Kinetics of spreading from solid particles on a water surface

Colloids and Surjnces. 66 (1992) 139- I47 Elsevier Science Publishers B.V., Amsterdam

Kinetics of spreading from solid particles on a water surface Ji-Ping Fang and P. Joos Department

UI A, University

of Chemistry,

(Received 20 January

1992; accepted

of Antwerp,

Universiteitsplein

I, B-2610

Wilrijk,

Belgium

13 April 1992)

Abstract The phcr.omenon observed when a drop of a relatively insoluble liqcid, e.g. oleic acid, is placed on a clean wz.t~cr surface, is that it quickly spreads out over the surface. For a solid particle placed on the water surface, however, the transfer of surfactant from the crystal on to the surface is rate controlling over the Marangoni effect. In the past, investigators described the spreading kinetics by a first-order reaction, and it was assumed that the spreading material behaved as an insoluble monolayer without any transfer to the aqueous bulk phase due to diffusion. In fact, the spreading kinetics are from the solid phase on to the water governed by two relaxation times, ox and TV. Ggnifying the transfer of surfactant surface, and the subsequent diffusion into the aqueous bulk phase. An equation is presented to describe the kinetics of this spreading from solid particles on the water surface, with r as an adjustable parameter (r = 2(~~/r~n)“~). The purpose of this paper is to take the diffusion into the water phase into full account. Keywords:

Diffusion;

kinetics;

relaxation;

spreading;

surfactant

particles;

Introduction

When an oil drop of a surfactant, e.g. oleic acid, is placed on a clean water surface, it quickly spreads out over the surface. Since the surface pressure near to the oil drop corresponds to the equilibrium spreading pressure (ESP), the spreading kinetics is governed by the Marangoni effect [l-3]. In this case, on the water surface infinitely close to the spreading drop, there is local equilibrium between the spreading bulk phase and the water surface. This means that the transfer of surface-active material from the spreading oil phase to the water surface is very fast. For a solid particle placed on the water surface, the transfer of surfaceactive material from the solid phase to the water surface is slow. As a result this transfer is rate Correspondence to: J.P. Fang. Dept. University of Antwerp, Universiteitsplein BeIgium. 0166-6622/92/$05.00

0

1992 -

of Chemistry, UIA, I, B-2610 Wilrijk,

Elsevier

Science

Publishers

transler.

controlling over the Marangoni effect. The kinetics of spreading from solid particles were first investigated long ago [4-G], and were described as first order. It was assumed that the spreading material behaved as an insolubie monolayer without any transfer to the aqueous bulk phase due to ditrusion. Indeed, for a homologous series of surfactants, e.g. long-chain fatty acids, the diffusion exchange becomes less important the larger the hydrocarbon chain. I-Iowever, the spreading rate also decreases with increasing chain length. In fact, the spreading kinetics are ruled by two relaxation times, TK and sD, signifying the transfer of surfactant from the solid phase on to the water surface and the subsequent diffusion into the aqueous bulk phase, and since both relaxation times decrease with increasing chain length it is not clear that diffusion may be neglected. ‘ ‘eer and Van den Tempel [7] considered, although in a somewhat different situation.. that in both processes, even for long-chain surfacB.V. All rights

reserved.

140

tants, the diffusion process cannot be neglected. Moreover, the transfer from the solid phase to the aqueous surface is governed by geometric conditions: the area R of the water surface and the perimeter C of the particle. Whereas the diffusion process does not depend on these conditions, it can be expected that for a particle with a small perimeter and spreading over a large area, diffusion may become more important. The purpose of this work is to take diffusion into the water phase into account.

the present one, in which the integration of the diffusion equation becomes very tedious. In these cases where the d’Rusion equations can be integrated, for instance, diffusion to a surface [9] or mass transfer across an interface [lo], the result is very well approximated by diffusion penetration theory. Since the bulk concentration is zero, Eqn (I) becomes dl-_=_~(~_r,)_~G_ dt

Jz

As a further approximation we assume a linear relationship between the subsurface concentration and the adsorption, giving

Theory

WC considered a solid particlc, say Iauric acid, placed on a clean water surface. The transfer from the solid phase on to the water surface is governed by first-order kinetics. Subsequently, the spread lauric acid diffuses into the aqueous phase. Hence the law of conservation of mass at the interface

gives (1) 0

where r is the adsorption of lauric acid, r the time, I-, the adsorption of lauric acid at equilibrium, corresponding to the equilibrium spreadhig pressure (BP), I; the rate constant for the transfer from the solid phase to the monolayer (k- ’ = rk, the transfer relaxation time), D the diffusion coefficient in the aqueous phase, C the concentration of lauric acid which diffuses from the surface to the bulk phase and is a function of Z, n;hd 2 the coordinate normal to the surface directed into the bulk phase. The concentration of lauric acid is zero for Z= z and is C’s, the subsurface concentration, at Z = 0. It should be recalled that the subsurface is the layer immediately below the surface and beIonging to the bulk phase. To avoid in?egrating the diffusion equation we approximate the diffusion flux in Eqn (I) using a diffusion pcnctration theory [S]. which is only an approximation, but is useful in situations such as

_rdC

c s-

dT

(3)

Equation (3) assumes, in a limited range, a linear relation between the adsorption and the concentration (according to the Henry isotherm). The expression dC/dT represents a thermodynamic !Jarameter obtainable from the equilibrium adsorption-concentration relationship. Equation (3) implies a local equilibrium between the subsurface and the surface. A more complex equation than Eqn (3), relating the adsorption with the concentration, is the Langmuir equation, where (I is the Langmuir constant, r = r” C/(a + C) requiring idea1 surface behaviour. Since for most present systems the requirement of ideal surface behaviour is not fu!filled, surfaceactivity coefficients have to be introduced into this Langmuir equation, making the mathematical analysis even more complicated. To grasp the essential features of the spreading process, we content ourselves, as an approximation. to use Eqn (3). Strictly, this equation is confined to a small concentration range, but we have shown experimentally [l I] that it gives suitable results over a wider range. Using Eqn (3), Eqn (2) becomes (4)

J.P. Fang. P. Joos/Colloids

where

TD,

Equation form

Surfaces 66 (1992) 139-117

the diffusion relaxation

(4) can be expressed

141

time, is given by

in a dimensionless

where x = kt and r is a function of the ratio of the relaxation times for the transfer and the diffusion processes: (7) where k-l = tk and n= 3.1415..., not to be confused with the surface pressure n. Considering that for t = 0 (x = 0) 1” = 0, integration of this differential equation (Eqn (7)) for r = 0 (rk <
(8)

and for r#O X

r=r..exp -(x+r&)

exp(u+r&)du

(9)

s 0

It is seen from Eqn (2) that for t --, ~9, the diffusion flux approaches zero and r+ r,; hence the ESP is measured as expected. Further, it is required that x I-

fl

hm exp -(x -t r&) x-+m 1

-E rte

7

exp (u -I- r&) J

du

and the surface pressure is, as in the oversimplification of Eqn (3), used here to make the mathematical treatment easier. As seen from the experimental record in Fig. 3 (see below), this does not hold over the entire range. At very low surface pressures, the surface pressure hardly increases with adsorption. As a bold approximation, this curve may be split into two regions: one where the surface pressure is zero with increasing adsorption, and another where there is a linear relation between the two. In the subsequent treatment of the data, only the second region is considered. It is expected that the experimental result described by Eqn (10) with r as an adjustable parameter. to be compared with other data. In Fig. 1 we have plotted n/n, as a function of x for different values of the parameter Y. It is seen that for r = 0.1, the curve approaches an exponential function as given in Eqn (8); this means that for increasing values of r, diffusion becomes more important. The increase in surface pressure is less. The parameter r is a function of the two relaxation times, 1k and TV, because for the increasing chain length of a homologous series, ~b and ok both increase, and ii is not clear how this parameter behaves, e.g. whether the diflusion can be neglected. The kinetic term - k(T - r,) in Eqn (1) is consistent with Langmuir adsorption kinetics from the YoIid phase. The rate of adsorption from the solid base on to the surface is proportional to the unoccupicl fraction (1 - r/rm) and to the concen-

= 1 J

0

This is confirmed by numerical evaluation of this function. We assume, further, a linear relation between the adsorption r and the surface pressure n: x

n/L/,

= exp -(x

+ r&)

exp(u + r&)

du

(10)

s 0

Again, a linear

relation

between

the adsorption

Fig. 1. The effect of the parameter r on the spreading kinetics of solid particles: --, according to Eqn (IO) with different values of r (r = 0. I (curve 1); r = I (curve 2); r = 2 (curve 3)): - - - , exponential (r = 0).

tration or mole fraction in the bulk which is now unity; the desorption rate is proportional to the occupied fraction:

where k, and k, are the adsorption and desorption rate constants respectively. Since at equilibrium r = I-, and dr/dt = 0, it follows that di_=dt

(k, + &)(r, l-’

_ r)

the surface:

where ‘1 is the bulk viscosity and v is the spreading velocity at the surface. As discussed before [2,3], this shearing stress is approximated by using the hydrodynamic penetration depth, 6 z (r~t/p)“~, where p is the density. In this way, after integration, we obtain the following equation:

(11)

l

n(0.

2

It is also known [I l] that the rate constant, k = (k, + kZ)/I_‘, depends on the spreading area R and the perimeter Z of the solid particle in contact with the liquid surface. Therefore we substitute

(12) where ,I,, which has the units mol cm - ’ s-l, is a flux not across the unit area but across the unit length. and is in some way to be compared with the exchange current for transfer-controlled electrode kinetics. Therefore we call Jo the exchange flux. We can now try to take the Marangoni effect into account. When the solid particle is put on to a clean water surface, near to this particle there is some surface-active material on the surface, but far away from this particle, the surface is still pure water. The surface-active material is conveyed over the surface by a surface tension gradient. If 4 is the distance f ,-,lvelled by the expanding monolayer, the surface tension (rr) gradient, as discussed before [2,3], is da -_= ds

beneath

fi(O, t) <

It should be noted that the parameter x is the coordinate along the surface, not to be confused with s = kr. The function I7(0, t) is the surface pressure at time r near the spreading phase (x = 0). This surface tension gradient is compensated by a shearing stress in the adjoining liquid layers

t)J;

dt = <2(i~p)“z

s

(13)

0

If the spreading phase is a liquid, J7(0, t) is no! a function of time and is equal to the ESP (J7,); this means that transfer from the liquid phase (oil drop) to the surface adjacent to the phase is fast and that local equilibrium is established between this phase and the surface near to this phase. Equation (13) then results in [=

4 1’2 n,l’2 t3,4 3 (VPY4

0

(14)

Previously, this equation has been verified by experiment and the agreement was quite good. Since these results are presented in a less accessible publication [12], they are reproduced here (see Fig. 2). For the present system, however, the surface pressure is time dependent. As will be explained in the experimental section this pressure is measured. Considering that during a time interval ti + l - ti, this surface pressure is constant ((n},), the integration of this equation gives (is)

where ii and ii+ 1 are the distances times ti and ti + I. respectively.

travelled during

Experimental

The products used in this investigation, i.e. high alcohols and high fatty acids, were of Analar grade

J.P. Fan_g. P. Joos/Colloids Surfaces 66 (1992) 139-147 Y km)

I

Fig. 2. The spreading of liquids: the distance 1; travelled by the expanding monolayer during time t: curve I. triolcin (0); curve 2. oleic acid (0); curve 3, decanol (+).

and were used as received. The tip of a Pasteur pipette was melted using a Bunsen burner in such a way that a glass ball was formed by gravity. The crystals were heated gently until they melted, and the tip of the Pasteur pipette was dipped in the melt and subsequently cooled. The perimeter of the crystal was measured, C = 0.95 cm. The tip was then brought into contact with a clean water surface in a Petri dish or a Langmuir trough in order to change the spreading area. The crystal that had condensed on the tip of the Pasteur pipette was removed from the surface after measurement, and was dried carefully with filter paper before further use. The surface tension was measured with a Wilhclmy p!ate connected to a Cahn electrobalance, and was monitored as a function of time using a strip chart recorder. As the solid surfactant touched the surface, the surface tension did not decrease at once; there was, in some cases (for the higher homologous compounds), a time lag during which the surface tension remained constant and equal to the surface tension of water.

143

After this, there was a decrease in the surface tension. At the beginning of this decrease the time was taken to be zero (see Fig. 3). The experimental set-up for measuring the distance travelled by the expanding monolayer was as that described previously [ 12,131. A long glass tube was cut along its length, placed in a horizontal position and filled with water. At one end of the glass tube was positioned a Wilhelmy plate, which was coupled to a force transducer with a microscale accessory (UC Gold cell, Statham), its output being connected to one channel of a strip chart recorder. Near to this plate (Wilhelmy plate 1) the solid material was placed on the surface as described. Another similar plate (Wilhelmy plate 2) was positioned at a variable distance t: from the first plate. This second plate was also coupled to a force transducer, the output then being connected to the second channel of the strip chart recorder. In this way, the time required for the expanding monolayer to travel a distance c is measured. The first Wilhelmy plate measures the surface pressure near the solid spreading phase, especially after this monolayer has travelled for a time t, where I?(O, t) is the surface pressure at s = 0 (see Fig. 4). Results and discussion

A thorough investigation of the spreading of lauric acid from the solid phase on to the water surface was made by changing the perimeter C (one, two and three rods were placed symmetrically

n.lmHm“l1

Fig. 3. The surface pressure II (mN m-I j of the spreading of lauric acid as a function of time r (s) (R = 184 cm’. and 1, is the time lag).

144

10

Fig. 4. The surface prcssurc II (mN m- ’ ) mc;~st~rcd hg two Wilhclmy plates which wcrc difTcrcnt distances rrom the solid particlc.

Fig. 6. As Fig. 5 (ESP. 71.7 mN m _ I: Z = 0.95 cm): curve I. !Z = 23.X C117’. Ii = 14.0. IO-’ S- ‘. I’ = (I.?5 (-1): CUTVC 2. R = 76.!! crnz. k = 7.1 - IW2 s-l. r = 0.38 (I): curve 3. .C?= I?0 cm2. k=5.6*IO-‘s-‘. r.=O.47 (a:‘); curve4. I?= 153cm’. k= I.h.IW’s-1. curve 5. R = 77-J cm’. )’ = 0.58 ( I-11: k=3.4-10-~s-‘. r = 0.75 (7); solid lines. according IO Eqn I IO).

on the water surface) and the area R. The results are given in Figs 5 and 6. It can be seen that all the curves arc described by Eqn (IO). The data arc compiled in TabIc I. The mean for the diffusion relaxation time is (r,,) = 98 s and this value is comparable to that for desorption from a spreading monolayer of lauric acid (rD = 123 s [I I]). If the perimeter is large and the area R is small, the rate constant k is large and consequently I* is small. This means that diffusion is of minor importance. The mean value of the cxchangc flux Jo, with r’ =7.10-‘0 is J0=4.7* IO-” mol cm - ‘,

mol cm-’ s- ‘. As seen in Figs 7 and 8, the results arc well described using this exchange flux. Finally, we considered the spreading kinetics for undecanoic. lauric and myristic acids. dodecanol and tetradecanol, for R = 64 cm’ and X = 0.95 cm. The results arc shown in Fig. 9 (and in Fig. 10 for myristic acid using a larger time scale) and Table 2 (r’ = 7 - lo-” mol cm-‘). The results are described by Eqn (10). For dodecanol the increase in surface pressure with time was exponential and diffusion can be neglected. The diffusion relaxation time for myristic acid obtained by desorption for

ni

TABLE Sprwding

1.0

,-1

c

\-

to

IV/i (s-l)

3.85 0.95 0.95 0.95 0.95

154 154 154 33.8 64 76 96

81 93 s4 145 57 174 to5

4.0 9.5 11.4 14.0 9.0 7.1 8.4

0.58 0.38 0.35 0.25 0.50 0.38 0.38

0.9J

I’0

103

0.95

I.52

93

0.95

lo

20

K:

tk.1

Fig. 5. The ratio or the surface prcssurc or sprcxding lauric xciu oi lime I isi (32= i5ii cm’: Hia. pioitcd as a iuncrion 21.7mNn1-‘):curvcI,~=O.95cm.k=4.6~lO-’s-’.1-=0.58 k=9.5*10-‘s-‘. r=O.38 (L): ( U 1: curve 2. .Y- t.9Ocm. k= 1?.4.I(i-‘S-l. r=0.35 (0); solid curve 3. Z=UScm. lines. acccxding to Eqn (IO).

ol’ lauric acid

(5)

I .90

0

kinetics R (cm’)

(cm) --

05

I

0.95

184

0.95

17-1

III 76

r

10’ J,,

(mol cm-’ 5.3 5.4 4.7 7.5 4.3 4.0 6.0

- ’

0.4:7

5.n

i:‘;

0.58

4.6

3.4

US8

4.6

3.4

0.70

5.7

s-l)

J.P. Fang. P. Joos/Colloids

Swfirces 66 (1992) 139- 147

145

2010’2

Fig. 9. The ratio of the spreading surbx prcssurc of diRerent fatty acids plotted as a function of time t (s) at constant arca (S2= 64 cm2) and constant perimctcr 0: = 0.95 cm): curve 1, undccanoic acid: k = 20 IO- 2 s- ‘, r = 0.65. ESP = 30.8 mNm-’ (A); curve2. lauric acid: k=9.0*10-‘s-l. r--OS. ESP = 21.7 mN m-’ (0): curve 3, myristic acid: k= o.7*lo-1 s-l, I’ =0.35, ESP= 10.2 mN m-l (Cl); solid lines. according to Eqn (IO). l

1

0

3

2

Etcm)

Fig. 7. The clTcct of the pcrimctcr L of the crystals on the spreading kinetics of lauric acid at constant arcn (Q = 154 cm2): -, according to Eqn (12). with Jo = 4.7. IO-” mol cm-’ se1 and ~‘=7~10-‘“m0lcm-‘.

monolayer is rD = 1890 s [l l], which is comparable with the present results. It can be seen that for both parameters T,, and Jo, there is a linear relationship between their logarithms and the chain length n. Hence

a spread

($ = 6.44 lo-' s, a = 1.55)

51)= zg exp all

where c1and /I are constants giving the dependence of the diffusion relaxation time and kinetic relaxation time with chain length, respectively. From this we obtain the parameter P: I’=

4Q -XC a

J

1

exp (B - r)n 2

The parameter r may be split up into a contribution depending on the experimental condition (Q/.Z)‘.‘“, and on a contribution proper to the system:

Jo = JE exp -/In (Jg = 3.40 10e3 mol cm-’

s-l, fi = 1.15)

which has a physical

kls”) )‘=

lSld'-

0

Q l’? 2.42 2’ -

l

meaning, whence

loa exp (- 0.2011)

lOlr?-

51d2-

t0

so

100

150

200

2'0 Rkm')

Firr.S. The

cITcct of the spreading surface arca C2on the sprcadkinctizs of laurlc acld at LuuaidS: pcrimcwr (,Y = 0.95 cm): --, according to Eqn (I& with Jo =4.7* I?-’ mol cm-’ s-’ and r’ = 7.10-l” mol cm-?. ing

_.

l-l&

Q, for mvristic acid using a larger 1L As rig. L

-,

according to Eqn (IO).

time

scak:

146

J.P. Fang. P. JoosiCotloids

TABLE 2 Spreading

3

Srwjkes

66 (1992) 139-147

Km1

kinetics of fatty acids and higher alcohols -

CadJO13

IO’k

chain length

;s:

Fatty ecids c,* C,, r--14

15 57 1485

r

(s-‘f

20.0 9.0 0.7

(mol cm-’

0.65 0.5 0.35

SO

toy .T,

s-‘)

9.4 4.3 0.33

tlighcr alcohols C,* C IJ

20 14

0 0.0

I

100

4.1

from which it is seen that the effects of the chain length on the diffusion relaxation time and on the reaction relaxation time compensate each other. An increase in chain length by four units results in a decrcasc in r, by a factor of 2. This equation seems useful for estimating the conditions under which, and for which materials, the diffusion process may or may not be neglected. The results for an expanding monolayer of undecanoic acid, giving the distance < travelled during a time t, are shown in Fig. ! 1, in which we have plotted the surface pressure II(O, I) measured by the first Wilhe~my plate. With these data using Eqn (1.5) we calculate the distance i as a function of time (intervals of 1 s were used). The result is shown as a solid line in Fig. 11 and good agreement between thecry and experiment is achieved. Since the surface pressure L!(O, t) increases with time, a steeper increase ir [ is expected than is given by the 0.75 power la! d. The change in surface pressure I7(0, t) with time is not to be compared directly with Eqn (10) since according to Eqn (1 I) the rate constant k depends on the area (< x b, where b is the width of the tube (b = 3.7 cm)} and is not constant. Moreover, the diffusion process is also more complicated since aiter time t the molecules in the monolayer have diffused in the underlying water phase, but at the beginning of the expanding monolayer, there is no diffusion at ah. Nevertheless, we have calculated the change in surface pressure with time using Eqn (IO), with rn = Ifi s and G!, the area of the tube (52 is the natural area}. :n this

S

10

t(s)

Fig. Il. Kinetics of the spreading ing to Eqn (15).

undecanoic

acid. -,

accord-

way, the calculated I7(0, t) curve is an underestimate, We see, however, in Fig. 12 that the experimental points are lower than those predicted by theory especially after a long period of time. The reason for this is still obscure.

IT (mN.m”l

I

W’

*

I

*

12345678

*

I

*

t

f

I

9

I

10

T(S)

Fig. 12. Surface pressure of undecanoic acid at plate 1 at s = 0, when the expanding monolayer has travelled a distance < during a time r: (0). experimental data obtained for every value of [; (x), expcrimcntal data !7(0, r) monitored at the first Wilhelmy plate: curve 1 (-- -), according to Eqn (10) (ESP= 21.7 mN m- ’ ); curve 2 (-), experimental data.

J. P. Fatly, P. Jms/Calloids

.Surfac
147

Acknowledgements

3 4

J.P. Fang is indebted to the Belgian government for a research grant during the course of this work. Thanks are due to Dr Guido Serrien for his help with some 0:’ the experiments.

5 6 7 8 9

R 4erenccs

Ii)

II L 2

D.G. Suciu. 0. Smigc!schi and 5. Ruckcnstcin. J. Colloid Inierface Sci., 33 ( l97Sj 520. J. Ahmad &nd R.S. Hansen, J. Colloid Interface Sci., 38 (1972) 601.

I2 I3

P. Joos and J. Pintens, J. Colloid Interface Sci.. 60 (1977) 507. A. Cary and E.K. Rideal, Proc. R. Sot. London, Ser. A. 109 (1925) 301. A. Roylance and T.G. Jones, J. Appl. Chem.. 9 (1959) 621. W.W. Mansfield, Aust. J. Appl. Sci., 12 (1959) 382. F.A. Veer and M. Van den Tcmpcl, I. Coiloid Intcrhcc Sci., 42 (1973) 418. R.B. Birt, W.E. Steward and E.M. Lightfood, Transport Phenomena, Wiley, New York, 1960. R. Van der Bogaert and P. Joe::. J. Phys. Chcm.. 83 (1979) 2244. J. Van Hunsel and P. Joos, Langmuir. 3 (1987) 1069. P. Joos and G. Bleys, Colloid Polym. Sci.. 261 (1983) 1038. P. Joos and E. Rillacrts, Lcvich Birthday Cot-X. Proc., 133, Oxford, June 1977. P. Joos and J. Van Hunscl, J. Colloid Intcrfacc Sci.. 106 (1985) 161.