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The effect of surface heterogeneity on pseudo-line tension and the flotation limit of fine particles
CO/lOidS clllc/ ~lW’27L’~‘s. 69 (I 992) 35-43 Elscvicr Scicncc Publishers B.V.. Amsterdam
35
The effect of surface heterogeneity on pseudo-line tension and the flotation limit of fine particles Jaroslaw Drelich and Jan D. MiYer Dep~tmvt of Meta!iurgicol Engineering, University
of’ Utah, Scrlt Lob
City, UT 841 I2. USA
(Received 27 April 1992; accepted 23 July 1992) Abstract The contact angles I! for wntcr at mcthylated quartz surfaces were mcnsured using the scssilc-drop technique to dctcrminc the advancing contact angle. and using a captive-bubble technique to examine the effect of bubble size on contact angle. No linearity bctwcen cos II and l/r. whcrc r is the drop base radius. was observed for these systems ;IS would be cspcctcd for an idcal system. In fact the pseudo-lint tension dccrcased with decreasing bubble size. Also, the deprcc of quartz mcthylation cffcctcd a change in the pseudo-line tension. The pseudo-line tension increased from 0.4. IO -” N to 3.3. IO-” N with an incrcasc in the fractional covcragc ol trimethylsilyl groups from 0.14 to 0.51 for large bubbles (bubble base diameter d > 0.34 mm). whcrcas the pseudo-lint tension dccrcascd from 2.2. IO”’ N to 0.8. IO-’ N with an incrcasc in fractional covcragc for small bubbles (bubble base diameter ri = 0.06-0.2 mm). The flotation limit of fine particles has been rc-cxamincd based on the cfFcct of bubble size on contact angle. and B new surlacc chemistry-limited relationship describing the minimum particle size which can bc ffoatcd is proposed: D,i,
3r,?yLv( I - cos 0)
=2 [
ApV’
1
1!3
whcrc I’, is the critical bubble (drop) radius for which thcrc is no cfTcctivc attachment bctwccn solid surracc and dispcrscd phase, yLv is the interfacial tension at the liquid-vapor interface, Ap is the density dilfcrence bctwccn the particle and the liquid and V is the bubble ascent velocity. K~_wwrcls: Contact
angle: flotation:
lint tension:
pseudo-line
tension.
Introduction
The effect of drop size on contact angle when the gravitational force can be neglected is now unques:ionuble in view of experimental results presented in the scientific literature [i-6]. A modified Young equation (Eqn (1)) has been accepted and is widely used as follows:
and vapor, respectively, 0 is the contact angle, ysLv is the line tension, K,, is the geodesic curvature which is equal to the reciprocal of the drop base radius (Key= I/r) for a spherica! drop setting on a flat and homogeneous solid surface. Thus, Eqn (I) can be expressed as follows:
;‘SV
;)sv - XiL = ;ILV cos
0+
L, S and V correspond
Cnrrc~sparlrlcrr~c~ 10: Engineering, USA.
University
0166-6622/92/505.00
to liquid, solid
J. Drelich, Dept. of Metallurgical of Utah, Salt Lake City, UT 84112,
(
;1SL
=;r,,cosB+~..
Elscvicr
Science
Publishers
(2)
I
(1)
)1SLV Kg5
where ~~~~~ psv and ysL are the interfacial tensions, subscripts
-
The line tension value can be determined txperiof the effect of drop mentally after examination (bubble) size on contact angle and is calculated from the slope of a plot of cos 0 vs l/r. However, the line tension values obtained experimentally B.V. All rights
rcscrvcd.
J. Drdidr.
36
from the contact angle/drop size relationship (1O-8-1O-5 N) arc much greater than those expected from theoretical estimations (IO- 121O- lo N) and those obtained experimentally using other techniques (IO- ‘o-IO-y N) [7,8], which has prompted considerable discussion in the literature [1,3,6]. More recently, it was found, contrary to Eqn (2), that them is no linear correlation between cos 0 and l/r over a wide range of drop size for different systems [6]. These observations suggest that the effect of drop size on contact angle cannot be interpreted in terms of a simple lint tension contribution according to the fact that rigs # I/I for non-ideal systems. Further, experimental problems make the determination of the true value of ~~~most difhcult. Surface imperfections, including heterogeneities and/or surface roughness, can contribute to the observed complexity in the relationship bctwecn cos 0 and I/r. Consequently, a new parameter, pseudo-line tension (;CJv), which includes the etrects of surface imperfections, was proposed to replace the thermodynamic line tension [6]:
and thus ;I&&, cos 0 = cos 0, - ~-
St)
(3)
I';'LV
or more universally
for heterogeneous
surfaces:
cosu=~f&osn,,-$ c .L;ISLV,, Qstt rLV = cos 0, -
?tt_V(f*.Si) ,”
1
rLV
where Xi represents the other variables affecting the pseudo-lint tension, U = 0, for r + ~13;-6, is the area fraction of the surface with contact angle O,,. The pseudo-line tension is a parameter which is equivalent to a hypothetical force which should be applied to the system to cause the observed changes in the contact angle for imperfect systems. This parameter, first postulated by Good and Koo [I],
J. D. Mi/lc~rjCo//oi(ls
Swfi~ces
69 (1992)
3.5-43
al1ov.s for the comparison of experimental results which, up to now, have been designated as actual line tensions. Inadequate design of experiments has not allowed for a detailed analysis of the pseudo-line tension and why the value of the pseudo-iine tension is two to five orders of magnitude greater than the theoretical line tension remains unsatisfactorily explained. Nevertheless, four factors were postulated in the literature to account for this lack of agreement between theory and experiment: surface hemrogeneity [ 1,9- 1 11, gravitational effect [ 121, solid strain in the vicinity of the triple line [ 131, and surface roughness [ 141. The gravitational effect should appear only for large drops with diameters above 5-10 mm, and the effect of solid strain at the three-phase contact Iine would seem to be important only for elastic solid surfaces. Our experim~ntai data show that the contribution from these two factors is negligible [ 151. Surface heterogeneities of:en have a basic contribution to the effect of drop (bubble) size on contact angle. When a drop is placed on a heterogeneous surface a corrugation of the three-phase contact line is observed; sometimes the effect can even be detected without a microscope. The corrugation of the three-phase contact line changes with drop size due to a difference in the resistance to deformation of hemispheres with varying diameter. The importance of this phenomenon in the interpretation of drop size/contact angle data was mentioned many years ago by Vesselovsky and Pertzov [9], then by Good and Koo Cl], and recently it has been considered by the research group directed by Neumann [lo, 1 I]. Nevertheless, analysis was restricted only to theoretical considerations and the dependencies derived still do not have any practical significance due to difficulties in the examination of the three-uhase contact ‘:ne on a microscopic scale. The data presented herein are probably the first experimental resuits which support the concept that surface heterogeneities have ir significant impact on pseudo-line tension, i.e. the relationship between contact angle and drop size. The particle size limit for the Rotation of small
J. Drclich. J. D. h~iiicr/Coiloids
Sut$~ccs 69 (1992) 35-43
37
particles has important practical significance in separation technology. Considerable experimental and theoretical research results have been reported in order t., evaluate the optimum particle size which can be floated effectively. Only the upper limit for the flotation of coarse particles can be successively predicted using several theoretical analyses [16-l S], whereas the flotation limit of fine particles is still unpredictable. A theoretical treatment of this problem was undertaken by Scheludko et al. [16], who considercd the process as the formation of a wetting perimeter, the three-phase contact line, on the basis of the capillarity theory. The minimum particle diameter which can be floated was found to be [16] L&in
=
2
[ dpPy,,(l
3YszLV - cos 0)
1 113
where V is the bubble ascent velocity and dp is the density difference between the particle and the liquid, Ralston and co-workers [l&l 91 presented experimental data for the flotation limit of fine hydrophobized quartz particles. The results did not correlate at all with the theoretical values calculated using Eqn (4), when an order of magnitude of line tension of lo- lo N was used. As has been mentioned previously factors other than the line tension, i.e. surface heterogeneities, are even more important in the analysis of the changes of the contact angle with drop (bubble) size. Automatically in these cases, the pseudo-line tension should replace the line tension in Eqn (4), and the dependence should be:
D win
=
1113
3r&?v
[
2 &I+,,(
1 -cos
0)
(5)
Even this formula, however, which describes real systems, has little practical significance due to difficulties in formulation of how the pseudo-line tension changes with change in bubble (drop) size, and thus y& = J (particle size) as is presented in the next section. Another form for the dependence Of Dmin can be derived from the same theoretical
background presented by Scheludko et al. [16]. The analysis of the work of expansion of a threephase contact leads to an equation describing the line tension as follows [16]: YSLV =
YLV(l
-
cop
WC
(61
where r, is the critical bubble (drop) base radius for which there is no effective attachment between solid surface and dispersed phase, 8 = 0. After substitution of Eqn (6) into Eqn (4) the flotation limit of fine particles is D min
3r,2J$_v(l -cos ApV’
1
e) 1’3
(7)
The critical value rc can be estimated from the effect of bubble (drop) size on contact angle as is presented later in this paper. Notice that neither the line tension nor the pseudo-line tension are required in Eqn (7). Contact angle measurements for the air-water-hydrophobized quartz systems will be presented for varying bubble size, and the results support the usefulness of Eqn (7). Experimental
procedure
Reagents awl ntaterials The basic chemicals used in our experiments were as follows: distilled water with specific conductivity less than 10 -6 S, pH 5.6 & 0.1, and surface tension yLv = 72.4 mN m-’ at 22°C; chlorotrimethylsilane (TMCS) with purity more than 98% Inc.); spectrograde cyclohexane (Mallinckrodt, (EM Science). Optical-grade quartz plates were purchased from Minarad Scientific Inc. Quartz wetkyiation The quartz plates were cleaned with toluene. distilled water, acetone, hydrochloric acid-ethanol mixture and distilled water, and were dried before methylaticn. The purity of the quartz surface was examined by measuring the contact angle for dis
38
tilled water. The quartz was accepted as being clean when the distilled water drop spread at the surface, i.e. having a zero contact angle. Such prepared quartz plates were immersed in a solution of TMCS in cyclohexane. A varying concentration of 8.5 - IO-‘-8.5 10m3 M TMCS in cyclohexane and different times of immersion from 1 to 14 h were used in the adsorption reaction experiments, stimulating changes in the surface coverage by TMCS, as suggested by Blake and Ralston [20]. Hydrophobized plates were washed with cyciohexane to remove unbonded residual ‘TMCS, and were then heated in a clean oven for 4 h at 110°C. l
The sessilc-drop technique was used in measurcments of the advancing contact angles for water. The measurements were performed with use of an NRL goniometer (Ramc-Hart, Inc.) with water drops which ranged in diameter from 3 to 4 mm. The captive-bubble technique was used in experiments for the examination of the bubble size effect on contact angle. A Zeiss ster.eoscopic microscope coupled with a camera was adopted to examine the dimensions of the bubbles, and the contact angle through water was measured from the photographs. Thirty to fifty bubbles with varying diameters were analysed for each system. Air bubbles were released into the water below the quartz plate using a syringe. Control of the bubble size was accomplished by changing the pressure for bubble formation (i.e. by increasing the volumetric flow rate of air into the water). Results and discussion Five quartz plates, each with a different degree of hydrophobization, were used in our experiments and advancing contact angles for the examined samples were determined to be 36, 46, 48, 60 and 73O, with an experimental error of 2-3”. The corresponding degrees of surface methylation were calculated based on the Cassie equation (Eqn (8))
and were found to be .I= 0.14, 0.23, 0.25, 0.37 and 0.5 1, respectively, for the advancing contact angles. cos 0 = f cos 0, + (I -.fl
cos o-,
(8)
where ,f‘ is the area fraction of the surface with contact angle 0, and (1 -j) is the ared fraction of surface with contact angle 0,. The contact angle between water and the clean quartz surface is 0, = 0” whereas the advancing contact angle of water for a surface with exposed methyl groups is 0, = 110 & 2” according to data reported in the literature [21]. The validity of the Cassie equation tor air-water-hydrophobized quartz systems was established by Crawford et al. [21-J The psetrtlo-lirw
tmsiorr
No linear correlation between cos 0 and l/j could be found for any of the surfaces (Fig. I; it should be noted that a given data point may represent as many as three distinct replicated measurements) which supports our previous discovery that the observed changes in the contact angle with drop (bubble) size cannot be explained by a simple line tension analysis [Cl. Of course the pseudo-line tension can be calculated from the slope of a plot of cos 0 =f‘( I/J-) but it changes for the entire range of bubble sizes. Two ranges of bubble size were selected for fuu;ther discussion; those data with i*> 0.17 mm and those data with I*= 0.03-0.1 mm. In each range the cos 0 vs 1/r relationship can be approximated as a linear one. On this basis the pseudo-line tensions are presented in Fig. 2 as a function of the degree of surface methylation. When the bubble size is large (I’> 0.17) the negative pseudo-line tension (the negative value means that the pseudo-line tension vector is directed away fron, the aqueous phase) is in the order of -lo-” N and increases with an increase in surface coverage by non-polar groups from -0.4. IO-” N to -3.3. IO-” N.Theopposite tendency is observed for the small bubbles (1’= 0.03-0.1 mm). In this case the pseudo-line tension is lower by one order of magnitude
J. Drdich.
.I. D. Mih,‘Cniloids
Srrrjirre.~
69
(1992) 35-43
0.6 0,=36+3”
0,=46+2”
f=O. 14
IO
b 20 I /r [I/mm]
f=O.23
I 30
0.44 0
40
IO
(b)
l-
20 7/r [I/mm]
‘I
0
0
0
0
0.6-
0,=60&2”
f=O.37
0,=48+2” f=‘=o.25
Q.40
30
L IO
I 20 I/r [I/mm]
30
0.4c
40
I
IQ
0
(d)
‘T------l 0.8
I
I
20 I/r [I/mm]
0
0,,=72+2”
f==QSl
0.4b (c) Fig. i. The cffcct of bubble
’
IO
t
20 l/r [I/mm]
size on contact
L
30
angle for mcthylatcd
I
40 quartz
surfaces.
I
30
40
A?ea Fraction Fig. 2. The pseudo-lint in
FiS. I
for
txvo
of TMS Groups
tensions calculated from the plots shown ranges or buhbtc six: r > 0.17 mm;
r = 0.03-O. I mm.
(-0.8. IO-’ to -?..2* IG-’ N) and decreases with an increase in the extent of methylation. Scattered data in the contact ang’s m:.:,surcments produced a wide confidence interval for the results especially for measurements with the smaller bubbles. The heterogeneity of the solid surface (the cxtcnt of methylation) is thus a very important factor affecting the correlation between bubble size and contact angle. A corrugation of the three-phase contact line was observed with a stereoscopic microscope and an example of this observation is presented in Fig. 3. Limited magnification and
AIR
)
._ ..: : ,.
Line
.’
‘.
Fig. 3. A corrugation of the three-phase contact line observed through the microscope for a methylatcd quartz surfarc (0,=72_c2”:j’=0.51).
resolution capabiIities of the optical equipment do not permit a more dctailcd analysis of the corrugation, and the shape of the three-phase contact line for different systems could not bc compared. Nevertheless, random distribution of polar and non-polar patches was obscrvcd. The water microdroplets remaineu non-uniformly at the quartz surface after a water film between air bubble and solid surface was disrupted (see Fig. 3). Nonuniform coverage of trimethylsilyl groups over the fused-silica surface has also been observed in another laboratory [22]. The dcpendcnce between cos 0 and the shape of the three-phase contact line with Iocal curvatures and contact angles was derived by Li et al. [l 11 as fo1lows: cos 0 =
R,(R,+R,+~)cos(I,+R,~cos02 iR, + R,)(R: R, +R2+~(l - (R, -I- Rz)(R,
+ I’)
-K)
(9)
+ ~‘);‘,_v
where K = ;1~~\~-,/;1~~v~. R is the local radius of curvature of the three-phase line and subscripts I and 2 correspond to two types of patches (in our case. I represents patches of quartz surface with adsorbed trimethylsilyl groups and 2 represents patches of clean quartz surface). As mentioned before, it is diflicult to examine the true shape of the curvature at the triple junction. The local radii of curvatures are expected to bc no larger than a few microns. Also, up to now it has been impossible to predict the changes in the radii of local curvatures with changes in bubble size. Consequently, Eqn (9) has little practical significance for us at the present time. Nevertheless, it should be emphasized that the predicted shape of a plot of cos 0 =f( 1/I-) for heterogeneous surfaces using Eqn (9) (see Ref. [I I]) agrees well with the shape observed from our experimental data. Our experimental effort has been focused on one system where the surface tension, ionic strength and pH of water were constant. Examination of different systems with different physico-chemical properties can provide quite different values of the
41
pseudo-line tension. The shape of the drop edge in the vicinity of’ the three-phase contact line (on a microscopic scale) is not predictable so far. Consequently, the probIem is even more complex than discussed in this paper. For example. factors as delicate as the surface charge density at a solid surface and the double-layer forces can affect the thermodynamic state of the three-phase system and thus the contact angle. However, a discussion of these factors is beyond the scope of this contribution.
Fig. 4. Determination
Our experimental res;its for the effect of bubble size on contact angle for methylated quartz surfaces also have important practical significance. As was proposed in the introduction, the flotation limit of fine particles derived by Scheludko et al. [ l6] should be better described using experimental results for the pseudo-line tension instead of the theoretical line tension, according to Eqn (5). Nevertheless, the determination of Dmin from our experimental results requires a numerical procedure if Eqn (5) is applied. A simpler way of determining D,,i,, is to use Eqn (7). The critical value of I’, can be estimated after extrapolation of a plot of cos 0 vs I/r or extrapolation of a plot of 0 vs r to 0 = 0. It was found that our experimental data fit a straight line in the case of 0 vs log r for bubbles with I’< 0.4 mm. An example of this extrapolation of our experimental data (data from Fig. i(b)) is presented in Fig. 4. Linear regression analysis was used to determine the critical va!ues of r, and computed values are collected in Table 1. The data obtained from extrapolation are always controversial and in practice the effect of bubble size on contact angle should be examined over as wide a range of bubble size as possible in order to eliminate extrapolation error. On this basis the flotation limit of fine particles was calculated using Eqn (7) with the following of water yLv = values: the surface tension 72.4mNm-1,dp=1650kgm-31and V=O.Zand 0.3 m s- ‘, as expected from the flotation experi-
TABLE
of the critical
1
2 3 4 5
size.
I
Critical values of bubble-methylatcd and the theoretical notation limit calculated from Eqn (7) Srtmple
bubble
0, (de@
36h 3 46 + 2 48 f 2 60 + 2 72 f?
./
0.14 0.23 0.25 0.37 0.51
rc (Pm)
0.85 0.30 0.29 0.09 0.01
quartz contac! radius (r,) of firm particles (D,i,)as
Dmin I’= 0.3 m/s
V = 0.3 m/s
15.4 9.0 9.0 4.7 I.2
II.7 6.9 6.9 3.4 0.9
ments of Crawford and Ralston [IS] and Blake and Ralston [ 191. The advancing contact angles B = OA were used in our calculations; however, the contact angles close to or equal to the receding contact angles are more important in the flotation process. Nevertheless, an error following from this assumption should be much smaller than the error in the determination of r,. Calculated values of Dmin based on contact angle measurements are presented in Fig. 5 together with the data of Ralston and co-workers [ 18,191 based on flotation experiments. There is even better agreement than would be expected. Some diflrerences would be expected since the fine particles examined in the flotation experiments [18,19] probably differ in roughness from our flat quartz plates used in the contact angle measurements.
w Crawford & Ralston
A Blake 8 Ralston
20 1
I
10
Am
I
01 0
Calculated 0 V~0.3 m/s 0 V--O.2 m/s
0 0
A 00 00
a 0
0.45 0.3 0.15 Area Fraction of TMS Groups
Fig. 5. The flotation
limit of mcthylatcd
qunrtz
porticlcs.
AIso, it might be expected that the distribution of adsorbed trimethylsilyl groups could dialer slightly. Finally, it should be emphasized that the hydrodynamic conditions in the flotation cell as well as surface forces were not considered in the derivation of Eqns (4), (5) and (7), as discussed in Ref. [I 61. Of course these factors can also contribute to the experimental variation. Summary
and conclusions
The contact angles of water have been measured on methylated quartz plates as a function of the bubble size. The contribution of surface heterogeneity to the effect of bubble size on contact angle is of pronounced importance as is evident from the experimental results presented herein. A corrugation of the bubble surface in the vicinity of the three-phase contact line was observed for airwater-methylated quartz systems, which affects the value of the pseudo-line tension. The pseudo-line tension decreased with a decrease in bubble-solid contact radius from 7 = 1.5 mm to I*= 0.03 mm for all systems which ranged in fractiona! coverage of trimethyIsily1 groups from 0.14 to 0.51, as calculated from the Cassie equation. The shape of the plot of cos 0 vs l/r changed with the extent of quartz methylation and the pseudo-line tension was found to increase with an
increase in fractional coverage of non-polar trimethylsilyl patches for bubbles whose base diameter was greater than 0.34 mm. The opposite trend was observed for small bubbles with a base diameter from 0.06 to 0.2 mm and the pseudo-line tension decreased with an increase in the extent of methylation of the quartz surface. The pseudo-line tension phenomer- ,n is of particular significance in the analysis of fine particle flotation. It is proposed that the flotation limit of fine quartz particles can be estimated from contact angle measurements using a modification of the theory developed by Scheludko et al. [ 161 which is described by Eqn (7). In this analysis the critical value of bubble-solid contact (v,) for which the contact angle is zero has to be determined. The values of I’, determined from our experimental data of contact angle versus bubble-solid contact diameter permit the prediction of the flotation limit of methylated quartz particles. Nevertheless, our contact angle measurements were limited to bubbles with base diameter larger than 0.06 mm, which. leads to some uncertainty in the determination of effort in contact angle I’,. Further experimental measurements for a wide range of bubbie sizes (including tiny bubbles) is necessary. Acknowledgment
The support of the U.S. Department of Energy, contract number DEFC21 SBMC26-1.68, is gratefully recognized. References R.J. Good
and
M.N.
Koo.
J. Colloid
In~crfacc
Sci.. 71
(1979) x3. J. Gaydos and A.W. Neumann, J. Coltoid Intcrrxc Sci.. 170 (1987) 76. D. Li ilnd A.W. Neumann, Colloids Surfaces. 43 (1990) 195 J.A. Wnllacc and S. Schurch. Colloids Sur~accs. 43 (1990 707. M. Yckta-Ford and A.B. Pontcr, J. Colloid Intcrriicc Sci. 176 (1988) 134. J. Drclich, J.D. Miller and J. Hupkn. J. Colloid Intcrfucc Sci.. in press. J. Mingins and A. Schcludko, J. Chcm. Sot., Farada! Trans. I. 75 (1979) I.
J. Drelich. 8
9 IO
1I 12
13 14
J. D. Millw/Colloids
Swfoces
69 (1992)
35-43
H.J. Schultze, Physico-Chemical Elementary Processes in Flotation. Else?;cr: Amsterdam, 1984. pp. 163-165. 253. VS. Vessclovsky and V.N. Pcrtzov. Zh. Fiz. Khim.. 8 (I 936) 245. L. Boruvka, J. Gaydos and A.W. Neumann. Colloids Surl-aces. 43 (1990) 307. D. Li, F.Y.H. Lin and A.W. Neumann, J. Colloid IntcrTace Sci 142 (1991) 224. J.
43
15 16 17 I8 19 20 21 22
J. Drclich and J.D. Miller. Part. Sci. Tcchnol.. submitted for publication. A. Scheludko. B.V. Toshcv and D.T. Bojadjicv. J. Chcm. Sot., Faraday Trans. I, 12 (1976) 2815. H.J. Schulzc, Int. J. Miner. Process., 4 (1977) 241. R. Crawford and J. Ralston, Int. J. Miner. Process.. 23 (1988) I. P. Blake and J. Ralston. Colloids Surfaces, 16 (1985) 41. P. Blake and J. Ralston, Colloids Surfaces, 15 (1985) 101. R. Crawford, L.K. Koopal and J. Ralston, Colloids Surfaces. 27 (1987) 57. M. Trau, B.S. Murray, K. Grant and F. Grieser, J. Colloid Interface Sci., 148 (1992) 182.
35
The effect of surface heterogeneity on pseudo-line tension and the flotation limit of fine particles Jaroslaw Drelich and Jan D. MiYer Dep~tmvt of Meta!iurgicol Engineering, University
of’ Utah, Scrlt Lob
City, UT 841 I2. USA
(Received 27 April 1992; accepted 23 July 1992) Abstract The contact angles I! for wntcr at mcthylated quartz surfaces were mcnsured using the scssilc-drop technique to dctcrminc the advancing contact angle. and using a captive-bubble technique to examine the effect of bubble size on contact angle. No linearity bctwcen cos II and l/r. whcrc r is the drop base radius. was observed for these systems ;IS would be cspcctcd for an idcal system. In fact the pseudo-lint tension dccrcased with decreasing bubble size. Also, the deprcc of quartz mcthylation cffcctcd a change in the pseudo-line tension. The pseudo-line tension increased from 0.4. IO -” N to 3.3. IO-” N with an incrcasc in the fractional covcragc ol trimethylsilyl groups from 0.14 to 0.51 for large bubbles (bubble base diameter d > 0.34 mm). whcrcas the pseudo-lint tension dccrcascd from 2.2. IO”’ N to 0.8. IO-’ N with an incrcasc in fractional covcragc for small bubbles (bubble base diameter ri = 0.06-0.2 mm). The flotation limit of fine particles has been rc-cxamincd based on the cfFcct of bubble size on contact angle. and B new surlacc chemistry-limited relationship describing the minimum particle size which can bc ffoatcd is proposed: D,i,
3r,?yLv( I - cos 0)
=2 [
ApV’
1
1!3
whcrc I’, is the critical bubble (drop) radius for which thcrc is no cfTcctivc attachment bctwccn solid surracc and dispcrscd phase, yLv is the interfacial tension at the liquid-vapor interface, Ap is the density dilfcrence bctwccn the particle and the liquid and V is the bubble ascent velocity. K~_wwrcls: Contact
angle: flotation:
lint tension:
pseudo-line
tension.
Introduction
The effect of drop size on contact angle when the gravitational force can be neglected is now unques:ionuble in view of experimental results presented in the scientific literature [i-6]. A modified Young equation (Eqn (1)) has been accepted and is widely used as follows:
and vapor, respectively, 0 is the contact angle, ysLv is the line tension, K,, is the geodesic curvature which is equal to the reciprocal of the drop base radius (Key= I/r) for a spherica! drop setting on a flat and homogeneous solid surface. Thus, Eqn (I) can be expressed as follows:
;‘SV
;)sv - XiL = ;ILV cos
0+
L, S and V correspond
Cnrrc~sparlrlcrr~c~ 10: Engineering, USA.
University
0166-6622/92/505.00
to liquid, solid
J. Drelich, Dept. of Metallurgical of Utah, Salt Lake City, UT 84112,
(
;1SL
=;r,,cosB+~..
Elscvicr
Science
Publishers
(2)
I
(1)
)1SLV Kg5
where ~~~~~ psv and ysL are the interfacial tensions, subscripts
-
The line tension value can be determined txperiof the effect of drop mentally after examination (bubble) size on contact angle and is calculated from the slope of a plot of cos 0 vs l/r. However, the line tension values obtained experimentally B.V. All rights
rcscrvcd.
J. Drdidr.
36
from the contact angle/drop size relationship (1O-8-1O-5 N) arc much greater than those expected from theoretical estimations (IO- 121O- lo N) and those obtained experimentally using other techniques (IO- ‘o-IO-y N) [7,8], which has prompted considerable discussion in the literature [1,3,6]. More recently, it was found, contrary to Eqn (2), that them is no linear correlation between cos 0 and l/r over a wide range of drop size for different systems [6]. These observations suggest that the effect of drop size on contact angle cannot be interpreted in terms of a simple lint tension contribution according to the fact that rigs # I/I for non-ideal systems. Further, experimental problems make the determination of the true value of ~~~most difhcult. Surface imperfections, including heterogeneities and/or surface roughness, can contribute to the observed complexity in the relationship bctwecn cos 0 and I/r. Consequently, a new parameter, pseudo-line tension (;CJv), which includes the etrects of surface imperfections, was proposed to replace the thermodynamic line tension [6]:
and thus ;I&&, cos 0 = cos 0, - ~-
St)
(3)
I';'LV
or more universally
for heterogeneous
surfaces:
cosu=~f&osn,,-$ c .L;ISLV,, Qstt rLV = cos 0, -
?tt_V(f*.Si) ,”
1
rLV
where Xi represents the other variables affecting the pseudo-lint tension, U = 0, for r + ~13;-6, is the area fraction of the surface with contact angle O,,. The pseudo-line tension is a parameter which is equivalent to a hypothetical force which should be applied to the system to cause the observed changes in the contact angle for imperfect systems. This parameter, first postulated by Good and Koo [I],
J. D. Mi/lc~rjCo//oi(ls
Swfi~ces
69 (1992)
3.5-43
al1ov.s for the comparison of experimental results which, up to now, have been designated as actual line tensions. Inadequate design of experiments has not allowed for a detailed analysis of the pseudo-line tension and why the value of the pseudo-iine tension is two to five orders of magnitude greater than the theoretical line tension remains unsatisfactorily explained. Nevertheless, four factors were postulated in the literature to account for this lack of agreement between theory and experiment: surface hemrogeneity [ 1,9- 1 11, gravitational effect [ 121, solid strain in the vicinity of the triple line [ 131, and surface roughness [ 141. The gravitational effect should appear only for large drops with diameters above 5-10 mm, and the effect of solid strain at the three-phase contact Iine would seem to be important only for elastic solid surfaces. Our experim~ntai data show that the contribution from these two factors is negligible [ 151. Surface heterogeneities of:en have a basic contribution to the effect of drop (bubble) size on contact angle. When a drop is placed on a heterogeneous surface a corrugation of the three-phase contact line is observed; sometimes the effect can even be detected without a microscope. The corrugation of the three-phase contact line changes with drop size due to a difference in the resistance to deformation of hemispheres with varying diameter. The importance of this phenomenon in the interpretation of drop size/contact angle data was mentioned many years ago by Vesselovsky and Pertzov [9], then by Good and Koo Cl], and recently it has been considered by the research group directed by Neumann [lo, 1 I]. Nevertheless, analysis was restricted only to theoretical considerations and the dependencies derived still do not have any practical significance due to difficulties in the examination of the three-uhase contact ‘:ne on a microscopic scale. The data presented herein are probably the first experimental resuits which support the concept that surface heterogeneities have ir significant impact on pseudo-line tension, i.e. the relationship between contact angle and drop size. The particle size limit for the Rotation of small
J. Drclich. J. D. h~iiicr/Coiloids
Sut$~ccs 69 (1992) 35-43
37
particles has important practical significance in separation technology. Considerable experimental and theoretical research results have been reported in order t., evaluate the optimum particle size which can be floated effectively. Only the upper limit for the flotation of coarse particles can be successively predicted using several theoretical analyses [16-l S], whereas the flotation limit of fine particles is still unpredictable. A theoretical treatment of this problem was undertaken by Scheludko et al. [16], who considercd the process as the formation of a wetting perimeter, the three-phase contact line, on the basis of the capillarity theory. The minimum particle diameter which can be floated was found to be [16] L&in
=
2
[ dpPy,,(l
3YszLV - cos 0)
1 113
where V is the bubble ascent velocity and dp is the density difference between the particle and the liquid, Ralston and co-workers [l&l 91 presented experimental data for the flotation limit of fine hydrophobized quartz particles. The results did not correlate at all with the theoretical values calculated using Eqn (4), when an order of magnitude of line tension of lo- lo N was used. As has been mentioned previously factors other than the line tension, i.e. surface heterogeneities, are even more important in the analysis of the changes of the contact angle with drop (bubble) size. Automatically in these cases, the pseudo-line tension should replace the line tension in Eqn (4), and the dependence should be:
D win
=
1113
3r&?v
[
2 &I+,,(
1 -cos
0)
(5)
Even this formula, however, which describes real systems, has little practical significance due to difficulties in formulation of how the pseudo-line tension changes with change in bubble (drop) size, and thus y& = J (particle size) as is presented in the next section. Another form for the dependence Of Dmin can be derived from the same theoretical
background presented by Scheludko et al. [16]. The analysis of the work of expansion of a threephase contact leads to an equation describing the line tension as follows [16]: YSLV =
YLV(l
-
cop
WC
(61
where r, is the critical bubble (drop) base radius for which there is no effective attachment between solid surface and dispersed phase, 8 = 0. After substitution of Eqn (6) into Eqn (4) the flotation limit of fine particles is D min
3r,2J$_v(l -cos ApV’
1
e) 1’3
(7)
The critical value rc can be estimated from the effect of bubble (drop) size on contact angle as is presented later in this paper. Notice that neither the line tension nor the pseudo-line tension are required in Eqn (7). Contact angle measurements for the air-water-hydrophobized quartz systems will be presented for varying bubble size, and the results support the usefulness of Eqn (7). Experimental
procedure
Reagents awl ntaterials The basic chemicals used in our experiments were as follows: distilled water with specific conductivity less than 10 -6 S, pH 5.6 & 0.1, and surface tension yLv = 72.4 mN m-’ at 22°C; chlorotrimethylsilane (TMCS) with purity more than 98% Inc.); spectrograde cyclohexane (Mallinckrodt, (EM Science). Optical-grade quartz plates were purchased from Minarad Scientific Inc. Quartz wetkyiation The quartz plates were cleaned with toluene. distilled water, acetone, hydrochloric acid-ethanol mixture and distilled water, and were dried before methylaticn. The purity of the quartz surface was examined by measuring the contact angle for dis
38
tilled water. The quartz was accepted as being clean when the distilled water drop spread at the surface, i.e. having a zero contact angle. Such prepared quartz plates were immersed in a solution of TMCS in cyclohexane. A varying concentration of 8.5 - IO-‘-8.5 10m3 M TMCS in cyclohexane and different times of immersion from 1 to 14 h were used in the adsorption reaction experiments, stimulating changes in the surface coverage by TMCS, as suggested by Blake and Ralston [20]. Hydrophobized plates were washed with cyciohexane to remove unbonded residual ‘TMCS, and were then heated in a clean oven for 4 h at 110°C. l
The sessilc-drop technique was used in measurcments of the advancing contact angles for water. The measurements were performed with use of an NRL goniometer (Ramc-Hart, Inc.) with water drops which ranged in diameter from 3 to 4 mm. The captive-bubble technique was used in experiments for the examination of the bubble size effect on contact angle. A Zeiss ster.eoscopic microscope coupled with a camera was adopted to examine the dimensions of the bubbles, and the contact angle through water was measured from the photographs. Thirty to fifty bubbles with varying diameters were analysed for each system. Air bubbles were released into the water below the quartz plate using a syringe. Control of the bubble size was accomplished by changing the pressure for bubble formation (i.e. by increasing the volumetric flow rate of air into the water). Results and discussion Five quartz plates, each with a different degree of hydrophobization, were used in our experiments and advancing contact angles for the examined samples were determined to be 36, 46, 48, 60 and 73O, with an experimental error of 2-3”. The corresponding degrees of surface methylation were calculated based on the Cassie equation (Eqn (8))
and were found to be .I= 0.14, 0.23, 0.25, 0.37 and 0.5 1, respectively, for the advancing contact angles. cos 0 = f cos 0, + (I -.fl
cos o-,
(8)
where ,f‘ is the area fraction of the surface with contact angle 0, and (1 -j) is the ared fraction of surface with contact angle 0,. The contact angle between water and the clean quartz surface is 0, = 0” whereas the advancing contact angle of water for a surface with exposed methyl groups is 0, = 110 & 2” according to data reported in the literature [21]. The validity of the Cassie equation tor air-water-hydrophobized quartz systems was established by Crawford et al. [21-J The psetrtlo-lirw
tmsiorr
No linear correlation between cos 0 and l/j could be found for any of the surfaces (Fig. I; it should be noted that a given data point may represent as many as three distinct replicated measurements) which supports our previous discovery that the observed changes in the contact angle with drop (bubble) size cannot be explained by a simple line tension analysis [Cl. Of course the pseudo-line tension can be calculated from the slope of a plot of cos 0 =f‘( I/J-) but it changes for the entire range of bubble sizes. Two ranges of bubble size were selected for fuu;ther discussion; those data with i*> 0.17 mm and those data with I*= 0.03-0.1 mm. In each range the cos 0 vs 1/r relationship can be approximated as a linear one. On this basis the pseudo-line tensions are presented in Fig. 2 as a function of the degree of surface methylation. When the bubble size is large (I’> 0.17) the negative pseudo-line tension (the negative value means that the pseudo-line tension vector is directed away fron, the aqueous phase) is in the order of -lo-” N and increases with an increase in surface coverage by non-polar groups from -0.4. IO-” N to -3.3. IO-” N.Theopposite tendency is observed for the small bubbles (1’= 0.03-0.1 mm). In this case the pseudo-line tension is lower by one order of magnitude
J. Drdich.
.I. D. Mih,‘Cniloids
Srrrjirre.~
69
(1992) 35-43
0.6 0,=36+3”
0,=46+2”
f=O. 14
IO
b 20 I /r [I/mm]
f=O.23
I 30
0.44 0
40
IO
(b)
l-
20 7/r [I/mm]
‘I
0
0
0
0
0.6-
0,=60&2”
f=O.37
0,=48+2” f=‘=o.25
Q.40
30
L IO
I 20 I/r [I/mm]
30
0.4c
40
I
IQ
0
(d)
‘T------l 0.8
I
I
20 I/r [I/mm]
0
0,,=72+2”
f==QSl
0.4b (c) Fig. i. The cffcct of bubble
’
IO
t
20 l/r [I/mm]
size on contact
L
30
angle for mcthylatcd
I
40 quartz
surfaces.
I
30
40
A?ea Fraction Fig. 2. The pseudo-lint in
FiS. I
for
txvo
of TMS Groups
tensions calculated from the plots shown ranges or buhbtc six: r > 0.17 mm;
r = 0.03-O. I mm.
(-0.8. IO-’ to -?..2* IG-’ N) and decreases with an increase in the extent of methylation. Scattered data in the contact ang’s m:.:,surcments produced a wide confidence interval for the results especially for measurements with the smaller bubbles. The heterogeneity of the solid surface (the cxtcnt of methylation) is thus a very important factor affecting the correlation between bubble size and contact angle. A corrugation of the three-phase contact line was observed with a stereoscopic microscope and an example of this observation is presented in Fig. 3. Limited magnification and
AIR
)
._ ..: : ,.
Line
.’
‘.
Fig. 3. A corrugation of the three-phase contact line observed through the microscope for a methylatcd quartz surfarc (0,=72_c2”:j’=0.51).
resolution capabiIities of the optical equipment do not permit a more dctailcd analysis of the corrugation, and the shape of the three-phase contact line for different systems could not bc compared. Nevertheless, random distribution of polar and non-polar patches was obscrvcd. The water microdroplets remaineu non-uniformly at the quartz surface after a water film between air bubble and solid surface was disrupted (see Fig. 3). Nonuniform coverage of trimethylsilyl groups over the fused-silica surface has also been observed in another laboratory [22]. The dcpendcnce between cos 0 and the shape of the three-phase contact line with Iocal curvatures and contact angles was derived by Li et al. [l 11 as fo1lows: cos 0 =
R,(R,+R,+~)cos(I,+R,~cos02 iR, + R,)(R: R, +R2+~(l - (R, -I- Rz)(R,
+ I’)
-K)
(9)
+ ~‘);‘,_v
where K = ;1~~\~-,/;1~~v~. R is the local radius of curvature of the three-phase line and subscripts I and 2 correspond to two types of patches (in our case. I represents patches of quartz surface with adsorbed trimethylsilyl groups and 2 represents patches of clean quartz surface). As mentioned before, it is diflicult to examine the true shape of the curvature at the triple junction. The local radii of curvatures are expected to bc no larger than a few microns. Also, up to now it has been impossible to predict the changes in the radii of local curvatures with changes in bubble size. Consequently, Eqn (9) has little practical significance for us at the present time. Nevertheless, it should be emphasized that the predicted shape of a plot of cos 0 =f( 1/I-) for heterogeneous surfaces using Eqn (9) (see Ref. [I I]) agrees well with the shape observed from our experimental data. Our experimental effort has been focused on one system where the surface tension, ionic strength and pH of water were constant. Examination of different systems with different physico-chemical properties can provide quite different values of the
41
pseudo-line tension. The shape of the drop edge in the vicinity of’ the three-phase contact line (on a microscopic scale) is not predictable so far. Consequently, the probIem is even more complex than discussed in this paper. For example. factors as delicate as the surface charge density at a solid surface and the double-layer forces can affect the thermodynamic state of the three-phase system and thus the contact angle. However, a discussion of these factors is beyond the scope of this contribution.
Fig. 4. Determination
Our experimental res;its for the effect of bubble size on contact angle for methylated quartz surfaces also have important practical significance. As was proposed in the introduction, the flotation limit of fine particles derived by Scheludko et al. [ l6] should be better described using experimental results for the pseudo-line tension instead of the theoretical line tension, according to Eqn (5). Nevertheless, the determination of Dmin from our experimental results requires a numerical procedure if Eqn (5) is applied. A simpler way of determining D,,i,, is to use Eqn (7). The critical value of I’, can be estimated after extrapolation of a plot of cos 0 vs I/r or extrapolation of a plot of 0 vs r to 0 = 0. It was found that our experimental data fit a straight line in the case of 0 vs log r for bubbles with I’< 0.4 mm. An example of this extrapolation of our experimental data (data from Fig. i(b)) is presented in Fig. 4. Linear regression analysis was used to determine the critical va!ues of r, and computed values are collected in Table 1. The data obtained from extrapolation are always controversial and in practice the effect of bubble size on contact angle should be examined over as wide a range of bubble size as possible in order to eliminate extrapolation error. On this basis the flotation limit of fine particles was calculated using Eqn (7) with the following of water yLv = values: the surface tension 72.4mNm-1,dp=1650kgm-31and V=O.Zand 0.3 m s- ‘, as expected from the flotation experi-
TABLE
of the critical
1
2 3 4 5
size.
I
Critical values of bubble-methylatcd and the theoretical notation limit calculated from Eqn (7) Srtmple
bubble
0, (de@
36h 3 46 + 2 48 f 2 60 + 2 72 f?
./
0.14 0.23 0.25 0.37 0.51
rc (Pm)
0.85 0.30 0.29 0.09 0.01
quartz contac! radius (r,) of firm particles (D,i,)as
Dmin I’= 0.3 m/s
V = 0.3 m/s
15.4 9.0 9.0 4.7 I.2
II.7 6.9 6.9 3.4 0.9
ments of Crawford and Ralston [IS] and Blake and Ralston [ 191. The advancing contact angles B = OA were used in our calculations; however, the contact angles close to or equal to the receding contact angles are more important in the flotation process. Nevertheless, an error following from this assumption should be much smaller than the error in the determination of r,. Calculated values of Dmin based on contact angle measurements are presented in Fig. 5 together with the data of Ralston and co-workers [ 18,191 based on flotation experiments. There is even better agreement than would be expected. Some diflrerences would be expected since the fine particles examined in the flotation experiments [18,19] probably differ in roughness from our flat quartz plates used in the contact angle measurements.
w Crawford & Ralston
A Blake 8 Ralston
20 1
I
10
Am
I
01 0
Calculated 0 V~0.3 m/s 0 V--O.2 m/s
0 0
A 00 00
a 0
0.45 0.3 0.15 Area Fraction of TMS Groups
Fig. 5. The flotation
limit of mcthylatcd
qunrtz
porticlcs.
AIso, it might be expected that the distribution of adsorbed trimethylsilyl groups could dialer slightly. Finally, it should be emphasized that the hydrodynamic conditions in the flotation cell as well as surface forces were not considered in the derivation of Eqns (4), (5) and (7), as discussed in Ref. [I 61. Of course these factors can also contribute to the experimental variation. Summary
and conclusions
The contact angles of water have been measured on methylated quartz plates as a function of the bubble size. The contribution of surface heterogeneity to the effect of bubble size on contact angle is of pronounced importance as is evident from the experimental results presented herein. A corrugation of the bubble surface in the vicinity of the three-phase contact line was observed for airwater-methylated quartz systems, which affects the value of the pseudo-line tension. The pseudo-line tension decreased with a decrease in bubble-solid contact radius from 7 = 1.5 mm to I*= 0.03 mm for all systems which ranged in fractiona! coverage of trimethyIsily1 groups from 0.14 to 0.51, as calculated from the Cassie equation. The shape of the plot of cos 0 vs l/r changed with the extent of quartz methylation and the pseudo-line tension was found to increase with an
increase in fractional coverage of non-polar trimethylsilyl patches for bubbles whose base diameter was greater than 0.34 mm. The opposite trend was observed for small bubbles with a base diameter from 0.06 to 0.2 mm and the pseudo-line tension decreased with an increase in the extent of methylation of the quartz surface. The pseudo-line tension phenomer- ,n is of particular significance in the analysis of fine particle flotation. It is proposed that the flotation limit of fine quartz particles can be estimated from contact angle measurements using a modification of the theory developed by Scheludko et al. [ 161 which is described by Eqn (7). In this analysis the critical value of bubble-solid contact (v,) for which the contact angle is zero has to be determined. The values of I’, determined from our experimental data of contact angle versus bubble-solid contact diameter permit the prediction of the flotation limit of methylated quartz particles. Nevertheless, our contact angle measurements were limited to bubbles with base diameter larger than 0.06 mm, which. leads to some uncertainty in the determination of effort in contact angle I’,. Further experimental measurements for a wide range of bubbie sizes (including tiny bubbles) is necessary. Acknowledgment
The support of the U.S. Department of Energy, contract number DEFC21 SBMC26-1.68, is gratefully recognized. References R.J. Good
and
M.N.
Koo.
J. Colloid
In~crfacc
Sci.. 71
(1979) x3. J. Gaydos and A.W. Neumann, J. Coltoid Intcrrxc Sci.. 170 (1987) 76. D. Li ilnd A.W. Neumann, Colloids Surfaces. 43 (1990) 195 J.A. Wnllacc and S. Schurch. Colloids Sur~accs. 43 (1990 707. M. Yckta-Ford and A.B. Pontcr, J. Colloid Intcrriicc Sci. 176 (1988) 134. J. Drclich, J.D. Miller and J. Hupkn. J. Colloid Intcrfucc Sci.. in press. J. Mingins and A. Schcludko, J. Chcm. Sot., Farada! Trans. I. 75 (1979) I.
J. Drelich. 8
9 IO
1I 12
13 14
J. D. Millw/Colloids
Swfoces
69 (1992)
35-43
H.J. Schultze, Physico-Chemical Elementary Processes in Flotation. Else?;cr: Amsterdam, 1984. pp. 163-165. 253. VS. Vessclovsky and V.N. Pcrtzov. Zh. Fiz. Khim.. 8 (I 936) 245. L. Boruvka, J. Gaydos and A.W. Neumann. Colloids Surl-aces. 43 (1990) 307. D. Li, F.Y.H. Lin and A.W. Neumann, J. Colloid IntcrTace Sci 142 (1991) 224. J.
43
15 16 17 I8 19 20 21 22
J. Drclich and J.D. Miller. Part. Sci. Tcchnol.. submitted for publication. A. Scheludko. B.V. Toshcv and D.T. Bojadjicv. J. Chcm. Sot., Faraday Trans. I, 12 (1976) 2815. H.J. Schulzc, Int. J. Miner. Process., 4 (1977) 241. R. Crawford and J. Ralston, Int. J. Miner. Process.. 23 (1988) I. P. Blake and J. Ralston. Colloids Surfaces, 16 (1985) 41. P. Blake and J. Ralston, Colloids Surfaces, 15 (1985) 101. R. Crawford, L.K. Koopal and J. Ralston, Colloids Surfaces. 27 (1987) 57. M. Trau, B.S. Murray, K. Grant and F. Grieser, J. Colloid Interface Sci., 148 (1992) 182.