Linear Compression of a Mixed Adsorbed Monolayer

JOURNAL OF COLLOID AND INTERFACE SCIENCE 182, 179–189 (1996)ARTICLE NO. 0449

Linear Compression of a Mixed Adsorbed Monolayer

PETER PETROV AND PAUL JOOS1

Department of Chemistry, University of Antwerp, U.I.A., Universiteitsplein 1, B-2610 Wilrijk, Belgium

Received November 15, 1995; accepted March 13, 1996

V Å V0 /dV

dtt Å V0[1 / at] [1]The theory for the compression of an adsorbed soluble mono-

layer was extended to systems containing two surfactants. Thistheory was confirmed experimentally for decanoic and dodecanoic

withacid mixtures. The parameters needed to describe the compressioncurves were estimated from the behavior of the pure systems.For doing this, solutions of decanoic and dodecanoic acid were

a Å 1V0

dV

dt.prepared in such a way that the equilibrium adsorptions were

nearly the same. Also, in the mixed systems, the sum of the adsorp-tion of both components were equal, allowing reasonable predic-

The parameter a is negative since the surface is compressedtion of the parameters needed to describe the experimental results.(dV /dt õ 0).The agreement between theory and the experiments is good. From

Since during a continuous surface deformation a convec-these experiments, we can obtain the surface elasticity for mixedtion current is generated, the convective diffusion equationmonolayers as a function of the effective time. q 1996 Academic

Press, Inc. must be considered. As argued before (2) for a compressedKey Words: mixed monolayers, compression of; mixed mono- surface the convective diffusion equation reads

layers, surface elasticity.

Ìci

Ìt/ uz

Ìci

ÌzÅ Di

Ì 2ci

Ìz 2 , [2]INTRODUCTION

where c is the concentration, t is time, z is the coordinateIn the previous paper (1) we considered the increase ofnormal to the surface and directed to the bulk, and D is thesurface pressure with time when an adsorbed monolayer,diffusion coefficient. u is the relative deformation rate de-initially in equilibrium, is compressed at a constant speed.fined asFrom these experiments, the elasticity of the monolayer is

obtained as a function of the effective time. In the presentpaper the behavior of the mixed monolayer is considered.

u Å d ln V

dtõ 0. [3]

The surface of an aqueous solution containing two surfac-tants, 1 and 2, is allowed to equilibrate, then it is compressedat a constant speed and the increase of the surface pressure The subscript i denotes surfactant 1 or 2.with time is monitored. During the compression the equilib- The diffusion equation [2] is now integrated as outlinedrium between surface and bulk is disturbed and surfactants before (1, 2) following Levich (3), and the result is1 and 2 will diffuse from the surface to the bulk.

Gif( t) 0 G ei 0 2 *

t

0

Gi df( t)THEORY

The surface of an aqueous solution containing two surfac- Å S4Di

p D1/2

*√t

0

[c 0i 0 c s

i (t 0 l)]d√l . [4]

tants is allowed to attain equilibrium, and after equilibrationit is compressed at a constant speed dV /dt , V being thesurface area. If before compression the surface area is V0 , Here Gi is the adsorption of component i , G e

i is the equilib-then the area as a function of time is given by: rium adsorption of component i , c 0

i is the bulk concentrationof component i , c s

i is the subsurface concentration of compo-1 To whom correspondence should be addressed. nent i . Functions f( t) and t are defined as

179 0021-9797/96 $18.00Copyright q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

AID JCIS 4316 / 6g13$$$$81 07-23-96 00:11:37 coidas AP: Colloid

180 PETROV AND JOOS

If we confine ourselves to small deviation from equilib-f( t) Å V0

VÅ 1

1 / at, [5] rium we linearise the jump in subsurface concentrations with

adsorptions. For a two surfactant system, the adsorption ofsurfactant i , Gi , depends on the concentration of surfactantt Å *

t

0

f 2( t)dt Å t

1 / at. [6]

1, c1 , and the concentration of surfactant 2, c2 ,

The integral in the l.h.s. of Eq. [4] can always been re- G1 Å f(c1 , c2) , [12]placed by a mean value,

or by inversion

*t

0

Gi df( t) Å »Gi …[ f( t) 0 1], [7]c1 Å f(G1 , G2) . [13]

and for this mean value we approximate byFrom this equation we obtain

»Gi … ÉGi / G e

i

2. [8] c s

i 0 c 0i

Å S Ìci

ÌGiDGj

(Gi 0 G ei ) / S Ìci

ÌGjDGi

(Gj 0 G ej ) . [14]

The convolution integral in the r.h.s. of Eq. [4] can beapproximated by considering that over the whole time do-main c 0

i 0 c si does not change much, with respect to

√t , Hence, explicitly, for both components Eqs. [11] and [14]

and is factorized out of the integral give

*√t

0

[c 0i 0 c s

i (t 0 l)]d√l É (c 0

i 0 c si )

√t . [9] DG1F1 / S4D1t

p D1/2S Ìc1

ÌG1DG2

GIn the past we made this approximation frequently: ( i) as

a long time approximation for the Ward and Tordai equation / DG2S4D1t

p D1/2S Ìc1

ÌG2DG1

Å G e1 [ f( t) 0 1],

(even for a growing drop) (4, 5) , ( ii ) to describe the surfacetension variation during expansion and compression of asurface with a constant dilatation rate (2, 6) , and (iii ) to

DG1S4D2t

p D1/2S Ìc2

ÌG1DG2

describe expansion with a linear speed (7). This approxima-tion allows us to avoid more complicated numerical calcula-tions and to present the results in a handy analytical expres-sion. It was found that with this approximation, the experi- / DG2F1 / S4D2t

p D1/2S Ìc2

ÌG2DG1

G Å G e2 [ f( t) 0 1],

mental results are reasonably described. Moreover, for somesituations, we compared this approximation with computer- [15]calculated solutions of the exact equation and it was foundthat this approximation was very reasonable (8) .

hereIn this way, taking into account Eqs. [7] and [9], Eq.[4] approximates to

DG1 Å G1 0 G e1 and DG2 Å G2 0 G e

2 . [16]

Gi 0 G ei / (Gi 0 2 »Gi …)[ f( t) 0 1]

If we denote the square roots of the diffusion relaxationtimes, t11 , t22 , t12 , and t21 ,Å S4Dit

p D1/2

(c 0i 0 c s

i ) , [10]

√t11 Å

1√D1S ÌG1

Ìc1DG2

;√t12 Å

1√D1S ÌG2

Ìc1DG1

;and with approximations expressed by Eq. [8]

Gi 0 G ei / S4Dit

p D1/2

(c si 0 c 0

i ) Å G ei [ f( t) 0 1]. [11]

√t22 Å

1√D2S ÌG2

Ìc2DG1

;√t21 Å

1√D2S ÌG1

Ìc2DG2

. [17]


181LINEAR COMPRESSION

Equations [15] are written as where we have defined e01 and e02 as

e01 Å G e1S ÌPÌG1

DG2

and e02 Å G e2S ÌPÌG2

DG1

. [22]DG1F1 / S 4tpt11

D1/2G / DG2S 4tpt12

D1/2

Å G e1 [ f( t) 0 1],

Considering Eq. [5] ,DG1S 4tpt21

D1/2

/ DG2F1 / S 4tpt22

D1/2G Å G e2 [ f( t) 0 1].

[18] f( t) 0 1 Å V0 0 V

VÅ 0 DV

V, [23]

From this set of equations we obtain DG1 and DG2 . and defining the elasticity of monolayers as

DG1Å [ f( t)0 1]G e1F1/ S4t

p D1/2

e Å 0 DP

DVV, [24]

Eq. [21] gives the elasticity of the monolayer as a function1 S 1√t22

0 1√t12

G e2

G e1DGYF1/ S4t

p D1/2 S 1√

t11

/ 1√t22D of t.

The elasticity defined in this way should not be confusedwith the Gibbs elasticity e0 , given by

/ 4tp S 1√

t11t22

0 1√t12t21

DG ,

e0 ÅdP

d(Gi

∑ Gi . [25]

DG2Å [ f( t)0 1]G e2F1/ S4t

p D1/2

The relation between both is

1 S 1√t11

0 1√t21

G e1

G e2DGYF1/ S4t

p D1/2S 1√

t11

/ 1√t22D e Å 0e0

d ln ( Gi

d ln V. [26]

Both elasticities are the same if/ 4tp S 1√

t11t22

0 1√t12t21

DG . [19]

d ln ( Gi

d ln VÅ 01, [27]

Since the increase of surface pressure depends on the in-crease of the adsorptions of components 1 and 2, which means that the mixed monolayer is insoluble. Equa-

tion [27] accounts for the change in adsorption during sur-face deformation due to the diffusion process (another relax-

DP Å S ÌPÌG1DG2

DG1 / S ÌPÌG2DG1

DG2 , [20] ation process as micellization, not considered here) . For asoluble monolayer e õ e0 .

If we define further,which allows us to obtain the jump of surface pressure DP.

e0 Å e01 / e02 , [28]

DP

f ( t) 0 1Å He01 / e02 / S4t

p D1/2FS e01√

t22

/ e02√t11D a1 Å S4

pD1/2FS e01√

t22

/ e02√t11D

0 SG e2

G e1

e01√t12

/ G e1

G e2

e02√t21DGJYH1 / S4t

p D1/2

0 SG e2

G e1

e01√t12

/ G e1

G e2

e02√t21DG , [29]

a2 Å S4pD

1/2S 1√t11

/ 1√t22D , [30]1 S 1√

t11

/ 1√t22D / 4t

p S 1√t11t22

0 1√t12t21

DJ , [21]


182 PETROV AND JOOS

FIG. 1. (a) Increase of surface pressure, DP, as a function of time, t , for compressed monolayers of decanoic acid (c 01 Å 2.8 1 1008 mol cm03)

and different compression rates: a Å 02.43 1 1002 s01 (s) ; a Å 01.44 1 1002 s01 (h) ; a Å 06.31 1 1003 s01 (n) . The solid lines are calculatedaccording to Eq. [34]. (b) Increase of surface pressure, DP, as a function of time, t , for compressed monolayers of decanoic acid (c 0

1 Å 2.8 1 1008

mol cm03) at lower compression rates: a Å 03.22 1 1003 s01 (s) ; a Å 01.63 1 1003 s01 (h) ; a Å 06.56 1 1004 s01 (n) ; a Å 03.30 1 1004 s01

(L) . The solid lines are calculated according to Eq. [34].

Equations [32] and [33] are general and valid for a com-a3 Å

4p S 1√

t11t22

0 1√t12t21

D , [31] pressed monolayer independent of how the surface is contin-uously compressed. The way the monolayer is compressedis taken into account by f( t) and t. For a linearly compressedwe obtainmonolayer these parameters are given by Eqs. [5] and [6].

If the monolayer is compressed with constant dilatationDP Å [ f( t) 0 1]e [32]rate u, as considered in a previous paper (2) ,

and

V Å V0e ut u õ 0e Å e0 / a1

√t

1 / a2

√t / a3t

. [33]f( t) Å e0ut



FIG. 2. (a) Increase of surface pressure, DP, as a function of time, t , for compressed monolayers of dodecanoic acid (c 02 Å 4 1 1009 mol cm03)

and different compression rates: a Å 02.52 1 1002 s01 (s) ; a Å 01.42 1 1002 s01 (h) ; a Å 06.26 1 1003 s01 (n) . The solid lines are calculatedaccording to Eq. [34]. (b) Increase of surface pressure, DP, as a function of time, t , for compressed monolayers of dodecanoic acid (c 0

2 Å 4 1 1009

mol cm03) at lower compression rates: a Å 03.23 1 1003 s01 (s) ; a Å 01.64 1 1003 s01 (h) ; a Å 06.60 1 1004 s01 (n) . The solid lines arecalculated according to Eq. [34].

and [31] and [17]. We report experiments with mixed mono-layers of decanoic and dodecanoic acid.

t Å 12ÉuÉ

(e 2ÉuÉt 0 1). EXPERIMENTAL

The apparatus was the same as previously described (9).Compression was started only if the surface tension of theIt is expected that Eqs. [32] and [33] describe the increase

of surface pressure with time for a linearly compressed solution was constant, indicating that equilibrium is attained.As surfactants we used decanoic and dodecanoic acid. Tomixed monolayer with four adjustable parameters. We re-

quire not only that these parameters be adjustable but also the aqueous solution some HCl (until approximately pH 2)was added to suppress the dissociation of the fatty acids.that they have the physical meaning expressed by Eqs. [28] –


184 PETROV AND JOOS

FIG. 3. Elasticity of decanoic acid monolayers, e, as a function of the square root of the effective time (Eq. [36]) . Summary of all compressionrates give in Figs. 1a and 1b: a Å 02.43 1 1002 s01 (l) ; a Å 01.44 1 1002 s01 (L) ; a Å 06.31 1 1003 s01 (s) ; a Å 03.22 1 1003 s01 (l) ; aÅ 01.63 1 1003 s01 (n) ; a Å 06.56 1 1004 s01 (/) ; a Å 03.30 1 1004 s01 (m) . The solid line is calculated with e0 Å 33.95 dyn cm01 and tD Å8.7 s.

Surfactants were of analar grade from Aldrich (Gold Label) . flask. This solution was stirred and the equilibration wasscarcely achieved after several days. Also the surface equili-The experiments were done at room temperature. Stock solu-

tions of decanoic acid (concentration 2.8 1 1008 mol cm03) bration of this solution requires a long time (several hours) .First the behavior of the each separate component (at theand dodecanoic acid (concentration 4 1 1009 mol cm03)

were prepared. Both solutions have nearly the same surface concentrations just mentioned) was investigated, to obtainGibbs elasticities and diffusion relaxation times for the puretension (se Å 61 dyn cm01) .

Solutions of dodecanoic acid were tedious to prepare be- components. Then different mixing ratios were investigated:( i) 75% of the stock solution of decanoic acid plus 25%cause the low solubility of this surfactant. A stock solution

was prepared by putting 8 mg dodecanoic acid in a 10-liter stock solution of dodecanoic acid (c1 Å 2.1 1 1008 mol

FIG. 4. Elasticity of dodecanoic acid monolayers, e, as a function of the square root of the effective time. Summary of all compression rates givein Figs. 2a and 2b: a Å 02.52 1 1002 s01 (n) ; a Å 01.42 1 1002 s01 (s) ; a Å 06.26 1 1003 s01 (l) ; a Å 03.23 1 1003 s01 (m) ; a Å 01.641 1003 s01 (L) ; a Å 06.60 1 1004 s01 (h) . The solid line is calculated with e0 Å 37.0 dyn cm01 and tD Å 356 s.



cm03 ; c2 Å 1009 mol cm03) , ( ii ) 50% decanoic acid plus50% dodecanoic acid (c1 Å 1.4 1 1008 mol cm03 ; c2 Å 21 1009 mol cm03) , and (iii ) 25% decanoic acid plus 75%dodecanoic acid (c1 Å 7 1 1009 mol cm03 ; c2 Å 3 1 1009

mol cm03) .

RESULTS AND DISCUSSION

First the behavior of the single systems was investigated.Some results for decanoic acid are given in Figs. 1a and 1b,and for dodecanoic acid in Figs. 2a and 2b. These resultsare analyzed using the following equation (1):

DP Å 0 e0at

1 / at

11 / [4t /ptD(1 / at)]1/2 . [34]

It is seen that for decanoic acid the experimental resultsare well described by this equation. For dodecanoic acid itdescribed only low jumps in surface pressure. This indicatesthat for decanoic acid the linearization we made to obtainEq. [34] applies over a considerable range, but for dodeca-noic acid this range is more restricted.

The experimental data for the one component surfactantsystems are fitted by Eq. [34], and care was taken to seethat this fit is good for small DP values where the approxi-mations made are believed to be reasonable. The parameterse0 and tD are obtained near equilibrium. Experimentally itis found that Eq. [34] describes the results in a wider range.If the surface pressure increase e0 increases and tD decreases,perhaps these two effects cancel each other. Deviations fromthis equation become apparent for the dodecanoic acid sys-tem, where the jumps in DP are larger.

The parameters used to describe the experiments are:

( i) for decanoic acid e0 Å 33.95 dyn cm01 and tD Å 8.7s. From these parameters we obtain the equilibrium adsorp-tion by the equation (10)

e0

√DtD Å

RTG 2e

c0

, [35]

giving Ge Å 5.02 1 10010 mol cm02 using a diffusion coeffi-cient D Å 5 1 1006 cm2 s01 .

( ii ) for dodecanoic acid e0 Å 37.0 dyn cm01 and tD Å356 s. With Eq. [36] we obtain an equilibrium adsorptionGe Å 5.01 1 10010 mol cm02 .

In Figs. 3 and 4 we have reported the elasticity of themonolayers as a function of the effective time, teff , definedas

teff Åt

1 / at, [36]

TA

BL

E1

Pre

dict

edK

inet

icP

aram

eter

sfo

rD

ecan

oic

Aci

d(1

)an

dD

odec

anoi

cA

cid

(2)

Mix

ture

s

c0 11

108

c0 21

109

»G1…1

1010

»G2…1

1010

e 01

e 02

e 0√ t

11

√ t22

√ t12

√ t21

a3

(mol

cm0

3 )(m

olcm

03 )

(mol

cm0

2 )(m

olcm

02 )

(dyn

cm0

1 )(d

yncm

01 )

(dyn

cm0

1 )(s

1/2 )

(s1/

2 )(s

1/2 )

(s1/

2 )a

1a

21

103

2.8

05.

00

33.9

5—

33.9

52.

95—

—`

—0.

38—

2.1

13.

751.

2525

.46

9.25

34.7

13.

8664

.52

4.17

80.0

00.

460.

311.

30

1.4

22.

502.

5016

.98

18.5

035

.48

5.58

35.7

26.

2540

.00

0.69

0.23

1.30

0.7

31.

253.

758.

9427

.75

36.2

410

.09

24.6

912

.51

26.6

70.

800.

161.

30

04

05.

0—

37.0

37.0

—18

.87

`—

—0.

06—


186 PETROV AND JOOS

FIG. 5. Compression curve for decanoic acid (c 01 Å 2.1 1 1008 mol cm03) and dodecanoic acid (c 0

2 Å 1009 mol cm03) mixtures for differentcompression rates: a Å 02.47 1 1002 s01 (s) ; a Å 01.42 1 1002 s01 (L) ; a Å 06.21 1 1003 s01 (n) ; a Å 03.19 1 1003 s01 (h) . The solidlines are calculated according to Eq. [33].

where the values calculated from experimental data are com-P Å 0RTG`lnS1 0 G

G`D 0 RTHS G

G`D2

, [38]pared with these ones calculated by the above-mentionedparameters (solid lines) .

For decanoic acid the equilibrium surface tension curvewith parameters G` Å 5.3 1 10010 mol cm02 , a1 Å 1.26 1is described by regular behavior (11),1008 mol cm03 , H1 Å 1.8 (12, 13). (Here G Å Ge ) . For theconcentration we used (2.8 1 1008 mol cm03) , we obtainfrom Eq. [37] for adsorption G Å 4.80 1 10010 mol cm02 ,

c

aÅ G

G` 0 GexpFHS1 0 2

G

G`DG , [37]


2 Å 2 1 1009 mol cm03) mixtures for differentcompression rates: a Å 02.54 1 1002 s01 (s) ; a Å 01.44 1 1002 s01 (L) ; a Å 06.30 1 1003 s01 (n) ; a Å 03.22 1 1003 s01 (h) . The solidlines are calculated according to Eq. [33].




2 Å 3 1 1009 mol cm03) mixtures for differentcompression rates: a Å 02.53 1 1002 s01 (s) ; a Å 01.45 1 1002 s01 (L) ; a Å 06.31 1 1003 s01 (n) ; a Å 03.23 1 1003 s01 (h) . The solidlines are calculated according to Eq. [33].

in good agreement with the value obtained from kinetic mea-e0 Å G

dP

dGÅ RT`FS G

G` 0 GD 0 2HS G

G`D2Gsurements.Moreover, from Eq. [37] we obtain

Å 87.7 dyn cm01 ,

which is much higher than the value obtained from kineticdc

dGÅ aG`

(G` 0 G)2 H1 0 2HF G

G` data. This disagreement was discussed before (1) , and atten-tion was drawn to this by Lucassen-Reynders et al. (14).

It is not easy to obtain the equilibrium surface tension–0 S G

G`D2GJexpFHS1 0 2G

G`DG . [39] concentration curve for dodecanoic acid, because the solu-tions are very diluted and require much time to be preparedand attain equilibrium. Therefore we have extrapolated fromThis gives for diffusion relaxation time (with D Å 5 1 1006

fatty acid homologues the parameters a and H , suggestingcm2 s01)regular behavior of the monolayers (13). For dodecanoicacid the parameters for regular behavior are G` Å 5.3 110010 mol cm02 , a2 Å 1.65 1 1009 mol cm03 , H2 Å 2.4,tD Å

1D S dG

dc D2

Å 1.09 s,and at the dodecanoic acid concentration we used (4 1 1009

mol cm03) the equilibrium surface tension was se Å 60.1which is much lower than the value obtained from kinetic dyn cm01 , in agreement with the experiments. With thesedata. Next, from Eq. [38] we obtain the Gibbs elasticity values we obtain an equilibrium adsorption Ge Å 5.07 1

10010 mol cm02 , in excellent agreement with the results ofkinetic data.TABLE 2

Moreover the diffusion relaxation times for decanoic acidAdsorptions of Decanoic Acid (1) and Dodecanoic Acid (2)(t1D Å 8.7 s) and dodecanoic acids (t2DÅ 356 s) are consis-in Mixed Systems Obtained from Eq. [46]tent with each other. If for simplicity we use Langmuir iso-

c01 1108 c0

2 1 109»G1… 11010

»G2… 11010 se therms, then at constant surface pressure, assuming that the(mol cm03) (mol cm03) (mol cm02) (mol cm02) (dyn cm01) diffusion coefficients are equal for both substances and also

that the saturation adsorptions are equal (G`1 Å G`

2 Å G`) ,2.8 0 4.8 0 60.8we obtain2.1 1 3.63 1.26 60.8

1.4 2 2.42 2.54 60.70.7 3 1.20 3.82 60.4 St2D

t1DD1/2

É a1

a2

.0 4 0 5.07 60.1


188 PETROV AND JOOS

FIG. 8. Calculated surface elasticity for mixed systems: ( —) c1 Å 2.8 1 1008 mol cm03 , c2 Å 0; (rrr) c1 Å 2.1 1 1008 mol cm03 , c2 Å 1 1 1009

mol cm03 ; (---) c1 Å 1.4 1 1008 mol cm03 , c2 Å 2 1 1009 mol cm03 ; (-r-r-r-) c1 Å 0.7 1 1008 mol cm03 , c2 Å 3 1 1009 mol cm03 ; (-rr-rr-) c1

Å 0, c2 Å 4 1 1009 mol cm03 .

We find (t2D /t1D)1/2 Å 6.40 and a1 /a2 Å 7.64. The agree- We assume that in our restricted concentration range, therelation between adsorption and concentration follow an ex-ment is reasonable and is a consequence of Traube’s rule.

It is no problem to adjust the different parameters in Eq. tended Langmuir isotherm:[33] for the mixture. However we want to have an estimate

c1 Å a *1G1

G` 0 G1 0 G2

; c2 Å a *2G2

G` 0 G1 0 G2

. [42]for them as a function of the kinetic data for pure systems.What we want to have is an estimate for parameters for

Here a *1 and a *2 are not Langmuir–von Szyszkovski constantsthe mixed system by predicting them from those of the oneas meant in Eq. [37] but just constants applicable in thiscomponent systems. For doing this the theory of Garrettrestricted concentration range. The values of a *1 and a *2 de-(15) can be followed. In this theory the behavior is predictedpend on the Langmuir–von Szyszkovski constants and onif the surface pressure of the pure and mixed systems are thethe activity coefficients. Closer inspection reveals that theysame as those in our experimental conditions. The method ofchange by a factor 2 over the whole mixing range.Garrett applies only for ideal systems where surface activity

From these relations we obtain the diffusion relaxationcoefficients are neglected. Nevertheless even for the nonidealtimes of the mixed system.system we are dealing with, this method also seems valid.

However, we present here another approach. √t11 Å

1

a *1√

D1

(G` 0 G1 0 G2) 2

(G` 0 G2);In our experimental situation the adsorption of the pure

systems for decanoic acid, G 01 (5.02 1 10010 mol cm02 from

kinetic data) , and dodecanoic acid, G 02 (5.01 1 10010 mol √

t22 Å1

a *2√

D2

(G` 0 G1 0 G2) 2

(G` 0 G1);

cm02 from kinetic data) , are very close. We assume that forour mixtures, the sum of the adsorptions of both componentsremains constant,

√t12 Å

1

a *1√

D1

(G` 0 G1 0 G2) 2

G1

;

G 01 Å G 0

2 Å G1 / G2 Å 5 1 10010 mol cm02 , [40] √t21 Å

1

a *2√

D2

(G` 0 G1 0 G2) 2

G2

. [43]and parameters e01 and e02 are approximated by

Returning now to the pure systems,

e01 ÅG1

G1 / G2

e 01 , e02 Å

G1

G1 / G2

e 02 , [41]

√t1D Å

1

a *1√

D1

(G` 0 G 01) 2

G`,

e 01 and e 0

2 being the Gibbs elasticity for the pure systems√t2D Å

1

a *2√

D2

(G` 0 G 02) 2

G`. [44]

(e 01 Å 33.95 dyn cm01 ; e 0

2 Å 37.0 dyn cm01) .



From Eqs. [43] and [44] we obtain c2

a2

Å G2

G` 0 G1 0 G2

1 expFH2S1 0 2G2

G`D 0 (H1 / H2)G1

G`G . [46]√t11 Å

√t1DG

`

G` 0 G2FG` 0 G1 0 G2

G` 0 G 01

G2

;

(The heat of mixing of decanoic with dodecanoic acid isassumed to be zero.) This set of equations allows us, for a

√t22 Å

√t2DG

`

G` 0 G1FG` 0 G1 0 G2

G` 0 G 02

G2

;given concentration in the mixture, to obtain G1 and G2 .Results are given in Table 2. It is seen that the calculatedvalues for adsorption are very close to the predicted ones.√

t12 Å√t1DG

`

G1FG` 0 G1 0 G2

G` 0 G 01

G2

; Consequently, the terms (G` 0 G1 0 G2 /G` 0 G 01) 2 and

(G` 0 G1 0 G2 /G` 0 G 02) 2 in Eqs. [45] can be omitted.

We can equally well predict the diffusion relaxation times,taking into account that the condition expressed by Eq. [40]

√t21 Å

√t2DG

`

G2FG` 0 G1 0 G2

G` 0 G 02

G2

. [45]is not met, by using Eqs. [45]. However, these new parame-ters give a bad fit for the compression curves, hence thecorrections (G` 0 G1 0 G2 /G` 0 G 0

1) 2 and (G` 0 G1 0 G2 /Since from the kinetic data of decanoic and dodecanoicG` 0 G 0

2) 2 make the predictions worse.acid at the used concentrations we obtained nearly the sameHence Eqs. [45] are in general not correct but give reason-adsorptions (G 0

1 Å 5.02 1 10010 mol cm02 and G 02 Å 5.01

able results if the condition [40] is met. We have chosen the1 10010 mol cm02) using Eq. [35], we can assume that formixed system in such a way that this condition is fulfilled.a mixture of these concentrations the condition given by Eq.

Finally, knowing the kinetic parameters a1 , a2 , a3 , e01 ,[40] is met and this allows us to simplify Eqs. [45]. Theand e02 , we can predict the elasticity of the monolayer forparameters predicted in this way are given in Table 1. Withmixed systems as a function of the effective time. Resultsthem the compression curves are calculated for differentare given in Fig. 8.mixing ratios using Eq. [33] and compared with experimen-

tal data. Some examples are given in Figs. 5, 6 and 7. For REFERENCESa high mixing ratio of dodecanoic/decanoic acid, we have

1. Petrov, P., and Joos, P., J. Colloid Interface Sci., to appear.monolayer collapse, not accounted for in our theory. Disre- 2. Joos, P., and Van Uffelen, M., Colloids Surf. A 100, 245 (1995).garding these collapse data, there is good agreement between 3. Levich, V. G., in ‘‘Physicochemical Hydrodynamics,’’ pp. 540–542.theory and experiment. Prentice Hall, Englewood Cliffs, NJ, 1962.

4. Rillaerts, E., and Joos, P., J. Phys. Chem. 86, 3471 (1982).To obtain the diffusion relaxation times of the mixed sys-5. Joos, P., and Van Hunsel, J., Colloids Surf. 33, 99 (1988).tems, we have assumed that condition [40] is met. We can6. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 107 (1994).give more evidence for it by considering the adsorption iso- 7. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 119 (1994).

therm for mixture. If both components follow regular behav- 8. Van Hunsel, J., Van Uffelen, M., and Joos, P., unpublished results.ior, the adsorption isotherm for the mixture is 9. Horozov, T., and Joos, P., J. Colloid Interface Sci. 173, 334 (1995).

10. Rillaerts, E., and Joos, P., J. Colloid Interface Sci. 88, 1 (1982).11. Lucassen-Reynders, E. H., and van den Tempel, M., ‘‘Proc. 4th. Int.

Congr. Surf. Act. Subst. Brussels, 1964,’’ Vol. II, p. 779. Gordon &c1

a1

Å G1

G` 0 G1 0 G2Breach, New York, 1964.

12. Marcipont, C., Ph.D. thesis, University of Antwerp, 1979.13. Li, B., Geeraerts, G., and Joos, P., Colloids Surf. A 88, 251 (1994).14. Lucassen-Reynders, E. H., Lucassen, J., Garrett, P. R., Giles, D., and1 expFH1S1 0 2

G1

G`D 0 (H1 / H2)G2

G`G ,Hollway, F., Adv. Chem. Ser. 144, 272 (1975).

15. Garrett, P., J. Chem. Soc. Faraday Trans. 1, 72, 2174 (1976).


Linear Compression of a Mixed Adsorbed Monolayer

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