A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions
A novel statistical model for the cooperative binding of monomeric ligands to a linear lattice is developed to study the interaction of ionic surfactant molecules with flexible polyion chain in dilute solution. Electrostatic binding of a ligand to a site on the polyion and hydrophobic associations b...
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irk-123456789-1534782019-06-15T01:27:52Z A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions Nishio, T. Shimizu, T. Yoshida, Sh. Minakata, A. A novel statistical model for the cooperative binding of monomeric ligands to a linear lattice is developed to study the interaction of ionic surfactant molecules with flexible polyion chain in dilute solution. Electrostatic binding of a ligand to a site on the polyion and hydrophobic associations between the neighboring bound ligands are assumed to be stochastic processes. Ligand association separated by several lattice points within defined width is introduced for the flexible polyion. Model calculations by the Monte Carlo method are carried out to investigate the binding behavior. The hypothesis on the ligand association and its width on the chain are of importance in determining critical aggregation concentration and binding isotherm. The results are reasonable for the interpretations of several surfactant-flexible polyion binding experiments. The implications of the approach are presented and discussed. Для вивчення взаємодiї молекул iонного сурфактанта з гнучким полiiонним ланцюжком у розведеному розчинi запропоновано нову статистичну модель для колективного зв’язування мономерних лiганд з лiнiйною граткою. Припускається, що електростатичне зв’язування лiганди з вузлом на полiiонi та гiдрофобнi зв’язки мiж сусiднiмi зв’язаними лiгандами є стохастичними процесами. Для гнучкого полiiона вводиться асоцiацiя лiганд, вiдокремлених декiлькома гратковими точками в межах визначеної ширини. Для того, щоб дослiдити поведiнку зв’язування, здiйсноються обчислення методом Монте Карло. Гiпотеза асоцiацiї лiганди та її ширини на ланцюжку є важливою для визначення критичної концентрацiї агрегацiї та iзотерми зв’язування. Результати є прийнятними для iнтерпретицiї декiлькох експериментiв по зв’язуванню сурфактант-гнучкий полiiон. Представлено та обговорено застосування методу. 2014 Article A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions / T. Nishio, T. Shimizu, Sh. Yoshida, A. Minakata // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43302: 1–11. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 36.20.Kd, 61.25.he, 82.60.Hc, 82.70.Uv, 87.10.Mn, 87.16.A- DOI:10.5488/CMP.17.43302 arXiv:1501.02340 http://dspace.nbuv.gov.ua/handle/123456789/153478 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A novel statistical model for the cooperative binding of monomeric ligands to a linear lattice is developed to study the interaction of ionic surfactant molecules with flexible polyion chain in dilute solution. Electrostatic binding of a ligand to a site on the polyion and hydrophobic associations between the neighboring bound ligands are assumed to be stochastic processes. Ligand association separated by several lattice points within defined width is introduced for the flexible polyion. Model calculations by the Monte Carlo method are carried out to investigate the binding behavior. The hypothesis on the ligand association and its width on the chain are of importance in determining critical aggregation concentration and binding isotherm. The results are reasonable for the interpretations of several surfactant-flexible polyion binding experiments. The implications of the approach are presented and discussed. |
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Nishio, T. Shimizu, T. Yoshida, Sh. Minakata, A. |
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Nishio, T. Shimizu, T. Yoshida, Sh. Minakata, A. A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions Condensed Matter Physics |
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Nishio, T. Shimizu, T. Yoshida, Sh. Minakata, A. |
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Nishio, T. |
| title |
A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
| title_short |
A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
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A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
| title_fullStr |
A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
| title_full_unstemmed |
A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
| title_sort |
model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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A model study of cooperative binding of ionic surfactants to oppositely charged flexible polyions / T. Nishio, T. Shimizu, Sh. Yoshida, A. Minakata // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43302: 1–11. — Бібліогр.: 29 назв. — англ. |
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Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-14T04:37:33Z |
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| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 4, 43302: 1–11
DOI: 10.5488/CMP.17.43302
http://www.icmp.lviv.ua/journal
A model study of cooperative binding of ionic
surfactants to oppositely charged flexible polyions
T. Nishio1, T. Shimizu2, Sh. Yoshida1, A. Minakata1
1 Department of Integrated Human Sciences (Physics), Hamamatsu University School of Medicine,
Hamamatsu 431-3192, Japan
2 Department of Electronic and Information System Engineering, Faculty of Science and Technology,
Hirosaki University, Hirosaki 036-8561, Japan
Received May 16, 2014, in final form September 26, 2014
A novel statistical model for the cooperative binding of monomeric ligands to a linear lattice is developed to
study the interaction of ionic surfactant molecules with flexible polyion chain in dilute solution. Electrostatic
binding of a ligand to a site on the polyion and hydrophobic associations between the neighboring bound
ligands are assumed to be stochastic processes. Ligand association separated by several lattice points within
defined width is introduced for the flexible polyion. Model calculations by the Monte Carlo method are carried
out to investigate the binding behavior. The hypothesis on the ligand association and its width on the chain are
of importance in determining critical aggregation concentration and binding isotherm. The results are reason-
able for the interpretations of several surfactant-flexible polyion binding experiments. The implications of the
approach are presented and discussed.
Key words: ionic surfactant-flexible polyion interaction, lattice of linear polyion, cooperative binding of
ligands, binding isotherm, association width, model calculation
PACS: 36.20.Kd, 61.25.he, 82.60.Hc, 82.70.Uv, 87.10.Mn, 87.16.A-
1. Introduction
Complex formation of surfactant molecules with polymer chain is one of the most important and
attractive subjects in colloid and polymer science. It is also useful in various fields of applied chemistry,
such as pharmaceutical chemistry, food science, cosmetic manufacture, and so on. Extensive studies have
been carried out on interaction of ionic surfactants with oppositely charged linear polyion, in particu-
lar [1–3].
One of the interesting subjects to be revealed in this field is the cooperative nature of surfactant
binding due to the hydrophobic interactions among bound surfactant molecules on the polyion chain.
To know about it, it is essential to understand the behavior of binding isotherms as well as the critical
aggregation concentration (CAC) which characterizes the onset of surfactant binding to polyion [4, 5]. The
cooperative nature is due to the side-by-side hydrophobic interactions (association) of the aliphatic tail of
the surfactant molecules bound to the polyion chain.
Concerning the theoretical approaches, simple cooperative site binding theory based on the Ising
model has been widely used to successfully analyze the binding isotherms in most cases of dilute solu-
tions of stiff polyions [6, 7]. In our previous works, the effects of steric hindrance and of intrapolymer
cooperative interaction between bound ligands across an unoccupied binding site, named skip-binding,
are investigated by expanding the simple Ising model. The effect of steric hindrance due to head group
size should be taken into account in some systems [8, 9]. Data fittings usingmatrixmethod to thismodified
binding model show that this approach is available to the analysis of binding isotherms to stiff polyions
[10]. Furthermore, it is important to consider the effect of polyion flexibility on the surfactant binding to
understand several experimental studies [11–13].
© T. Nishio, T. Shimizu, Sh. Yoshida, A. Minakata, 2014 43302-1
http://dx.doi.org/10.5488/CMP.17.43302
http://www.icmp.lviv.ua/journal
T. Nishio et al.
However, there seem to be still at least two shortcomings in these models at the moment. Firstly, the
aliphatic tail of the bound surfactant molecules has some degree of freedom with its binding orientation:
this should lead the neighboring interactions between the bound surfactants to be stochastic process
essentially. Secondly, the hydrophobic association between the bound surfactants separated by several
sites could substantially take place in the ionic surfactant-flexible polyion system. These two new points
of view should be taken into account in the site binding model treatment.
In the improved model, the polyion molecule, on which bound monomeric surfactant molecules as-
sociate with each other, is also assumed to be a linear lattice chain. Here, electrostatic ligand binding
to a lattice point and the hydrophobic association between bound ligands are assumed to be stochas-
tic processes according to the free energy change. In addition, the associations between bound ligands
are acceptable, when their separation is within the given range on the linear lattice. This expansion of
the binding model is necessary for a sufficiently flexible polyion chain. The calculation procedure using
random numbers is employed to obtain the binding isotherms of the system.
In this paper, the typical calculations of the parameter dependence are attempted for the interpreta-
tion of the binding isotherm of ionic surfactants to the flexible polyion with standard Monte Carlo (MC)
algorithm. The results of this simple one-dimensional model calculation show an apparent increase of
the surfactant binding, and is reasonable for the interpretation of the characteristics observed in the ex-
perimental studies. The expansion of the scheme on the association of a pair of the ligands is essential
for the binding to the flexible polyion. The present approach for a new lattice model should be feasible
to analyze the experimental results of the system of ionic surfactant-flexible polyion.
2. Theory
2.1. The model and parameters
In this model, a ligand is not necessary to interact with the ligands on the adjacent binding sites,
differently from the ordinary Ising models including our previous ones. It is based on the assumption
that the electrostatic interaction permits the ligand bindings in spatially different orientations around
the polyion, although the model does not explicitly include the electrostatic interaction. Another signifi-
cant point of the model is the introduction of the association across the multiple sites. The characteristic
parameter association width n, mentioned later on, is introduced to the model to reflect the persistence
length of a polyion. Namely, the more flexible the polyion backbone is, the larger n the polyion has.
These assumptions enable the association between ligands, separated by several (occupied and/or non-
occupied) binding sites on the polyion. This brings a partially two- or three-dimensional character into
the one-dimensional lattice, and improves the behavior of ligand associations which is more suitable for
analysis.
A linear chain consisting of n equivalent charged lattice points (binding sites) is regarded as a model
of a flexible polyion. A monomeric ligand (oppositely charged ionic surfactant molecule) can bind to one
of the sites with the binding free energy change ∆εb, where the intrinsic binding free energy change is
∆ε0
b. The main part of the change ∆ε
0
b is due to electrostatic energy change between a charged site and
an ionic head group of the surfactant. Bound ligand is capable of stochastically associating with other
bound ligands with the association free energy change ∆εa primarily due to the hydrophobic interac-
tions between the aliphatic tails. This association is assumed to cover the bound ligand on the n-th neigh-
boring lattice points on both sides due to polyion chain collapse. The free energy change contains the
contribution from the polyion deformation. The number n, dependent on polymer configuration, poly-
mer flexibility, and surfactant size, is called ‘association width’ in this manuscript. In the case of n Ê 2,
the association is possible between the bound ligand separated by lattice point(s). One ligand can simply
associate with two ligands on both sides of the polyion chain in the present model, as shown in figure 1.
The pair association between the bound ligands is chosen to be the nearest neighbor within the asso-
ciation width n. In this model, electrostatic ligand binding without hydrophobic association is permitted
even within the association width. This association does not prevent the associations by other ligands
between the pair (see figure 1). In our previous work, cooperative ligand association across only one un-
occupied site (skip-binding) was introduced for the surfactant association on the relatively stiff chain [10].
43302-2
Binding of ionic surfactants to flexible polyions
Figure 1. Schematic representation of the surfactant binding to the sites on the flexible polyion in the
present model. The possible pair associations between bound ligands in the case of ‘association width’
n = 4 are indicated by thick arrows between aliphatic tails of surfactant molecules.
The assumption in the present model is expanded more suitably for the flexible polyion chain, although
the formation of very long loops is ignored and the change of the model makes the analytical calculation
difficult.
The other quantities and the parameters used in thismanuscript are defined as,Cf, free ligand concen-
tration, θ, binding fraction of sites (degree of binding), K , intrinsic binding constant, and u, cooperativity
parameter where σ= 1/u was used as a cooperativity parameter in our previous papers.
In the present model, five (one ligand-free and four bound) states of a binding site are assumed on
each lattice point. Their symbols and definitions are
f, there is no bound ligand, i.e., the site is ligand-free,
b0, there is a bound ligand which does not associate with other ligands,
bL, there is a bound ligand which associates with a ligand on left side,
bR, there is a bound ligand which associates with a ligand on right side, and
b2, there is a bound ligand which associates with two ligands on both sides.
A bound ligandwithout associated ligandswithin thewidth n is in state b0. If the ligand has associable
ligands, its state is stochastically determined according to their energy change in the MC trials. The states
and their transitions are schematically illustrated in figure 2.
The fraction of sum of bound states (binding fraction) θ is represented as,
θ = [b0]+ [bL]+ [bR]+ [b2] , (2.1)
where the symbol with brackets means their fraction on the polyion and obviously [bL] = [bR] since
the binding with left-hand and right-hand ligands is equivalent. The relations of K and u with these
concentrations and the above mentioned free energy changes are expressed as,
K = [b0]
[f]Cf
= exp
(−∆ε0
b/kBT
)
, (2.2)
and
u = 1
σ
= exp(−∆εa/kBT ) , (2.3)
respectively, where kB is the Boltzmann constant and T is absolute temperature.
43302-3
T. Nishio et al.
Figure 2. Schematic representation of five states (f, b0, bL, bR and b2) of a binding site (small closed circle)
on the polyion chain with surfactant and their associations (small arrows). Large arrows with a fraction
indicate the directions of the state transitions and their probabilities of the MC trial, respectively.
The formation of nano- and microparticles is one of the interesting subjects of the surfactant-polyion
[14]. The serial cluster of bound surfactant molecules with the intrapolymer interactions can be treated in
the present model. A series of bound states with associating surfactants on a polyion chain is considered
as a ‘cluster’. The cluster, whose size (number of associating ligands) is more than one, is serial b2 states
having bL and bR states on both ends, such as (bRb2 · · ·b2bL). The b0 state without association is also
assumed to be the smallest cluster whose size is one. In figure 1, there are three clusters containing an
isolated surfactant. Then, the fraction of the cluster is given by [b0]+ [bL]. The mean size of the cluster lC
is defined as,
lC = θ
[b0]+ [bL]
, (2.4)
where lC Ê 1. By a simple idea, at a full binding state θ = 1, the following relation is assumed
u = [bL]
[b0]
= [bR]
[b0]
= [b2]
[bL]
= [b2]
[bR]
.
It is expected that the lC value tends to u+1when θ approaches unity. The distribution of the cluster size
can be obtained as well.
2.2. Model calculation
In the MC process, the state transition of each site is decided by the probability function with the
change of its energy using a pseudo-random number (figure 2). The free energy change of only ligand
binding without association between bound ligands (f → b0) is
∆εb =∆ε0
b −kBT ln(Cf) =−kBT ln(KCf) , (2.5)
which is dependent on the concentration of free ligand. The free energy change with association between
a pair of bound ligands (b0 → bL or bL and bL or bR → b2) is
∆εa =−kBT lnu. (2.6)
The acceptance of state change trials is determined according to the standard Metropolis algorithm using
the probability function of the energy change of the system [15, 16].
43302-4
Binding of ionic surfactants to flexible polyions
For example, if the state of the site is b0, the probabilities of the MC trial for the transition to the states
f, bL and bR are 1/2, 1/4, and 1/4, respectively. In the process from b0 to bL, if the bound ligand which
can associate to the objective ligand in the left-hand side within width n is found, the state transition
is decided in the above mentioned manner. If no associable ligand is found, the process is discarded.
Other probabilities of the process and the manner of decision are similarly given in order to maintain the
detailed balance (see figure 2).
The number of MC step per site is chosen within the range from 105
to 106
at a given KCf (∆εb ) value
so that the convergence of the quantities is achieved in the early stage (mostly within a few percent of
half MC steps). After half discarded MC steps, the quantities are finally evaluated by summing up in the
later MC steps.
All f state is usually adopted as an initial state of sites for a series of calculations. A final state at each
KCf value is used as an initial state of the next point. The MC trials are carried out in the order of the
lattice site number within a single loop. The results are unaffected even if the site is randomly chosen, or
an initial state is changed.
In the standard calculations, the lattice has N = 1000 sites under the periodic boundary condition.
When N = 5000, no difference of the result is observed. The generalized Fibonacci method is used as a
random number generator [17].
2.3. Analytical solutions
In the classical cooperative model without various more realistic effect, we have the well-known
equation of binding fraction by Satake and Yang [7], expressed as,
θ = 1
2
[
1− 1− s√
(1− s)2 +4σs
]
= 1
2
[
1− 1−K uCf√
(1−K uCf)2 +4KCf
]
, (2.7)
where well-used parameter s is defined as, s = K uCf [7]. When u = 1, the relation means simple binding,
as
θ = KCf
1+KCf
. (2.8)
The binding isotherm by our previous scheme with a matrix method is also presented in the case of
n = 2 in figure 4 and figure 5 [10].
3. Results and discussion
3.1. Dependence on the cooperativity parameter u
First, the fundamental properties of our model system are presented as compared with the classical
cooperative bindings of equation (2.7). The dependence of binding isotherm on the cooperativity param-
eter u is presented at the association width n = 1 in figure 3. In this case, the association with the nearest
neighbor is only considered. At u = 1, the MC calculation curve shows a higher affinity (lower CAC) and
cooperativity than the simple binding curve of equation (2.8) due to the present binding schemewith four
bound states. With an increase of the parameter u, the difference between both curves becomes smaller.
At u = 100, the differences of θ values are within ±10% at most points.
In figure 4, the dependence on the cooperativity parameter u is shown in the case of n = 4. Associ-
ations between the surfactants separated by binding sites within the width n = 4 reduce the CAC values
to about one third of the classical model. The association with larger u value results in higher affinity
and asymmetrical binding isotherms in the case of the ligand association across the binding sites. In the
high u cases, most of the bound surfactants provide associations with the neighboring ones in the range
θ < 0.5. Then, the bindings become relatively weak in high θ range. At u = 100, the MC result shows a
slight two-phase binding curve with a shoulder at around θ = 0.5, corresponding to the rearrangement of
clusters (shown below).
The binding isotherm with a sharp rise and gradual saturation are frequently observed in the experi-
ments of surfactant binding to a flexible polyion[4, 18]. A factor deforming binding isotherm is exhibited
43302-5
T. Nishio et al.
Figure 3. Dependence of binding isotherm on the cooperativity parameter u at association width n = 1.
Symbols, MC calculations, lines, curves by the Satake-Yang equation, equation (2.7). u; 1, 3.33 (1/u = 0.3),
10, 33.3 (1/u = 0.03), and 100, from right to left, respectively. N = 1000 under the periodic boundary
condition.
Figure 4. The dependence of binding isotherm on the cooperativity parameter u at association width
n = 4. the others are the same as in figure 3.
from our model calculations. In the previous works, we showed that the two-phase binding isotherm is
interpreted by the hypothesis of ligand association across a site and/or of steric hindrance. In the present
model, we can observe the similar feature only in the restricted (n = 2−8 and high u) cases.
3.2. Dependence on the association width n
Dependence of the binding isotherm on the association width n is shown in figure 5 in the case of
u = 10. With an increase of the association width n, the binding becomes stronger and the curve clearly
shows non-symmetrical feature with a sharp rise and relatively gradual saturation. In the cases of n = 2
or 4, cooperative binding looks steep in low θ range, but in larger n (n Ê 8) cases, the cooperativity of the
isotherm becomes low.
In a more cooperative case of u = 100, the dependence of the curves on n is complicated as shown
in figure 6. The distances between the binding isotherms are slightly wider than those in figure 5. When
n = 2 or 4, a strong two-phase feature appears with a shoulder on the binding isotherm. In large n (n Ê 8)
cases, the binding isotherm becomes monotonous with a very sharp rise.
The shoulders of binding isotherms in figure 4 and figure 6 are interpreted as follows. In the cases of
n = 2−4 at high cooperativity parameter for low binding fraction (θ < 0.5), the ligands tend to entropically
43302-6
Binding of ionic surfactants to flexible polyions
Figure 5. The dependence of binding isotherm on the association width n at cooperativity parameter
u = 10. n; 1 (×), 2 (square), 4 (triangle), 8 (cross), 16 (circle), and 32 (inverted triangle), from right to left,
respectively. N = 1000 under the periodic boundary condition. Full line is drawn by the Satake-Yang’s
theory. Broken line indicates the isotherm of n = 2 case calculated in our previous scheme (β1a = β1b =
β1 = 1, γ= 1, σ= 1/u = 0.1) [10].
Figure 6. The dependence of binding isotherm on the association width n at u = 100. The others are the
same as in figure 5. Broken line indicates the isotherm of n = 2 case calculated in our previous scheme
(β1a =β1b =β1 = 1, γ= 1, σ= 1/u = 0.01).
bind at intervals, leading to associations across unoccupied site(s). When the binding fraction becomes
around 0.5, the ligands begin to bind to the remaining unoccupied sites. However, these bindings are
suppressed because the newly bound ligand cannot sufficiently associate with the neighboring ligands. In
the higher binding range, recombination of the associations occurs, represented by complicated cluster
size changes shown in figure 8. Such a transition of the situation of the ligands makes a shoulder on
the binding isotherms. In the cases of n = 5− 8 at high cooperativity parameter, the binding isotherm
frequently shows a stepwise increase of θ. In the cases of still larger n, the transition becomes obscure
due to the extensive possibilities of association formations.
The isotherms in our previous scheme can be calculated only in the n = 2 case using the matrix
method (broken line in figure 5 and figure 6). Although the curves and CAC values resemble those of
n = 2 in the model, their rising tendency is more gradual in high θ range than those of the present model
due to the incorporation of the flexibility in the present model.
It has been observed that the long tail of the surfactant molecule enhances the binding affinity, and
leads to low CAC [5, 18, 19]. In these cases, the long surfactant may induce an increase in association
43302-7
T. Nishio et al.
region (association width n in the model). This can effect the shape of the binding isotherm as well as
the CAC value. The effect of molecular geometry of surfactant must be reconsidered from this point of
view [11, 20].
3.3. End effect (Dependence on the lattice length)
The above MC results are obtained under the periodic boundary condition of N = 1000, i.e., by con-
necting both ends of the lattice. The end effect and the dependence of the polyion (lattice) length are
examined without the periodic boundary condition. Figure 7 shows a typical effect of small N values on
the isotherms under non-periodic condition at u = 100 and n = 4, compared with N = 1000 case under
the periodic boundary condition.
Figure 7. The dependence of binding isotherm on the number of lattice points N at u = 100 and n = 4
without periodic boundary condition. N ; (closed symbols) 1000 (triangle), 300 (inverted triangle), 100
(right-hand triangle, dark), 30 (right-hand triangle, light), and 10 (square), from left to right, respectively.
Open triangle; N = 1000 case under periodic boundary condition.
In the case of N = 1000, the end effect cannot be detected, and in the cases of N Ê 100 the effect is
relatively small. In N < 100 cases, however, the shift of the binding curve is significant. The decrease in
N makes the binding curve smooth and gradual. The end effect is larger in larger n and/or larger u cases
(data not shown).
The effect of polyion size on the surfactant-polyion system has been investigated by several research
groups [21–25]. With a decrease of the polymer chain length, an increase of CAC and a decrease of coop-
erativity have been observed [21, 22]. The reduction of the binding with a decrease of the polyion length
can be interpreted by this calculation.
3.4. Cluster size
In figure 8, the changes of mean cluster size lc with the surfactant binding are shown with the same
parameters as in figure 6. The onset of self-assembly is coincident with the CAC depending on n value.
In cases of n = 2, 4, and 8, the curves show up and down in a complex manner. The cluster size distribu-
tions are monotonous due to the one-dimensional lattice system (data not shown). In cases of n = 2−8,
ligands separated on the lattice induce aggregations in low θ range with collapse of the polyion chain.
The rearrangements of the association between ligands occur with an increase of the ligand binding in
the middle θ range. In cases of larger n, the rearrangement is more gradual and spread to wider θ range.
This unexpected behavior may be important to understand the binding of the surfactant to the flexible
polyion.
In the model calculation, large u value leads to high lc. It has been shown that a large cluster size
(aggregation number) corresponds to a high cooperativity [26]. The dependence of the cooperativity on
the charge density of the polyion may be interpreted in the present idea.
43302-8
Binding of ionic surfactants to flexible polyions
Figure 8. The dependence of variation in mean cluster size lc on the association width n at cooperativity
parameter u = 100. Symbols are the same as in figure 6.
In the present model, one ligand can simply associate with two ligands on both sides of the polyion
chain. More complicated models showing phase transition, should be suitable for the coil-globule transi-
tion by introducing additional hypotheses [27, 28].
3.5. Comparison with the experiment
Shimizu reported the bindings of cationic surfactants by several flexible poly(carboxylic acid)s using
a potentiometric technique [29]. Most of the binding isotherms show a sharp onset followed by gradual
saturation. The gradual slope in a range of higher θ means that highly ligand association suppresses a
new binding, although the interpretation due to the electrostatic factor is also likely.
The tentative comparison of the calculated binding curve with an experimental one is executable,
although the precise data-fitting in the whole range is difficult. Figure 9 shows the binding isotherm of
the dodecylpyridinium ion (DP+) by poly acrylic acid (PAA) for α= 1.0 at 0.01 m NaCl [29] with the corre-
spondent calculated curve. In the low surfactant concentration range, reasonable agreement is achieved
with the association width n = 3 and estimated cooperativity parameter σ= 1/u = 0.003. The high coop-
Figure 9. Comparison of the experimental isotherm of dodecylpyridinium ion to poly acrylic acid for
α = 1.0 (large squares) by Shimizu [29] with the correspondent calculated values (small filled squares).
Experimental conditions; T = 30 °C, mNaCl = 0.01 mol · kg−1
H2O, and mp = 0.915 mN. Parameters of
the model calculation; N = 1000, n = 3, σ = 1/u = 0.003, and logK = 0.66 (∆ε0
b = −3.8 kJ/mol, ∆εa =
−14.6 kJ/mol).
43302-9
T. Nishio et al.
erativity and the association within the third neighboring ligands lead to this binding profile. The slight
shoulder of the experimental isotherm in the higher range may be due to additional weak associations of
the ligands, as observed in other isotherms [29].
4. Conclusions
The objective of the study is to analyze the essential characteristics of the system using a simple
model. It seems more suitable rather than molecular simulation studies. The present model is examined
to investigate the bindingmechanism of ionic surfactants to a flexible polyion in dilute solution by theMC
calculation. Introducing the non-cooperative binding and the ligand association separated by multiple
lattice points, we can show their effects as new factors that determine the binding isotherm. Charges
on the polyion should play an appreciable role in increasing the local concentration of the surfactant
molecules.
In this study, a simple model is examined by the MC calculation to investigate the binding mechanism
of ionic surfactants to a flexible polyion in dilute solution. Charges on the polyion play a role in increas-
ing the local concentration of the surfactant molecules. Introducing the ligand association separated by
lattice points within the association width n in the stochastic processes, we can show its effect as a new
factor that determines the binding isotherm.
In the case at high u of n = 1, the results of the present model converge to those of a classical coop-
erative model. With making the width n double, the binding affinity looks approximately twofold, i.e.,
reducing CAC by half. In the range of n = 2−8, two-phase or stepwise nature of the isotherm is observed
due to the complicated rearrangement of the surfactant cluster on the polyion chain, in particular, in the
high u cases. The short polyion chain also lowers the affinity and their cooperativity in the model due to
the end effect.
The model of cluster formation is so simple that the results are rather limited at the present stage.
However, our approach suggests the essential points of the effect of polyion flexibility on the binding
isotherm reflecting the freedom of ligand binding and of their association.
Acknowledgements
The calculations in this study were carried out using the workstation in the information technology
center of the Hamamatsu University School of Medicine.
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Модельне дослiдження колективного зв’язування iонних
сурфактантiв з протилежно зарядженими
гнучкими полiiонами
Т. Нiшiо1, Т.Шiмiзу2,Ш. Йошiда1, А.Мiнакати1
1 Вiддiл об’єднаних наук про людину (фiзика),Медична школа унiверситету м. Хамацу,
Хамацу 431-3192, Японiя
2 Вiддiл iнженерiї електронних та iнформацiйних систем, факультет природничих наук i технологiї,
Унiверситет м. Хiросакi, Хiросакi 036-8561, Японiя
Для вивчення взаємодiї молекул iонного сурфактанта з гнучким полiiонним ланцюжком у розведено-
му розчинi запропоновано нову статистичну модель для колективного зв’язування мономерних лiганд
з лiнiйною граткою. Припускається, що електростатичне зв’язування лiганди з вузлом на полiiонi та гi-
дрофобнi зв’язки мiж сусiднiмi зв’язаними лiгандами є стохастичними процесами. Для гнучкого полiiона
вводиться асоцiацiя лiганд, вiдокремлених декiлькома гратковими точками в межах визначеної ширини.
Для того,щоб дослiдити поведiнку зв’язування, здiйсноються обчислення методом Монте Карло. Гiпотеза
асоцiацiї лiганди та її ширини на ланцюжку є важливою для визначення критичної концентрацiї агрегацiї
та iзотерми зв’язування. Результати є прийнятними для iнтерпретицiї декiлькох експериментiв по зв’язу-
ванню сурфактант-гнучкий полiiон. Представлено та обговорено застосування методу.
Ключовi слова: взаємодiя iонний сурфактант-гнучкий полiiон, гратка лiнiйного полiiона, колективне
зв’язування лiганд, iзотерми зв’язування,ширина асоцiацiї, модельне обчислення
43302-11
http://dx.doi.org/10.1016/j.jcis.2008.01.016
http://dx.doi.org/10.1063/1.1699114
http://dx.doi.org/10.1016/0301-4622(91)85026-M
http://dx.doi.org/10.1016/j.colsurfa.2008.12.008
http://dx.doi.org/10.1016/S0927-7757(98)00819-X
http://dx.doi.org/10.1021/j150654a002
http://dx.doi.org/10.1021/jp971198l
http://dx.doi.org/10.1007/s003960050206
http://dx.doi.org/10.1021/jp022541b
http://dx.doi.org/10.1021/jp051272x
http://dx.doi.org/10.1016/j.jcis.2007.05.035
http://dx.doi.org/10.1021/jp953637r
http://dx.doi.org/10.1021/ja00145a003
http://dx.doi.org/10.1103/PhysRevE.76.041807
http://dx.doi.org/10.1016/0927-7757(94)02986-5
Introduction
Theory
The model and parameters
Model calculation
Analytical solutions
Results and discussion
Dependence on the cooperativity parameter u
Dependence on the association width n
End effect (Dependence on the lattice length)
Cluster size
Comparison with the experiment
Conclusions
|