CHINA PARTICUOLOGY Vol. 3, No. 4, 208-212, 2005

ADSORPTION OF NANO-PARTICLES ON BUBBLE SURFACE IN NANO-PARTICLE SUSPENSION Buxuan Wang*, Chunhui Li and Xiaofeng Peng

Laboratory of Phase-Change and Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua University, Beijing 100084, P. R. China

*Author to whom correspondence should be addressed. E-mail: bxwang@mail.tsinghua.edu.cn

Abstract The adsorption of nano-particles on bubble surface is discussed for saturated boiling on thin wire of nano-particle suspensions. Owing to the decrease of surface tension for suspensions, the nano-particles tend to adsorb on the bubble surface to decrease the Gibbs free energy for stability, and meanwhile the velocity of nano-particles would be smaller than that of bubble growth. The long-range van der Waals force existing between “water particles” and nano-particles is considered the attractive force between the nano-particles and the bubble surface. Thus, the nano-particles would attach on the bubble surface if the particle-surface distance is smaller than its critical value. The distribution of nano-particles on the bubble surface and in the adjacent region is also investigated. Keywords nano-particle suspension, attachment, critical distance

1. Introduction Due to its widespread applications, boiling has been

extensively investigated, for a long time, with much atten-tion on boiling on thin wires (Lee, 1998; Gold et al., 2002; Shen et al., 1997). Wang et al. (2002; 2003; 2004) reported a series of experiments on subcooled boiling of pure liquids on thin Pt wires, including analysis of such novel phe-nomena as bubble sweeping, jet flow, and so on. Even newer is the boiling of nano-particle suspensions. Vassallo et al. (2004) conducted an experimental investigation on boiling heat transfer of silica nano- and micro-particle suspensions on a 0.4 mm NiCr wire at atmospheric pres-sure, and compared the results with those of pure water. A marked increase in CHF (critical heat flux) was observed for both nano- and micro-particle suspensions, though little difference was noticed in heat transfer prior to CHF.

The present authors (Li et al., 2004) recently presented experimental results on subcooled boiling of aqueous suspensions of 25 nm SiO2 on Pt wires. Bubble overlap-ping/clustering was commonly observed for nano-particle suspensions at low heat fluxes, probably resulting from the increase of bubble mass and attractive force between the nano-particles due to the adsorption of nano-particles on the bubble surface. However, the phenomena were still not well understood.

Research is limited on adsorption in the field of boiling, though it has been well investigated in flotation, which has been employed since a century ago as an important separation technique in the mining industry to separate and recover valuable minerals. The change of mineral mass in a given volume of flotation cell per unit time is affected by the transport process including macro-turbu-lence, convection of fluid and settling of solids on the one hand, and the flotation including collision, attachment and detachment between particles and bubbles on the other.

which was first introduced by Sven-Nilsson in 1934, is defined as the minimum time required for the film to reach a critical thickness to be ruptured spontaneously to form a stable multi-phase attachment in a solution (Schulze & Gottschalk, 1981). Among the many investigations on in-duction time, Hewitt et al. (1995) considered it to be equal to the time for a particle to slide around an air-bubble from the point of contact to the point where it stopped moving. Schulze and Gottschalk (1979) conducted a single air-bubble flotation test and noted that the induction time decreased with increasing contact angle and decreasing bubble size, and it decreased initially and then increased with increasing particle size for a specified bubble size.

Derjaguin and Dukhin (1961) first described bub-ble-particle interaction by considering surface forces. Be-fore a particle could adhere on the surface of a bubble, it has to pass through three distinct zones: the hydrodynamic, the diffusiophoretic and the wetting zone. In the wetting zone, three surface forces, namely, van der Waals, elec-trostatic and structural, need to be considered. In a later publication, Derjaguin and Dukhin (1979) suggested that small particles can directly adhere onto the surface of a bubble without penetrating the wetting film. The fine parti-cles could be held on a bubble surface by relatively weak forces such as van der Waals and electrostatic, because they are not subject to significant tearing-off forces due to their small inertia. Yoon (2000) attributed the difficulty in floating fine particles to the low probability of bub-ble-particle collision while for coarse particles, to detach-ment.

It was clear from the above review that research on flo-tation was confined to air bubbles, and the size of particles was almost micrometer until recently. In contrast, for the boiling of nano-particle suspensions, bubbles come from phase change and the size of particles was nanometer. During boiling, many nano-particles around the bubble provide a high possibility of their being attached onto the “Induction time” as an important concept in flotation,

Wang, Li & Peng: Adsorption of Nano-particles on Bubble Surface in Nano-particle Suspension

209

trate the attachment process and to describe the possible distribution of nano-particles on the bubble surface. This research will focus on the saturated boiling of aqueous suspensions of 25 nm SiO2 without considering the coa-lescence of nano-particles.

2. Basic Consideration When nano-particles are assumed evenly dispersed in a

base liquid, as shown in Fig. 1, the center distance be-tween nearby particles, a, could be expressed as (Li et al., 2004),

3p p p6a d cMρ= π , (1)

where pd is the diameter of the nano-particles, pρ is the

density of the nano-particles, pM is molecular weight of

SiO2, and c is the molar volume concentration of nano-particles.

a

a

a

Fig. 1 The ideal distribution of nano-particles in suspensions.

2.1 Change of Gibbs function It is well known that the system Gibbs free energy must

be kept at its minimal value to maintain stability. The components in solutions on the bubble surface could be altered to influence the system Gibbs free energy, that is, the particle would be concentrated near the surface to decrease the system Gibbs free energy if the addition of particles could decrease the surface tension.

The surface tension of aqueous 25 nm SiO2 suspen-sions is determined by measuring the height 0h of liquid rising in the capillary tube (inner radius 0.80 mmr = ) as shown in Fig. 2. To account for the concave liquid surface, the capillary rise, lh , is corrected from the measured value

0h as 2 3

l 0 30 0

0.1288 0.13123r r rh h

h h= + − + , (2)

from which the surface tension, σ , can be derived as

l( )2cos

gr hρσθ

Δ＝ , (3)

where ρΔ expresses the density difference between liquid and vapor, g is the gravity acceleration, and θ is the contact angle.

Fig. 2 Measurement of surface tension.

The measured surface tension of pure water was 73.98 mN⋅m-1, which is consistent with the handbook (Lide, 1990) value of 71.96 mN⋅m-1, with a relative deviation of 2.80%. The measured results of surface tension for aque-ous 25 nm SiO2 suspensions are illustrated in Fig. 3, showing that the surface tensions of nano-particle sus-pensions are all lower than that of pure water and de-crease with increasing SiO2 concentration. This indicates that adsorption of nano-particles would decrease the Gibbs free energy and is a spontaneous process.

Pure water

0wt%

0.05wt%

0.20wt% 0.35wt%

0.50wt%

60

65

70

75

Surfa

ce te

nsio

n/(m

N⋅m

-1)

Fig. 3 Surface tension of water and suspensions.

2.2 Bubble growth rate The heat flux bgq needed for bubble growth can be ex-

pressed as 2

bg b g lg b4 dq r h rρ= π , (4)

where br is the radius of the bubble, gρ is the vapor

density, lgh is the latent heat for phase change from liquid

r R′

θ

0h

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to vapor. Assume the heat flux supplied from the electri-cally heated Pt wire of length b2r beneath the bubble is completely used for bubble growth, as shown in Fig. 4, and then the heat applied to bubble growth, q , during the time interval dt would be

b Pt Pt2 ( )dq r d q t= π , (5)

where Ptd is the diameter of the wire, Ptq the heat flux per unit surface area of the Pt wire.

b2r

Ptd

Fig. 4 The heat flux needed for bubble growing.

Combining Eqs. (4) and (5) yields

b b2 d dr r C t= , (6)

where Pt Pt g lgC d q hρ= . For b cr r= at 0t = , integration of

Eq. (6) gives 2

b cr Ct r= + . (7)

Thus, the bubble growth rate is obtained as b

bg 2c

dd 2r Cvt Ct r

−= − =

+. (8)

Typically, for 4Pt 10 md −＝ , 2

Pt 1.10 MW mq −= ⋅ , 3 1

g g1 1.6736 m kgv ρ −= = ⋅ , 6c 10 mr −= ,

1lg 2256.6 kJ kgh −= ⋅ ,

the initial bubble growth rate as 0t → would reach 40.8 m⋅s-1. The time of bubble growth is thus limited to just a few thousands of a second, and the estimated bubble growth rate is 1 1

bg0.14 m s 13.5 m sv− −⋅ ≤ ≤ ⋅ for the dura-

tion of 7 310 s 10 st− −≤ ≤ .

2.3 Velocity of nano-particles in motion Before a nano-particle is attached to a bubble surface, it

moves in the surrounding liquid with velocity lv . Assume

the Brownian collision velocity of liquid molecules to be Bv ,

and then the velocity of a nano-particle p B lv v v= + . From

DLVO theory (Fu et al., 2001), the mean Brownian velocity of nano-particles can be expressed as,

Bp

/6

kTv td η

=π

, (9)

where 23 11.38 10 J Kk − −= × ⋅ is the Boltzman number, η the dynamic viscosity, T temperature. For 373 KT = ,

9p 25 10 md −= × , 628.8 10 Pa sη −= × ⋅ , the predicted

Brownian velocities are listed in Table 1. Apparently, the Brownian velocity is two orders of magnitude smaller than the corresponding velocity of bubble surface (which is equal to the bubble growth rate).

Table 1 Brownian velocity of nano-particles at the different time

t/s 10-8 10-7 10-6 10-5 10-4 10-3

v /(mm⋅s-1) 195 62 19.5 6.2 1.95 0.62

The fluid velocity field around a bubble was calculated using a Fluent software (Wang, 2004). For the case of bubble diameter 0.06 mm, temperature of Pt wire 380 K, and temperature of water 372 K, the predicted fluid velocity field is shown in Fig. 5, indicating that the liquid velocity is the highest at top of the bubble, e.g., about 0.03–0.06 m⋅s-1, which decreases with increasing diameter. The bubble growth rate in the time range of 7 310 s 10 st− −≤ ≤ is computed to be 0.14 m⋅s-1≤vbg≤13.5 m⋅s-1, which is much higher than that of the nano-particles, 0.03–0.06 m⋅s-1, and thus, the nano-particles are expected to be pushed by the bubble surface to move with the growing bubble.

Fig. 5 Fluid velocity field around a bubble.

The difference in velocities indicates that nano-particles would be easily adsorbed on the bubble surface, that is, the decrease in interfacial tension for nano-particle sus-pensions and the difference in velocity for bubble growth and nano-particle movement in the liquid, provide substan-tial evidence of nano-particle adsorptive attachment on the bubble surface.

2.4 Criterion of adsorption The attachment process of nano-particles on a bubble

surface is considered similar to that discussed by Der-jaguin and Dukhin (1961; 1979), i.e., nano-particles might be directly attached on the bubble surface by a relatively weak force. The long-range van der Waals force is there-fore considered the attractive force in this paper. According to the DLVO theory (Fu et al., 2001), the attractive energy between particles aV is expressed as:

Wang, Li & Peng: Adsorption of Nano-particles on Bubble Surface in Nano-particle Suspension

211

2

a 2 2 2

1 1 2[ 2ln ]12 2 2 1 2 1A X XV

X X X X X X+

= − + ++ + + + +

, (10)

where A is Hamaker constant, and p p3

p p

16

a dX

d cMρ− π

= = −

is the dispersion parameter. The attractive force between particles is

a aa 2 2 3

p

d d dd d d 6 ( 2) ( 1)V V X AFa X a d X X X

= − = − =+ +

. (11)

If the 25 nm SiO2 particles can be attached onto a bub-ble surface, the attractive force on the bubble surface must be large enough to overcome the effect of gravity. If the bubble surface is assumed to consist of “water particles”, as shown in Fig. 6, with the same diameter as nano-particles, the nano-particles will be attached on the bubble surface only when the attractive force between “water particles” and nano-particles is larger than gravity, that is, according to the following critical condition,

310a-10 p p2 2 3

p6 ( 2) ( 1) 6AF d g

d X X Xρπ

= =+ +

, (12)

where A10 is the Hamaker constant between “water parti-cles” and nano-particles, A00 is the Hamaker constant of “water particle”, A11 is the Hamaker constant of nano-particle, which is expressed for the particles with the same diameter and spacing distance as:

0.510 11 00( )A A A= . (13)

The computed critical value is

cr 6.23X = or

cr cr p( 1) 181nma X d= + = (14)

a

Bubble

Fig. 6 Interaction between “water particles” and nano-particles.

When the distance between nano-particles and bubble interface is below this critical value, the nano-particles can overcome gravity, and hence be attached onto the bubble surface.

3. Distribution Behavior A bubble would push away nano-particles during its

formation and growth. The number of nano-particles in-volved in the volume displaced by the bubble is calculated according to an ideal distribution as

3b p

3p p

d cMN

d ρ= , (15)

where bd is the diameter of bubble. When the nano-particles are attached to the bubble surface as a mono-layer, the total interfacial area covered by the nano-particles will be

2p

p 4d

S Nπ

= , (16)

Compared to the bubble surface area, bS 2

b bS d= π , (17) the following ratio s between these two surfaces can be written

p b p

b p p4S d cM

sS d ρ

= = . (18)

For 4b 5 10 md −= × , 9

p 25 10 md −= × , 3p 2600 kg mρ −= ⋅ ,

1p 60 kg kmolM −= ⋅ , the values of s for different concen-

trations calculated from Eq. (18) are illustrated in Table 2.

Table 2 s for different concentration suspensions

c / wt% 0.05 0.20 0.35 0.50

s 1.0 3.8 6.7 9.6

The above table shows that for a 0.05wt% suspension, the nano-particles might be firmly attached onto the bubble surface as a mono-layer, and for 0.20wt% and 0.35wt% suspensions the nano-particles might be firmly attached onto the bubble surface and might move along with the bubble. For 0.50wt% suspension, however, only the nano-particles located at a distance smaller than the criti-cal distance from the bubble surface will be attached on the bubble surface, while those outside the critical range might be loosely dispersed near the bubble surface.

4. Conclusions (1) The surface tension of nano-particle suspensions is

smaller than that of pure water, and decreases with increasing concentration, resulting in the attachment of nano-particles onto bubble surface during boiling to decrease the system Gibb’s free energy. The velocity of bubble growth is higher than that of Brownian movement of nano-particles surrounding the bubbles. These findings provide substantial evidence of the at-tachment of nano-particles on the bubble surface.

(2) The long-range van der Waals force, attracting nano- particles to “water particles” on the bubble surface would favor the attachment of nano-particles onto the bubble surface. A critical distance to overcome gravity is proposed to ascertain the adsorption of nano-particles on bubble surface when the spacing between the par-ticles and the bubble is smaller than the critical dis-tance.

(3) When this spacing is smaller than the critical distance,

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the nano-particles can be attached onto the bubble surface and move along with the bubble, while for spacing greater than this critical distance, the nano- particles cannot move with the bubble.

Acknowledgement This project is supported by the National Natural Science

Foundation of China (Grant No. 59995553).

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Manuscript received March 4, 2005 and accepted July 6, 2005.