arXiv:cond-mat/0407637v1 [cond-mat.stat-mech] 23 Jul 2004
with Monovalent Salt
Paulo S. Kuhn
Departamento de F´ ısica, Instituto de F´ ısica e Matem´ atica
Universidade Federal de Pelotas, Pelotas, RS, Brasil
Marcia C. Barbosa
Instituto de F´ ısica,
Universidade Federal do Rio Grande do Sul
Caixa Postal 15051, 91501-970,
Porto Alegre, RS, Brasil∗
(Dated: February 2, 2008)
We present a model for describing flexible polyelectrolytes in a good solvent and in the presence
of monovalent salt . The molecule composed by N monomers is characterized by the end to end
distance Re = b(Z − 1)γand the number of associated counterions n. At high temperatures the
polyelectrolyte behaves as a neutral polymer (γ = 0.588). Decreasing the temperature, the macro-
molecule changes from this extended configuration(γ = 0.588) to a stretched form (γ ≈ 1). At
even lower temperatures, above the Manning condensation threshold, the polyelectrolyte collapses
(γ ≈ 0.3). Our results show good agreement with simulations.
PACS numbers: Valid PACS appear here
Polyelectrolytes represent a very interesting class of
materials. Biological systems abound with polyelec-
trolytes. Two of the most well known of all polymers,
DNA and RNA, are charged polymers . Besides,
a large class of synthetic polyelectrolytes are present in
the chemical industry. For instance, polyacrylic acid is
the main ingredient for diapers  and dispersions of
copolymers of acrylamide or methacrylamide and acrylic
or methacrylic acid are fundamental for cleaning water
. Even though the tremendous interest in polyelec-
trolyte, unlike neutral polymers , the understanding
of the behavior of electrically charged macromolecules is
still rather poor.
The contrast between our understanding of charged
and neutral polymers results from the long range na-
ture of the electrostatic interactions that introduces new
length and time scales that render the analytical descrip-
tion very complicated . The presence of charges intro-
duces more than one new length scale making the scaling
theories used for no longer as simple and applicable. Be-
sides the length scale associated with the strength of the
Coulomb interaction, there is also lengths associated with
the mean separation of the counterions required by the
charge neutrality. The association of these free ions to
the polyion, renormalizing its charge, is another factor
that has to be taken into account.
Therefore, simulations seems to be a good technique to
overcome these major difficulties. Recent molecular sim-
ulations of salt-free systems - were able to obtain
∗Electronic address: firstname.lastname@example.org
the end to end distance of strong electrolytes below Man-
ning parameter . The picture provided by these sim-
ulations have shown to be more complicated than early
analytic theories have predicted -. Past theoreti-
cal works tended to neglect entropy. For stiff chains, such
as DNA, entropy is an small contribution to the free en-
ergy and, in principle, can be disregarded. In contrast,
for flexible polyelectrolytes, treating entropy along with
with the Coulomb interactions is essential. Acknowledg-
ing that, one of us developed a theory for describing the
thermodynamics of salt-free polyelectrolytes in a good
solvent. Within this approach both entropy and electro-
static interactions are taken into account . The
end to end distance and pressure calculated using this
method show good agreement with simulations .
The addition of salt is a key factor both in biologi-
cal and industrial applications. However, including new
charged species in the system, adds another scale to
this already very complicated problem. In this work we
develop a simple theory for studying a dilute polyelec-
trolyte solution in the presence of monovalent salt in the
framework of three theories: Debye and Bjerrum for elec-
trolytes -, Manning condensation  and Flory
elasticity for polymers .
Unlike neutral polymers, the polyelectrolyte in salt so-
lution exhibits three distinct behaviors. At high tem-
peratures the chain is extended.
is decreased, the electrostatic repulsion between the
monomers along the polyion becomes relevant and the
chain stretches. At even lower temperatures, above the
Manning condensation threshold, counterions condense
into the polyion. The polymer chain net charge decreases
and the polyion collapses. The addition of salt screen the
electrostatic interactions and the polymer contracts. Our
results show a good quantitative agreement with simula-
As the temperature
FIG. 1: The System
In section II, our model is presented in detail. In sec-
tion III, the free energy of the system is constructed and
minimized with respect to two parameters: number of
ions associated to the polyion and end to end distance.
The results are presented in section IV. Conclusions end
We consider a dilute polyelectrolyte solution of con-
centration ρp (see Figure 1). The chains are immersed
in a continuum solvent with dielectric constant D and
monovalent salt. There are Z charged groups of diam-
eter σ along the polymer chain. b = 21/6σ is the dis-
tance between adjacent monomers. The total charge of
the completely ionized molecule is −Zq < 0, where q is
the proton charge. The counterions that neutralizes the
solution have charge +q, diameter σ and concentration
ρm= Zρp. The salt ions have charge +q or −q, diame-
ter σ and density ρs. For simplicity the positive ions of
salt will be refereed as counterions and the negative as
ρ+= Z(1 − m)ρp+ ρs
are the density of counterions and coions. The inverse
Debye screening length κ is defined in the usual way
by κ =√4π ρ1λB, where λB = β q2/D is the Bjerrum
length, the distance between two ions where the electro-
static energy equals the thermal energy, ρ1= ρ++ ρ−,
and β = 1/kBT. The reduced density for species j is
The Manning charge parameter is  ξ = β q2/Db.
Hence ξ = σ/bT∗and ξ = λB/b.
j= ρjσ3, the reduced temperature is T∗= Dσ/βq2.
macroion does not assume any preferential geometry.
Hence, the distance between monomers i and j is rep-
resented by rij = b|i − j|γas usual . The end to end
distance Reis the distance between monomers 1 and Z
and it is represented by:
Re= b(Z − 1)γ.(3)
The electrostatic interaction between the chain and
the counterions leads the formation of a complexes made
of one macromolecule and n associated microions. At a
given temperature T, monomer density ρm= Z/V and
salt density ρs, the system reaches the equilibrium char-
acterized by a number of associated counterions, m and
the end to end distance characterized by γ.
III.THE HELMHOLTZ FREE ENERGY
The system is, therefore, composed by complexes made
of one polyion and n counterions, free counterions and
free coions. The end to end distance of the complex is Re.
The equilibrium configuration is found by minimizing the
Helmholtz free energy density with respect to n and Re.
The free energy density, f = F/Npis composed by three
f = fELEC+ fHC+ fENT
where fELEC contains all the electrostatic interactions,
fHChas the hard core contribution between the different
species and fENT has the entropic contributions for the
free energy density.
The electrostatic free energy density is splited into:
fELEC= fpc+ fff+ fdi
where fpc accounts for the electrostatic interaction be-
tween the polyelectrolyte chain and the free electrolyte,
ffftakes care of the electrostatic interaction between the
counterions e coions and fdi includes the dipole-ion in-
teraction between the dipole pairs along the chain and
both charged monomers and free counterions and coions.
The hard core free energy density is divided into two
contributions given by:
fHC= fhc+ fcs
where fhc takes care of hard core interaction between
monomers along the chain and fcsincludes the hard core
interaction between free ions. Here cs represents the free
energy of Carnahan-Starling, the hard core term between
Finally, the free energy density terms that contains
entropic contributions are:
fENT= fel+ fig
where felis the elastic free energy of the flexible chain,
fig is the ideal gas free energy for the mixture of the
different species present in the solution.
A.The polyion-free ions electrostatic free energy
The electrostatic free energy density between the com-
plex made of a polyelectrolyte and n counterions associ-
ated to it and the free ions in the solution can be derived
in the framework of the Debye-Huckel theory yielding (
see appendix A):
where pz= −1 + m = −1 + Z/n and
I = 2
dx(Z − x)e−κr(x)− 1
where r(x) = b xγ.
These ideas have been successful in describing charged
systems, including simple electrolyte solutions [30, 31],
charged rods [32, 33], charged spherical colloids , and
flexible charged chains [23, 24, 35]. In particular, the
binding isotherms of DNA with surfactant molecules have
been found in good agreement with experiment .
The integral I might also be writen using the incom-
plete gamma function , namely
dte−ttα−1, Reα > 0. (10)
The suffix inc is used to avoid confusion with the expo-
nent γ. In this case, we get
(κb)1/γγγinc(−1 + 1/γ,κbZγ)
(κb)2/γγγinc(−1 + 2/γ,κbZγ)
b(1 − γ)+
b(2 − γ).(12)
B. Free ion-free ion electrostatic free energy
For the interaction between free ions we employ the
Debye-H¨ uckel free energy for electrolytes [26, 27],
?ln(1 + κσ) − κσ + (κσ)2/2?. (13)
C.The dipole-ion electrostatic free energy density
When a counterion associate to a negative monomer
of the polyelectrolyte, it forms dipole.
tion between these dipoles formed by association and
the monopoles consisting of free counterions and coions
and non-associated monomers along the chain is given by
the usual dipole-ion interaction originally derived for the
coulomb gas and given by [30, 31]:
T∗Z m(σ/a2)3x2ω2(x) ,(14)
where x = ¯ κa2, ¯ κσ =
, ¯ ρ∗
−= Z (1−m)σ3/Ve,
e/3, a2= 1.1619σ, and
ω2(x) = 3[ln(1 + x + x2/3) − x + x2/6]/x4.(15)
Note that we use here ¯ κ in order to take the interac-
tions between the dipoles formed on the chain and the
charged monomers and free ions. It is essential for the
attractive forces that lead to collapse of the chain at low
temperatures. The density ¯ ρ∗
monomers not associated in the volume Veoccupied by
−is the density of charged
D.The excluded volume free energy densities fhc
The polyelectrolyte will be consider to be in a good sol-
vent. Therefore, hard-core repulsion between monomers
is approximated by a virial coefficient from Flory-de
Gennes theory [5, 6, 38, 39],
2W1¯ ρ ,(16)
where W1 = 4πσ3/3 is the second virial coefficient for
hard spheres of diameter σ, and ¯ ρ = Z/Ve with Ve =
The hard-core repulsion between free ions is approxi-
mated by the Carnahan-Starling free energy ,
βfcs= V ρ1y
4 − 3y
(1 − y)2,(17)
where y = πρ∗
free positive and negative ions.
1/6 is the volume fraction occupied by the
E.The elastic free energy density fel
We are considering here flexible polyelectrolytes. The
entropic contribution of the elastic free energy density is
the same of a neutral polymer and it is given by [5, 35,
41, 42, 43]
2(α2− 1) − 3lnα ,(18)
where α = Re/R0, where R0 = b√Z − 1 is the end to
end distance of a polymer in a Θ solvent.
F.The ideal gas free energy density fig
Another important entropic contribution is due to the
mixing of the different species, complexes, counterions
and coions. The free energy density associated with the
mixing of ideal particles is given by [30, 31]:
− ρ−] + lnρpΛ3− 1 + βfc
where Λ = h/√2πmkBT is the thermal wavelength. fc
includes the internal degrees of freedom of the complex
made of a polyelectrolyte and the associated ions and it
is given by:
+Λ3− ρ++ ρ−lnρ∗
ig= Z(1 − m)[lnZ(1 − m)
Veζ2Λ6− 1] − Z[lnZ
VeΛ3− 1] + βfint
Since the monomers along the chain are mobile, the
first two brackets in Eq. (20) account for the entropy of
mixing of charged monomers and dipoles along the chain.
The third bracket discounts the overcounting since the
entropy of a neutral polymer was already accounted in
Eq. (18). The last term is the internal partition function
of the complex in its internal reference frame. The in-
ternal partition function of the dipoles, ζ2, is given by:
r2eσ/rT∗dr ≡ K(T) ,(21)
where RBjis chosen according to Bjerrum to be σ/2T∗,
the location of the minimum of the integrand. Thus the
association constant is different from zero for T∗< 0.5.
Above this temperature there is no counterion association
and no dipole formation on the chain. We have then
[28, 30, 31]
K(T) = 4πσ3Q0e1/T∗T∗,(22)
6T∗4e−1/T∗[Ei(1/T∗) − Ei(2) + e2]
The internal partition function of a n-complex con-
tains: one electrostatic term due to the interaction be-
tween the monomers and associated ions and one term
due to the different ways n ions can associate into Z
monomers in a polyions. The additions of these two
= ξ p2
(Z − i)
+ Z m ln m + Z (1 − m) ln(1 − m) ,
with r(i) = biγ.
IV.RESULTS AND CONCLUSIONS
The behavior of the polyelectrolyte under the variation
of temperature is illustrated in Figure 2. At high tem-
peratures ( low Bjerrum length) the end to end distance
approaches the one of a neutral polymer in a good sol-
vent and Re= b(Z − 1)0.588. For Z = 32 and b = 21/6σ
the end to end distance is given by Re≈ 8.45σ. As the
temperature decreases, the electrostatic interactions be-
come dominant, the polymer stretches and the end to
end distance is given by γ ≈ 1. Above the Manning
condensation threshold, the counterions associate to the
monomers along the chain, decreasing the electrostatic
interactions. The elastic energy becomes dominant and
the polymer collapses. Our results are compared with
simulations . The actually salt density used for that
simulation is not specified in the manuscript but is within
the interval ρ∗
s= 0.001 and ρ∗
Figures 3 and 4 show that for a given Z, the radius of
end to end distance, Redepends only on the value of the
bare inverse of screening length, κ0=
and not on the density of polyelectrolytes. In Figure 3
the renormalized end to end distance given by
is ploted against lnκ0σ for Z = 32,64 for varius densities
and for λB= 0.83σ (ξ = 0.75). The decrease of the end
to end distance is, in this case, due to the screending of
the salt present in the solution. Note that here there is no
condensation. The renormalized end to end distance, R∗,
was constructed as follows. R was divided by the end to
end distance of an equivalent neutral polymer in a good
solvent R0≈ bZ0.588. In order to keep both Z = 32 and
Z = 64 in the same scale, N was also divided by Z = 32.
Our results agree with simulations performed in the scale
In Figure 4 we show R∗for λB= 3.2σ(ξ = 2.9). In this
case, since the system is above the Manning threshold,
condensation is present and the polyelectrolyte should
form a more compact form than for λB = 0.83σ (ξ =
0.75). Comparison with Figure 3 shows that for lnκ0σ >
−0.5 larger λBhas larger R. Part of this effect is due to
plotting versus lnκ0σ. For the same ρmand ρs, the end
to end distance is surely smaller for λB= 0.83σ than for
λB= 3.2σ(ξ = 2.9) as indicated by Figure 2.
In summary, we studied the end to end distance of
polyelectrolytes with added salt. We found that R for
a fixed Z and λB depends only on the amount of salt
in solution. In this case, the main effect of the elec-
trolytes is to screen the interactions and decreasing R.
For fixed values of Z, ρp and ρs, as the temperature is
decreased, R exhibits three distinct behaviors: stretched,
extended and collapsed. This three regimes are related
to the stretched polymer, fully extended polyelectrolyte
and collapsed polymer behavior.
FIG. 2: Plot of end to end distance as a function
of λB for Z = 32, ρ∗
to bottom ρs = 0 − 0.008. The circles are the
simulational results extracted from ref. .
m = 0.001, and from top
−4 −3−2 −101
distance R∗as a function of lnκσ for Z = 32
( solid line ) and Z = 64 ( dashed line), for
m= 0.0001, 0.001,0.01,0.02 and ξ = 0.75. The
points for different densities fall in the same line.
The symbols are the simulation results for the
(squares) extracted from ref. .
Plot of the renormalized end to end
m and Z = 32 (circles) and Z = 64
We thanks Paulo Krebs and Eduardo Henriques for
the use of the computational resources at the Universi-
dade Federal of Pelotas. This work was supported by the
brazilian science agencies CNPq, FINEP, and Fapergs.
−3 −2.5 −2−1.5−1 −0.50 0.51
FIG. 4: Same as in Figure 3 but for ξ = 2.9.
APPENDIX A: CALCULATION OF fpc
In order to compute the free energy density associated
with the polyion-free ions interaction, let us start by cal-
culating the electrostatic potential at a point P due to
the charged chains and electrolytes. Each chain has a
linear charge density λ′= −(Z − n)q/L. The coordi-
nates on the molecule are denote by a prime, and the
coordinates of the point P where we want to evaluate
the electrostatic potential are P = (x,y,z). The electro-
static potential at a point P located in an ionic solution,
due to a charge element λ′ds′on the molecule, is given
by the Debye-H¨ uckel expression,
where r is the distance of the charge element to the point
P. Integrating the expression above, the electrostatic
total potential at P becomes:
where L = b(Z − 1) is the contour length. The electro-
static potential at P due to the chain when there is no
electrolyte is given by:
the suffix pr meaning proper.
The potential due to the ionic solution discounting the
chain is given by ψ ≡ φ − φpr. The free energy is then
evaluated by using the charging process of Debye [26, 27]
in this potential.
The electrostatic energy of a charge element dq at
(x,y,z) in the potential ψ is
dU = dq(x,y,z)ψ(x,y,z) .(A4)
The electrostatic energy of the entire molecule is obtained
integrating the above expression, what gives
where r = r(s − s′) = r(s′− s) denotes the distance
between line elements ds and ds′, both on the molecule.
The factor 1/2 is included to avoid double counting. Now
we substitute q by ζq in U and integrate in ζ from 0 to
1. We obtain
ζdζ = 2U(1
2− 0) = U .
Fpc= U =
We now change from s to a dimensionless variable t, with
bdt . (A8)
r(t − t′)
Changing variables again, x = t − t′, and integrating by
parts, the free energy becomes
I ,I = 2
dx(Z − x)e−κr(x)− 1
Defining pz = −1 + m and substituting λ = (−1 +
m)q/b = pzq/b, we may write
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