PAPER
www.rsc.org/pccp | Physical Chemistry Chemical Physics
Dielectric scaling in polyelectrolyte solutions with different solvent
quality in the dilute concentration regime
F. Bordi,a C. Cametti,a S. Sennato,a S. Zuzzi,a S. Doub and R. H. Colbyb
Received 20th April 2006, Accepted 20th June 2006
First published as an Advance Article on the web 3rd July 2006
DOI: 10.1039/b605624e
In this note, we present a set of radiowave dielectric spectroscopy measurements of two dilute,
differently-charged polyelectrolyte solutions, under different solvent conditions. We have found
that both the dielectric strength, De, and the relaxation time, tion, of the dielectric relaxation
process associated with the counterion polarization along a length scale of the order of the
correlation length obey the scaling laws with the polyion concentration, according to the Ito
model. This is verified with good accuracy independently of the quality of the solvent, which has
been varied from poor to good solvent conditions. This finding supports evidence to the fact that,
in dilute solutions, the counterion polarization is independent of the polyion concentration, in
spite of what occurs at the semi-dilute concentrations.
I.
Introduction
Polyelectrolyes are charged macromolecules carrying many
ionizable groups. Under appropriate conditions, these groups
dissociate, leaving charged groups on the polyion chain and
counterions in the solution.1 These systems have very interesting properties and show a very complex phenomenology
due to the delicate balance between attractive hydrophobic
and long-range electrostatic repulsive interactions.2 The conformation of the polyion chain depends on the quality of the
solvent and, as predicted by Dobrynin and Rubinstein,3,4 in
poor solvent media, polyelectrolytes take up a pearl-necklace
structure, where globular beads are linked by flexible chain
segments. The combining effects of the solvent quality and the
counterion condensation on the overall polyelectrolyte properties are not completely understood and a complete description
of these systems has yet to be developed. Computer simulations5–7 have demonstrated that, in semi-dilute solutions and
in poor solvent conditions, two different regimes can occur: a
string-controlled and a bead-controlled regime, and counterion condensation can favor different scaling laws, associated
with a non-monotonic dependence of the chain size on the
polyion concentration.
In a recent paper,8 we have investigated the dielectric
behavior of a polyelectrolyte with different degrees of ionization in solvents of different qualities, from good to poor. The
analysis was confined to semi-dilute systems and the dielectric
data were discussed within the scaling theory of polyelectrolyte
solutions developed by Dobrynin et al.3,9 In particular, we
found that, in this concentration regime, different scaling laws
apply depending on the quality of the solvent. In good solvent
a
Dipartimento di Fisica, Universita’ di Roma ‘‘La Sapienza’’, Piazzale
A. Moro 5, I-00185 Rome, Italy and INFM-CRS SOFT, Unita’ di
Roma, Rome, Italy
b
Department of Materials Science and Engineering, The Pennsylvania
State University, 121 Steidle Building University Park PA 16802,
USA
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condition, the dielectric increment, De, and the relaxation time,
tion, scale according to the relationships
De4/3tion1 B c
(1)
whereas, in poor solvent condition, the scaling law
De4/3t1tion1/3 B c1/3
(2)
holds, where c is the polyion concentration and t the solvent
quality.
In this note, we extend the dielectric analysis of the same
polyelectrolyte chains to dilute systems, where the chain–chain
interaction is weak and one deals, effectively, with single-chain
properties. We consider the polarization mechanism associated with the counterion fluctuation in the neighborhood
of the polyion chain, which causes a dielectric dispersion
falling in the frequency range from approximately 100 kHz
to 100 MHz. The counterion fluctuation is confined within
lengths of the order of the correlation length x0. We investigated two differently-charged poly(N-methyl-2-vinyl pyridinium chloride) (PMVP–Cl) polyions, possessing a carbonbased backbone for which water is a poor solvent and ethylene
glycol (EG) is a good solvent.10,11 Our results clearly show that
both the dielectric parameters, i.e., the dielectric increment De
and the relaxation time tion, follow the scaling laws predicted
for a counterion polarization mechanism occurring over a
length scale of the order of the correlation length.
II. Theory
Consider a polymer chain with degree of polymerization N,
monomer size b and numerical monomer concentration c, in a
solvent with dielectric constant em. The chains are partially
charged because of their charged side groups incorporated by
a quaternization procedure. The total number of ionizable
groups per chain is Nc = QN, where Q is the degree of
quaternization. According to the classic Manning theory of
counterion condensation,12 the counterions will either partially condense on the ionized polyion groups in order to
Phys. Chem. Chem. Phys., 2006, 8, 3653–3658 | 3653
reduce the electrostatic interactions between monomers or are
mobile in the solution, depending on whether the charge
density parameter x = lB/beff is greater than or less than a
critical value xc, (xc = 1, for monovalent counterions). Here,
lB = e2/emkBT is the Bjerrum length, defined as the distance
where the thermal energy kBT exactly compensates the Coulombic interactions, and beff is the effective average charge
spacing along the polyion. When x > xc = 1, it is assumed
that sufficient counterions will condense on the polyion chain
to renormalize the reduced charge parameter, x, to unity. For
our polyelectrolyte systems, the Manning parameter is above
the critical value (x = 1.3 in water and x = 2.7 in EG) and
counterion condensation is expected. In this case, a fraction
(1 f) = 1 1/x of the charged monomers is neutralized by
counterion condensation and f is the fraction of monomers
that are effectively charged.
The dielectric relaxation investigated in this note is associated with the electrical polarization induced by counterion
fluctuation along a length scale of the order of the correlation
length, x0. This relaxation falls in the frequency range 100 kHz
to 100 MHz. The origin of this dispersion has been ascribed to
the presence of some kind of potential barriers that may exist
along the polymer chain, on a short time scale.13 These
barriers define the average distances, b, over which counterions can fluctuate. As pointed out by Odijk,14 since x0 measures the mean distance between contact points in the semidilute system, one can suppose that the correlation length
equals b. This could give a physical interpretation to the ad hoc
potential barriers proposed by van der Touw and Mandel.13,15,16 According to Ito et al.,17,18 in the dilute system
(c o c*), free counterions can polarize by free diffusion (in a
3-D space) to a scale of the order of the distance between
chains, Rcm,
tion R2cm
6D
ð3Þ
and De is simply the product of the number density of free
counterions, fc, and their polarizability aion, which is determined by the square of the ion charge (e for uni-valent
counterions) and the polarization distance Rcm,
e2 R2cm
fcem lB R2cm
kB T
De fcaion fc
ð4Þ
In dilute systems, for any solvent quality, Rcm scales with c
according to
1=3
N
ð5Þ
Rcm c
resulting in values for tion and De given by
tion B N2/3c2/3
De B fN
2/3 1/3
c
(6)
(7)
According to the above scaling laws, we expect that, in the
dilute regime, the scaling exponents do not change as the
quality of the solvent is changed.
3654 | Phys. Chem. Chem. Phys., 2006, 8, 3653–3658
III. Experimental
A. Materials
The polyions of PMVP–Cl, were prepared from the neutral
parent dry polymer Poly(2-vinyl-pyridine) (Mw = 364 kD),
(P2VP), by means of a quaternization procedure. P2VP dissolved in N-dimethyl formamide (DMF) was quaternized with
dimethyl sulfate (on the pyridine nitrogen), and the methyl
sulfate counteranion was substituted by Cl upon addition of
NaCl. By means of this procedure, two different samples with
two different degrees of quaternization (Q = 0.17 and Q =
0.55) were prepared. The polydispersity index of the samples is
estimated to be Mw/Mn = 1.06 and the degree of quaternization Q of the partially quaternized polymers was determined
by using sodium chloride and silver nitrate according to the
standard counterion titration technique (conductimetric titration with AgNO3).
For solvents, we employed mixed EG–water solutions at
five different molar fractions X = [EG]/([EG] þ [H2O])
(X = 0, 0.25, 0.50, 0.75, 1). The polymer concentration was
varied in the range covering the dilute regime (from 0.01 to
2 mg ml-1). All the dilutions were prepared using deionized
Q-quality water (electrical conductivity, s, less than 1 106
mho cm-1 at room temperature) and highly-purified EG
(Sigma Chem. Co.). All the dielectric measurements were
carried out at 25.0 0.1 1C.
B. Dielectric measurements
The dielectric and conductimetric spectra of partially-charged
PMVP–Cl polymers have been measured in the frequency
range from 1 kHz to 2 GHz by means of two radio-frequency
impedance analyzers, Hewlett-Packard model 4294A (in the
frequency range from 1 kHz to 10 MHz) and model 4291A (in
the frequency range from 1 MHz to 2 GHz). The measurement
cell consists of a short section of a cylindrical coaxial line, with
a characteristic impedance of 50 O, directly connected to the
input of the meter by means of a precision APC7 connector.
The input impedance Z*(o) (magnitude |Z| and phase angle
j) was converted to the complex dielectric constant e*(o)
through an appropriate lumped element electrical circuit. The
electrical and geometrical cell constants were determined by
measuring the complex impedance of the cell filled with
standard electrolyte solutions of known permittivity and
electrical conductivity. Details of the dielectric cell and the
calibration procedure have been reported elsewhere.19,20
C. Analysis of the dielectric spectra
Fig. 1 shows typical dielectric and conductimetric spectra of
55% PMVP–Cl polyions in water at selected concentrations.
Polyions in mixed EG–water solutions behave similarly. As
can be seen, due to the relatively high dc electrical conductivity, s0, of the polyion solutions, the low-frequency region of
the dielectric spectra is dominated by the electrode polarization effect, caused by a charge accumulation at the electrode–solution interface, leading to the formation of an electrical
double layer.
In the presence of a relevant electrode polarization effect,
the electrical properties of the electrode–solution interface can
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to the procedure outlined above, have been analyzed by a
Cole–Cole relaxation function22 to which has been added a
Debye relaxation function that models the orientational
polarization of the solvent phase, according to
es e1
e1 e1H2 O
s0
ð9Þ
e ðoÞ ¼ e1 þ
þ
þ
ð1 þ ðiotion Þb Þ 1 þ iotH2 O ie0 o
Fig. 1 A: The permittivity, e 0 , of 55% PMVP–Cl polymer in water
(poor solvent condition) as a function of frequency of the applied
external electric field, at three different polyion concentrations. (’):
0.06 mg ml1; (K): 0.03 mg ml1; (m): 0.01 mg ml1. The increase of
the permittivity as the frequency is lowered is due to the electrode
polarization effect. The full lines are the values corrected for the
electrode polarization effect on the basis of a Constant Phase Angle
(CPA) element (see ref. 18 for details). B: The electrical conductivity,
s. Symbols as for A.
be represented, in terms of a lumped circuit element, by a
Constant Phase Angle (CPA) element. The ‘‘total’’ complex
dielectric constant e*tot(o) has been written as the sum of two
contributions
e*tot(o) = A(io)g þ e*(o)
(8)
where e*(o) arises from the effective polarization mechanisms
in the sample. The parameters A and g have been derived from
the low-frequency region of the spectra, where the electrode
polarization contribution exceeds the effect associated with the
polarization induced by the polyions by order of magnitude.
The correction procedure of the raw data to take into account
of the electrode polarization effects by means of a CPA
element has been discussed in detail elsewhere.21 Values of A
and g depend on the bulk ionic conductivity of the solvent
phase. They range between 2.5 106 and 3.5 107 s for A and
between 1.3 and 1.7 for g. The linear regression coefficient of
the fit, for all the samples investigated, is consistently better
than r = 0.9998.
The dielectric and conductimetric spectra of polyelectrolyte
solutions in the frequency range between 10 kHz to 2 GHz,
once corrected for the electrode polarization effect according
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Here, De = es eN, tion and eN are the dielectric increment,
the relaxation time and the high-frequency limit of the permittivity associated with the dielectric process driven by the
polyion, respectively. The distribution of the relaxation time is
taken into account by the parameter b and e0 and o are the
dielectric constant of free space and the angular frequency of
the applied electric field. The dc conductivity of the polyion
solution is denoted by s0.
The de-convolution of the whole dielectric spectrum into its
components (electrode polarization effect, relaxation associated with the polyions and orientational polarization of the
bulk solution at the higher frequencies investigated, occurring
at partially overlapping frequency ranges) may be difficult and
an accurate procedure must be implemented in order to obtain
meaningful results.
In particularly, the localization of the relaxation associated
with the counterion polarization detected by separately observing the frequency dependencies of the permittivity e 0 (o) and
the dielectric loss e00 (o) = [s(o) s0]/e0o may be difficult,
particularly when the effects are relatively small. To overcome
this, we have performed a multi-step fitting procedure based
on the Marquardt algorithm for complex functions,23 which
furnishes the values of the dielectric parameters De, tion and b.
The algorithm minimizes the relative residuals between theoretical curves and experimental data according to
R(e,s) = (S(eth eexp)2/Se2exp þ S(sth sexp)2/Ss2exp)2 (10)
where e and s are the permittivity and the electrical conductivity and the subscripts th and exp stand for theoretical and
experimental values, respectively.
In detail, since the shape of the dielectric loss spectrum,
e00 diel(o), strongly depends on the value of the dc electrical
conductivity s0 to be subtracted from the total loss s(o)/e0o,
we have made a preliminary simultaneous fit of e 0 (o) and the
total loss s(o)/e0o with five free parameters De, tion, b, eN, s0,
with the only constraint that all parameters should be >0. The
value of s0 thus obtained is then subtracted from the measured
conductivity s(o), the dielectric loss e00 diel(o) is evaluated and a
new set of parameters from the simultaneous fit of both the
permittivity e 0 (o) and the dielectric loss e00 diel(o) are obtained.
This procedure is iterated until a reasonable minimization is
reached and the dielectric parameters De and tion converge to
stable values. It is worth noting that the same set of parameters describe both the real part, the permittivity e 0 (o), and
the imaginary part, e00 diel(o), of the complex dielectric constant
e*(o). This strongly supports the reliability of the dielectric
data we present. A further check of the quality of the fit is the
values we obtain for the parameters s0 and eN. Their values,
within the derived uncertainties, agree with the values measured directly in the high-frequency limit for eN and in the
low-frequency limit for s0. Even though the fitting algorithm
works with five free parameters, their number is actually
Phys. Chem. Chem. Phys., 2006, 8, 3653–3658 | 3655
reduced to three (De, tion and b). The uncertainties on the
dielectric parameters De and tion derived from the non-linear
least-squares minimization vary between 10 and 20% according to the quality of the solvent and the polyion concentration.
In pure EG, at the lowest polymer concentration employed,
the dielectric strength De is less than 1 or 2 dielectric units and
the uncertainties on the dielectric parameters are appreciably
larger.
The quality of the overall procedure used can be judged
from Fig. 2 and 3, which shows two typical dielectric spectra in
the case of 55% PMVP–Cl polyion in water and 17%
PMVP–Cl, in X = 0.25 water–EG mixture, respectively,
together with a Cole–Cole plot (e 0 (o), e00 diel(o)). The counter-
ion relaxation is well evidenced and well separated from the
electrode polarization effect (deviation from the right-hand
side of the Cole–Cole arc) and the orientational relaxation of
the aqueous phase (deviations from the left-hand side of the
Cole–Cole arc). As can be seen, the calculated values on the
basis of eqn (7) give a reasonable agreement with the experimental data (in the frequency range where the counterion
relaxation works) and, moreover, allow the dielectric
parameters De and tion to be determined with a reasonable
accuracy.
As far as the spectra in Fig. 2 and 3 are concerned, in the
most favorable case, when the dielectric strength is relatively
high (Fig. 2, 55% PMVP–Cl in water, C = 0.06 mg ml1), the
set of dielectric parameters derived from the fitting procedure
Fig. 2 A typical dielectric and conductimetric spectrum of 55%
PMVP–Cl polyions in water (poor solvent) at a concentration of
0.06 mg ml1. A: the permittivity e 0 (o) as a function of the frequency;
B: the dielectric loss e00 (o) as a function of the frequency; (C): the
Cole–Cole plot showing the dielectric relaxation together with the
high-frequency tail of the electrode polarization effect (on the right)
and the low-frequency tail of the relaxation associated with the
orientational polarization of the aqueous phase (on the left). In all
the panels, the full line represents the calculated values based on the
described fitting procedure (see text).
Fig. 3 A typical dielectric and conductimetric spectrum of 17%
PMVP–Cl polyions in a water–EG mixture (X = 0.25) at a concentration of 0.55 mg ml1. A: the permittivity e 0 (o) as a function of the
frequency; B: the dielectric loss e00 (o) as a function of the frequency; C:
the Cole–Cole plot showing the dielectric relaxation together with the
high-frequency tail of the electrode polarization effect (on the right)
and the low-frequency tail of the relaxation associated with the
orientational polarization of the aqueous phase (on the left). In all
the panels, the full line represents the calculated values based on the
described fitting procedure (see text).
3656 | Phys. Chem. Chem. Phys., 2006, 8, 3653–3658
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Fig. 4 A: The dielectric increment De of 55% PMVP–Cl as a function
of c (in mg ml1) in different EG–water mixtures in the dilute regime:
(’): X = 0 (pure water); (K): X = 0.75; (m): X = 1 (pure EG). Full
lines represent the dependence De B ca with the scaling exponent
a = 0.33. B: The relaxation time tion of 55% PMVP–Cl as a function
of c. Symbols as for A. Full lines represent the dependence tion B ca
with the scaling exponent a = 0.66.
is De = 9.1 0.6, nion = 1/(2ption) = 370 35 kHz, eN =
78.2 0.3 and b = 0.29 0.02 and in the most unfavorable
case, when the dielectric strength is relatively small (Fig. 3,
17% PMVP–Cl in water–EG, X = 0.25, C = 0.55 mg ml1),
the dielectric parameters are De = 2.4 0.8, nion = 1/(2ption)
= 490 60 kHz, eN = 77.5 0.3 and b = 0.28 0.02.
IV. Results and discussion
In dilute systems, since the distance between chains varies with
c according to Rcm B c1/3 regardless of the solvent quality,
the scaling model predicts that De and tion scales with c
according to eqn (4) and (5), with scaling exponents equal to
a = 1/3 and 2/3, respectively.
Fig. 4 and 5 show the measured data together with the
expected scaling laws for both 55% PMVP–Cl and 17%
PMVP–Cl polyions in solvents of different qualities, from
water (poor solvent) to EG (good solvent). As can be seen, a
reasonable good agreement is obtained and the straight lines,
which represent the expected results using the scaling laws,
intersect all the experimental data within the respective error
bars. The investigated polyion concentrations are confined to
the dilute regime, below the overlap concentration, c*. Experimentally, the point at which a polymer solution becomes
semi-dilute can be determined24 by plotting the log of the
specific viscosity, Zsp, as a function of the polymer concentration. For independent, non-interacting polymers, dilute-soluThis journal is
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Fig. 5 (A): The dielectric increment De of 17% PMVP–Cl as a
function of the polyion concentration c in different EG–water mixtures
in the dilute regime: (’): X = 0 (pure water); (K): X = 0.25; (m):
X = 0.5; (E): X = 0.75; (.): X = 1 (pure EG). Full lines represent
the dependence De B ca with the scaling exponent a = 0.33. B:
The relaxation time tion of 17% PMVP–Cl as a function of the
polyion concentration c. Symbols as in A. Full lines represent the
dependence De B ca with the scaling exponent a = 0.66.
tion theories predict a slope of the order of unity. In the
present case, for the degree of quaternization Q = 0.55, the
overlap concentration is about c* = 0.05 mg ml1 and c* =
0.2 mg ml1 for polymers in water and in EG, respectively and,
for the degree of quaternization Q = 0.17, this concentration
is approximately c* = 0.2–0.3 mg ml1 for polymers in both
EG and in water.25 In the case of 55% PMVP–Cl, these values
agree remarkably well with those estimated from dielectric
measurements, where the scaling laws given in eqn (4) and (5)
hold, whereas it appears that the linear dependencies of the
dielectric parameters, (De and tion), extend above this regime
in the case of 17% PMVP–Cl. It must be noted that, in this
latter case, the dielectric relaxation strengths are relatively
small, of the order of only few unities, making their estimates
less reliable. Moreover, differences between the expected scaling laws for dilute and semi-dilute systems are progressively
reduced, as the dielectric strength becomes smaller and smaller
when the solvent approaches the poor solvent condition.
However, at higher concentrations, also for the 17%
PMVP–Cl polymer, a semi-dilute regime prevails and a completely different dielectric behavior is evidenced.8
It is worth noting that the dielectric behavior obeys the
scaling laws for any solvent we have investigated.
The situation is completely different in semi-dilute systems,
where the conformation of the polymer chain changes from a
Phys. Chem. Chem. Phys., 2006, 8, 3653–3658 | 3657
necklace model in water (poor solvent) to a random arrangement of blobs in EG (good solvent). We have discussed this
complex phenomenology in detail elsewhere,8 where we
showed that different scaling exponents hold, according to
the quality of the solvent. Moreover, in the semi-dilute concentration regime, these exponents change continuously as the
quality of the solvent is progressively changed, from poor to
good condition.
On the contrary, in the dilute solution, as pointed out by
Chang and Yethiraj,26 the number of condensed counterions is
not a strong function of the solvent quality, at least for values
of the solvent quality parameter that are not very low. This
means that, as long as the chain conformation does not change
appreciably, i.e., the chain does not collapse into a beadnecklace structure, a simple description, that we think still
captures the essential physics, ignores the detailed role of
counterion close to the chain and is simply based on counterion fluctuation on lengths of the order of the correlation
length. This finding may appear unusual (or unexpected) since,
in the poor solvent conditions, counterion condensation may
take place on strings, on beads or on the whole necklace with
different modalities, i.e., governed by the Manning condition
or by the size and the charge of each bead.27 On the contrary, a
simple counterion fluctuation over a length of the order of the
distance Rcm between chains seems adequate to also describe
the dielectric properties of hydrophobic polyelectrolytes, in the
dilute concentration regime.
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