2014-RSC-Advances_AZ-RH_CMC-Frequency

Electric polarizability dispersion of alumina
particles with adsorbed carboxymethyl cellulose
Alexandar M. Zhivkov* and Rosen P. Hristov
When the linear charge density of a polyelectrolyte chain reaches some critically high level, an electrostatic
adsorption of part of the counterions appears, that is the so-called counterion condensation. There are
contradictory opinions in the literature about the condensed counterion mobility along the polymer
chain: the analytical theories predict some mobility, but the experimental research does not give an
unequivocal answer. The present experimental investigation aims to verify the reports for the condensed
counterion migration in a sinusoidal electric field. By using electric light scattering we investigate an
aqueous suspension of g-alumina particles after the complete adsorption of carboxymethyl cellulose on
their surface; the probable migration of condensed counterions must contribute to the electric
polarizability (when the frequency is under a given critical value) leading to a higher degree of particle
orientation in the applied electric field. We compare the frequency dependences of the polarizability at
two polyelectrolyte concentrations in the suspension: under and above the recharging point (appearing
due to adsorption of the negatively charged polyelectrolyte on the positively charged surface) where the
total polarizability is equal, but the ratio between the quantities of the diffuse and the condensed
counterions is different. A procedure for determination of the counterion shares is invented; it uses the
measured electrophoretic mobility and the calculated fraction of the condensed ions. The results
indicate the absence of the polarizability component caused by the condensed counterions; i.e. they do
not manifest their presence when an external electric field is applied. We have concluded that the
condensed ions are immobile in a sinusoidal field with moderate intensity in the frequency range of 10
Hz to 1 MHz.
1. Introduction
Polyelectrolytes immersed in aqueous solution have a high
linear charge density due to the ionized groups of the identical
polymer units.1,2
Because of the high electric potential around
the polyelectrolyte chain, the local concentration of the coun-
terions is very high and part of them are bound electrostatically
to the polymer charged groups forming temporary group-ion
pairs. The phenomenon is known as counterion condensation
and it appears when the electrostatic energy is higher than the
thermal energy kT. As a result the total charge density of the
chain decreases, but it remains charged. This excessive charge
is equal to that of the diffuse counterions in the surrounding
ionic atmosphere.
Several approaches have been developed to describe the
counterion condensation as a function of the linear charge
density;3
most of them are based on the cylindrical model of the
polyelectrolyte chain and Poisson–Boltzmann equation.4
The
cylindrical cell (Katchalsky’s) model5–7
considers a cylindrical
polyion in the center of a coaxial cylinder cell; the mean-eld
approximation gives the radial density prole at the equilib-
rium distribution of point-size diffuse counterions. A conne-
ment of this model is that an exact analytical solution of the
non-linear Poisson–Boltzmann equation is possible only for a
straight innite cylinder; in other cases a numerical solution is
needed. The modication of the model by introducing an
additional cylinder allows the distribution of small counterions
with denite radius and valency to be obtained; an example is
the numerical analysis showing the selective adsorption of
simple ions.8,9
The activity coefficient of monovalent counter-
ions determined by their local concentration (analytically
calculated for a salt-free solution and numerically at an 1 : 1
added salt) employing the cell model of rod-like polyelectrolyte
is in good agreement with that computed by Monte-Carlo
simulations and with experimental data for aqueous solutions
of sodium salt of carboxymethylcellulose (NaCMC) for the cases
of salt-free and added NaCl.10
In another approach the counterions are separated into
several categories with different phase states in order to derive a
simpler analytical solution. In the two-state model of Oosawa–
Manning1,11
the counterions are considered as condensed
(localized inside a narrow potential valley around polymer
backbone) and ‘free’ counterions (freely moving outside thisInstitute of Physical Chemistry, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,
bl. 11, Soa 1113, Bulgaria. E-mail: zhivkov_ipc@doctor.bg
Cite this: RSC Adv., 2014, 4, 2715
Received 25th January 2013
Accepted 30th September 2013
DOI: 10.1039/c3ra40431e
www.rsc.org/advances
This journal is © The Royal Society of Chemistry 2014 RSC Adv., 2014, 4, 2715–2728 | 2715
RSC Advances
PAPER

region). The ion-binding12,13
and ion-localization14,15
models
also separate counterions as bound and free; in the rst model
the condensed counterions form temporary pairs with the
polymer groups and in the second one they are freely moving
inside the chain volume. Oosawa has also proposed a three-
state model distinguishing three categories of counterions:
condensed, loose and free, accepting that the fractions of the
last two (non-condensed) are equal.16
In another three-state
model the counterions are separated as condensed and free
moving inside and outside the chain volume.17
The phase-state
models are useful approximations allowing evaluation of the
condensed counterion fraction, but they do not give the distri-
bution of the diffuse counterions around the chain.
The simplest solution giving the condensed ion fraction is
provided by the Manning’s two-state theory.18–20
It is developed on
a model considering the polyelectrolyte chain as an innitive
straight polyion with average distance b between the charges
surrounded by a coaxial cylinder on which surface the charges are
uniformly smeared; the electrolyte concentration in the medium
is vanishingly small (salt-free solution).11
Condensation appears
when the mean distance between two neighbour charges of
the polyion is smaller than the Bjerrum length lB (the critical
length under which ion pairs appear in concentrated electrolyte
solution; lB z 0.7 nm at room temperature); i.e. when the
dimensionless charge parameter x h lB/b is higher than 1. The
fraction 4 of the chain’s charges neutralized by the condensed
counterions with valency z is equal to: 1 À 1/zx ¼ 1 À b/zlB.
When the degree of dissociation a of a weak polyelectrolyte
increases, the linear charge density bÀ1
¼ f (a) increases linearly
until x reaches unity and then bÀ1
forms a plateau at x > 1.
Namely, according to Manning’s approximation the effective
charge density bÀ1
has its maximal value at x $ 1 and does not
increase more, although the chain’s own charge continue to
increase with a; this constancy of the total charge is due to
counterion condensation.21
Likewise, the condensed ion fraction
4 is determined by the ratio between the electrostatic $ e2
/3b and
thermal kT energies: x ¼ e2
/4p303bkT (where e is the elementary
charge and 303 is the dielectric permittivity). Thus, Manning’s
theory connects the counterion condensation with the chemical
properties of the polyelectrolyte chain in a very simple manner
and allows the separation of the polyelectrolytes as weakly
charged (x < 1, all counterions are diffuse) and strongly charged
(x > 1, a part 4 of the counterions are condensed).
Both the cylindrical cell and the phase-state models assumed
a uniform distribution of the charges along the polyion; which
is admissible because the respective theories consider only the
static electric properties of the polyelectrolytes. On the contrary,
the electric polarizability theories consider the dynamic prop-
erties in terms of counterion mobility. Oosawa’s uctuation
theory predicts the thermally induced migration of counterions
along the chain backbone and as a result spontaneous dipoles
appear.22,23
When an external electric eld is applied, the counterion
migration is directed and this leads to a large electric polari-
zation and accordingly to a large dielectric increment D3. The
polarizability g ¼ gk
À gt
has components along and across the
chain, but the transverse component gt
is negligible due to the
small diameter of the polyelectrolyte chain in comparison with its
length. The magnitude of the longitudinal component gk
is
proportional to the number of counterions and the squared
migration distance. The polarizability dispersion in alternating
electric elds is determined by the counterion mobility and the
half-period; the character relaxation frequency nch is dened by a
twofold decreasing of g z gk
. The counterion polarizability is an
object of numerous theoretical and experimental researches. The
theories are based on different models: cylindrical particles with
dened length,24–27
an innite cylinder with continuum charge
distribution,1
and an innite one-dimensional lattice.28,29
Opinions about the contribution of the diffuse and
condensed counterions to the polarizability are discrepant. The
polarization of diffuse counterions at low eld strength30
or
condensed ions at high eld strength31
are theoretically
described by Manning. On the contrary, according to Taka-
shima the condensed counterions do not participate in the
polarization.32
Most theories do not distinguish the contribu-
tion of the diffuse and condensed counterions to the longitu-
dinal polarizability component. However, it is intuitively clear
that the condensed and diffuse counterions have different
mobilities and their contribution to gk
in the dispersion region
should be different. Something more, the discrete distribution
of the polyelectrolyte charges suggests that the mobility of the
diffuse counterions should depend on their distance from the
polyion’s surface because of their migration in potential
troughs along the chain. Monte-Carlo simulations conrm this
assumption.33–35
Nevertheless, in recent theories developed for a discrete
charge line36
and long cylinders37
the origin of the induced
dipole moment is postulated to be the polarization of the
condensed counterions; the possible contribution of the diffuse
counterions is neglected. Comparison of the theoretical results
with experimental ones shows that the effective diffusion coef-
cient of the counterions is equal to that of the free ions in the
bulk;36
this result is interpreted by Manning as evidence for
condensed counterion polarization, but in our understanding it
reveals that the polarizability of polyelectrolytes in solution is
due to the diffuse counterions.
Minakata’s stochastic theory has an advantage in describing
the uctuation polarizability along a one-dimensional lattice
and taking into account the interaction between condensed
counterions.38
The counterions are assigned to be bound to the
individual sites or to the potential troughs along the lattice;
these two states correspond to condensed and diffuse coun-
terions. The theory shows that the counterion polarizability
depends on the potential barrier which they have to overcome,
jumping from one trough to the next, so the diffuse counterions
have higher mobility and give the main contribution at high
frequency n; the condensed counterions polarization should
appear at lower n. Thus the theory proves the possibility of
distinguishing the contributions of the condensed and diffuse
counterions by their relaxation frequencies nch and increments
D3, both determined by the dispersion dependence g(n).
In this way the participation of both the diffuse and
condensed counterions in the electric polarization of the poly-
electrolytes in solution is theoretically validated. Experimentally
2716 | RSC Adv., 2014, 4, 2715–2728 This journal is © The Royal Society of Chemistry 2014
RSC Advances Paper

their contribution can be distinguished by the measuring of
g(n) in the sinusoidal electric eld. The usually employed
techniques are dielectric spectroscopy39–41
and electric conduc-
tance42,43
due to their experimental simplicity. Both the
methods are based on diminishing the effective electric eld
owing to the counterion polarization in polyelectrolyte solution.
Dielectric spectroscopy allows the distinguishing of the
different kinds of polarizations by their relaxation frequencies
(due to its extremely wide frequency range technically avail-
able). The origin of the low-frequency permittivity increment is
assigned to be condensed counterion polarization and that of
the middle-frequency – polarization of the ‘free’ counterions. In
the rst case the effective diffusion coefficient is estimated to be
two orders lower than that of the free ions in the bulk.44
Both the
dielectric spectroscopy and the conductivity are experimentally
restricted: the rst to low ionic strength (because of the active
current and the electrode polarization) and the second to low
frequency range.
The electro-optical methods (birefringence, dichroism and
light scattering) are based on the alteration of the optical
properties of the solution with the application of an external
electric eld. The apparent electro-optical effect (EOE) is caused
by the orientation of the polyions.45,46
This technique has
principal advantages because the bulk does not give contribu-
tions to the measuring signal (in contradiction to the permit-
tivity and conductivity), which allows increasing of the
sensitivity by lengthening the optical path and working at lower
polymer concentrations. An additional advantage is the possi-
bility of applying high voltage electric elds.
The interpretation of electro-optical data is relatively easy for
rigid or short semi-exible chains with a contour length Lc
shorter than its persistent length lp, like DNA fragments,47
which allows the determination of the lp (ref. 48) and the time
constant of the chain bending.49
In the case of long chains
having the conformation of a random coil (when Lc [ lp) the
orientation of their segments is accompanied by deformation of
the coil and the values of g, Dg and nch are inuenced by the
mechanical and hydrodynamical properties of the poly-
electrolyte chain, which strongly complicates the interpretation
of the experimental results.50,51
A possible solution to the problem is immobilization of the
chains by adsorption on a solid surface and then exploring the
electrical properties of the polyelectrolyte layer. The electro-
optical technique is applicable when the polymer adsorption is
on non-spherical colloid particles. In this case the optical signal
is due to the particles, but the degree of their orientation is
determined by the electric properties of the polymer layer. An
additional advantage is the relatively low eld strength required
to orientate submicron particles in comparison with free poly-
electrolyte chains.
Extensive electro-optical investigations of polyelectrolyte
covered colloid particles have been made by Radeva et al. in the
last twenty years.52,53
The experiments show a lower-frequency
shi of the high frequency branch of the dispersion curve
(accordingly nch) in comparison with bare particles. This fact is
interpreted as evidence that the particles orientation is due to
condensed counterion polarization in the polymer layer; a
participation of other kind of counterions is not considered.
This inference is made on the assumption that the condensed
counterions are mobile, but less than the diffuse ones. This
interpretation is reasonable, but not convincing enough
because such a low frequency shi of nch is observed also in the
absence of polyelectrolytes, for instance, in water suspensions
of bacteria with the addition of ethanol.54
We accept the conclusions of Radeva et al. as a hypothesis
and try to obtain additional experimental evidence for
condensed counterion polarization. As a starting assumption
we accept that the electro-orientation of polyelectrolyte coated
particles is due to the polarization of all counterions in the
polymer layer (the particle’s volume polarization is negligible)
and the components of the torque are additive. To estimate the
contribution of the condensed and diffuse counterions it is
necessary to evaluate the ratio between their quantities, and for
that we employ Manning’s theory.
In this work we distinguish three states of the counterions
electrostatically associated to particles with adsorbed poly-
electrolyte: condensed (immobile during the existence of the
ion-group pairs), loosely bound (mobile in the vicinity around
the adsorbed polyelectrolyte chains) and diffuse (counterions in
the ionic atmosphere surrounding the particle). The last two
kinds of counterions are designated as ‘free’. Respectively, we
distinguish three components of the counterion polarizability
caused by condensed, loose and diffuse counterions: gc, gl, and
gd, every one with its own magnitude and relaxation frequency.
The fraction of the condensed counterions (the ratio of their
quantity to the sum of the three kinds of counterions) we
denominate as a share, considering the presence of the parti-
cle’s diffuse counterions.
For investigation of the three components of counterion
polarizability we measure electro-optically the degree of particle
orientation F(E, n) (a function of the eld strength E and
frequency n), depending on the polyelectrolyte surface concen-
tration. We assume that the magnitudes of gc and gl are
proportional to the amount of the polyelectrolyte adsorbed on
every single particle and the total polarizability g is due to the
counterions in all three states. We equate the measured value of
g only to the counterion polarizability, taking into account that
the contributions of other kinds of polarization as Maxwell–
Wagner (counterion migration normally to the surface) and the
volume (electronic, atomic) are negligible.55
Thus, the torque in
the electric eld has three components: the rst two caused by
the condensed and loose counterion migration along the
adsorbed polyelectrolyte chains and the third one by the diffuse
counterion migration in parallel to the particle surface.
In several consecutive works we investigated the electric
polarizability of carboxymethyl cellulose (CMC) adsorbed on
alumina colloid particles (g-Al2O3). The polymer was chosen
because of its electrical and mechanical properties:56
pH-
dependent dissociable COOH groups and chain conformational
rigidity conditioned by strongly hindered rotation around the
C–O–C bonds between the neighbouring glucose rings (char-
acteristic of the b-conguration of the 1,4-linkage in the cellu-
lose backbone). A measure for the chain rigidity is its persistent
length lp; for CMC in aqueous solution lp z 17 nm.57
The value
Paper RSC Advances

of lp is an order higher than the chain diameter; this ratio is one
of the conditions for using Manning’s theory (developed for a
straight cylinder). The rigidity of the CMC chains prevents a
high concentration of polymer segments in the adsorption
layer, and diminish by that the electrostatic interaction between
them. In this work the polarizability was studied at pH 6 where
the COOH groups of CMC are almost fully dissociated;57
this
ensured maximal counterion condensation due to the high
linear charge density.56
Particles of metal oxide are used as an adsorbent because the
oxide surface has a relatively low surface charge density even at
maximal ionization,58
which conditions fewer numbers of
contacts between the charged groups of the chain and the
surface and together with the image-charge effect strongly
diminishing the condensed ion release from ion-group pairs (in
our case the release of Na+
form the COOÀ
Na+
) with an
adsorption of highly charged polyelectrolytes on a weakly
charged substrate of low-dielectric permittivity.59–61
The
alumina particles are chosen because the point of zero charge of
the g-Al2O3 surface is at pH 8.2–8.5,62,63
so at pH 6 they are
positively charged,64
which conditions the electrostatic adsorp-
tion of the negatively charged CMC chains. The alumina parti-
cles are stable aggregates of 20 nm spheroidal nanoparticles
coalesced in the process of their synthesis by pyrolysis; as a
result the alumina particles have a submicron size, irregular
nonspherical form and an uneven surface. The nonspherical
form allows particle orientation in an electric eld, and the
surface roughness together with the chain rigidity decreases the
number of contacts of the adsorbed CMC with the alumina
surface, which additionally diminishes the counterion release
from COOÀ
Na+
pairs.
In our previous article65
we indicated that the condensed
counterions do not contribute to the induced dipole moment in
a sinusoidal electric eld with frequency of 1 kHz and intensity
up to 0.5 kV cmÀ1
. This conclusion resulted from the similar
dependences of the degree of orientation F $ gE2
and the
electrophoretic mobility uel on the CMC concentration (CCMC) in
the suspension; otherwise, the concentration dependences
g(CCMC) and uel(CCMC) should have different courses because of
altering the ratio between the diffuse and condensed counter-
ions with CMC adsorption. The inference is made on the
presumption that the condensed counterions should increase
g, but do not inuence uel. The assertion that uel is determined
only by the diffuse counterions is experimentally validated;2
one
piece of evidence is the presence of plateau of uel(a) at x > 1
indicating that the increasing counterion condensation with
the degree of dissociation a does not inuence uel, and also
other properties of polyelectrolyte solutions.21
Unlike the previous work,65
here we investigate g at two CMC
concentrations (respectively, under and above the recharging
point) where the total electric polarizabilities g are equal but the
condensed ion quantities differ 14 times. Because of the
different ratios between the counterions in the three states their
contributions to g have to be different: the diffuse counterions
contribute predominantly at the lower CCMC, but the condensed
and loosely bound ions are supposed to have the main contri-
bution at higher CCMC values. Due to the different mobilities of
the three-state counterions their polarization must be man-
ifested at different frequencies, which is why we investigate the
polarizability within a frequency range 10 Hz to 1 MHz.
2. Materials and methods
2.1. Materials
A sodium salt of carboxymethyl cellulose (NaCMC) with a degree
of substitution of 1.2 and molar mass Mw ¼ 250 kg molÀ1
(952
monomers per chain and contour length Lc ¼ 490 nm (ref. 56))
dissolved in triply distilled water was investigated. Thus, on
average 80% of the glucose monomeric units have one carbox-
ymethyl residue attached, 20% have two such residues and
almost all carboxyl residues are ionized at pH 6.57
Gamma-
aluminum oxide (g-Al2O3, Degussa) particles (stable aggregates
of 20 nm spheroidal nanoparticles) with irregular form and
mean size 0.3 mm dispersed in triply distilled water were used as
an adsorbent. The suspension was treated with an ultrasonic
disintegrator (Techpan, Poland) for 20 sec; the procedure
decreases the light scattering intensity by 5% and increases the
steady-state electro-optical effect (EOE) by 20% because of the
destruction of the biggest aggregates. The possible presence of
Ti-nanoparticles generated from the titanium tip does not
inuence the electric properties of the alumina surface, an
indication for which is the found equality of EOEs for particles
dispersed in sonicated or non-solicited triply distilled water.
The nal suspensions with 0.05 g dmÀ3
g-Al2O3 were prepared
by the mixing of the alumina suspension and the NaCMC
aqueous solution with continuous stirring for 30 min at 20
C.
This time is enough to nish the polyelectrolyte adsorption, as
we found out by measuring the kinetics of steady-state EOE at
1 kHz, 380 V cmÀ1
. The pH of the suspension was controlled
before and aer the electro-optical measurements; its value
was pH 6.0.
Because Manning’s theory was developed for the case of salt-
free solutions, we do not add NaCl to compensate for the
difference in the ionic strength at different concentrations of
NaCMC because we found out that the ionic strength at the
highest CCMC used inuences insignicantly the value of g.
2.2. Electrophoresis
The electrophoretic mobility uel ¼ la/tE was determined by
measuring the time t for a particle passing a xed distance la in
d.c. eld with strength E using Mark II apparatus (Rank
Brothers, UK) with a at quartz cell at 20
C. The uel practically
does not depend on the non-adsorbed CMC because of the low
polymer concentration not increasing signicantly the viscosity
h of the medium. The mobility uel ¼ (2330/h)f (a/d)z is deter-
mined by the electrokinetic potential z, bulk viscosity h,
dielectric permittivity 330 and the function f (a/d) ¼ 1/2–1/3 of
the ratio of the radius a of a spherical dielectric particle and the
thickness d of its electric double layer according to Smo-
luchowski–Hückel–Henry’s equation.66
The measured uel ¼
1.80 Â 10À8
m2
sÀ1
VÀ1
of bare particles at pH 6.0 and ionic
strength 2.5 Â 10À4
mol dmÀ3
corresponds to z ¼ 32 mV; the
RSC Advances Paper

surface charge density is 1.3 Â 10À3
C mÀ2
(0.8 elementary
charges per 100 nm2
) according to the Guye–Chapman theory.
2.3. Electric light scattering theory
In the Rayleigh–Debye–Gans approximation67
the light scat-
tering intensity I0 at a random orientation of the independent
particles of a disperse system is determined by the function of
internal interference (form-factor) P(q) at a scattering angle q:68
I0 ¼ kicHMP(q), (1)
where ki is the instrumental constant determined by the scat-
tering volume and the solid angle of the photoreceiver; c is the
weight concentration of the dispersed substance; H is the
optical constant of the suspension, dened by the refractive
indexes of the particles n1 and the medium n0 at the wavelength
in vacuum l0; M is the particle mass.
When an electric eld is applied to the suspension, the light
scattering intensity changes up to IE. The effect is caused by
alteration of the intraparticle light interference because of
particles orientation. The electro-optical effect (EOE) DI ¼ IE À
I0 is dened by the functions of internal interference at a certain
degree of orientation P(q, F) and at random orientation P(q):69
DI ¼ kicHM[P(q, F) À P(q)]. (2)
The value of P(q, F) is determined by the orientation degree
F(g, E, T, t) (varying from 0 at random orientation to 1 at full
orientation) which is a function of the electric polarizability g,
the electric eld strength E, the temperature T and the time t:
P(q, F) ¼ P(q) + A(KL) Â F(g, E, T, t), (3)
where the optical functions P(q) and A(KL) are determined by the
form and the relative size L/l of the particles with length L at
wavelength in the medium l ¼ l0/n0; KL ¼ 2p(L/l)sin(q/2).
The relative EOE DI/I0 does not depend on ki, c, H, M; it is
dened only by P(q, F) and P(q). So, DI/I0 can be presented as a
product of the optical function [A(KL)/P(q)] (determined by the
particle geometry) and the orientation function F(g, E) (deter-
mined by the induced dipole gE).
In the process of orientation and disorientation P(q, F)
changes with the time t; then the transient EOE is:
DIt/I0 ¼ [P(q, F)/P(q)] À 1 ¼ [A(KL)/P(q)] Â F(g, E, T, t). (4)
The EOE decay aer switching off the electric eld is dened
by the rotational diffusion coefficient Dr, respectively by the
relaxation time sr ¼ 1/6Dr (the time for e fold decreasing of DI
starting from its stationary value DIs). The EOE decay in the case
of a monodisperse suspension is mono-exponential for particles
with axial symmetry:70
DIt ¼ DIs exp(À6Drt) ¼ DIs exp(Àt/sr), (5)
where DIs and DIt are the values of EOE at the steady-state and at
the moment t aer eld switching out, respectively.
In the steady-state F(g, E, T) depends only on the ratio
between the orientation energy gE2
and the energy of random
motion kT. Then the EOE at low degrees of orientation (gE2
(
kT) is:46
DIs/I0 ¼ [A(KL)/P(q)] Â (gE2
/15kT). (6)
2.4. Electric light scattering experiment
The EOE was measured at q ¼ 90
by computerized home-made
apparatus whose optical scheme is described in ref. 46. The
electro-optical cell is made of glass and platinum electrodes
with surface areas 1 cm2
, an interelectrode distance of 2.6 mm
and volume of 10 ml. To the cell were applied impulses of
sinusoidal a.c. voltage up to 140 V, with a frequency of 10 Hz to
1 MHz, generated by a functional generator Wavetek-185 and
amplied by a wide band amplier Krohn-Hite-7500. The
concentration dependences of both I0 and DIs were linear up
to 0.3 g dmÀ3
g-Al2O3, which guarantees that at the chosen
concentration of 0.1 g dmÀ3
g-Al2O3 the alumina particles
scatter the light as independent centers and the contribution of
the multiple scattering is omissible. The values of I0 and DIs
(at 1 kHz) were measured at the beginning, during and at the
end of the electro-optical experiment; the constancy of I0 and
DIs was used as an indication for nished CMC adsorption and
the absence of particle aggregation.
3. Results and discussion
3.1. Electric charge density of CMC
The distance between glucose units in the CMC chain is 0.515
nm,71
the mean distance between the COOH groups at a degree
of substitution DS ¼ 1.20 is equal to 0.429 nm; so the dimen-
sionless charge density parameter is x0 ¼ lB/b0 ¼ 1.66 at
full dissociation (determined by the Bjerrum length lB ¼
e2
/4p303kT ¼ 0.71 nm at 20
C and the mean distance between
two neighbour charges b0 ¼ 0.43 nm). At pH 6.0 the degree of
dissociation is a z 0.95 (in 0.01 mol dmÀ3
NaCl at DS ¼ 1.2),57
so b ¼ b0/a z 0.45 nm, x ¼ alB/b0 z 1.58 and the fraction of
condensed Na+
counterions is equal to 4 ¼ 1 À xÀ1
z 0.37
according to Manning’s approximation. Consequently, in
solution the free CMC chain remains negatively charged and
about 63% of its own charges at pH 6 are compensated by
diffuse counterions.
This value is approximate because Manning’s analytical
solution is derived for salt-free semidilute polyelectrolyte solu-
tions of rod-like polyions assuming sinh j z j. The numerical
calculations in the frame of the two-state model show that the
curve (1 À 4)x ¼ f (x) has a curvature at x z 1 but continues to
increase with a slower slope, decreasing with x.3
The absence of
plateau at x $ 1 means that the counterion condensation is
smaller than that predicted by Manning’s theory, the deviation
depends on the polyelectrolyte volume fraction. Nevertheless,
(1 À 4)x ¼ 1 at x z 2 in a wide range of polymer concentrations;
i.e. at x z 1.6 (CMC solution at pH 6) the condensed
Paper RSC Advances

counterions fraction is almost equal to that predicted by
Manning’s theory.
When the CMC is adsorbed on the particles surface two
additional effects inuence the condensation of the ions in
opposite directions. The rst is the strengthened electrostatic
eld in the vicinity of the COOÀ
groups owing to the increased
concentration of the negative charges in the polymer layer
and leading to higher local concentration of the cations (Na+
and H+
), as a result the condensation increases although the
degree of dissociation a somewhat decreases. The H+
/Na+
competition decreases the Na+
condensation because of
decreasing the linear charge density with a, but this effect
is negligible in our conditions (pH 6 and CCMC z 10À3
to
10À2
g dmÀ3
) due to the 3–4 order higher Na+
concentration
than that of the H+
. The second effect of the CMC adsorption is
the formation of pairs between COOÀ
groups and the positive
charged centers of the alumina surface causing condensed Na+
release from COOÀ
Na+
pairs.
However, both effects of the polyelectrolyte adsorption have
to be weakly manifested in the case of a high molecular CMC
(forming loops above the particle surface) due to the high chain
rigidity not allowing formation of a dense polymer layer and
decreasing the share of the segments lying on the surface. The
few condensed ions realized at CMC adsorption are conditioned
mainly by the big difference in the charge density of the poly-
electrolyte chain56
and the oxide surface (the measured elec-
trophoretic mobility of bare particles corresponds to 12 nm
mean distance between two neighbour charges instead of
0.45 nm for CMC) and additionally by the uneven surface of the
alumina particles (formed by 20 nm spheroidal nanoparticles);
both the factors diminish the number of contacts between
the COOÀ
groups and the protonated centers in the hydrated
layer on the oxide surface.58
Another factor is the low surface
concentration of the polyelectrolyte. The CMC concentrations
used are 3 and 43 times lower than CCMC of saturation,
respectively at the higher and the lower CCMC chosen for the
measuring of the polarizability.
Due to the above reasons we assume that the adsorption of
CMC on alumina particles at pH 6 does not signicantly change
the degree of Na+
condensation compared to that on the free
chains in the bulk. So, the above estimated condensed coun-
terion share is an approximate value certifying the presence of
the condensed counterions in the CMC adsorption layer.
3.2. Polymer concentration dependence
Because of the opposite charge of CMC and g-Al2O3 at pH 6, the
adsorption of the polyelectrolyte chains on the alumina surface
causes a decrease of the total charge before reaching the
recharging (isoelectric) point and increasing aer it. The total
charge of every Al2O3–CMC particle determines its diffuse
counterion quantity and the adsorbed polyelectrolyte amount
determines its condensed and loose counterion quantity; thus,
the CMC adsorption changes the ratio between the three types
of counterions. The aim is to nd two different CMC concen-
trations (C1 and C2) with equal polarizability (g1 ¼ g2), but with
different fractions of the ‘free’ (diffuse and loose) and the
condensed counterions. So, C1 and C2 should be chosen before
and aer the recharging point; the values of g1 and g2 can be
found out from the dependence of the steady-state EOE (DIs/I0)
on CCMC.
In Fig. 1 the concentration dependence DIs/I0 ¼ f (CCMC) is
shown (CCMC is the total NaCMC concentration in the suspen-
sion); the DIs was measured at n ¼ 1 kHz, E ¼ 380 V cmÀ1
. The
decreasing (le) part of the curve is linear; this is evidence that
all of the added CMC is adsorbed on the particles. This fact
corresponds with the well known ability of the polyelectrolytes
to complete irreversible adsorption on oppositely charged
surfaces at low occupation density.58
The slope of the increasing
(right) part of the curve is much less than the decreasing one
(the x axis scale of the two parts is 50 times different). The fact is
an indication for incomplete CMC adsorption on already
recharged particles owing to the electrostatic repulsion of the
non-adsorbed chains; an indication for this is the plateau
appearing in the same concentration range.
Extrapolation of the le and right parts of the curve in Fig. 1
to DIs/I0 ¼ 0 gives CCMC ¼ 1 Â 10À3
g dmÀ3
and 3 Â 10À3
g
dmÀ3
, respectively; so, the mean value CCMC ¼ 2 Â 10À3
g dmÀ3
can be assumed as the CMC concentration where g ¼ 0. This
value practically coincides with the isoelectric point (uel ¼ 0 at
CCMC ¼ 3 Â 10À3
g dmÀ3
) obtained by electrophoresis.65
The coincidence of the polyelectrolyte concentrations at
which g ¼ 0 and uel ¼ 0 is an argument against the condensed
counterion polarization. The inference is based on the experi-
mental fact that in the d.c. eld the condensed ions do not
contribute to uel.2
The last fact means that in the isoelectric
point the polarizability of the CMC–alumina particles should be
zero (g ¼ rÀgdr + gl ¼ 0, taking into account the opposite sign
and the equal quantities of the diffuse and loose counterions) if
condensed Na+
are immobile in sinusoidal a.c. eld, but the
polarizability should be non-zero (g ¼ rÀgdr + gl + gc 0) if they
participate in the polarization. However the obtained experi-
mental evidence (uel ¼ g ¼ 0 at the same CCMC) is not reliable
enough because in the CCMC range (1–3) Â 10À3
g dmÀ3
the
CMC–alumina particles are not single but aggregated. So, it is
Fig. 1 Dependence of the steady-state EOE (DIs/I0) on NaCMC
concentration (CCMC) in g-Al2O3 suspension at field strength E ¼ 380 V
cmÀ1
and frequency n ¼ 1 kHz.
RSC Advances Paper

not excluded that the condensed counterions have some
contribution to the polarizability (gc s 0), but the aggregation
in the isoelectric point masks it.
3.3. Field strength dependence
The equality of DIs/I0 at C1 and C2 (Fig. 1) indicates g1 z g2, but
this inference is correct only at low degrees of orientation at
both CCMC, i.e. when the orientation energy gE2
is small in
comparison with the energy of the thermal motion kT. Criterion
for gE2
( kT is a linearity of the function DIs(E2
) (where DIs $
gE2
); so, the eld dependences must be measured at these two
CCMC where the total particle charge is opposite – positive at C1
and negative at C2 (due to the over-equivalent polyelectrolyte
adsorption).
Fig. 2 represents the eld strength dependences at n ¼ 1 kHz
of four suspensions of alumina particles in different media:
water (C0) and CMC solutions at concentrations: C1 ¼ 7 Â
10À4
g dmÀ3
(below the isoelectric point), C2 ¼ 1 Â 10À2
g dmÀ3
(above it) and C3 ¼ 5 Â 10À2
g dmÀ3
(on the plateau of the
concentration dependence DI(CCMC)). The light scattering
intensity at random orientations I0 was found to be equal at the
four CCMC, which is evidence for the absence of aggregation.
The equality of I0 at both C0 (bare particles) and C3 (saturated
CMC adsorption) indicates that the presence of the poly-
electrolyte layer on the particle surface does not change their
optical properties; so, the optical function [A(KL)/P(q)] in eqn (4)
has the same value in all the studded suspensions.
The linearity of the graphs in Fig. 2 shows that the degree of
orientation is low in the range DIs/I0 z 0–0.1, so g $ (DIs/I0)/E2
(eqn (6)) in the whole CMC concentration range (Fig. 1). The
equal slope of the lines 1 and 2 is evidence that g1 z g2 at C1
and C2.
The comparison of g at different CCMC values may not be
correct if the increasing concentration of Na+
with CCMC (a
result of using a sodium salt of CMC) is not taken into account
because the ionic strength growth diminishes the torque,
accordingly the slope (DIs/I0)/E2
. So, the contribution of the
three-state counterions to the polarizability could be different at
different NaCMC concentrations, which requires the estimation
of Na+
concentration and its inuence on g. For NaCMC with
DS ¼ 1.2 at C2 ¼ 1 Â 10À2
g dmÀ3
the concentration of the Na+
ions is 2 Â 10À6
M,56
which corresponds to 5 Â 10À6
M NaCl. As
the experiments have shown, the addition of NaCl at such a
concentration to the suspension at C0 (alumina particles in
triple distilled water) does not lead to a measurable decrease in
the slope of DIs/I0 ¼ f (E2
) at 1 kHz. Hence, the different Na+
concentrations at C1 and C2 (C1 ( C2) does not inuence the
measured value of g and the contribution of the counterions in
all three states.
3.4. Aggregation stability
The interfacial polarizability g is determined by both surface
electric properties and particle geometry (size and shape): g h
gk
À gt
is dened as the difference between its longitudinal gk
and the transverse gt
components. To use the slope (DIs/I0)/E2
of the eld strength dependence as a measure of counterion
polarization, the particle geometry has to be unchanged
because g strongly increases with the long axis of the particles:
g $ Ln
, where n ¼ 2–3 (depending on the axis ratio and the ionic
strength).72
Therefore, the aggregation should be excluded as a cause for
the particle geometry changing. The aggregation possibility
increases not only at CCMC where the complete polyelectrolyte
adsorption leads to zero total charge (the isoelectric point), but
at higher CCMC values as well because of passing through this
point in the process of CMC adsorption before the surface
getting overcharged. An aggregation of the already recharged
particles is also possible owing to the unhomogeneous
Fig. 2 Dependence of the EOE (DIs/I0) on the squared field strength
(E2
) of g-Al2O3 particles in water (line 0) and aqueous NaCMC solutions
with concentrations (g dmÀ3
): 7 Â 10À4
(line 1), 1 Â 10À2
(line 2) and 5
Â 10À2
(line 3) at n ¼ 1 kHz.
Fig. 3 Dependence of transitional EOE (DIt/I0) on the time (t) (after
orientation in a sinusoidal electric impulse with strength E ¼ 54 kV mÀ1
)
in g-Al2O3 suspension with water (curve 0) or NaCMC solution with
concentrations: 7 Â 10À4
g dmÀ3
(curve 1) and 1 Â 10À2
g dmÀ3
(curve
2). Insert. Dependence of the light scattering intensity (I0) at angle q ¼
90
on the concentration (CCMC) of NaCMC in alumina suspension
with concentration c ¼ 0.05 g dmÀ3
at random particles orientation.
Paper RSC Advances

distribution of the adsorbed CMC chains on their surface, or the
inter-particle bridging by polymer chains.73,74
The dependence of the scattered light intensity at chaotic
orientation I0 $ MP(q) (eqn (1)) on the mass M and the relative
size L/l of the particles can be used to study their aggregation,
although the determination of M is sometimes complicated by
the nonlinear dependence of P(q) on the particle geometry
(usually P(q) decreases with L, but weakly when L is commen-
surable with l at q ¼ 90
). The resulting dependence I0(M, L)
increases at aggregation because the function I0(M) is signi-
cantly stronger than I0(L). So, the constancy of I0/c ¼ f (CCMC) is a
reliable criterion for the absence of aggregation.
The concentration dependence I0(CCMC) (the insert in Fig. 3)
shows that signicant aggregation can be observed only around
the recharging point, whereas at C1 and C2 the particles remain
single aer CMC adsorption. The little increase of I0 with CCMC
out of the aggregation region is an indication for an increment
of M due to adsorbed CMC chains, but the small slope of the
line shows that polymer contribution to the single particle mass
is negligible.
The decay of the EOE (eqn (5)) is the most sensitive criterion
for aggregation due to the strong dependence of the rotation
diffusion coefficient Dr $ 1/L3
on the particle long axis L.
Fig. 3 represents the EOE decay in a semilogarithmic scale
aer reaching a low degree of orientation in the steady-state; at
that condition the contribution of the possible aggregates is
maximal due to the strong dependence of g on the particle size
(g $ L2–3
) and the weaker dependence of the orientation-optical
function on L/l (eqn (4)). The insignicant curvature of the
relaxation curve without polyelectrolyte (bare particles, curve 0)
is an indication of the small polydispersity of the initial
suspension. This fact is due to the ultrasonic treatment of the
suspension (before adding the CMC) destroying the big alumina
particles (actually aggregates of nanoparticles); an indication
for this was the nding of a 5% decrease of I0 and 20% increase
of DIs/I0.
The relaxation time sr increases with CCMC from s0 ¼ 2.0 ms
(at C0 ¼ 0) to s2 ¼ 2.3 ms (at C2 ¼ 1 Â 10À2
g dmÀ3
NaCMC)
(Fig. 3). If this increase is owing to aggregation, it should cause
bending of the relaxation curve (the slope must be smaller in
its last part), but the unchanged curvature of the curves at C1
(curve 1) and C2 (curve 2) (down to DIt/DIs z 0.1) shows that the
polydispersity remains unaltered aer CMC adsorption. The
result allows the conclusion that the found 14% increase of sr at
CMC adsorption (equivalent to 4% particles size growth) is
caused by a rise of the friction on the solid–liquid interface. The
absence of aggregation during the recharging process (at C2) is
due to the low concentration of particles and fast polyelectrolyte
adsorption; both factors lead to recharging before the collision
of two neighbour particles.
3.5. Condensed counterions share
At CMC adsorption the ratio between the condensed and ‘free’
(diffuse and loose) counterions alters because the quantity of
the condensed and loose counterions increases with the
amount of adsorbed polyelectrolyte and that of the diffuse
counterions decreases before the isoelectric point and increases
aer it, reecting the addition of negative charges (the COOÀ
groups not making group-ion pairs with Na+
) to the positive
particle surface or to the already negative CMC-particles,
respectively. To estimate the contribution of the condensed
counterion to the polarizability it is needed to estimate their
share at C1 and C2 (where g1 z g2). For that purpose the ratio
C2/C1 z 14 can be used on the condition that all of the added
CMC is adsorbed on the particles. This supposition is proven at
C1 by the linearity of the concentration dependence DIs(CCMC)
before the recharging point (Fig. 1), but the lower slope aer it
(taking into account the 50 times different CCMC scale) suggests
that the most CMC chains remain non-adsorbed at C2.
The problem with the unknown adsorbed CMC amount on
the recharged particles can be resolved using the electropho-
retic mobility alteration Duel ¼ up À u0 (in the presence and
absence of polyelectrolyte, respectively) as a measure for the
added negative charges (uncompensated COOÀ
groups) to the
liquid–solid interface. The value of uel is determined by
the charges immovably attached to the particle (including the
condensed counterions);2,75
so, uel is proportional to the ‘free’
(diffuse and loose) counterions. The condensed counterions
share can be estimated by Duel and the condensed ions fraction
4 ¼ [COOÀ
Na+
]/[COOÀ
] (the ratio between the compensated to
all dissociated carboxylic groups); the condensed/loose Na+
counterions ratio is equal to 4/(1 À 4).
The surface concentration of uncompensated COOÀ
groups
(1 À 4)Cs is proportional to the mobility decrement Duel ¼ (u1 À
u0) before the isoelectric point and to the sum of the decrement
and increment aer it: Duel ¼ À (ru2r + u0). The adsorbed
CMC amount is proportional to |Duel|/(1 À 4), so the
condensed, loose and diffuse counterion quantities are propor-
tional to 4|Duel|/(1 À 4), |Duel| and u0, respectively; uel is
proportional to the ‘free’ counterion quantity (the sum of
Table 1 Electrophoretic mobility uel of the alumina particles and the counterion share at polymer concentration CCMC in the suspension
CCMC [g dmÀ3
] uel Â 108
[m2
VÀ1
sÀ1
] Duel Â 108
[m2
VÀ1
sÀ1
]
Counterion share
Condensed Loose Diffuse
C0 0 +1.94 0 0 0 1
C1 7 Â 10À4
+0.34 À1.60 0.21 0.36 0.43
IEP 3 Â 10À3
0 À1.94 0.23 0.39 0.39
C2 1 Â 10À2
À1.12 À3.06 0.26 0.45 0.29
C3 5 Â 10À2
À1.71 À3.65 0.28 0.47 0.25
RSC Advances Paper

oppositely charged diffuse and loose ions). The shares of the
three kind counterions are equal to:
(condensed) [4|Duel|/(1 À 4)]/[u0 + |Duel| + 4|Duel|/(1 À 4)] (7a)
(loose) |Duel|/[u0 + |Duel| + 4|Duel|/(1 À 4)] (7b)
(diffuse) u0/[u0 + |Duel| + 4|Duel|/(1 À 4)] (7c)
The electrophoretic measurements have given steady-state
mobility uel (in units 10À8
m2
VÀ1
sÀ1
) at NaCMC concentration
CCMC (g dmÀ3
) in the suspension: u0 ¼ 1.94 at C0 (water
medium); u1 ¼ 0.34 at C1 ¼ 7 Â 10À4
(complete adsorption);
u2 ¼ À1.12 at C2 ¼ 1 Â 10À2
(incomplete adsorption), and u3 ¼
À1.71 at C3 ¼ 5 Â 10À2
(saturated adsorption). Respectively,
Duel (10À8
m2
VÀ1
sÀ1
) is equal to Du1 ¼ À1.60, Du2 ¼ À3.06, and
Du3 ¼ À3.65.
In Table 1 the shares of condensed, loose and diffuse
counterions (eqn (7)) are presented assuming that 4 ¼ 0.37
(Section 3.1) is the same for the CMC chains in solution and in
the adsorption layer. The quantity of the condensed counter-
ions at C2 and C3 is 1.9 and 2.3 times higher than that at C1,
respectively.
The presence of both loose and diffuse counterions (opposite
by sign and equal by quantity in the isoelectric point) is justied
by their spatial separation because of the inhomogeneous
adsorption and low degree of surface occupation. The physical
picture is motivated by the high values of the linear charge
density, rigidity and contour length of the CMC chains leading
to the inhomogeneous distribution of the polymer segments,
accordingly their negative charges on the positive particle surface.
That is the difference from the case of small ions adsorption on a
smooth surface; in the last case the ‘free’ counterion quantity
(equivalent of the sum of loose and diffuse counterions at poly-
electrolyte adsorption) is zero in the isoelectric point.
The above values are estimated disregarding the mobility
reduction owing to the hydrodynamic friction of the adsorbed
CMC chains. The inuence of this effect is different before and
aer the isoelectric point: at C1 the condensed ions quantity is
overstated because both the CMC charge and hydrodynamic
friction diminish u1, but the estimation is lowered at C2 where
these two factors have an opposite effect on u2. So, the real
condensed counterions share is lower at C1 and higher at C2 in
comparison with the above estimated values. Thus, the igno-
rance of the polymer layer friction paradoxically favors the goal
of our research: we aim to compare the electro-optical behav-
iour of CMC-particles with equal polarizability, but different
condensed ion quantities.
The above estimations indicate that the condensed coun-
terion quantity is more than 2 times higher at C2 than at C1, but
their shares are commensurable.
3.6. Frequency dependence
3.6.1. Polarizability dispersion. The denition of the
counterions as diffuse, loose and condensed allows the dis-
tinguishing of three polarizability components: gd, gl and gc;
each of them with parallel gk
and perpendicular gt
components to the long particle axis. The electro-orientational
effect is determined by the polarizability anisotropy g ¼ gk
À
gt
; so, the geometry of the particles determines the value of g,
but the absence of aggregation (Section 3.4) allows the ignoring
of this dependence. The insignicant Na+
concentration also
neglects the dependence of g on the ionic strength (Section 3.3).
So, at the conditions of our experiment the contribution of gd,
gl and gc depends only on the quantity and valency of the
counterions, their mobility and the distance of migration for a
half-period of sinusoidal eld. The last two parameters deter-
mine the polarizability relaxation time si and accordingly the
characteristic frequency nch ¼ 1/2psi corresponding to a twofold
decreasing of g.
For diffuse counterions sd $ a2
/Di where a is the particle’s long
size;46
and sl $ l2
/Di for loosely bound counterions driing on a
distance l along a linear polyion;28,22
the Di is the ions diffusion
coefficient. So, nch for the diffuse gd and loose gl components
should be considerably different when a [ l. Assuming that l is
equal to the polymer segment length (about 30 nm for CMC57
) we
roughly have a/l z 10 (a z 0.3 mm for the alumina particles) and
accordingly nl/nd z 100. So, the expected two orders difference
in nch allows the distinguishing of the two dispersions corre-
sponding to the polarization of the diffuse and loose counterions.
Because of the lower mobility of the condensed counterions
their nch should be considerably lower than that of the loose ones.
The difference in the quantities of the diffuse and bound
(loose and condensed) counterions (Section 3.5) supposes their
different contributions to g at n ( nch.
Fig. 4 shows the dependencies of the steady-state EOE DIs/I0
on n in the range 10 Hz to 1 MHz at E ¼ 1 kV cmÀ1
. As Fig. 2
indicates, at this eld strength the degree orientation F(g, E) is
low (gE2
( kT) and the orientation-optical effect is a linear
function of F(gE2
). Therefore, g $ DIs/E2
(taking into account
that I0 z const, Fig. 3); that proportion is valid at all frequencies
because the F(g, E) is not higher than at 1 kHz (the plateau of
DIs/I0 ¼ f (n), Fig. 4).
Fig. 4 Semi-logarithmic dependence of the steady-state EOE (DIs/I0)
on the frequency (n) of the applied electric field with strength E ¼ 100
kV mÀ1
in a suspension of g-Al2O3 particles in water (curve 1) and
NaCMC solution at concentrations (g dmÀ3
): 4 Â 10À4
(curve 2); C1 ¼ 7
Â 10À4
(curve 3); C2 ¼ 1 Â 10À2
(curve 4) and C3 ¼ 5 Â 10À2
(curve 5).
Paper RSC Advances

Fig. 4 shows that the dispersion curves DIs(n) have a kilohertz
plateau and hertz and submegahertz branches. The most
evident difference between the curves is the value of g $ DIs/I0
in the plateau, which reects the quantity of the three kinds of
counterions at different adsorbed CMC amounts, accordingly
the polarizability components gd, gl and gc of the total polar-
izability g ¼ |Àgd + gl + gc|. Below the recharging point
the diffuse anions (counterions to the alumina surface)
predominate over the Na+
cations (coming with the adsorbed
CMC): |Àgd| |gl + gc| and g decreases with CCMC. At recharged
particles |Àgd| |gl + gc| and g increases with CCMC.
The values of DIs/I0 give the ratio g3/g2 z 1.4 in the plateau at
C3 and C2 (Fig. 4). The estimated values of g ¼ |Àgd + gl + gc|
according to the electrophoretic mobility |uel| (Table 1) give the
ratios gu3/gu2 ¼ 1.19 at gc 0 and 1.38 at gc ¼ 0. The last ratio
practically coincides to the electro-optically measured g3/g2,
which is an indication that the condensed counterions do not
participate in the kilohertz polarization.
Information about the relaxation times sd, sl and sc of the
three components of g could be derived from the low and high
frequency branches of the dispersion curve DIs(n) where nch
depends on the effective diffusion coefficients Dd, Dl, Dc and the
migration distances a and l, respectively for the diffuse, loose
and condensed counterions. To compare adequately DIs(n) at
different CCMC on the next two gures the dispersion curves are
normalized to unity at 1 kHz.
3.6.2. Low frequency dispersion. The condensed ion
polarization should appear as an additional component gc
(decreasing or increasing g, respectively before and aer the
recharging point) in the low-frequency (hertz) branch of DIs(n) if
a2
/Dd l2
/Dc; so if Dd [ Dc taking into account that a/l z 10.
Fig. 5 shows that in the hertz range the counterion contri-
bution is not the same as in the kilohertz plateau where g1 z g2.
The fact that DIs2 DIs1 (in the plateau DIs2 z DIs1, Fig. 4) could
be interpreted as an indication for the polarization of
condensed counterions taking into account that their share is
higher at C2 than at C1 (Table 1). But this interpretation is not
corroborated by the form of DIs(n): they do not show any addi-
tional dispersion (a curve’s shoulder) indicating contribution of
the counterions with different si. A coincidence of si of the
diffuse and condensed counterions (sd ¼ sc if Dd ¼ 100Dc at a/
l ¼ 10) is a small probability.
3.6.3. High frequency dispersion. The submegahertz
branches of the normalized dispersion curves are shown in
Fig. 6. The mean nch z 0.2 MHz at C1 (curve 1) lies in
the frequency range theoretically predicted76
for diffuse
counterions polarization (taking into account the dependence
nch $ 1/a2
on the size a of the alumina particles). This suggests
that the electro-orientational effect at a low adsorbed CMC
amount is determined basically by the diffuse counterions
surrounding the particle as a whole.
Fig. 6 shows two dispersions in the submegahertz branch of
g1(n) at C1: nm1 z 60 kHz for the middle dispersion and nh1 for
the high dispersion. The curve g2(n) at C2 also shows two obvi-
ously delineated dispersions with relaxation frequencies: nm2 z
40 kHz and nh2 z 0.3 MHz. The curve g3(n) at C3 (fully saturated
CMC adsorption) shows only middle dispersion: nm3 z 4 kHz.
The estimation of the relaxation time si of loosely bound
counterions by Oosawa’s equation si ¼ l2
/4p2
Di z 1.16 Â 10À8
s
gives nch z 13.7 MHz if the migration distance l ¼ 30 nm is
equal to the Kuhn segment length LK of a free CMC poly-
electrolyte chain57
and Di ¼ 1.96 Â 10À9
m2
sÀ1
is the diffusion
coefficient of free Na+
. If we assume that the counterions
migrate at a distance l ¼ 50 nm equal to the unperturbed radius
of gyration Rg ¼ (NK/6)À1/2
LK for a chain with contour length
Lc ¼ 505 nm (CMC with M ¼ 250 kg molÀ1
) having NK z 17
statistical segments with length LK ¼ 30 nm,56
than si z 3.2 Â
10À8
s and nch z 4.9 MHz.
But the experiment shows that nh2 z 0.3 MHz and g2(n) falls
to zero at 0.5 MHz (curve 2 in Fig. 6). The displacement of the nch
from the megahertz region to the submegahertz one means a
drastic decreasing of Di, or increasing of l from the macromo-
lecular scale to the particle one. So, the counterions behave
diffusely (to the recharged particles), but not as loosely bound
Fig. 5 Low-frequency dependence of EOE (DIs/I0) at the frequency (n)
at E ¼ 54 kV mÀ1
in a suspension of g-Al2O3 particles in water (curve 0)
and NaCMC solution at concentrations: C1 ¼ 7 Â 10À4
g dmÀ3
(curve 1)
and C2 ¼ 1 Â 10À2
g dmÀ3
(curve 2).
Fig. 6 High frequency dependence of EOE (DIs/I0) on the frequency
(n) at E ¼ 54 kV mÀ1
in suspension of g-Al2O3 particles in water (curve
0) and NaCMC solution at concentrations (g dmÀ3
): C1 ¼ 7 Â 10À4
(curve 1); C2 ¼ 1 Â 10À2
(curve 2) and C3 ¼ 5 Â 10À2
(curve 3).
RSC Advances Paper

(to the adsorbed CMC chains) accepting that the Di of the free
counterions cannot be strongly changed. This means that the
counterions classication as diffuse and loose is well-dened
only at a low degree of surface covering with polyelectrolyte
when the adsorbed macromolecules have a mosaic structure.
The decreasing of nch (Fig. 6) indicates an increase of si with
the adsorbed CMC amount; that diminishes DIs (relative to DIp
on the kilohertz plateau) in the region of a high-frequency
branch of g(n): at 0.5 MHz DIs/Ip is about 0.7 for bare particles,
0.4 at C1 and almost zero at C2. The increase of si can be partly
explained by the replacement of the faster OHÀ
anions with the
slower Na+
cations (the mobility u+ $ 1/DNa+ of free Na+
is 3.9
times less than the uÀ of OHÀ
) as predominant counterions
with CMC adsorption. We consider the si increase as caused
mainly by some reduction of the effective diffusion coefficient
Di and increase of the hydrodynamic friction in the polymer
layer. As a result the electroosmotic ow around the particle
surface (the cause for the particles orientation77–80
) is dimin-
ished, which leads to a low degree of orientation.
3.6.4. Middle frequency dispersion. The principal distinc-
tion of the dispersion curve g(n) at incomplete CMC adsorption
is the shoulder dividing its submegahertz branch into middle-
and high-frequency dispersions with increments Dgm and Dgh,
respectively (Fig. 6). The relaxation frequencies of the middle-
frequency dispersion gm(n) are nm1 z 60 kHz at C1 and nm2 z
40 kHz at C2; the increment Dgm2 is somewhat higher at
recharged particles than Dgm1 before the recharging. The ratio
Dgm2/Dgm1 z 1.2 is very close to the ratio of the condensed
counterion shares (equal to 1.24 at C2 and C1, Table 1); this
coincidence suggests that the middle-frequency polarization
has its origin in the condensed counterion motion along the
polymer segments. This is why we start the interpretation with
this hypothesis. Below we try to verify it considering the
mobility and migration distance of the counterions.
The effective diffusion coefficient Di determining the
mobility uÆ ¼ zeDi/kT of counterions with charge ze along the
polyelectrolyte chain is the rst criterion allowing us to distin-
guish the three kind of counterions. The relaxation time of the
middle-frequency dispersion g2m(n) (Fig. 6) is s2m ¼ 1/2pn2m z
4.0 ms which corresponds to Di ¼ 5.7 Â 10À10
m2
sÀ1
according
to the Oosawa equation s ¼ l2
/4p2
Di at the migration distance
l ¼ 0.3 mm (equal to the long size L of the alumina particles);
s1m z 2.7 ms and Di ¼ 8.6 Â 10À10
m2
sÀ1
at C1. The experi-
mentally found DNa for bound Na+
in CMC solution81
is 33-fold
smaller than DNa ¼ 1.96 Â 10À9
m2
sÀ1
of free Na+
. The only 2–3-
fold decrease in Di obtained in our experiment argues against
the condensed ions polarization.
The migration distance used in the above estimations
correspond to the particle length L; i.e. the counterions migrate
along the particle surface. If the counterions migrate along the
polyelectrolyte chain (where its curvature is less than 90
), then
the distance must be commensurable with the persistent length
lp z 15 nm; so the critical frequency nm $ Di/lp
2
should lie in the
megahertz range (taking into account that L/lp z 20 and
assuming that Di is not drastically reduced). However the
experiment shows that nm is in the kilohertz range; hence the
migration distance corresponds to the L, but not to the lp. Thus
the found kilohertz value of nm appears as a second argument
against the condensed counterion polarization, considering
that strongly bound ions should migrate at a distance z lp
without possibly being able to leave the polyelectrolyte chain.
An additional criterion is the dependence of the middle-
frequency dispersion on the polymer concentration: the nm
should increase with CCMC because the length of the chain
segments that lie in parallel to the particle surface have a
tendency to decrease with the density of the polyelectrolyte
layer. But Fig. 6 shows just the opposite: the nm moves from
60 kHz at C1 to 40 kHz at C2 and further to 4 kHz at C3. Thus, the
low-frequency shi of nm with CCMC supports the above made
inference that the condensed counterions do not originate from
the polarization at kilohertz frequencies.
The above arguments allow the middle-frequency dispersion
gm(n) to be attributed to the ‘free’ counterions (predominantly
diffuse anions at C1 and loosely bound cations at C2) whose
mobility is somewhat reduced because of interactions with the
chains’ charges at their migration through the regions occupied
by the polyelectrolyte chains (forming a quasi two-dimensional
mosaic structure on the surface). That is the difference from the
high-frequency dispersion gh(n), which could be attributed to
the counterions migrating out of these regions. That supposi-
tion is corroborated by the low-frequency displacement of nm
(from nm1 z 60 kHz down to nm3 z 4 kHz) accompanied with
some decreasing of the increment Dgh/gp of the high-frequency
dispersion (from Dgh1/gp z 0.5 to Dgh3/gp z 0.3) when the
number of adsorbed chains increases with CCMC increasing
from C1 to C3 (Fig. 6).
3.6.5. Origin of the counterions polarization. The above
results allow the conclusion that the condensed counterions do
not participate in the polarization at moderate eld strength
and that the origin of the kilohertz polarization is the migration
of ‘free’ counterions surrounding the CMC-covered particle.
This inference is conrmed by the coincidence of the ‘electro-
phoretic’ ratio g3/g2 ¼ 1.38 (where g ¼ |gl À gd| is estimated by
the counterion shares in the absence of condensed ion polari-
zation, Table 1) with the electro-optical ratio gp3/gp2 z 1.4
(measured in the kilohertz plateau at C3 and C2, Fig. 2 and 4).
With regards to the migration distance the difference
between the diffuse and loose counterions disappears at a high
degree of surface coverage. The loosely bound counterions then
behave diffusely, migrating a distance commensurable with the
particle length, which leads to the appearance of nm in a kilo-
hertz range and to the disappearance of the megahertz polari-
zation (g2 z 0 at n 0.5 MHz).
The observed low-frequency displacement of nch with CCMC
increasing can be explained with slowing of the electro-osmotic
ow (orienting the particle) caused by two factors. The rst is
the diminished counterion mobility of the ‘free’ counterions in
the polyelectrolyte layer where the COOÀ
groups play a role of
electrostatic traps for Na+
cations and thus decreasing the
effective diffusion coefficient Di, accordingly the velocity of
counterion migration in electric eld with given strength. The
second factor is the increased hydrodynamic friction accom-
panying the polymer layer growing with the adsorbed CMC
amount. Both the Di reduction and the friction magnication
Paper RSC Advances

lead to a decrease of the nm and an increase of the relative
increment Dgm/gp in the frequency range where the counterion
polarization shows the middle-frequency dispersion. Hence, the
low-frequency displacement of the nm (the relaxation frequency
attributed to the counterion polarization in the polyelectrolyte
layer) and the diminishing of the polarizability gp (in the kilo-
hertz plateau) with the CCMC increasing are results of the elec-
tro-osmotic ow attenuation because of increasing of the
adsorbed CMC amount and polymer layer density.
3.7. Comparison with the literature
To decrease the differences in the experimental conditions with
the works of Radeva et al.82,83
we use the same experimental
technique (electric light scattering under 90
) and the same
polymer sample (NaCMC with molecular mass Mw z 250 kg
molÀ1
). The difference consists only in the colloid particles: we
chose aluminum oxide (achromic in the optical spectrum)
instead of b-ferrioxide (coloured in the visible range). Owing to
the light absorbance in a ferrioxide suspension two electro-
optical phenomena with different orientation-optical functions
appear – scattering and dichroism. This circumstance is not
taken into account by Radeva et al., although it is a prerequisite
for an incorrect interpretation.
The studies of Radeva et al.52,53
on polyelectrolyte polarizability
are done with recharged particles at saturated adsorption; they
do not concern the polarization of the counterions under the
recharging point, although just this region (where |gd| gl) is the
most informative due to the opposite sign of the polarizability
components gc and |Àgd + gl|, attributed to the condensed and
‘free’ (diffuse and loose) counterions, respectively.
Our results for the recharged particles are similar to those in
the cited references, but the conclusion about the origin of the
kilohertz polarization is quite different. Radeva’s interpretation
is that the electro-orientational effect is due to the condensed
counterion polarization; the inference is based on estimation of
nch at a migration distance l, equal to the contour length Lc ¼
0.5 mm of the CMC used.83
This estimation is wrong because it is
based on the presumption that the adsorbed chain is a straight
polyion with length Lc. The polyelectrolyte chain can be
considered as rod-like (both free in solution and adsorbed on
a surface) only when Lc is smaller than the persistent length
lp (at low ionic strength lp z 15–17 nm for CMC with DS ¼
1.2)56
. Because Lc/lp z 30, the calculated relaxation frequency
(nch $ lÀ2
, where l is the migration distance) is three orders
less at l ¼ Lc than at l ¼ lp. Owing to the occasion that the Lc z
0.5 mm is commensurable with the particles length L z 0.3
mm, the cited authors have mistakenly attributed the exper-
imental value of nch z 4 kHz (ref. 82) to the condensed ions
polarization instead to the ‘free’ ones.
4. Conclusion
The interface electrical polarizability of alumina particles with
adsorbed CMC is estimated electro-optically by the degree of
orientation in sinusoidal electric eld. The polarization is
assumed to have three components caused by the counterions
migration in parallel to the particle surface; respectively, three
kinds of counterions are considered: diffuse anions (counter-
ions to the positive particle surface), and Na+
cations in two
physical states (loosely bound to the negatively charged CMC
chains and condensed on them). A procedure for determination
of the shares of three kinds of counterions (diffuse, loose and
condensed) of arbitrary colloid particles with adsorbed semi-
exible polyelectrolyte chains is invented; it uses the measured
electrophoretic mobility and the fraction of the condensed
counterions calculated for free chains in polymer solution.
The results allow the conclusion that the electro-orienta-
tional affect of alumina particles with adsorbed CMC at
moderate eld strength is due to the polarization of the ‘free’
counterions (diffuse and loosely bound) and that the condensed
ions do not participate in the whole frequency range 10 Hz to
1 MHz. The conclusion can be extended to higher frequencies
due to the fact that the steady-state EOE falls to zero at n
0.5 MHz in the case of recharged particles. The revealed
migration distance of the counterions is commensurable with
the particle size. This nding suggests that the behaviour of the
diffuse and loosely bound counterions is analogical in regard to
the electro-orientational effect in a sinusoidal electric eld. The
value of the polarizability in the kilohertz plateau is determined
by the quantities of the diffuse anions and the loosely bound
cations, considering the opposite directions of their migration.
The found two dispersions found in the kilohertz region
indicate the presence of counterions with different mobilities:
normal and slowed down; the effective diffusion coefficient of
the latter is reduced. The counterion deceleration can be
explained by interactions with the polyelectrolyte charges at
ions migration through the regions of adsorbed chains. The
found decreasing of the relaxation frequency with CMC
adsorption amount is presumably an effect of reducing the
effective diffusion coefficient and the increase of the hydrody-
namic friction in the polymer layer.
Finally, taking into account the literature data it is possible
to extend the conclusions to other semi-exible polyelectrolytes
adsorbed on colloid particles with low surface charge density.
This generalization means that the interface electric polariza-
tion of colloid particles covered with polyelectrolytes is deter-
mined by the ‘free’ counterions migrating along the surface on a
distance commensurable with the particle length; the presence
of the polymer layer leads to a diminishing of the polarizability
and low-frequency shi of its dispersion.
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