Markov Chain Model

1
Markov Chain Model of Mixed Surfactant Systems
II. Critical micelle concentration, solution and
micelle composition
R. Slavchov1, G. S. Georgiev2 University of
Sofia, Faculty of Chemistry 1 Department of
physical and theoretical chemistry 2 Laboratory
of Water-Soluble Polymers, Polyelectrolytes and
Biopolymers E-mail fhri_at_chem.uni-sofia.bg
2
The model
Micellization as a Markov chain process. Solution
of a single surfactant S1
Monomer in the solution
K11
S1 S1 S1S1
Micelle active center
Dyad active center
S1 S1S1
3
The model
Micellization as a Markov chain process. Solution
of two surfactants S1 and S2
Kii
S1 S1 S1S1
K12
S1 S2 S1S2
K21
S2 S1 S2S1
K22
S2 S2 S2S2
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The model
Candidates for active centers
Linear aggregates
S2

S1
S1
S2S2
In the micelle molar part
In the solution monomer concentration molar part
c S1 S2
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The model
Candidates for active centers
Disc-shaped micelles
In the micelle molar part
In the solution monomer concentration molar part
c S1 S2
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The model
Candidates for active centers
In the micelle molar part
In the solution monomer concentration molar part
c S1 S2
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The results
Solution of two surfactants S1 and S2
S1 S1 S1S1
S1 S2 S1S2
S2 S1 S2S1
S2 S2 S2S2
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The results
Solution of two surfactants the c(x) dependence
Only one parameter
9
The results
Solution of two surfactants the y(x) dependence
Conditional probabilities for alteration/conservat
ion of the active centers
A second parameter needed for y(x)
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The results
Solution of two surfactants closing the system
At given total concentration C with total
composition X1there are 3 unknown quantities c,
x1, y1
The mass balance
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Relation to RS model
Regular solution model as a special case of MC
Markov chain
Regular solution
The c(x) dependencies are approximately equal when
The y(x) dependencies are approximately equal
when additionally
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Relation to RS model
RS as a special case of MC the c(x) dependence
Markov chain
bRS 2
c mM
Regular solution
bRS 1
bRS 0
bRS -1.5
bRS -10
x1
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Relation to RS model
RS as a special case of MC the y(x) dependence
Markov chain
y1
Regular solution
bRS -10
bRS 1
x1
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Relation to RS model
RS as a special case of MC the y(x) dependence
Markov chain
y1
Regular solution
bRS -1.5
bRS 2
x1
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Analysis
c(x) at different g
The solution for c
Eventual extrema
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Analysis
c(x) at different g
At g?0 (bg ? -8) full synergism case, c 0 at
any x1.
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Analysis
c(x) at different g
c mM
g0.5
g0.01
x1
At 0 lt g lt 1 (bg lt 0) synergetic interactions, c
lt cid If gltcmc2/cmc1 a single minimum
(condition for synergism).
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Analysis
c(x) at different g
c mM
g1
x1
At g1 (bg 0) ideal solution,
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Analysis
c(x) at different g
c mM
g5
x1
At g gt 1 (bg gt 0) antagonistic interactions, c gt
cid If ggtcmc1/cmc2 a single maximum (condition
for antagonism).
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Analysis
c(x) at different g
c mM
g20
x1
At g gt 1 (bg gt 0) antagonistic interactions, c gt
cid If ggtcmc1/cmc2 a single maximum (condition
for antagonism).
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Analysis
c(x) at different g
c mM
.
x1
At g?8 (bg ? 8) immiscibility,
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Comparison with experimental results for cmc(X)
standard deviations from the experiment
type (nonionic/ionic/zwiterionic)
whichs better
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Comparison with experimental results for cmc(X)
Results 6 wins for regular solution model5
wins for Markov chain model In average, Markov
chain is fitting cmc with 0.2 better Models
are describing cmc(X) equally well
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Comparison with experimental results for y(x)
cmc(X) data determination of g and b
Results One parametric models cannot describe
both cmc(X) and y(x) in most cases. Markov
model can.
y(x) data determination of K12
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Example with mixture of C8E5 and C8TAB
DErrico et al., PCCP, 2002
experimental data
cmc mm
MC model
y1
RS model
1 par. MC
x1
x1
g0.44
K129.39 mm-1
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Example with mixture of C8E5 and C8TAB
DErrico et al., PCCP, 2002
The mass balance
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Example with mixture of C8E5 and C8TAB
DErrico et al., PCCP, 2002
y1, x1
extrapolation for y(cmc)?
y1(C)
X10.15
x1(C)
C mm
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Analysis
c(Cgtgtcmc)
c mM
Behavior with increasing the total concentration
C
cmc(X1)
g22 (bRS 2) K12 1 mM-1 X10.25
x1
y1
c in antagonistic systems can exhibit a maximum
y1 X1
x1(Cgtgtcmc)
x1
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Analysis
x1, y1
x1
Behavior with increasing the total concentration
C
X1
g22 (bRS 2) K12 1 mM-1 X10.25
y1
C mM
c mM
This maximum can be at C much larger than
cmc. It is an alternative explanation of the
proposed by some authors second cmc.
C mM
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Analysis
Behavior with increasing the total concentration
C Gharibi et al., 2000-2006
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Summary
This study proposes an alternative to the
currently most used RS model for mixed
micellization. This new model is based on Markov
chain mechanism. Markov model - has one
fitting parameter for c(x) and one more for
y(x) - is leading to analytical functions for
both c(x) and y(x) - can be modified in one
parametric MC model, which is nearly equivalent
to RS for a large interval of bRS values -
describes cmc(X) data as good as RS -
describes y(x) data where RS fails - has A LOT
of unrealized capacity.
E-mail fhri_at_chem.uni-sofia.bg