Characterization of the wettability of solid particles by film flotation 2. Theoretical analysis

145

ColEoids und Surfaces, 60 (1991) 145-169 Elsevier Science Publishers B.V., Amsterdam

Characterization of the wettability film flotation 2. Theoretical analysis

of solid particles by

Jianh Diao and D.W. Fuerstenau Department of Materials Science and Mineral Engineering, Berkeley, CA 94720, USA

Uniuersity of Califurniu,

(Received 17 December 1990; accepted 22 April 1991)

Abstract This paper presents a model of the fi’im flotation process, which is a useful technique for characterizing the wetting behavior of particulates, especially those with heterogeneous surfaces. Two different approaches were used, one involving the total and the other the differential free energy for the process of transferring particles from a liquid/ vapor interface into the liquid phase in film flotation. Analysis shows that interfacial anergy has dominant control over the behavior of particles in film flotation over a wide range of particle size and solid density. This study confirms theoretically that the critical wetting surface tension of a particle is equal to the value of the surface tension of a wetting liquid at which particles are just imbibed into the wetting liquid.

INTRODUCTION

Film flotation is a technique developed in our laboratories to characterize the wetting behavior of an assembly of particles, particularly those having heterogeneous surfaces [l-3].Film flotation has proved to be very useful for characteri?iag coal particles because it yields statistical information on the distribuSon of surface hydrophobicity and surface heterogeneity of coal particles and offers a means for correlating surface properties with the behavior of coal particles in various kinds of processes 13-53. In a preceding paper [6], we presented experimental results of how various factors affect film flotation response. By using film flotation and contact angle measurements to study the wetting behavior of such homogeneous hydrophobic particles as elemental sulfur, silver iodide and methylated glass/quartz, we verified that particles are imbibed into the wetting liquid during film flotation only when the critical wetting surface ten&i >n of the particles is equal to or higher than the surface 0166-6622/91/$03.50

0

1991 Elsevier

Science Publishers

B.V. All rights reserved.

146

tension of the wetting liquid. Our results showed that fi.!m flotation is sensitive primarily to the hydrophobicity and heterogeneity of particles and that such factors as particle size and particle density have a negligible effect on the results over a fairly wide range. In order to explain these findings and t-0 gain further insight into phenomena underlying the film flotation technique, mathematical modeling of the process was carried out, using spherical geometry for simplicity. The results presented in this paper show which properties of particles control their behavior in film flotation and why the results of film flotation can be used to evaluate the critical wetting surface tension of solid particles. DESCRIPTION

OF FILM FLOTATION

Film flotation is an experimental technique designed ~1. :lineate the surface tension of a liquid that will just wet a solid particle. In conducting a film flotation experiment, closely-sized particles (for example 100 x 150 pm) are sprinkled onto the surface of the wetting liquid (such as an aqueous methanol solution) and the fraction of particles that sink into the liquid is determined. Depending on the wetting characteristics of the material and the surface tension of the test liquid, the particles either remain at the liquid/vapor interface or are immediately engulfed into the liquid. At a particular surface tension, those particles that do not sink into the wetting liquid are considered to be lyophobic or hydrophobic (non-wetted), while those that are imbibed into the liquid are lyophilic or hydrophilic (wetted). After performing a film flotation test, the lyophobic and lyophilic fractions are recovered, dried, and weighed. The percentage by weight of the particles that refer to the lyophobic fraction for each solution can then be plotted versus the corresponding surface tension in order to construct the cumulative distribution of wetting surface tension for the sample. From this distribution curve, four parameters for defining the wetting characteristics of the particulate samples can be determined: the mean critical wetting surface tension jj,, the minimum and the maximum wetting surface tension, and the standard deviation of the wetting surface tension [3,5]. In film flotation, four states are involved when a particle is transferred from the vapor phase into the liquid phase, as shown schematically in Fig. 1. These four states can be defined as foJ.lows: the parti& :e released in the vapor phase over the State I liquid surface from a distance h. State ?I the particle reaches the liquid surface after attaining some maximum velocity before it is wet.

Particle

Vapor

Phase

-!L.Eila STATE

II

STATE

R

7

Liquid

Phase

III

0

STATE

IV

Fig. 1. Schematic representzkion of the four states involved in film flotation transfer of a particle from the vapor phase into the liquid phase.

for the

the liquid wets the particle and dissipates its kinetic energy, forming a three-phase wetting line. State IV the particle is completely wetted and sinks into the liquid. For a single film flotation run, separation is achieved between particles that sink into the liquid and those that remain on the surface of the liquid. The portion of the particles that sink is defined as the lyophilic (hydrophilic) fraction and that which floats as the lyophobic (hydrophobic) fraction. In terms of the four steps outlined above, a particle is considered lyophilic if it passes through all four states, that is I-IV. A lyophobic particle, however, does not go beyond State III. Whether a particle exhibits lyophobicity or lyophilicity at a given wetting surface tension in film flotation depends on such factors as its surface energy (wettability), size and density. The following energy analysis will delineate the most important properties of t5e particle that determine its transfer from the vapor phase into the liquid phase in film flotation. State III -

INTEGRAL

ENERGY

ANALYSIS

When a particle transfers from the vapor phase to the liquid phase during film flotation (Fig. l), the contributions to the change in free energy are the following: the change in kinetic energy of the free falling particle from dG, -

148

State I to State II. This energy is associated with the velocity of the falling particle. dGP the change in potential energy between State II and State IV. This energy is associated with gravity and buoyancy forces. LOG, the change in inter-facial energy due to adhesion, immersion and spreading. This energy is associated with replacing the solid/vapor interface of the particle by the equivalent area of solid/liquid interface. The total free energy change, AG,, for a particle that moves from State I to State IV is the sum of the kinetic, potential and interfacial energies:

A%

(1)

=AG,,+AGp+AG,

For a particle being dropped and immersed just below the interface, AG, must be less than zero. However, because the change in interfacial energy during this process is associated with three different kinds of wetting processes, the particle may not immerse into the liquid phase spontaneously even though the free energy change is negative for complete immersion of the particle. Generally, wettir_g processes are classified as adhesional wetting, immersional wetting and spreading wetting, depending on the detailed process involved [73. In adhesional wetting, a unit of solid surface and a unit of liquid surface disappear upon the formation of a unit of solid/ liquid interface; in immersional wetting, a unit of solid surface disappears with the formation of a unit of solid/liquid interface; and in spreading wetting, a unit of solid surface disappears with the simultaneous formation of a unit of liquid surface and a unit of solid/liquid interface. The change in the interfacial energy, AG,, is the sum of the surface free energy change associated with adhesional (AG,), immersional (AG;) and spreading wetting (AG,):

AG,=

AG, + AGi + AG,

(2)

from the On a unit area basis, AG,, AG, and AG, can be calculated interfacial energies at the solid/vapor (ysv), solid/liquid (ysL) and liquid/ vapor (yEv) interfaces using the following relationships: (3)

&a=-Ysv+YsL-YLV AGi

=

AG =

-YSV -ysv+

+YSL

YsL+ YLV

(4)

(5)

149

Substituting Ysv -

YSL =

the Young

-yLT/ cos

equation

(6)

8

into Eqns (3), (4) and (5) yields: dG, = - yLv(cos 8 + 1)

(7)

dGi = -y~v

(3)

COS

dG, = -yLV(cos

8

8 - 1)

(9)

Thus, the change in the interfacial energy can be calculated in terms of two measurable quantities: the surface tension of the liquid (yLv) and the equilibrium contact angle (8). Equations (7), (8) and (9) clearly show that adhesional wetting is spontaneous (dGa < 0) when the contact angle is less than X30”, immersional wetting is spontaneous when the contact angle is less than 90”, and spreading wetting is spontaneous when 8 = 0. Among these wetting processes, spreading wetting requires greatest energy and, therefore, it is a control step in determining the transfer of a particle from the vapor phase into the liquid phase. The purpose of this integral energy analysis is to determine the contribution of the changes in the kinetic, potential and interfacial energies to the total change in free energy when a particle transfers from the vapor phase to the liquid phase in film flotation. The conditions at which a particle can transfer fro*m the vapor phase to the liquid phase spontaneously will be delineated in the differential free energy analysis in the section that follows. Assuming that the particle is spherical with a diameter D, the three components of the total free energy change can be calculated from the following expressions:

AG,=

AGp

-

=

p(P, -

(10)

Pvkh

&v +

AG1 = - nD2yLv cos 8

PL) 1

g (12)

where ps, pv and pL are the densities of the solid, vapor and liquid, respectively, g is the acceleration constant due to gravity,and h is the distance between the center of the particles at State I and State II. Considering the particular case of a particle falling onto an aqueous methanol solution for a system having the following set of parameters: D = 125 pm, ps = 1.6 g cmB3, pv = 0, pL = 0.95 g cmB3, yLv = 40 mN m-‘, 8 = 45” and h = 0.5 cm, the absolute value of each energy contribution

150

was calculated and the results are shown in Table 1. These calculations show that the interfacial energy is by far the main component of the total free energy change involved. in the process. Further theoretical analysis of this model was carried out by varying the values of the contact angle (O), the particle diameter (D) and the density of the solid particle (pS). The results of these calculations are presented in Figs 2, 3 and 4 where the free energy change is plotted as a function of contact angle, particle density, and particle diameter, respectively. Figure 2 shows that the interfacial energy change for particle imbibition becomes less negative as the contact angle increases and that both. the kinetic and potential energie, = are independent of the contact angle. TABLE

1

Energy distribution

of the particle

in film flotation

Energy component

Absolute

AGK

8.02 0.14 138.84 147.00.

AG

AG AG

value (J)

Percent of total energy

- 10 - ’ ’

5.4

- lo- ’ l -10 - ’ ’

0.1 94.5 100.0

lo-

ll

I

-& -

E v, 0.6

*

s

25

0.4

E > 4o 5 z c9

3 0.2

i-

0

0

5

-(AGK+h~P) I

15

a

I

I

I

30

45

60

75

CONTACT

ANGLE,

I

g

iz

90"

degrees

Fig. 2. The effect of contact angle on the energy change associated with the film flotation of spherical particles 125 pm in diameter and specific gravity of 1.6 with a wetting liquid having a surface tension of 40 mN m- l.

151 %

150

z ‘- a 125

E

-AGl

-

B-

6

/ACT

-0 -

x

?I u

iO0

75

L 2

50

E 0, g

-

E 5

--(AG~+AG~)

250

I 2

1

DENSllY

I

I

I

3

4

5

OF PARTICLE,

6

g/cm3

Fig.3.The effect of particle

density on the energy change associated with the film flotation of a spherical particle having a diameter of 125 pm and a contact angle of 45” with a wetting liquid having a surface tension of 40 mN m-*.

10-

E .c lO-6 1

ti 0 i=

3

to -10

E a, & lo-l2

E

zi 10 -14 30

60

PARTICLE

120

DIAMETER,

240

480

pm

Fig. 4. The effect of partkle diameter on the energy change associated with the film flotation of spherical particles of specific gravity I .6 and contact angle of 45” using a wetting liquid with a surface tension of 40 mN m-r.

152

These calculations also show that the inter-facial energy is the main component of the total free energy change (more than 85%) for conditions when contact angles are less than 75”. The interfacial energy term loses its dominant position only when 8 is larger than 87”. Figure 3 shows that the interfacial energy is independent of the particle density, but that the kinetic and potential energies increase linearly with density. The contribution of interfacial energy to the total free energy change is greater than 80% even at densities up to 6 g cmB3. Figure 4 shows that there is a linear relationship of dG,, d GK and dGP with particle size when the quantities are plotted on a log-log scale. The contribution of interfacial energy to the total energy is 96% for 53 pm particles and is 82% for 425 pm particles, which is the normal particle size range in froth flotation. These calculations confirm our experimental results, which showed that the surface hydrophobicity of particles is far more important than their size and density in determining partitioning of particles in film flotation [6,8]. DIFFERENTIAL

FREE ENERGY

ANALYSIS

In the foregoing energy analysis of film flotation, we showed that the interfacial energy predominates over the kinetic and potential energy components of the total energy change for the process. To better understand the film flotation technique, a differential free energy analysis was carried out for a spherical particle residing at the liquid/vapor interface. From this analysis, the minimum contact angle required for a particle residing on the surface of the wetting liquid is calculated and the critical wetting surface tension of that particle is estimated. The mathematical treatment of the equilibrium position of particles at liquid/vapor interfaces has traditionally been undertaken by means of a force analysis [g-11] that predicts the equilibrium when the net vertical force acting on the particle is zero. The general free energy analysis for the equilibrium position of cylindrical and spherical particles at the liquid/vapor interface has been developed by Neumann and co-workers [12,13]. The Helmholtz principle was used in their analysis to predict the equilibrium if the free energy has an absolute minimum. The main advantage of the free energy analysis is that it allows the study of the energy balance of particles that have heterogeneous surfaces. It does not, however, elucidate the difference in free energy change associated with wetting different parts of the particle surface. Accordingly, in the preser I’t work, a differential free energy analysis was developed in terms of a di.mensionless free energy that was obtained by dividing the increment of free energy by the product of the increment

153

of wetted surface area and the surface tension of the liquid. This analysis also allows the use of the general free energy concept on a differential area to predict the equilibrium position of particles at the liquid/vapor interface. That is, when the change in free energy is when the change is positive, negative, immersion is spontaneous; immersion is non-spontaneous; and when the change is equal to zero, the equilibrium state is reached. Figure 5 is a schematic representation of a spherical particle residing at the liquid/vapor interface in State III of the film flotation process. Considering a small change in wetted surface area, dA, the corresponding differential change in total free energy, ddG,, is related to the +Eerential changes in kinetic (ddGK), potential (ddG,) and interfacial (ddGI) free energy by: ddG,=ddG,+ddG,+ddG,

03)

Not only because the kinetic energy is negligibly small under the conditions of film flotation compared to the other two terms but also after a particle hits the surface of the liquid its kinetic energy is further dissipated and so ddG, x 0. Using elementary trigonometric relations, ddGP can be written as: ddGr=--zD3

r (~s-~v)-~(l-cos#)2(2+cos~)(~L-~v)

+13sin2+ZO(p,-~p,)gdL

1

gsin#dL

L

O<#
(1.9

where # is a position coordinate angle that locates the three-phase ‘ine on the solid rface, dL is the arc length and Z,, is the vertical dist: ice

Fig. 5. Schematic interface.

representation

of a spherical

particle

residing

at the liquid/vapor

154

between the three-phase contact line and the horizontal liquid surface. Z0 is positive when the interface is deflected downward, and negative in the reverse situation (capillary rise). For particle sizes involved in normal Rotation, that is for D < 600 pm, the hydrostatic pressure term [D sin2 #I ZO(pL - pv)g dL] is negligibly small compared to the other two terms [14].Thus, ddGp=

- iD3

[

A(1 - cos +)“(2 + cos 4

(Ps-Pv)--

Neglecting the influence of the meniscus (dA) upon immersion, ddG, is given by:

dAG1= (YSL- YSV)dA -

YL~

~0s

$)(pL

-

p,,)

on the surface

1 g

sin 4 dL (15)

area change

(16)

4 dA

Substituting Eqn (6) and the trigonometric into Eqn (16: ‘?lds

relation

dA = nD sin # dL

ddG, = - nD sin # yLxV (cos 8 + cos #) dL

(17)

Equation

(17) reduces

to:

dAG&D

sin C#I dL) = - ~~~(~0s

dAG,l(nD

s.in 4 dL) = - yLv cos 0 = AG,

dAG,/(nD

sin 4 dL) = - yLv(cos 0 - I) = AG,

0 + I) =

AG,

4-0

(18)

U-9) # +x

(20)

where AGa, AGj and AG, are the free energy changes per unit area for adhesional, immersional and spreading wetting, respectively. The conditions delineated by Eqns (18), (19) and (20) indicate that the lower half of the spherical surface (0 -C + K 742) is wetted by both adhesional and immersional wetting, while the upper half of the spherical surface (Z/Z c + -C n) is wetted by the combination of immersional and spreading wetting. In any wetting process, AG, -C AG, -C AGs. For conslete wetting to occur, the sufficient condition dG, 6 0 should be satisfied. This implies that the total interfacial free energy change AG, = AG, + AG, + AG, < 0. However, AG, < 0 is only a necessary condition and hence cannot by itself satisfy the condition for complete wetting. Only if a particle satisfies AG, < 0 (and not AG, < 0 alone) can It be spontaneously immersed into the liquid. This is the reason why particles having a contact angle less than 90” reside at liquid surface even though the total energy change is negative for complete immersion into the liquid. The total free energy change (dAG,) associated with the wetting of

155

a differential surface dA is the sum of ddGP and dd GI given by Eqns (15) and (16) respectively: ddGT = - ZD sin 4 yLv(cos 0 + cos 4) dL - 5D3

[

(ps-pv)-

~(l-cos#)2(2+cos

#)(pL-pv)

1

gsin4d-L

OS#
(21)

Dividing Eqn (21) by the differential wetted surface area dA (= nD sin # dL) and by the liquid surface tension yLvresults in a dimensionless total free energy change, dG+ .

LiGT = - (cos 8 + cos #) -

g$.[

(Ps-Pd-

+

+xX

#G2(2+ cas MPL

-

pv)

1

J

e-59

This is a general equation for calculating the dimensionless total free energy change associated with different states of the particle at the liquid/vapor interface. When dG,* > 0, particles reside on the liquid surface; when dGT -C 0, particles are immersed into the liquid phase. At equilibrium, dGF is zero, that is (cos 0 + cos t&J + -

D”

6YLV

x

g

1(Ps-Pv)-$-

cos #,I’@ -I- cos 4e)lPL

-

=0

pv)

(23)

1

The equilibrium position angle, #=, can be calculated from Eqn (23). A minimum contact angle, 0mill,which supports a spherical particle floating on a liquid surface can be calculated from the equation derived by Scheludko et al. [lo]. COS

Omin=

1

D2 - -3y g(Ps -

PL)

(24)

LV

For given values of yLv, D, ps and pL, a particle with a contact angle smaller than 0minwill be engulfed into the liquid phase. Again, taking a particle falling onto an aqueous methanol solution as an example and using the same set of parameters as berbre (D = 125 urn, ps = 1.6 g cmB3, pv = 0, pL = 0.95 g cmm3, yLv= 40 mN m-’ and 8 = 45”), the equilibrium position angle calculated from Eqn (23) is 135”. This means that the particle will be immersed into the liquid phase spontaneously until its position at the interface is that 4 = 135”.

156

Further immersion is impossible unless external energy is applied. The minimum contact angle of the particle calculated using Eqn (24) is 2.3”. This indicates that if a particle under the foregoing conditions htis a contact angle less than 2.3”, it will be completely immersed into the liquid spontaneously. Zisman [15] defined the critical wetting surface tension of a solid, yC, LS the surface tension of a liquid which forms a zero _ontact angle on the solid, that is, the condition where cos 8 = 1. Empirically, he related cos 8 with yC and yLv by cos 6 == 1 - b(yLJ.- y,)

(25)

where the constant b ranged from 0.03 to 0.04 for various polymers 1151. The average value of b for 14 coals was found by Parekh and Aplan [16] to be 0.026. In order to estimate the difference between the critical wetting surface tension of a particle (yC) and the surface tension of the liquid (y&) at which the particle sinks into the liquid in film flotation, we rewrite Eqn (25) as follows y&

-

yc

=

1-

COS

Omin

b

When 0min= 2.3” and assuming b = 0.026, $& - yC calculated using Eqn (26) is 0.031 mN m- ‘. Since such a small value is negligible for practical film flotation, the critical wetting surface tension of coal particles can be taken to be the surface tension of the liquid at which the particle sinks, that is, yEv is eqllal to yC. Further theoretical analysis of this model was carried out by varying the values of 4, 8, D, ps, pL and yLv in Eqns (22), (24) and (261. Figure 6, which gives the dimensionless free energy change as a function of position angle for various contact angles ranging from 0 to 110”, shows that at constant 4 the dimensionless free energy change is higher for large values of 9, which means that imbibition of particles with small 8 is thermodynamically more favorable. At a constant contact angle, the dimensionless free energy change increases with increasing position angle and finally becomes positive. Therefore, the immersion process tends to become thermodynamically impossible and hence the particle resides at the liquid/vapor interface. At dG$ = 0, the particle reaches equilibrium and the corresponding position angle is c$=. As seen from Fig. 6, particles having smaller contact angles have larger equilibrium position angles. Only when 8 is close to zeA-o, the condition dGT < 0 is satisfied at (3e x 180”. The values of the minimum contact angle and the difftirence between

157

EQUILIBRIUM

-2.5

1 0

I 30

LINE

I 60

POSITION

I 90

ANGLE,

15’0

1

120

1

180

degrees

Fig. 6. The dimensionless free energy change of a spherical liquid/vapor interface as a function of the position angle.

particle

residing

at the

~~~~and yC calculated from Eqns (24) and (26) are given in Tables 2-5 for different values of D, ps, pL and yLv. It is seen from the:se results that eminincreases with D and ps, and decreases with pL and yLv. The value of 8, varies from zero to 7.9” for particle sizes ranging from 53 to 325 pm, for particle densities between 1 and 6 g cm - 3, liquid densities between 0.8 and 1.6 g cmm3, and yLv between 20 and 80 mN m-‘. over the same ranges of particle size, particle density, liquid density and TABLE

2

Minimum contact angle (O,,,) and the error involved in the evaiuation of the critical wetting surface tension by film flotation of spherical particIes with density 1.6 g cme3 of different sizes residing at the surface of a liquid having a surface tension of 40mNm-’ Particle size (pm)

kin (deg.)

YZV

53 75 106 150 212 300 425

1.0 1.4 2.0 2.8 4.0 5.6 7.9

0.01 0.01 0.02 0.05 0.09 0.18 0.37

- yc

(mN m-l)

158 TABLE 3 Minimum contact angle (e,i,) and the error involved in the evaluation of the critical wetting surface tension by film flotation of spherical particles of diameter 125 pm of different densities residing at the surface of a liquid having a surface tension of 4r!mNm-’ Particle density (g cmm3)

&A, (deg.)

r&--r,

1

0.6 3.0 4.2 5.1 5.8 6.5

0.00 0.05 0.10 0.15 0.20 0.25

2 3 4 5 6

(mNm_‘1

TABLE 4 Minimum contact angle (emin) and the error involved in the evaluation of the critical wetting surface tension by film flotation of spherical particles of diameter 125 pm with density 1.6 g cm- ’ residing at the surface of liquids having different densities but the same surface tension (40 mN m-l) Liquid density (g cm - 3,

kin (deg.)

0.8 0.9 1.0 I.2 1.4 1.6

2.6 2.4 2.2 1.8 1.3 0.0

TABLE

0.04 0.03 0.03 0*02 0.01 0.00

5

Minimum contact angle (em,,) and the error involved i.n the evaluation of the critical wetting surface tension by film flotation of spherical particles of diameter 125 pm with density 1.6 g cmW3 residing at the surface of liquids having different surface tensions Surface tension (mN m- 1>

emin (deg.)

20 30 40 50 60 70 80

3.3 2.7 2.3 2.1 1.9 1.8 I.6

0.06 0.04 0.03 G.03 0.02 0.02 0.02

159

liquid surface tension,, the values of yzv - yC calculated using Eqn (26) are between zero and 0.37 mN m-‘. Furthermore, if the shape of a particle is cubic, the value of Y;fjv- yC is much smaller than the value for spherical particles. For a cubic particle with al: edge length D, the minimum contact angle, which can support its Boating on a liquid surface, can be calculated from the equation sin @min=

D2 (Ps-- PLk’ 4YLV

(28)

-

Over the same ranges of particle size, particle density, liquid density and liquid surface tension used for the spherical particle, the values Of e&n for the cu’bic particles vary from zero to 0.4” and the corresponding vaiues of J& - y, are between zero and 0.001 mN m-l. Such a small value of y& - yCfor both cubic and spherical particles is negligible for all practical purposes. This value is also smaller than the error involved in determining the critical wetting surface tension using the Zisman plot [S] from contact angle measurements on a Aat surface. Therefore, the critical wetting surface tension of a particle can be taken to be equal to the surface tension of the wetting liquid at vlhich t’he part&Se is engulfed and the effect of particle size and density can be considered negligible for practical film flotation. These predictions are confirmed by the film flotation results reported in our compa.nion paper [6]* SUMMARY

AND CONCLUSIONS

Film fiotation is a technique for characterizing the wetting behavior (hydrophobic) and of particulate samples, in which the lyophobic lyophilic (hydrophilic) particles are partitioned by systematically varying the surface tension of the wetting liquid, t uch as using aqueous methanol solutions of various compositions. In this paper, a model of the film flotation process was developed utilizing both the integral (total) free energy and the differential free energy for the transfer of a particle from the vapor phase into the liquid phase. These analyses show that interfacial energy is far more dominant than kinetic energy and Fotential energy in controlling the behavior of a particle in film flotation over the wide range of particle sizes and densities normally encountered in wetting systems. This study confirmed theoretically that the value of the critical wetting surface tension of a particle can be taken to be equal to the value of the surface tension of the wetting liquid at which the particle is imbibed, and the effect of particle size and particle density on the critical wetting surface

160

tension determined from film flotation is negligible for practical purposes. The model also suggests the necessary and sufficient conditions for particles to be imbibed into the liquid phase. ACKNOWLEDGMENTS

The authors wish to acknowledge the U.S. Department of Energy, Pittsburgh Energy Technology Center, grant No. DE-FG22-84PC70776 and DE-FG22-86PC90507 for the support of this research. REFERENCES 1

2 3 4 5 6 7 8

9 I.0 11 12 13 14 15 16

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