P.A. Kralchevsky, K.D. Danov, N.D. Denkov, N.C. Christov, K.P. Ananthapadmanabhan,* and Alex Lips*

Laboratory of Chemical Physics and Engineering, Faculty of Chemistry, University of Sofia, Bulgaria, and *Unilever R&D, Trumbull, CT, USA

Plenary Lecture at the 59th Divisional Meeting on Colloid and Interface Chemistry, The Chemical Society of Japan

Hokkaido University, Sapporo, Japan, September 13-15, 2006

The Four Kinetic Regimes of Adsorption from Micellar Surfactant Solutions

Effect of micelle diffusion and disassembly on the dynamic surface tension

(1) The interfacial expansion gives rise to surfactant adsorption and to decrease in the monomer concentration near the interface;

(2) This leads to micelle decomposition and to diffusion of micelles.

{Fast micellar process} =

{Exchange of monomers

between the micelles}(Rate constant km)

{Slow micellar process} =

{Decomposition of a “critical-size micelle”

to monomers}.

(Rate constant kS)

General Set of Equations (Aniansson & Wall, 1974)

Multi-Step Micellization:

,...)4,3,2(11 =↔++

−− sAAA s

k

ks

s

s

∑∞

=

−−=⋅∇+3

211 2

dd

ssJJ

tc

I

,...)4,3,2(d

d1 =−=⋅∇+ + sJJ

tc

ssss I

Diffusion and Reaction Fluxes:

,...)3,2,1( =∇−= scD sssI

,...)4,3,2(11 =−≡ −−

+ sckcckJ sssss

The process is theoretically described by a system

containing tens of kinetic equations, which is

inconvenient for applications. For this reason, one of the basic problems of micellar kinetics is how to simplify the general set

of equations without loosing the adequacy and correctness of the theoretical description.

Introduction of a Model (Gaussian) Micelle Size Distribution

(i) Region of the monomers and oligomers, Ωo (1 ≤ s ≤ no)

(ii) Region of the rare aggregates, Ωr (no < s < nr)

(iii) Region of the abundant micelles, Ωm (s ≥ nr)

)(]2

)(exp[2 r2

2m nsmsC

cs ≥−

−=σσπ

New: (1) We do not assume σ = const.

(2) We do not use of the quasi-equilibrium approximation: local chemical equilibrium between micelles and monomers.

(3) The derived general equations are nonlinear: applicable to both large and smallperturbations.

Aniansson & Wall, 1974

∑

∑

∑

≥

≥

≥

−=

=

=

r

r

r

)(1

1

22

m

2

m

m

nss

nss

nss

cmsC

scC

m

cC

σ

Reduction of the Problem to 4 Equations for 4 Unknown Functions

Monomer concentr.; Total micelle conc.; Mean aggreg. number; Polydispersity

c1(r,t), Cm(r,t), m(r,t), σ(r,t)

m,0r11

dd JJn

tc

−−=⋅∇+ I

Jt

C=⋅∇+ m,0

m

dd I

m,0rm,1m )(dd JJnmCt

+=⋅∇+ I

m,1m,02rm,2m

22 2])[(dd JJJnCmt

+−=⋅∇++ Iσ

Nonlinear expressions for the fluxes J, Jm,0, and Jm,1 are derived.

)2,1,0(r

m, =≡ ∑≥

isns

si

i II

)1,0(,r

r m, =≡≡ ∑>

iJsJJJns

si

in

Relaxation of a Spatially Uniform Perturbation

(C – jump; T – jump; P – jump: bulk relaxation methods)

Our purpose is to see what are the predictions of the model for this type of perturbations.

Dimensionless perturb. in: Total mic. conc.; Mean aggreg. number; Polydispersity

eq

p

eq

p

eq,m

p,m ;;σσ

ξξξ σ ≡≡≡mm

CC

mc

Linearization of the problem:

System of three linear equationsfor ξc, ξm, and ξσ σξλδ

σ,,,0)(

,,mcia jijij

mcj==−∑

=A homogeneous system has a nontrivial solution only if its determinant is equal to zero

⇒ characteristic equation

I1, I2, I3 – invariants of the matrix (aij); three eigenvalues ⇒ three relaxation times:

τc , τm , τσ.

0322

13 =−+− III λλλ

στ

λ ,,;11

mmcj

tk jjj =−≡−≡

The Three Micellar Characteristic Relaxation Times

στ

λ ,,;11

mmcj

tk jjj =−≡−≡

⎟⎟⎠

⎞⎜⎜⎝

⎛

++≈

eq2eq

eqeqS /1

11m

mmk

tc βσβ

β

CMCCMCtot −

≡Cβ

2eq

m21

σσk

t≈

2eqeq

2eqeq

m1

σβσ

mm

ktm

+≈

Dimensionless micelle concentration

tc is the characteristic time of the slow process, that is the relaxation of the total micellar concentration Cm ; kS – rate constant of the slow process (Aniansson-Wall)

tm is the first characteristic time of the fast process, related to the relaxation of the mean micellar aggregation number, m ; km – rate constant of the fast process (Aniansson-Wall)

tσ is the second characteristic time of the fast process, related to the relaxation of the micellar polydispersity, σ ; (new!) km – rate constant of the fast process.

The expressions for tc and tm by Aniansson & Wall are confirmed!

Numerical Results for Typical Parameter Values:

Dimensionless relax. time

Exact Approximate expressions

Low micelle concentration: β = 1

τc 4.12 × 103 4.11 × 103

τm 1.87 × 101 1.87 × 101

τσ 1.21 × 101 1.21 × 101

High micelle concentration: β = 100τc 1.18 × 105 1.18 × 105

τm 6.43 × 10−1 6.43 × 10−1

τσ 1.25 × 101 1.25 × 101

Dimensionless micelle concentration:

τc – relaxation time of micelle concentration (of the slow process, Aniansson & Wall)

τm – relaxation time of mean aggreg. number (of the fast process, Aniansson & Wall)

τσ – relaxation time of micelle polydispersity (new effect, predicted by present theory)

CMCCMCtot −

≡Cβ

στττ >> mc

:ionsconcentratmicelleLow

mc τττ σ >>:ionsconcentratmicelleHigh

7mSeqeq 10/;5;60 −=== kkm σ

(1) The Relaxation of a Perturbation in Micelle Concentration, Cm, is Governed only by the Slow Micellar Time, tc

),()/exp()( σξ tttttAt mcccc >>−≈

Hence, if the relaxation of micelle concentration is measured, only the Slow Micellar Time, tc , could be determined.

Asymptotic Expressions:

Note, however, that a perturbation in Cm perturbs also m and σ !

( Ac – amplitude of the perturbation in Cm )

The three basic physical parameters, Cm, m and σ , are perturbed.

The Relaxation of a Perturbation in Micelle Mean Aggregation Number is Governed by both the Slow and Fast Times, tc and tm

Hence, if the relaxation of micelle mean aggregation number is measured, then both the slow and fast micellar times, tc and tm ,

could be determined.

Note that a perturbation in m perturbs σ , but does not affect Cm

)/exp()()/exp()(eqeq

mm

cmcm

cm ttm

AAttm

At −++−−≈βτβτξ

( Ac and Am – amplitudes of the perturbation in Cm and m )

The Relaxation of a Perturbation in Micelle Polydispersity is Governed by the two Fast Relaxation Times, tm and tσ

Hence, if the relaxation of micelle polydispersity is measured, then the two fast micellar times, tm and tσ, could be determined.

Note that a perturbation in polydispersity σ does not affect Cm and m.

)]/exp()/[exp(2

)(

)/exp()(

2eq

eq

eqσ

σ

σ

σσσ

σβτ

ξ

tttttt

tmm

AA

ttAt

mm

mcm −−−

−++

−≈

Summary of Part 1: Micellar Relaxation Processes in the Bulk

The theoretical analysis implies:

(A) The relaxation of the three basic parameters, the micelle concentration,

Cm, the mean aggregation number, m, and the polydispersity, σ, are

characterized by three distinct relaxation times: tc, tm, and tσ .

(B) The first two of them, tc and tm, coincide with the conventional slow and

fast micellar relaxation times .

(C) The third relaxation time, tσ, is close to tm for low micelle concentrations,

but at high micelle concentrations we have tc > tσ > tm. .

(D) The relaxation of Cm is affected by tc alone.

(E) The relaxation of m is affected by both tm and tc .

(F) The the relaxation of σ is affected by tσ and tm .

Simple, but accurate analytical expressions are available:

(1) For calculation of the three relaxation times;

(2) For describing the evolution of a micellar system.

[K.D. Danov, P.A. Kralchevsky, et al., Adv. Colloid Interface Sci. 119 (2006) 1-16]

Next step: Investigation of the problem about the kinetics of adsorption from

micellar solutions, and the respective dynamic surface tension (Part 2).

Part 2: Theoretical modeling of adsorption from micellarsolutions at quiescent and expanding surfaces

Main questions to be answered:

• Why in different cases different kinetic regimes are observed ?

(a) diffusion – limited kinetics: [ Δσ ∝ t –1/2 ];

(b) reaction – limited kinetics: [ Δσ ∝ exp(– t /τ ) ].

• Which of the two very different theoretical expressions for the effective diffusivity of a micellar solution is correct ?

(a) by J. Lucassen (1975): Deff = D1(1 + β meq)(1 + β meqDm/D1)

(b) by Paul Joos (1988): Deff = D1(1 + β )(1 + β Dm/D1)

Parameters of the Adsorption Process

ζ – dimensionless distance; τ – dimensionless time; D1 – diffusivity of the monomers; ks – rate constant of the slow micellar process; km – rate constant of the fast process.

−− ≡≡

≡≡

m1

2a

mS1

2a

s

2a

1

a

;

;;1

kDh

KkDh

K

thD

zh

τζ

)lengthadsorption(

)( eq1

a ch

∂Γ∂

≡

Dimensionless Perturbations

)(]2

)(exp[2 r2

2m nsmsC

cs ≥−

−=σσπ

ppeq

eq,1ap

p2eq

eq,1ap,m

p

ap,1

p

a1 )0(

;)0(

;)0(

;)0(

σσ

ξσ

ξβ

ξξ σ Γ≡

Γ≡

Γ≡

Γ≡

chm

chC

hc

hmc

ξ1 – dimensionless perturbation in the concentration of monomers, c1;ξc – dimensionless perturbation in the micelle concentration, Cm; ξm – dimensionless perturbation in the micelle mean aggregation number, m;

ξσ – dimensionless perturbation in the micelle polydispersity, σ .

General System of Kinetic Equations (from Part 1)

ms SmK

SK

wm ϕβ

ϕσζξ

τξ

eq

mseqeq2

12

1 )( −−−∂∂

=∂∂

scc K

B ϕβζ

ξτξ s

2

2

m +∂

∂=

∂∂

msmm Kmw

KB ϕσ

ϕσβζ

ξτ

ξ2eq

m

eq

eqs2

2

m +−∂∂

=∂

∂

σσσ ξ

σϕ

σϕ

βζξ

τξ

2eq

m2eq

meq2s2

2

m2

22)1( KKm

wKB ms −−−+∂

∂=

∂∂

(surfactant monomers)

(concentration of micelles)

(micelle mean aggregation number)

(polydispersity)

ϕs – dimensionless reaction flux of the slow relaxation process; ϕm – dimensionless reaction flux of the fast relaxation process.

CMC/)CMC( tot −= Cβ

1

mm D

DB =

Reaction fluxes from the slow and fast relaxation processes

mm ξξϕ −≈ 1

mcs wmwm ξσξξσϕ eqeq1eqeq )( +−−≈

For ξ1 = ξm , we obtain ϕm = 0

(criterion for equilibrium with respect to the fast micellization process)

For ξ1 = ξc = ξm , we obtain ϕs = 0

(criterion for equilibrium with respect to the slow micellization process)

Method of solution of the general system of linear partial differential equations: Laplace transform, solving the equations, and numerical reverse Laplace transform

Numerical Results: Typical Relaxation Curves (Four kinetic regimes of adsorption: AB, BC, CD, and DE)

)()0()(

0,1eq

eq τξγγγγ

=−

−t

⇒ ξ1,0 describes the relaxation of the

surface tension γ (t).

Four differentrelaxation regimes: AB, BC, CD, DE

(B) ξ1 = ξm, then ϕm = 0 ⇒ equilibrated fast micellar process;

(D) ξ1 = ξc = ξm ⇒ ϕs = ϕm = 0 ⇒ equilibrated fast and slow micellar processes.

Analytical Expressions for the Relaxation in Different Regimes

Two exponential regimes (AB and CD) with relaxation times τF and τC ;

Two inverse-square-root regimes (BC & DE) with relaxation times τBC and τDE.

σθθτ 2111

F−=

m

cθτ =C

θc – dimensionless relaxation time of the slow process;

θm and θσ – dm.less relaxation time of the fast process.

(1) Kinetic Regime AB

∫∞

+++−+

−−

−=

022

F2

22

F

F

F

F0,1

~d~)/1~(

~)]~(exp[2)

21exp(1 τ

ττττττ

ττ

πτξ s

ss

2/1

FF

m

eqF )41(,

211

τθθβτ

σ+≡−=≡ s

Km

m

)sasymptotictimeshort(...)(21 2/10,1 −++−= τ

πτξ

)sasymptotictimelong(...)2

1exp(

1 F

F

F0,1 −+

−−

−= τξ

ss

s

The regime AB(exp) was observed by P. Joos for Triton X-100, inclined plate method.

(Depending on the surfactant and experiment. method, different regimes are observed!)

The fast micellar process governs the adsorption kinetics [exp(t/τF)]

F

F 12

1τ

≈−s

(2) Kinetic Regime BC

The regime BC was observed by us for SDS with the maximum bubble pressure method (MBPM), and by Makievski et al. for Triton X-100, MBPM again.

For σeq2/meq ≈ 1, the expression for DBC reduces to that proposed by Joos (1988).

ξ1,0 ≈ ξm,0

Diffusion control – the fastmicellar process is equilibrated, whereas the effect of the slow

process is negligible [t–1/2]

...)( 2/1BC0,1 +=

τπτ

ξ

)1)(1(1

1

m

eq

2eq

eq

2eq

1

BC

BC DD

mmDD βσβσ

τ++==

(inverse-square-root time dependence)

(relaxation time τBC, & effective diffusivity DBC)

(3) Kinetic Regime CD

The slow micellar process governs the adsorption kinetics

[exp(t/τS)]

(exponential relaxation)

(the relaxation time, τC, coincides with the characteristic time of the slow micellar process)

)exp()(C

2/1BC0,1 τ

ττπ

τξ −=

s3eq

2eq

C Kmcβσ

θτ ≈≡

)()0()(

0,1eq

eq τξγγγγ

=−

−t

(4) Kinetic Regime DE

The Lucassen equation was unsuccessfully tried by Joos et al. to fit data for Brij 58(strip method). It turns out that the data by Joos et al. (1988) correspond to the

regime BC, which has not been identified at that time.

Diffusion control – both the fast and slow micellar processes are

equilibrated [t–1/2]

...)( 2/1DE0,1 +=

τπτξ (inverse-square-root-of-time dependence; Lucassen 1975)

(Expression for the Relaxation Time & Diffusivity))1)(1(1

1

m

eq

2eq

2eq

eq

2eq

2eq

1

DE

DE DD

mm

mm

DD +

++

+≡=σ

βσ

βτ

Combined Expression for the whole BCDE region

Analyzing the basic system of equations we arrived at the following combined formula for the whole region BCDE:

)exp()()(C

2/1BC2/1DE0,1 τ

ττπ

ττπ

τξ −+=

)1)(1(1

1

m

eq

2eq

eq

2eq

BC DD

mmβσβσ

τ++=)1)(1(1

1

m

eq

2eq

2eq

eq

2eq

2eq

DE DD

mm

mm +

++

+≡σ

βσ

βτ

BCDE2eq

2eq 1100/ ττσ <<⇒>>≈m (easy to distinguish regime BC from DE)

[Details in: K.D. Danov, P.A. Kralchevsky, et al., Adv. Colloid Interface Sci, 119 (2006) 17-33.]

Difference between the exponential regimes

AB and CD

)exp()(C

2/1BC0,1 τ

ττπ

τξ −=

Exponential regime CD:

1/τF increases with micelle concentration;

1/τC decreases with micelle concentration

Exponential regime AB:

)exp(2

2

FF0,1 τ

ττ

ξ −+

=

Surfactant vs. Methods in Relation to the Relaxation Regime

Fast surfactant = {surfactant that adsorbs quickly}:

Regimes CD and DE: Difficult for detection because ξ1,0 has become very small for these regimes; in principle, these regimes could be detected for very fast surfactants by very slow and sensitive methods.

Fast method = {method that measures early surface age}:

Regime AB:

Can be detected for “slow”surfactants by “fast” methods. Example: Triton X-100 by inclined plate method.

Regime BC:

Can be detected for “fast”surfactants by “slow” methods. Example: SDS by MBPM.

Rudimentary Kinetic Diagram at Low Micelle Concentration (β ≈ 1) and/or at Smaller Difference between the Rates of the Fast and

Slow Micellization Processes (smaller Km/Ks)

The three micellar relaxation times, θc , θmand θσ , are close to each other.

Points A, B, and C coincide.

)sasymptotictimeshort(

...)(21 2/10,1

−

++−= τπτξ

(the same as in regime AB)

)exp(2

2

CC0,1 τ

ττ

ξ −+

=

The long-time asymptotics is the same as in regime AB but τF → τC

The effect of the fast relaxation process disappearsat low surfactant concentrations (β ≈ 1 )

Methods with StationaryInterfacial Expansion:

Micellar Kineticsconst.

d)(d

dd1)( ==≡

tt

tA

At αα&

αθ &1

2a

Dh

≡ θ – dimensionlessexpansion rate

The kinetic regimes are:

(AB) θ /(KAB + θ )1/2 kinetic regime governed by the fastmicellization process;

(BC) θ 1/2– diffusion-limitedregime at equilibrated fast process but negligible slow process;

(CD) θ /(KCD + θ )1/2 kineticregime governed by the slow micellization process;

(DE) θ 1/2– diffusion-limitedregime at equilibrated both the fast and slow micellization processes.

[Details in: K.D. Danov, P.A. Kralchevsky, et al., Colloids & Surfaces A, 282-283 (2006) 143-161]

Comparison of Theory and Experiment

Example 1: Dynamic surface tension of Brij 58 measured by the strip method (Paul Joos)

Theory:2/1

BC

2eq

eq )2

(CMC D

kT απσσ&Γ

=−

CMC = 1.0 × 10−8 mol/cm3

Γeq = 2.70 × 10−10 mol/cm2 (at CMC)

From the slope of the plot

σ − σ eq vs. at fixed β, one determines the apparent diffusivityDBC(β).

2/1α&Different curves correspond to different micelle concentrations, β

β = (Ctot − CMC)/CMC0 5 10 15 20

DB

C /D

1

0

20

40

60

80

100 Brij 58, Strip methodu = 1.06, Bm = 0.243

Comparison of Theory and Experiment (Continued)

Example 1: Dynamic surface tension of Brij 58 measured by the strip method (Paul Joos)

From the fit of the data, one determines:

For Brij 58, meq = 70 ; hence the polydispersity of the micelles is σeq = 8.6

The obtained reasonable values confirm that the kinetic regime is BC

)1)(1( m1BC BuuDD ββ ++≡

06.1/ eq2eq =≡ mu σ

)valuereasonable(243.0/ 1mm == DDB

The Overflowing Cylinder Method (OFC) C.D. Bain et al.

Langmuir 2004, 20, 4436-4445

The Overflowing Cylinder Method

.dd1)( const

tA

At =≡α&

Another method with stationary interfacial expansion

Adsorption (rather than surface tension) is detected by ellipsometry

or neutron reflection

0.0 0.5 1.0 1.5 2.0 2.5

Ads

orpt

ion,

Γ (

μmol

/m2 )

1.5

2.0

2.5

3.0

3.5

4.0

4.5C14TAB + 100 mM NaBr

Overflowing cylinderu = 1.2; Bm = 0.23

Y (s−1/2)

Γeq

Example 2: Dynamic surface tension of C14TAB by the Overflowing Cylinder Method(Colin Bain et al.; the adsorption Γ is directly measured by ellipsometry)

The data points are not for the same β !

Theory:

Y = 0 (i.e. = 0), gives the equilibrium

adsorption at CMC, Γeq.

The best fit Γ vs. Y corresponds to

Bm = Dm/D1 = 0.23 and u = σeq2/meq = 1.2

meq = 80; from u = 1.2 we determine that the polydispersity of the C14TAB micelles is

σeq = 9.8. The obtained values of Γeq. Bm and σeq are reasonable.

This confirms that the kinetic regime is BC. [τdif = ha2/D1 is the characteristic diffusion time]

Yeq2/1

difeq )( Γ−Γ=Γ τ

2/1m )]}1)(1(2/[{ BuuY ββαπ ++≡ &

α&

1.5 mM SDS, 128 mM NaCl

t−1/2 (s−1/2)

0 2 4 6 8 10

Dyn

amic

sur

face

tens

ion,

mN

/m

30

35

40

45

50

55

60

Kruss BP 2Setup - Horozov

Explanation:

Different time-dependence, A(t), of the bubble surface area for different apparatuses.

Part 3: Application of the Maximum Bubble Pressure MethodProblem: Different tensiometers - different results for the dynamic surface tension.

This is difference is demonstrated with our data for two apparatuses.The data are plotted as DST vs. t–1/2:

Solution of the Problem Suggested by the Experiment

The experiment indicates that A(td) is independent of:

1. Bubbling period, tage

2. Surfactant concentration3. Surfactant type

(General validity?)

The theory indicates that in most cases γ(t) depends on a constant parameter, λ = integral of A(t), rather than on the function A(t).

A(t) (the apparatus function) can be determined only by cinematography;

λ (the apparatus constant) can be determined also by MBPM (much easier!)

Below we check whether λ is independent of tage, surfactant type and concentration.

Expanding Surface vs. Immobile Surface

2ageu2/1

2eq

0, /,)(

λπ

λλ γγ ttcD

kTss =

Γ==

∞

∫ −≡

1

0d

0

d

d2/1

1d])([

dd

)(1 t

AtA

tττλ

)1(,ˆd)ˆ( d10

d20

d2d

==≡ ∫ ttA

tAt

τττ

(1) The whole effect of the interfacial expansion is

incorporated in λ ;

(2) tu (universal surface age), is the age of an (initially clean) immobile surface with the sameγ as that registered by the MBPM tensiometer.

2/1u

0,eq2/1

age

0,eq2/1

ageeq )()()( t

s

t

s

t

s γγγ γλ

γγγ +=+=+=

c∞ – bulk surfactant concentration; Γeq – equilibrium adsorption; γeq – equilibrium surface tension; D – surfactant diffusivity;

sγ,0 – the value of sγ for an immobile interface.

Example 1:

Comparison of data obtained by MBPM (expanding bubbles) with data for immobile bubbles (IB) for SDS + 100 mM NaCl.

For MBPM λ = 6.074 is determined by integration of the experimental A(t) curve (for our apparatus);For IB we have λ = 1 (no expansion)

tu = tage/λ2 is used to plot the MBPM data in (b) (λ2 ≈ 37)

Example 2:

Comparison of data obtained by MBPM (expanding bubbles) with data from Wilhelmy plate method for Na N-Cocoylglycinate.

Nominal surface age, tage1/2 (s1/2)

0 2 4 6 8 10

Surf

ace

tens

ion,

γ

(mN

/m)

30

40

50

60

70 1.5 mM NaCG + 100 mM NaCl

MBPM

Wilhelmy plate

Universal surface age, tu1/2 (s1/2)

0 2 4 6 8 10

Surf

ace

tens

ion,

γ

(mN

/m)

30

40

50

60

70 1.5 mM NaCG + 100 mM NaCl

MBPM

Wilhelmy plate

mN/m01.043.24eq

2/1u0,

0,eq

±=

++=

γ

γγγ

γ

tasFor Wilhelmy plate (immobile

surface): tu = tage (λ = 1)For MBPM (expanding surface): tu = tage/λ2 (λ = 6.07).

The excellent fit with the theoretical dependence evidences diffusion-limitedadsorption kinetics.

Two ways to determine the apparatus constant, λ:

(1) By integration of the experimental A(t) curve;

(2) By MBPM experiments, λ = sγ/sγ,0 .

(Compare the values of λ obtained in the two ways!)

Procedure:

(1) γ (tage) curves are obtained by MBPM and fitted with the dependence:

Thus sγ is determined.

(2) Next, sγ,0 is calculated from fits of equilibrium surface tension isotherms.

(3) λ = sγ/sγ,0

2/1age

eq )(ta

s

++= γγγ

∞

Γ=

cD

kTs 2/1

2eq

0, )(πγ

CSDS(mM)

Γeq(μmol/m2)

sγ,0(mN.m−1.s1/2)

sγ(mN.m−1.s1/2)

λ = sγ/sγ,0

SDS + 10 mM NaCl

0.1 0.91 0.553 3.36 6.077

0.5 2.65 0.979 5.95 6.078

1 3.32 0.796 4.83 6.066

2 3.76 0.545 3.31 6.073

3 3.96 0.423 2.57 6.075

SDS + 100 mM NaCl

0.2 3.43 4.560 27.7 6.074

0.5 3.86 2.316 14.0 6.046

0.75 4.00 1.658 10.1 6.091

1 4.08 1.299 7.89 6.075

Average: λ = 6.07 ± 0.01Theory: λ = 6.074

CDTAB(mM)

Γeq(μmol/m2)

sγ,0(mN.m−1.s1/2)

sγ(mN.m−1.s1/2)

λ = sγ/sγ,0

DTAB + 5 mM NaBr

1 1.69 0.222 1.35 6.083

2 2.36 0.236 1.43 6.057

3 2.68 0.217 1.32 6.079

5 3.01 0.178 1.08 6.060

7 3.18 0.150 0.912 6.075

10 3.33 0.122 0.738 6.066

12 3.40 0.108 0.657 6.069

DTAB + 100 mM NaBr

2 3.26 0.438 2.66 6.068

3 3.35 0.312 1.89 6.068

4 3.41 0.244 1.48 6.075

Average: λ = 6.07 ± 0.01Theory: λ = 6.074

Conclusions: The results confirm the concept about the apparatus constant:

(1) λ is the same for all concentrations of a given surfactant and electrolyte;

(2) λ is the same for SDS and DTAB;

(3) λ is calculated from sγ determined of the data fits for 10 ms < tage < 40 s

tu = tage/λ2

λ2 = (6.07)2 ≈ 37Hence, in terms of tu the

MBP method is much (37 times) faster.

tu = tage/λ2, accounts for the surface expansion, and for

this reason tu gives the physically correct surface

age.

Details in: Christov et al., Langmuir 22 (2006) 7528.

Plot of the dimensionless effective diffusivity of micellar solutions, Deff/D, vs. β

2

2CMC,eff

γ

γ

s

sD

D=

D – diffusivity of surfactant monomers.

The data for γ(t) are fitted well with the

expression for diffusion-limited

adsorption:

2/1age

eq )(ta

s

++= γγγ

Hence, the kinetic regime is either BC or DE. The data complies with BC!

)1)(1( m

eq

2eq

eq

2eqBCeff β

σβ

σD

DmmD

DD

D++==

Maximum Bubble Pressure Method (MBPM)

Plot of Plot of σeq2 vs. β calculated from the data for Deff/D

Kinetic regime BC

Rudimentary kinetic regime

meq – micelle mean aggregation

number;

σeq – micelle polydispersity

Values σeq

2/meq < 2are reasonable.

For example, for β = 10 and meq = 70 we

obtain:

σeq = 7.2 (SDS)

σeq = 9.5 (DTAB)Reasonable parameter values ⇒ the kinetic regime is BC

(MBPM)

Summary and Conclusions

The theory indicates the presence of four different kinetic regimes of adsorption from micellar surfactant solutions:

(1) Regime AB: the fast micellar process governs the adsorption kinetics [exp(t/τF)]

(2) Regime BC: diffusion control – the fast micellar process is equilibrated, whereas the effect of the slow process is negligible [t–1/2].

(3) Regime CD: the slow micellar process governs the adsorption kinetics [exp(t/τS)]

(4) Regime DE: diffusion control – both the fast and slow micellar processes are equilibrated [t–1/2].

(5) MBPM: The determination of the apparatus constant, λ, for a given tensiometer allows one to characterize a given surfactant solution with a universal dynamic surface tension curve, γ(t).

1. K.D. Danov, P.A. Kralchevsky, N.D. Denkov, K.P. Ananthapadmanabhan, and A. Lips, “Mass Transport in Micellar Surfactant Solutions: 1. Relaxation of Micelle Concentration, Aggregation Number and Polydispersity”, Adv. Colloid Interface Sci. 119 (2006) 1-16.

2. K.D. Danov, P.A. Kralchevsky, N.D. Denkov, K.P. Ananthapadmanabhan, and A. Lips, “Mass Transport in Micellar Surfactant Solutions: 2. Theoretical Modeling of Adsorption at a Quiescent Interface”, Adv. Colloid Interface Sci. 119 (2006) 17-33.

3. K.D. Danov, P.A. Kralchevsky, K.P. Ananthapadmanabhan, and A. Lips, “Micellar Surfactant Solutions: Dynamics of Adsorption at Fluid Interfaces Subjected to Stationary Expansion”, Colloids & Surfaces A, 282-283 (2006) 143-161.

4. N.C. Christov, K.D. Danov, P.A. Kralchevsky, K.P. Ananthapadmanabhan, and A. Lips, “The Maximum Bubble Pressure Method: Universal Surface Age and Transport Mechanisms in Surfactant Solutions”, Langmuir 22 (2006) 7528-7542.

5. K.D. Danov, P.A. Kralchevsky, K.P. Ananthapadmanabhan, and A. Lips, “Influence of Electrolytes on the Dynamic Surface Tension of Ionic Surfactant Solutions: Expanding and Immobile Interfaces”, J. Colloid Interface Sci. (2006) in press.

The results are published in the following papers: