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Adsorption kinetics with surface dilatation 5. Transient surface tensions of an adsorbed monolayer during compression at a constant dilatation rate
Colloids and Surfaces A: Physicochemical and Engineering Aspects 100 (1995) 245-253
ELSEVIER
COLLOIDS AND SURFACES
A
Adsorption kinetics with surface dilatation 5. Transient surface tensions of an adsorbed monolayer during compression at a constant dilatation rate P. Joos *, M. Van Uffelen Department of Chemistry,
University of Antwerp, Universiteitsplein
1, B-261 0 Wilrijk, Belgium
Received 22 February 1995; accepted 17 March 1995
Abstract In the present article the integration of the convective diffusion equation for compression at a constant rate is presented, which enables us to describe the course of surface tension with time for the desorption adsorbed monolayer. The theory is evaluated by compression experiments performed with decanoic Triton X-45. Keyucords:
Adsorbed
monolayer;
Adsorption
dilatation rate of an acid and
kinetics; Surface dilatation
1. Introduction Van Voorst Vader et al. [l] studied the dynamic surface tension of an adsorbed monolayer, initially in equilibrium with the bulk solution, by expanding the monolayer at a constant dilatation rate 8. They have showed that after a transient period of rising surface tension, a steady state is obtained, where the amount of surface active material removed by expansion is compensated by diffusion from the bulk. In the theory of van Voorst Vader only this steady state situation is analyzed. The transient surface tension also contains valuable information about the adsorption process. Therefore, the convective diffusion equation for expansion needs to be integrated [2-41 following the procedure outlined by Levich [S]. Experimentally, the monolayer is expanded at a constant dilatation rate using a logarithmic spiral.
* Corresponding author. 0927.7757,/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0927-7757(95)03167-7
Alternative methods have since become available: the strip method [6], the dynamic capillary method [7] and adsorption onto an expanding drop [S-lo] or bubble [ 111. If we consider compression experiments at a constant dilatation rate, the convective diffusion equation for an expanding surface does not apply [ 12,131. Arguments for this will be given later on. The integration of the convective diffusion equation for compression is more difficult than was the case for expansion, since the boundary condition for the conservation of mass at the interface cannot be expressed as a total differential. In the present paper, a solution for this problem is given. In order to check the theoretical results, this solution is evaluated by means of experimental data.
P. Joos, M. Van UffelenjColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
246
2. Theory
face will decrease with time:
2.1. The convective &fusion equation for expansion
a44 4 < at
and compression
A surface at equilibrium with the bulk solution, with initial area 9,, is expanded or compressed at a constant dilatation rate I!?,defined as 8=
Hence, the left-hand side of Eq. (2) is negative. Qualitative consideration of the concentration-distance profile shows that it is curved downwards (see Fig. l(b)):
wt, 4 <
d ln Q(t)
o
az2
dt
where Sz is the area and t is the time. For a surface expansion, 8 > 0, and for a surface compression,
e < 0.
Continuous deformation of a surface implies a convection current near to the surface, which enhances the diffusion process, and therefore must be taken into account in the diffusion equation. As van Voorst Vader et al. Cl] have shown for expansion, this convective diffusion equation is
a4 4 4 -_
o
8z
at
a2444 -a44 4 = D ~ az2 az
d>O
In Eq. (2) the time derivative of the concentration and the convective term have the same signs as the diffusion term, which means that the convection helps the diffusion process to attain thermodynamic equilibrium. Therefore, Eq. (2) is correct for an expanding surface. Let us now apply Eq. (2) to a compressed monolayer (8 < 0). The sub-surface concentration c, for this kind of disturbance is higher than the bulk
(2)
where c is the concentration, z is the coordinate normal to the surface and D is the diffusion coefficient. In the case of a surface compression, Eq. (2) gives erroneous results for both theory and experiments. Therefore, for a compressed monolayer:
a4 t, 4 +
Bz
at
wt, 4 = a2444 D
aZ
a22
e
(3)
The validity of Eq. (2) can be tested for a surface expansion and a surface compression as follows. If we consider a surface element which is expanded ((3> 0), the sub-surface concentration c, is less than the bulk concentration cO. For z > 0, the concentration gradient i3c(t, z)/az > 0 and the fluid moves from the bulk to the surface. As a result, the convection term in Eq. (2) is -8Z-
aa 4 <
to
CO
cs
o
aZ
On expansion, the concentration
near to the sur-
Fig. 1. Schematic representation of the concentration profile as a function of z and t: (a) for a compressed surface element (0 < 0), and (b) for an expanded surface element (0 > 0).
P. Joos. M. Van UffelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
concentration cO, so for z > 0, &(t, z)/~z < 0, and the fluid now moves away from the surface into the bulk solution. Hence, the sign of the convection term remains unaffected in comparison with the expanded surface. However, on compression the concentration near to the surface will increase with time: dc(t,z) >.
at and the concentration-distance upwards (see Fig. l(a)):
profile is curved
a2c(t, z) > o
a22 Thus, the convection term and the diffusion term have opposite signs, indicating that the convection should counteract the diffusion process by accumulating tensio-active molecules at the surface. This is obviously incorrect: Eq. (2) is inconsistent with a compressed surface. Consequently, for a compressed surface we use Eq. (3). This argument can be supplemented by considering the diffusion penetration depth 6nc. For a surface which is expanded at a constant dilatation rate 0, integration of Eq. (2) shows that the diffusion penetration depth is [4] 6,,=
$(1-e-2ez) [
l/2 e>o 1
rium with the bulk. The system will try to return to its equilibrium state by diffusion from the surface into the bulk. Instead of measuring the transient surface pressures with time, the surface pressure can be kept constant by further compression of the surface. Consequently, the change in surface area can be measured as a function of time. This kind of experiment was introduced by Ter Minassian-Saraga [ 141 for a slightly spread soluble monolayer. Calculating for a hypothetical desorption process without convection and starting from Eq. (2), it emerges that convection slows down the diffusion process [4]. Hence, we are confident that Eq. (3) should be applicable for a compressed surface. 2.2. Integration of the convective diflusion equation for compression (0 < 0) For integration of Eq. (3), we introduce, according to the method of Levich [S], a first variable x = zf(t)
(6)
where f(t) is an unknown function of time. In this way, Eq. (3) becomes aa, 4 T+z[~+sf(t)]%$
(4) =
If we compare 6,, with 6n, the diffusion penetration depth without convection (0 = 0) 6, = (7&y
247
(5)
we see that
D
a2444
Tf
2
(t)
We choose f(t) in such a way that the convection term cancels out: df(t)
,,+ef(t)=o (except for t = 0, when both are zero), indicating that convection speeds up the diffusion process. However, for a compressed surface (0 < 0), the diffusion penetration depth with convection, according to Eq. (2), is larger than that without convection, which is in contradiction with the fact that convection enhances the diffusion process. Therefore, Eq. (2) is only applicable for expansion. As a third argument, we consider sudden compression of the surface, which is initially in equilib-
(8)
Since 0 is constant, it follows from Eqs. (1) and (8) that f(t)=e-et=52,/i2(t)
(9)
and Eq. (7) reduces to
aa,XI = a244xl -f2(t) D
at
a2
We now eliminate
(10)
f2(t) by introducing a second
248
P. Joos, M. Van UflelenlCoNoids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
variable z: dc(z, x) ~_
az
introduce a new function 4 = d(r, T(z)):
a7 = at
D
d24?xl --jj+2w
(11) Hence, Eq. (18) can be written as
which is defined as ;
(19)
(12)
= j-2(t)
Therefore, Eq. ( 11) results in the ordinary diffusion equation: (13)
4 -_=
(20)
dz
Using Eqs. (13) and (20), the problem is reduced to that of Ward and Tordai [ 151. The result is .1/z 4 = 2(D/7~)“~
[co - c,(z - A)] di”2 !
where c is now a function of the variables x and z. In order to integrate this equation with the initial condition that at t = 0, c = co, we choose z such that for t= 0,z = 0:
z=
f’(t)dt
Using Eq. (9), we obtain the following expression for z:
(15) The bounary condition is the law for conservation of mass at the interface: dt
where J. is the integration variable. However, we still do not have an expression for 4. Therefore, we solve Eq. (19) with the requirement that at t=O(z=O), c$=O:
4=
0
dr( t)
0
(14)
s
+
s s f(t) w4
0
where r the adsorption. This equation must also be expressed as a function of the new variables x and 7. Thus, considering Eq. (S), we obtain
dt
f(t)
dz
s T
4 = f( 0-w -
r, - 2
r(7) v(t)
(23)
0
of Eqs. (23) and (21) yields p D
r(z)f(t)=r,+2
0 n
112
s
[co- c,(z-A)] d/I”2
0
dt
Using Eqs. (6) and (12), Eq. ( 17) reduces to
dW) ,f(t)-r(r)?
0
1 df(t) dz
-------_(t)=D dz
(22)
On partial integration, considering that at t= 0 (z = 0), f(t) = 1 and T(z) = r,, the equilibrium adsorption, corresponding to the bulk concentration cO, gives
Combination
dT(z) dz
m) W(t)
-
(16)
W(t) = D
(21)
= D
(y)
0
(18)
As can be seen, the left-hand side of Eq. (18)cannot be written as a total differential. Therefore, we
Since f(t) and z are known functions of t, Eq. (24) predicts the adsorption as a function of time. The numerical integration must be performed using a computer. Formally, the expression for adsorption in Eq. (24) is different from that for an expanding
249
P. Joos, M. Van UffelenlColloids Surfaces A: Physicochem. Emg. Aspects 100 (1995) 245-253
imental
surface by the integral
jump
in surface pressure:
417(r) = n(z) - z& = g
(T(z) - r,)
(31)
we obtain
0
In order to avoid the numerical evaluation of Eq. (24) by computer, we consider only small deviations from equilibrium [2,3,13]. In this situation, we may assume a linear relationship between concentration and adsorption: co - c,(z - A)=
$;(re - T(z -A))
eo[2-~]uv-11 AZZ(z) =
T(z)f( t) = r, + (4r/7rrrJ1’2 (r, - T(r))
where co is the Gibbs
elasticity,
Now, if (T(z)) gives
z T(Z) and 1 CC(4r/nrn)1’2,
AL’(z) = ~,2~~(7r/4r)~‘~ [f( t) - l] Hence, using the Gibbs 1,2
time, given by
_
~(r(4) -dn D1’2 dc
z
s
r(z) df( t) = W(3) U-(t) - 11
as a mean value:
(28)
Eq. (32)
(34)
equation
(RTZ-,2/CoLP2)
RTT2 = e co (~/4W2
(27) can be expressed
as
(35)
we can write Eq. (34) as
AIT
This last integral
defined
(33)
EOTD -
relaxation
1 + (4r/712,)“2
(25)
and, because at long times co - c,(z - 1) and therefore also r, - T(z - A) stays more or less constant, we may factorize these terms out of the integral:
where rn is the diffusion
(32)
Cf(t) - 1I
(36)
For a surface compression at a constant dilatation rate, bearing in mind Eqs. (9) and (15), Eq. (36) becomes
AU(z) = !?$i
(g>‘;’
tanh”f!$
0
whence
Eq. (26) becomes
r(r)f(t)
= r, + (4r/nrn)1’2(& + 2V(r))
[f(r)
= An, -T(r)) -
11
(29)
(37)
where [ 3,4]
A17 co
or alternatively
tanh’12 m ( 2 I
1’2 (38)
AT(z) = T(r) - r,
=
(42-$3f(1)-11
(30)
1 + (42/~2n)1’2
Since the difference in adsorption for small deviations from equilibrium is proportional to the exper-
since for t + GO,a steady state is obtained. Eqs. (37) and (38) describe the variation of the jump in surface pressure with time for a surface compression at a constant dilatation rate. We have previously obtained the same expression for a surface expansion at a constant dilatation rate
~31.
250
P. Joos, M. Van Uffelen/Colloids
Surfaces A: Physicochem.
3. Materials and methods
The materials used in this investigation were decanoic acid (analar grade 99 + %) obtained from Aldrich, and Triton X-45 (a nonionic surfactant: octylphenol-polyethylene glycol ether, containing approximately five ethoxy groups; a molecular weight of 427 is assumed) purchased from Serva. In order to be sure we were dealing with the acid form of decanoic acid (regular behaviour [ 163) in the experiments, the pH was kept lower than 4. Therefore, we worked with a 0.01 N HCl + 0.09 N NaCl solution, both products having been obtained from Merck. Thoroughly triple-distilled water was used and all experiments were performed at room temperature 22 f 1 “C. A surface, initially in equilibrium with the bulk solution, was compressed at a constant rate of dilatation 0. The variation of surface pressure with time was measured using a Wilhelmy plate, connected to a force transducer (Statham Gould cell), and the output signal was recorded on a strip chart recorder. For the compression of the surface, we employed the experimental set up described by van Voorst Vader et al. [ 11, using a logarithmic spiral, which was constructed in such a way to give an exponential decrease in the surface area.
4. Results and discussion
Eng. Aspects 100 (1995) 245-2.53
1 <<(4z/rcrn)“2 are fulfilled, we plotted the experimental jump in surface pressure 617(t) as a function of
From the slope, we calculated a value for L7, according to Eq. (37). Some of the experimental results are presented in Figs. 2-6. An(mN/m)
Fig. 2. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 10m7mol cm -‘.. the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) 1f?1=0.0066s-r, (A17,),,,,, = 5.6mNm-r; (x) (0l= 0.0026 s-r, (A17m)cxptl. = 3.2 mN m-‘. An(mN/m) I"""
’
”
”
'1
Experiments were performed to verify our theory concerning the transient surface tensions of a compressed adsorbed monolayer at a constant dilatation rate. 4.1. Decanoic acid Compression experiments were performed with the logarithmic spiral for four concentrations of decanoic acid: lo-’ mol cme3, 7 x lo-* mol cmm3, 5 x lops mol cme3 and 3 x lop8 mol cmP3 (the experiments with c,, = 7 x lop8 mol cmF3 were performed by De Keyser [17]). The following problem arose in the interpretation of the experimental data: steady state is often not reached when the logarithmic spiral has come to an end. Because for decanoic acid the conditions (T(z)) x T(z) and
Fig. 3. Results of experiments with the logarithmic spiral for decanoic acid, c,=7 x lo-* mol cn-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) [@I= 0.0066 s-r, (A17m)cxptl. = 13.1 mN m-r; (AZ7m)cxpt,,= 10.1 mN m-r; (A) 161= (x) )0/=0.0026s-r, 0.013 s-r, (A17,),,,,, = 8.5 mN m-r.
251
P. Joos, M. Van Uffelen/Colloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253 ATclmNlm) I 301
AnimNlml
1
I
I
I
::
10
100
Fig. 4. Results of experiments with the logarithmic spiral for decanoic acid, cc, = 7 x lo-* mol cmm3: the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) JBJ = 0.0066 s-r, (417,),,,,,, = 6.9 mN m-l; (x) 101=0.0026s~‘, (AZZW),,,,,,=5.4mNm-‘; (A) lfIl= 0.0013 s-r, (A17,),,,,,, = 4.4 mN m-l.
200
300
400
500t(s)
Fig. 6. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 3 x lo-* molcm-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates, The theoretical curve is calculated according to Eq. (37): (+) l0l= 0.026 s-l, (417,),,,,, = 18.9 mN m-‘; (0) 161=0.013 s-l, (Al7&,,, = 13.7 mN m-r; (x) 101= 0.0066 s-r, (AlI,),,,,,, = 10.4 mN m-r; (0) 161= 0.0026 s-r, (ALr,),,pt,. = 7.1 mN m-l.
tion rates, this does not present any problems, since the minor components adsorb at large times. The experimental jumps in surface tension in the steady state are summarized in Table 1. If we compare them with the theoretical values calculated according to Eq. (38) with D = 5 x lO-‘j cm2 s-l, we can see that in the absence of adjustable parameters, the experimental values are of the right order of magnitude.
AnimN/ml
ISI
4.2. Triton X-45 50
100
150
250
IfsI
Fig. 5. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 5 x lo-* mol cm-? the jump in surface pressure Ail is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) 181= 0.026 s-r, (A17,),,,,,, = 14.4 mN m-r; (x) ISI =0.013 s-r, (Al7,),,,,,, = 10.1 mNm_‘; (0) 181= 0.0066 s-r, (AL’,),,,,,. = 8.2 mN m-r.
For low dilatation rates, we observe that the experimental points do not tend towards a steadystate surface tension. This is due to the fact that the time scale of the desorption of the surfactant under study coincides with the desorption of minor components, which are adsorbed onto the surface during the equilibrium process. For higher dilata-
Experiments with the logarithmic spiral were carried out for two concentrations of Triton X-45: lop8 mol cm-j and 5 x 10e9 mol cmm3. Interpretation of the experiments was performed in exactly the same was as for decanoic acid, since for Triton X-45 the conditions (T(z)) z T(z) and 1 K (4z/7cz,) are also fulfilled. Some of the experimental results are shown in Figs. 7 and 8. The resulting parameters are reported in Table2 (with D=5 x 10-6cm2s-1).
5. Conclusions
As a general conclusion, we can say that the theory presented to describe the transient surface
P. Joos, M. Van UflelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
252
Table 1 Decanoic acid - logarithmic spiral. Jump in surface at the steady state 417, for different dilatation rates
Arc(mN/m) pressure
/
I
I
I
I
-I
30c
14
(Anco hxptl.
An,
(s-l)
(mN m-‘)
from Eq. (38) (mN rn-‘)
1. c0 = 1O-7 mol cm-’ 2.62 x 1o-3 6.55 x 1O-3 1.31 x 1o-2 2.62 x lo-’
3.2 5.6 7.7 11.1
2.0 3.1 4.4 6.2
2. c,=7x 1.31 x 2.62 x 6.55 x 1.31 x 2.62 x 6.55 x
10-8molcm~3 1o-3 4.4 1O-3 5.4 lo-’ 6.9 1o-2 8.5 lo-* 10.1 lo-’ 13.1
1.9 2.1 4.3 6.0 8.5 13.4
3. c,=5x 2.62 x 6.55 x 1.31 x 2.62 x
10-8molcm~3 1O-3 6.2 1O-3 8.2 1o-2 10.1 lo-’ 14.4
3.6 5.8 8.2 11.5
4. c,=3 2.62 6.55 1.31 6.55
x 10~8molcm~3 1O-3 7.1 1O-3 10.4 1o-2 13.7 lo-’ 18.9
5.6 8.9 12.5 17.7
x x x x
ArcimNlm)
I 20
10
100
pressures resulting from surface compression at a constant dilatation rate is corroborated by experimental data.
200
300
400
500 tls)
Fig. 8. Results of experiments with the logarithmic spiral for Triton X-45, cO= 5 x 10m9 mol cmm3: the jump in surface pressure A17 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (A) 181=0.026 s-l, (A17,),,,,, = 26.9 mN m-l; (0) l%l = 0.013 s-1, (AIZm)cxptl.= 23.7 mN m-l; (x) I%1= (0) 181= 0.0026 s-l, 0.0066 s 1, (AlT,),,,,,, = 19.2 mN m-‘; (AZ7m)expt,.= 14.8 mN m-l.
Table 2 Triton X-45 - logarithmic spiral. Jump in surface the steady state Al7, for different dilatation rates
pressure
IQ1 (s-l)
(417, L,,,. (mN m-‘)
An, from Eq. (38) (mN m-‘)
1. c0 = lo-’ mol cmm3 2.62 x 1O-3 6.55 x 1O-3 1.31 x 1o-2
10.4 14.2 17.5
9.3 14.7 20.8
2. c0 = 5 2.62 x 6.55 x 1.31 x 2.62 x
Fig. 7. Results of experiments with the logarithmic spiral for Triton X-45, c0 = 10-s mol cm-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) [%I=0.013 s-l, (AI7_),,,,,, = 17.5 mNm_‘; (x) 181= (0) l%l = 0.0026 s-‘, 0.0066 s - 1, (A17,),,,,,, = 14.2 mN m-‘; (AZZm),,,,,, = 10.4 mN m-‘.
-I
x 10e9 mol cmm3 1O-3 14.8 10m3 19.2 1om2 23.7 lo-’ 26.9
at
10.32 14.09 17.27 20.55
References Cl1 F. van Voorst Vader, Th.F. Erkens and M. van den Tempel, Trans. Faraday Sot., 60 (1964) 1170. c21 M. Van Uffelen and P. Joos, Colloids Surfaces A: Physicochem. Eng. Aspects, 85 (1994) 107. Surfaces A: c31 M. Van Uffelen and P. Joos, Colloids Physicochem. Eng. Aspects, 85 (1994) 119. of Antwerp, t-41M. Van Uffelen, Ph.D. Thesis, University 1994. Hydrodynamics, Prenticec51 V.G. Levich, Physicochemical Hall, New York, 1962.
P. Joos, M. Van UffelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253 [6]
E. Rillaerts and P. Joos, J. Colloid Interface Sci., 88 (1982) 1. [7] E. Rillaerts, J. Van Havenbergh and P. Joos, Bull. Sot. Chim. Belg., 90 (1981) 197. [S] V.B. Levich, B.I. Khaikin and E.D. Belokolos, Elektrochimija, 1 (1965) 1273. [9] R. Miller, Colloid Polym. Sci., 258 (1980) 179. [lo] J. Van Hunsel, G. Bleys and P. Joos, J. Colloid Interface Sci., 114 (1986) 432. [ 1 l] V.B. Fainerman, Kolloidn. Zh., 41 (1979) 111.
253
[ 121 P. Joos and M. Van Uffelen, J. Colloid Interface Sci., 155 (1993) 271. [ 131 M. Van Uffelen and P. Joos, J. Colloid Interface Sci., 158 (1993) 452. [ 141 L. Ter Minassian-Saraga, J. Colloid Interface Sci., 11 (1956) 398. [15] A.F.H. WardandL.Tordai,J.Chem.Phys., 14(1946)453. [ 161 Ch. Marcipont, Thesis, University of Antwerp, 1979. [ 171 P. De Keyser, Ph.D. thesis, University of Antwerp, 1983.
ELSEVIER
COLLOIDS AND SURFACES
A
Adsorption kinetics with surface dilatation 5. Transient surface tensions of an adsorbed monolayer during compression at a constant dilatation rate P. Joos *, M. Van Uffelen Department of Chemistry,
University of Antwerp, Universiteitsplein
1, B-261 0 Wilrijk, Belgium
Received 22 February 1995; accepted 17 March 1995
Abstract In the present article the integration of the convective diffusion equation for compression at a constant rate is presented, which enables us to describe the course of surface tension with time for the desorption adsorbed monolayer. The theory is evaluated by compression experiments performed with decanoic Triton X-45. Keyucords:
Adsorbed
monolayer;
Adsorption
dilatation rate of an acid and
kinetics; Surface dilatation
1. Introduction Van Voorst Vader et al. [l] studied the dynamic surface tension of an adsorbed monolayer, initially in equilibrium with the bulk solution, by expanding the monolayer at a constant dilatation rate 8. They have showed that after a transient period of rising surface tension, a steady state is obtained, where the amount of surface active material removed by expansion is compensated by diffusion from the bulk. In the theory of van Voorst Vader only this steady state situation is analyzed. The transient surface tension also contains valuable information about the adsorption process. Therefore, the convective diffusion equation for expansion needs to be integrated [2-41 following the procedure outlined by Levich [S]. Experimentally, the monolayer is expanded at a constant dilatation rate using a logarithmic spiral.
* Corresponding author. 0927.7757,/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0927-7757(95)03167-7
Alternative methods have since become available: the strip method [6], the dynamic capillary method [7] and adsorption onto an expanding drop [S-lo] or bubble [ 111. If we consider compression experiments at a constant dilatation rate, the convective diffusion equation for an expanding surface does not apply [ 12,131. Arguments for this will be given later on. The integration of the convective diffusion equation for compression is more difficult than was the case for expansion, since the boundary condition for the conservation of mass at the interface cannot be expressed as a total differential. In the present paper, a solution for this problem is given. In order to check the theoretical results, this solution is evaluated by means of experimental data.
P. Joos, M. Van UffelenjColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
246
2. Theory
face will decrease with time:
2.1. The convective &fusion equation for expansion
a44 4 < at
and compression
A surface at equilibrium with the bulk solution, with initial area 9,, is expanded or compressed at a constant dilatation rate I!?,defined as 8=
Hence, the left-hand side of Eq. (2) is negative. Qualitative consideration of the concentration-distance profile shows that it is curved downwards (see Fig. l(b)):
wt, 4 <
d ln Q(t)
o
az2
dt
where Sz is the area and t is the time. For a surface expansion, 8 > 0, and for a surface compression,
e < 0.
Continuous deformation of a surface implies a convection current near to the surface, which enhances the diffusion process, and therefore must be taken into account in the diffusion equation. As van Voorst Vader et al. Cl] have shown for expansion, this convective diffusion equation is
a4 4 4 -_
o
8z
at
a2444 -a44 4 = D ~ az2 az
d>O
In Eq. (2) the time derivative of the concentration and the convective term have the same signs as the diffusion term, which means that the convection helps the diffusion process to attain thermodynamic equilibrium. Therefore, Eq. (2) is correct for an expanding surface. Let us now apply Eq. (2) to a compressed monolayer (8 < 0). The sub-surface concentration c, for this kind of disturbance is higher than the bulk
(2)
where c is the concentration, z is the coordinate normal to the surface and D is the diffusion coefficient. In the case of a surface compression, Eq. (2) gives erroneous results for both theory and experiments. Therefore, for a compressed monolayer:
a4 t, 4 +
Bz
at
wt, 4 = a2444 D
aZ
a22
e
(3)
The validity of Eq. (2) can be tested for a surface expansion and a surface compression as follows. If we consider a surface element which is expanded ((3> 0), the sub-surface concentration c, is less than the bulk concentration cO. For z > 0, the concentration gradient i3c(t, z)/az > 0 and the fluid moves from the bulk to the surface. As a result, the convection term in Eq. (2) is -8Z-
aa 4 <
to
CO
cs
o
aZ
On expansion, the concentration
near to the sur-
Fig. 1. Schematic representation of the concentration profile as a function of z and t: (a) for a compressed surface element (0 < 0), and (b) for an expanded surface element (0 > 0).
P. Joos. M. Van UffelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
concentration cO, so for z > 0, &(t, z)/~z < 0, and the fluid now moves away from the surface into the bulk solution. Hence, the sign of the convection term remains unaffected in comparison with the expanded surface. However, on compression the concentration near to the surface will increase with time: dc(t,z) >.
at and the concentration-distance upwards (see Fig. l(a)):
profile is curved
a2c(t, z) > o
a22 Thus, the convection term and the diffusion term have opposite signs, indicating that the convection should counteract the diffusion process by accumulating tensio-active molecules at the surface. This is obviously incorrect: Eq. (2) is inconsistent with a compressed surface. Consequently, for a compressed surface we use Eq. (3). This argument can be supplemented by considering the diffusion penetration depth 6nc. For a surface which is expanded at a constant dilatation rate 0, integration of Eq. (2) shows that the diffusion penetration depth is [4] 6,,=
$(1-e-2ez) [
l/2 e>o 1
rium with the bulk. The system will try to return to its equilibrium state by diffusion from the surface into the bulk. Instead of measuring the transient surface pressures with time, the surface pressure can be kept constant by further compression of the surface. Consequently, the change in surface area can be measured as a function of time. This kind of experiment was introduced by Ter Minassian-Saraga [ 141 for a slightly spread soluble monolayer. Calculating for a hypothetical desorption process without convection and starting from Eq. (2), it emerges that convection slows down the diffusion process [4]. Hence, we are confident that Eq. (3) should be applicable for a compressed surface. 2.2. Integration of the convective diflusion equation for compression (0 < 0) For integration of Eq. (3), we introduce, according to the method of Levich [S], a first variable x = zf(t)
(6)
where f(t) is an unknown function of time. In this way, Eq. (3) becomes aa, 4 T+z[~+sf(t)]%$
(4) =
If we compare 6,, with 6n, the diffusion penetration depth without convection (0 = 0) 6, = (7&y
247
(5)
we see that
D
a2444
Tf
2
(t)
We choose f(t) in such a way that the convection term cancels out: df(t)
,,+ef(t)=o (except for t = 0, when both are zero), indicating that convection speeds up the diffusion process. However, for a compressed surface (0 < 0), the diffusion penetration depth with convection, according to Eq. (2), is larger than that without convection, which is in contradiction with the fact that convection enhances the diffusion process. Therefore, Eq. (2) is only applicable for expansion. As a third argument, we consider sudden compression of the surface, which is initially in equilib-
(8)
Since 0 is constant, it follows from Eqs. (1) and (8) that f(t)=e-et=52,/i2(t)
(9)
and Eq. (7) reduces to
aa,XI = a244xl -f2(t) D
at
a2
We now eliminate
(10)
f2(t) by introducing a second
248
P. Joos, M. Van UflelenlCoNoids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
variable z: dc(z, x) ~_
az
introduce a new function 4 = d(r, T(z)):
a7 = at
D
d24?xl --jj+2w
(11) Hence, Eq. (18) can be written as
which is defined as ;
(19)
(12)
= j-2(t)
Therefore, Eq. ( 11) results in the ordinary diffusion equation: (13)
4 -_=
(20)
dz
Using Eqs. (13) and (20), the problem is reduced to that of Ward and Tordai [ 151. The result is .1/z 4 = 2(D/7~)“~
[co - c,(z - A)] di”2 !
where c is now a function of the variables x and z. In order to integrate this equation with the initial condition that at t = 0, c = co, we choose z such that for t= 0,z = 0:
z=
f’(t)dt
Using Eq. (9), we obtain the following expression for z:
(15) The bounary condition is the law for conservation of mass at the interface: dt
where J. is the integration variable. However, we still do not have an expression for 4. Therefore, we solve Eq. (19) with the requirement that at t=O(z=O), c$=O:
4=
0
dr( t)
0
(14)
s
+
s s f(t) w4
0
where r the adsorption. This equation must also be expressed as a function of the new variables x and 7. Thus, considering Eq. (S), we obtain
dt
f(t)
dz
s T
4 = f( 0-w -
r, - 2
r(7) v(t)
(23)
0
of Eqs. (23) and (21) yields p D
r(z)f(t)=r,+2
0 n
112
s
[co- c,(z-A)] d/I”2
0
dt
Using Eqs. (6) and (12), Eq. ( 17) reduces to
dW) ,f(t)-r(r)?
0
1 df(t) dz
-------_(t)=D dz
(22)
On partial integration, considering that at t= 0 (z = 0), f(t) = 1 and T(z) = r,, the equilibrium adsorption, corresponding to the bulk concentration cO, gives
Combination
dT(z) dz
m) W(t)
-
(16)
W(t) = D
(21)
= D
(y)
0
(18)
As can be seen, the left-hand side of Eq. (18)cannot be written as a total differential. Therefore, we
Since f(t) and z are known functions of t, Eq. (24) predicts the adsorption as a function of time. The numerical integration must be performed using a computer. Formally, the expression for adsorption in Eq. (24) is different from that for an expanding
249
P. Joos, M. Van UffelenlColloids Surfaces A: Physicochem. Emg. Aspects 100 (1995) 245-253
imental
surface by the integral
jump
in surface pressure:
417(r) = n(z) - z& = g
(T(z) - r,)
(31)
we obtain
0
In order to avoid the numerical evaluation of Eq. (24) by computer, we consider only small deviations from equilibrium [2,3,13]. In this situation, we may assume a linear relationship between concentration and adsorption: co - c,(z - A)=
$;(re - T(z -A))
eo[2-~]uv-11 AZZ(z) =
T(z)f( t) = r, + (4r/7rrrJ1’2 (r, - T(r))
where co is the Gibbs
elasticity,
Now, if (T(z)) gives
z T(Z) and 1 CC(4r/nrn)1’2,
AL’(z) = ~,2~~(7r/4r)~‘~ [f( t) - l] Hence, using the Gibbs 1,2
time, given by
_
~(r(4) -dn D1’2 dc
z
s
r(z) df( t) = W(3) U-(t) - 11
as a mean value:
(28)
Eq. (32)
(34)
equation
(RTZ-,2/CoLP2)
RTT2 = e co (~/4W2
(27) can be expressed
as
(35)
we can write Eq. (34) as
AIT
This last integral
defined
(33)
EOTD -
relaxation
1 + (4r/712,)“2
(25)
and, because at long times co - c,(z - 1) and therefore also r, - T(z - A) stays more or less constant, we may factorize these terms out of the integral:
where rn is the diffusion
(32)
Cf(t) - 1I
(36)
For a surface compression at a constant dilatation rate, bearing in mind Eqs. (9) and (15), Eq. (36) becomes
AU(z) = !?$i
(g>‘;’
tanh”f!$
0
whence
Eq. (26) becomes
r(r)f(t)
= r, + (4r/nrn)1’2(& + 2V(r))
[f(r)
= An, -T(r)) -
11
(29)
(37)
where [ 3,4]
A17 co
or alternatively
tanh’12 m ( 2 I
1’2 (38)
AT(z) = T(r) - r,
=
(42-$3f(1)-11
(30)
1 + (42/~2n)1’2
Since the difference in adsorption for small deviations from equilibrium is proportional to the exper-
since for t + GO,a steady state is obtained. Eqs. (37) and (38) describe the variation of the jump in surface pressure with time for a surface compression at a constant dilatation rate. We have previously obtained the same expression for a surface expansion at a constant dilatation rate
~31.
250
P. Joos, M. Van Uffelen/Colloids
Surfaces A: Physicochem.
3. Materials and methods
The materials used in this investigation were decanoic acid (analar grade 99 + %) obtained from Aldrich, and Triton X-45 (a nonionic surfactant: octylphenol-polyethylene glycol ether, containing approximately five ethoxy groups; a molecular weight of 427 is assumed) purchased from Serva. In order to be sure we were dealing with the acid form of decanoic acid (regular behaviour [ 163) in the experiments, the pH was kept lower than 4. Therefore, we worked with a 0.01 N HCl + 0.09 N NaCl solution, both products having been obtained from Merck. Thoroughly triple-distilled water was used and all experiments were performed at room temperature 22 f 1 “C. A surface, initially in equilibrium with the bulk solution, was compressed at a constant rate of dilatation 0. The variation of surface pressure with time was measured using a Wilhelmy plate, connected to a force transducer (Statham Gould cell), and the output signal was recorded on a strip chart recorder. For the compression of the surface, we employed the experimental set up described by van Voorst Vader et al. [ 11, using a logarithmic spiral, which was constructed in such a way to give an exponential decrease in the surface area.
4. Results and discussion
Eng. Aspects 100 (1995) 245-2.53
1 <<(4z/rcrn)“2 are fulfilled, we plotted the experimental jump in surface pressure 617(t) as a function of
From the slope, we calculated a value for L7, according to Eq. (37). Some of the experimental results are presented in Figs. 2-6. An(mN/m)
Fig. 2. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 10m7mol cm -‘.. the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) 1f?1=0.0066s-r, (A17,),,,,, = 5.6mNm-r; (x) (0l= 0.0026 s-r, (A17m)cxptl. = 3.2 mN m-‘. An(mN/m) I"""
’
”
”
'1
Experiments were performed to verify our theory concerning the transient surface tensions of a compressed adsorbed monolayer at a constant dilatation rate. 4.1. Decanoic acid Compression experiments were performed with the logarithmic spiral for four concentrations of decanoic acid: lo-’ mol cme3, 7 x lo-* mol cmm3, 5 x lops mol cme3 and 3 x lop8 mol cmP3 (the experiments with c,, = 7 x lop8 mol cmF3 were performed by De Keyser [17]). The following problem arose in the interpretation of the experimental data: steady state is often not reached when the logarithmic spiral has come to an end. Because for decanoic acid the conditions (T(z)) x T(z) and
Fig. 3. Results of experiments with the logarithmic spiral for decanoic acid, c,=7 x lo-* mol cn-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) [@I= 0.0066 s-r, (A17m)cxptl. = 13.1 mN m-r; (AZ7m)cxpt,,= 10.1 mN m-r; (A) 161= (x) )0/=0.0026s-r, 0.013 s-r, (A17,),,,,, = 8.5 mN m-r.
251
P. Joos, M. Van Uffelen/Colloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253 ATclmNlm) I 301
AnimNlml
1
I
I
I
::
10
100
Fig. 4. Results of experiments with the logarithmic spiral for decanoic acid, cc, = 7 x lo-* mol cmm3: the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) JBJ = 0.0066 s-r, (417,),,,,,, = 6.9 mN m-l; (x) 101=0.0026s~‘, (AZZW),,,,,,=5.4mNm-‘; (A) lfIl= 0.0013 s-r, (A17,),,,,,, = 4.4 mN m-l.
200
300
400
500t(s)
Fig. 6. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 3 x lo-* molcm-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates, The theoretical curve is calculated according to Eq. (37): (+) l0l= 0.026 s-l, (417,),,,,, = 18.9 mN m-‘; (0) 161=0.013 s-l, (Al7&,,, = 13.7 mN m-r; (x) 101= 0.0066 s-r, (AlI,),,,,,, = 10.4 mN m-r; (0) 161= 0.0026 s-r, (ALr,),,pt,. = 7.1 mN m-l.
tion rates, this does not present any problems, since the minor components adsorb at large times. The experimental jumps in surface tension in the steady state are summarized in Table 1. If we compare them with the theoretical values calculated according to Eq. (38) with D = 5 x lO-‘j cm2 s-l, we can see that in the absence of adjustable parameters, the experimental values are of the right order of magnitude.
AnimN/ml
ISI
4.2. Triton X-45 50
100
150
250
IfsI
Fig. 5. Results of experiments with the logarithmic spiral for decanoic acid, c0 = 5 x lo-* mol cm-? the jump in surface pressure Ail is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) 181= 0.026 s-r, (A17,),,,,,, = 14.4 mN m-r; (x) ISI =0.013 s-r, (Al7,),,,,,, = 10.1 mNm_‘; (0) 181= 0.0066 s-r, (AL’,),,,,,. = 8.2 mN m-r.
For low dilatation rates, we observe that the experimental points do not tend towards a steadystate surface tension. This is due to the fact that the time scale of the desorption of the surfactant under study coincides with the desorption of minor components, which are adsorbed onto the surface during the equilibrium process. For higher dilata-
Experiments with the logarithmic spiral were carried out for two concentrations of Triton X-45: lop8 mol cm-j and 5 x 10e9 mol cmm3. Interpretation of the experiments was performed in exactly the same was as for decanoic acid, since for Triton X-45 the conditions (T(z)) z T(z) and 1 K (4z/7cz,) are also fulfilled. Some of the experimental results are shown in Figs. 7 and 8. The resulting parameters are reported in Table2 (with D=5 x 10-6cm2s-1).
5. Conclusions
As a general conclusion, we can say that the theory presented to describe the transient surface
P. Joos, M. Van UflelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253
252
Table 1 Decanoic acid - logarithmic spiral. Jump in surface at the steady state 417, for different dilatation rates
Arc(mN/m) pressure
/
I
I
I
I
-I
30c
14
(Anco hxptl.
An,
(s-l)
(mN m-‘)
from Eq. (38) (mN rn-‘)
1. c0 = 1O-7 mol cm-’ 2.62 x 1o-3 6.55 x 1O-3 1.31 x 1o-2 2.62 x lo-’
3.2 5.6 7.7 11.1
2.0 3.1 4.4 6.2
2. c,=7x 1.31 x 2.62 x 6.55 x 1.31 x 2.62 x 6.55 x
10-8molcm~3 1o-3 4.4 1O-3 5.4 lo-’ 6.9 1o-2 8.5 lo-* 10.1 lo-’ 13.1
1.9 2.1 4.3 6.0 8.5 13.4
3. c,=5x 2.62 x 6.55 x 1.31 x 2.62 x
10-8molcm~3 1O-3 6.2 1O-3 8.2 1o-2 10.1 lo-’ 14.4
3.6 5.8 8.2 11.5
4. c,=3 2.62 6.55 1.31 6.55
x 10~8molcm~3 1O-3 7.1 1O-3 10.4 1o-2 13.7 lo-’ 18.9
5.6 8.9 12.5 17.7
x x x x
ArcimNlm)
I 20
10
100
pressures resulting from surface compression at a constant dilatation rate is corroborated by experimental data.
200
300
400
500 tls)
Fig. 8. Results of experiments with the logarithmic spiral for Triton X-45, cO= 5 x 10m9 mol cmm3: the jump in surface pressure A17 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (A) 181=0.026 s-l, (A17,),,,,, = 26.9 mN m-l; (0) l%l = 0.013 s-1, (AIZm)cxptl.= 23.7 mN m-l; (x) I%1= (0) 181= 0.0026 s-l, 0.0066 s 1, (AlT,),,,,,, = 19.2 mN m-‘; (AZ7m)expt,.= 14.8 mN m-l.
Table 2 Triton X-45 - logarithmic spiral. Jump in surface the steady state Al7, for different dilatation rates
pressure
IQ1 (s-l)
(417, L,,,. (mN m-‘)
An, from Eq. (38) (mN m-‘)
1. c0 = lo-’ mol cmm3 2.62 x 1O-3 6.55 x 1O-3 1.31 x 1o-2
10.4 14.2 17.5
9.3 14.7 20.8
2. c0 = 5 2.62 x 6.55 x 1.31 x 2.62 x
Fig. 7. Results of experiments with the logarithmic spiral for Triton X-45, c0 = 10-s mol cm-? the jump in surface pressure Al7 is plotted as a function of time for different dilatation rates. The theoretical curve is calculated according to Eq. (37): (0) [%I=0.013 s-l, (AI7_),,,,,, = 17.5 mNm_‘; (x) 181= (0) l%l = 0.0026 s-‘, 0.0066 s - 1, (A17,),,,,,, = 14.2 mN m-‘; (AZZm),,,,,, = 10.4 mN m-‘.
-I
x 10e9 mol cmm3 1O-3 14.8 10m3 19.2 1om2 23.7 lo-’ 26.9
at
10.32 14.09 17.27 20.55
References Cl1 F. van Voorst Vader, Th.F. Erkens and M. van den Tempel, Trans. Faraday Sot., 60 (1964) 1170. c21 M. Van Uffelen and P. Joos, Colloids Surfaces A: Physicochem. Eng. Aspects, 85 (1994) 107. Surfaces A: c31 M. Van Uffelen and P. Joos, Colloids Physicochem. Eng. Aspects, 85 (1994) 119. of Antwerp, t-41M. Van Uffelen, Ph.D. Thesis, University 1994. Hydrodynamics, Prenticec51 V.G. Levich, Physicochemical Hall, New York, 1962.
P. Joos, M. Van UffelenlColloids Surfaces A: Physicochem. Eng. Aspects 100 (1995) 245-253 [6]
E. Rillaerts and P. Joos, J. Colloid Interface Sci., 88 (1982) 1. [7] E. Rillaerts, J. Van Havenbergh and P. Joos, Bull. Sot. Chim. Belg., 90 (1981) 197. [S] V.B. Levich, B.I. Khaikin and E.D. Belokolos, Elektrochimija, 1 (1965) 1273. [9] R. Miller, Colloid Polym. Sci., 258 (1980) 179. [lo] J. Van Hunsel, G. Bleys and P. Joos, J. Colloid Interface Sci., 114 (1986) 432. [ 1 l] V.B. Fainerman, Kolloidn. Zh., 41 (1979) 111.
253
[ 121 P. Joos and M. Van Uffelen, J. Colloid Interface Sci., 155 (1993) 271. [ 131 M. Van Uffelen and P. Joos, J. Colloid Interface Sci., 158 (1993) 452. [ 141 L. Ter Minassian-Saraga, J. Colloid Interface Sci., 11 (1956) 398. [15] A.F.H. WardandL.Tordai,J.Chem.Phys., 14(1946)453. [ 161 Ch. Marcipont, Thesis, University of Antwerp, 1979. [ 171 P. De Keyser, Ph.D. thesis, University of Antwerp, 1983.