m on l ine ) r e p l a c e d by 0(^*^*) ( a s obse r v e d a t t h e a ppa r e n t c o m m on

l ine a s s e e n by a n e xp e r im e n ta l i s t w i th , s a y , 10 x m a gn i f i c a t i on ) .

3 . 3 . 3 A R e v i e w o f C o a l e s c e n c e ( w i t h J . D . C h e n )

T he r a t e a t w h ic h d r ops o r bubb le s , su spe nde d in a l i qu id , c oa l e sc e i s impor

t a n t t o t h e p r e p a r a t i o n a n d s t a b i l i t y of e m u l s ions , of f oa ms , a n d of d i spe r s ions ;

t o l i qu id - l i q u id e x t r a c t i on s ; t o t h e f o r m a t ion o f a n o il ba nk du r in g th e d i s

pl ac em en t of oi l f rom a rese rvo ir rock ; to m in er a l f lota tion.

O n a sma l l e r s c a l e , w he n tw o d r ops ( o r bubb le s ) a r e f o r c e d to a pp r oa c h one

a n o t he r i n a l i qu id pha s e o r w he n a d r op i s d r ive n th r o ug h a li qu id pha s e t o a

so l id sur face , a th in l iqu id f i lm forms be tween the two in te r faces and begins to

d r a in . A s t he t h i c kne s s o f t he d r a in ing f i lm be c ome s su f f i c i e n t ly sma l l ( a bou t

100 n m ) , th e ef fec ts o f th e cor rec t ion for in t e rm ole cu la r fo rces (Sec t . 2 .2 ) an d of

e l e c t r o s t a t i c doub le - l a ye r f o r c e s be c ome s ign i f i c a n t . D e pe nd ing upon the s ign


51/102


and the magnitude of the disjoining pressure attr ibutable to the correction

for intermolecular forces and the repulsive force of electrostatic double-layer

forces, there may be a critical thickness at which the film becomes unstable,

ruptures and coalescence occurs. (See Exercise 2.2.4-1 for the definition of the

disjoining pressure.)

In examining prior studies, it is helpful to consider separately those per

tinent to the early stage of thinning in which the effects of any disjoining

pressure are negligible and those describing the latter stage of thinning in

which the disjoining pressure may be controlling. Since the literature is ex

tensive, we will also separate studies of coalescence on solid planes from those

at fluid-fluid interfaces. (Comprehensive reviews of thin films are given by

Kitchener

[280],

Sheludko

[281],

Clunie et al.

[282],

Buscall and Ottewill

[283],

and Ivanov

[284].

Reviews concerned with the drainage and stability of thin

liquid films are given by Liem and Woods

[285],

Ivanov and Jain

[286],

Jain

et al.

[287],

Ivanov

[288],

Ivanov and Dimitrov

[289],

and Maldarelli and Jain

[290].)

In i t ia l s tage of th inning: a t a so l id p lane For th e mo m ent, let us

confine our attention to the initial stage of thinning in which the effects of the

correction for intermolecular forces and of electrostatic forces can be neglected.

When a drop or bubble approaches a solid plane, the thin liquid film

formed is dimpled as the result of the radial pressure distribution developed

[291-297].

It is only in the limit as the approach velocity tends to zero that

the dimple disappears.

Prankel and Mysels [298] have developed approximate expressions for the

thickness of the dimpled film at the center and at the rim or barrier ring as

functions of time. Their expression for the thinning rate at the rim is nearly

equal to that predicted by the simple analysis of Reynolds [299] for two plane

parallel discs and it is in reasonable ag reemen t w ith some of th e me asure m ents

of Platikanov [294, 300, 301]. Their predicted thinning rate at the center of

the film is lower than that seen experimentally [294, 300, 301]. However, it

should be kept in mind that their result contains an adjustable parameter (an

initial time) that has no physical significance.

Hartland [295] developed a more detailed model for the evolution of the

thinning fi lm assuming that the shape of the drop beyond the rim did not

change with time and that the configuration of the fiuid-fluid interface at the

center was a spherical cap. He proposed that the initial film profile be taken

directly from experimental data.

H ar tla nd an d Ro binso n [302] assum ed th a t the fluid-fluid interface co nsists

of two parabolas, th e radius of curvatu re at the apex varying with t ime in th e

central parabola and a constant in the peripheral parabola. A priori knowledge

was required of the radial position outside the dimple rim at which the film

pressure equaled the hydrostatic pressure.

For the case of a small spherical drop or bubble approaching a solid plane,

the development of Lin and Slattery [300] is an improvement over those of


52/102


Hartland [295] and of Hartland and Robinson

[302].

Since the initial and

bo un da ry con ditions are more comp lete, less a priori expe rime ntal information

is required. The results are in reasonable agreement with Platikanov [294]'s

data for the early stage of thinning in which any effects of the correction for

intermolecular forces and of electrostatic forces can generally be neglected.

(Since the theory is for small spherical drops, it cannot be compared with the

data of Hartland [295] for large drops.)

Initial stage of thinning: at a fluidfluid interface Let us continue

to confine our attention to the initial stage of thinning in which the effect

of the correction for intermoleculax forces and of electrostatic forces can be

neglected.

As a drop (or bubble) is forced to approach a fluid-fluid interface, the

minimum film thickness is initially at the center. As thinning proceeds, the

minimum film thickness moves to the rim or barrier ring [303-305] and a

dimpled film is formed. Allan et al. [303] found that the dimpling developed

wh en th e film was 0.3 to 1.2 jim thick. T he rim ra diu s is a function of tim e

[303-305].

Princen [306] extended the Prankel and Mysels [298] theory to estimate

the thinning rate both at the center and at the rim as a small drop (or bubble)

approaches a fluid-fluid interface. His prediction for the thinning rate at the

rim is again nearly eq ual to th at given by the simple analysis of Reyn olds

[299].

But, just as in the Prankel and Mysels [298] theory, there is an adjustable

parameter (an initial time) that has no physical significance.

Ha rtlan d [307] developed a more de tailed analysis to predict film thickness

as a function b ot h of tim e and of radial position. He assum ed th at bo th fluid-

fluid interfaces were equidistant from a spherical equilibrium surface at all

t imes and that the fi lm shape immediately outside the rim was independent

of time. The initial film profile had to be given experimentally.

Lin and Slattery [23] developed a more complete hydrodynamic theory for

th e thinn ing of a liquid film between a small, nearly spherical drop an d a fluid-

fluid interface that included an a priori estimate of the initial proflle. Their

theory is in reasonable agreement with data of Woods and Burrill

[308],

B urrill

and Woods

[309],

and Liem and Woods [310] for the early stage of thinning in

which any effects of London-van der Waals forces and of electrostatic forces

generally can be neglected. (Because their theory assumes that the drop is

small and nearly spherical, it could not be comp ared w ith Ha rtla nd [311, 312,

313] data for larger drops.)

Barber and Hartland [314] included the effects of the interfacial viscosities

in describing the thinning of a liquid film bounded by partially mobile parallel

planes. The impact of their argument is diminished by the manner in which

they combine the effects of the interfacial viscosities with that of an interfacial

tension gradient.


53/102


Latter s tage of th inning: a t a so l id p lane Exp erim entally we find

that, when a small bubble is pressed against a solid plane, the dimpled hq-

uid film formed may either rupture at the rim [293, 296] or become a flat,

(app arently ) stable film [291, 293, 315-321].

When the thickness of the draining film becomes sufficiently small (about

1000

A),

the effects of the London-van der Waals forces and of electrostatic

double-layer forces become significant. Depending upon the properties of the

system and the thickness of the film on the solid, the London-van der Waals

forces can contribute either a positive or negative component to the disjoining

pressure [174, 321 , 322]. An electros tatic dou ble layer also ma y con tribu te

either a positive or negative component to the disjoining pressure. A positive

disjoining pressure will slow the rate of thinning at the rim and stabilize the

thinning film. A negative disjoining pressure will enhance the rate of thinning

at the rim and destabilize the film.

When London-van der Waals forces were dominant, with t ime the fi lm

either ruptures or becomes flat. Platikanov [294] and Blake [321] showed ex

perimentally that a positive London-van der Waals disjoining pressure can be

responsible for the formation of a flat, equilibrium wetting film on a solid sur

face. Schulze [323] found that, for an aqueous film of dodecylamine between a

bub ble and a qua rtz surface a t a high KCl concen tration (0.01 to 0.1 N) , the

eff'ects of the electrostatic double-layer forces are negligible compared with

those of the London-van der Waals forces. If the electrostatic double-layer

disjoining pressure is positive, the wetting film thickness decreases with in

creasing electrolyte concentration. Read and Kitchener [316] showed that for

a dilute K Cl aqueou s film (< 10~^ N) on a silica surface thicker th an 300 A,

the electrostatic double-layer forces dominate. As the KCl concentration in

creases from 2 X 10~^ to 10~^ N, they found th at th e w etting film thickness

decreases from 1200 to 800 A due to th e decrease in the e lectrostatic double-

layer repulsion. Similarly, Schulze and Cichos [320] found that the thickness of

the aqueous wetting film between a bubble and a quartz plate decreases with

increasing KCl concentration in the aqueous film. The wetting fllm thickness

varied from almost 500 A in 10 ^ N K Cl solution to abo ut 100 A in 0.1 N KCl

solution. The y concluded t ha t, for film thickness greater t ha n 200 A, the con

tribution of London-van der Waals forces to the disjoining pressure could be

neglected in aqueous electrolyte solutions of less th an 0.01 N conc entration .

A positive contribution to the disjoining pressure by electrostatic double-

layer forces can be changed to a negative one by the adsorption in one of the

interfaces of an ion having the opposite charge. For example, a stable aqueous

wetting film may be formed in the absence of an ionic surfactant, but the film

may rupture in its presence [323-325].

When the contributions to the disjoining pressure of both the London-

van der Waals forces and the electrostatic double-layer forces are negative,

the dimpled film always ruptures, and the rupture thickness decreases with

increasing elec trolyte c onc entr atio n [296, 318, 323].


54/102


W he n these two com pone nts to th e disjoining pressure have different signs,

th e dimpled film will either ru ptu re or become flat, depe nding u po n which con

tribution is dominant. Schulze and Cichos [324] explained that the existence

of a 0.1 N KC l aque ous film on a q ua rtz surface is du e to th e ex istence of a

positive London-van der Waals disjoining pressure. They also explained the

ru pt ur e cause d by t he ad dit ion of AICI3 is due t o th e a dso rptio n of Al *

ions in the quartz surface, resulting in a negative electrostatic double-layer

contribution to the disjoining pressure.

Because of mathematical difficulties, especially when using the dimpled

film model, only London-van der Waals forces have been taken into account

in most analyses of the stability of thin liquid films on solid planes. Only Jain

and Ruckenstein [326] and Chen [327] have included the effect of electrostatic

double-layer forces in their disjoining pressure. Chen [327] has discussed the

interaction of positive and negative components to the disjoining pressure on

the dynamics of a dimpled film. The numerical results are consistent with the

experimental observations mentioned above.

Jain and Ruckenstein [326, 328] studied the stability of an unbounded,

stagnant, plane, liquid film. Jain and Ivanov [329] considered a ring-shaped

film. Their results suggest that the critical thickness decreases as the rim

radius increases and as the width of the ring decreases. Blake and Kitchener

[318] found experimentally that films of smaller diameter were more stable t

applications of the differential balances to momentum transfer

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