Full text of DOI:10.1007/s101890050009

Eur. Phys. J. E 1, 75–86 (2000)
THE EUROPEAN
PHYSICAL JOURNAL E
EDP Sciences
Societa` Italiana di Fisica
Springer-Verlag 2000
c
ꢀ
Solubility of highly charged anionic polyelectrolytes
in presence of multivalent cations:
Specific interaction effect
I. Sabbagh and M. Delsantia
CEA-Service de Chimie Mol´eculaire, CE-Saclay, 91191 Gif-sur-Yvette Cedex, France
Received 3 March 1999 and Received in final form 2 September 1999
Abstract. Studies performed on strong polyelectrolytes and on
a weak polyelectrolyte, sodium
poly(acrylate), show that their stability in presence of multivalent cations depends on the chemical nature
of the charged side groups of the polymer. For sulfonate groups (SO−3 ) or sulfate groups (OSO3−) phase
separation generally occurs in presence of inorganic cations of valency 3 (as La3+) or larger and a resolu-
bilization takes place at high salt concentration. The interactions of the polyelectrolyte with multivalent
cations are of electrostatic origin and the phase diagrams are weakly dependent on the chemical nature of
the polymer backbone and on the specificity of the counterions. For acrylate groups, (COO−), the phase
separation was observed with inorganic cations of valency 2 (as Ca2+) or larger without resolubilization
at high salt concentration. The phase separation is due to a chemical association between cations and
acrylate groups of two neighboring monomers of the same chain. This chemical association creates a hy-
drophobic complex by dehydrating both monomer and cation. With organic trivalent cation, as spermidine
+H3N(CH2)4NH+2 (CH2)3NH3+, where no chemical association occurs with the charged side groups COO−
or SO−3 of the polyelectrolyte, similar phase diagrams were observed whatever was the polyelectrolyte with
a resolubilization at high trivalent cation concentration.
PACS. 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling – 61.20.Qg Structure
of associated liquids: electrolytes, molten salts, etc. – 83.70.Hq Heterogeneous liquids: suspensions,
dispersions, emulsions, pastes, slurries, foams, block copolymers, etc.
1 Introduction
centration) and the L-precipitation type (low critical salt
concentration). The H-precipitation occurs at high salt
concentration threshold and weakly depends on the poly-
mer concentration. The L-precipitation type appears at
low multivalent salt concentrations, and the salt concen-
tration needed for phase separation is proportional to
polyelectrolyte concentration.
Anionic polyelectrolytes possess ionizable groups, which
dissociate in aqueous media to give negatively charged
polymer chains and positively charged counterions. The
repulsion between charged monomers tends to expand the
polymer chain. The high charge density on the polymer
backbone produces a high electrostatic potential around
it and a fraction of counterions are consequently located
in the immediate vicinity of the polymer chain lead-
ing to a phase separation at low salt concentration, if
counterions are multivalent. This precipitation of poly-
electrolytes by multivalent salt, usually called “salting
out”, has been studied for long time by several au-
thors [1–13], but still remains poorly understood. The
stability of the polyelectrolyte solution depends on the
counterion valency and the characteristics of the poly-
mer: nature of backbone (hydrophobic, hydrophilic, rigid,
flexible), distance between charged groups, and nature
of charged groups (carboxylate, sulfonate, sulfate, phos-
phate). Two types of phase diagram have been observed [2]
namely the H-precipitation type (high critical salt con-
The present study addresses the nature of the spe-
cific interactions between highly charged polyelectrolytes
and counterions, especially for L-type diagrams. Phase
diagrams of different highly charged polyelectrolytes, in
presence of different multivalent cations, were established.
Two general mechanisms of precipitation were found,
which depend on the nature of the charged groups and of
the counterions. In the case of sodium polyacrylate, our
interest was focused on calcium ion, for its broad range
of applications, and on the cobalt ion which offers the ad-
vantage that its interaction with polyelectrolytes can be
studied by absorption spectrometry.
More specifically, in Section 2 (theoretical considera-
tion) the point on electrostatic condensation and chemi-
cal association are made. In this section we also recall the
electrostatic model [9], introduced to describe solubility
a
e-mail: Delsant@scm.saclay.cea.fr

76
The European Physical Journal E
of the sodium polystyrene sulfonate in presence of z-valent tively. This description is only qualitative and a more
counterions, and we generalize this model when chemi- quantitative way to evaluate the electrostatic conden-
cal association between counterions and charged groups of sation is obtained with the Poisson-Boltzmann models
the polyelectrolyte can occur. In Section 3, experimental [19–23].
conditions are given. Section 4 is devoted to the presen-
Up to now, chemical association between counterions
tation of different types of phase diagram obtained for a and charged groups of polyelectrolyte was assumed to
large variety of polyelectrolytes and of counterions. At the be absent. Charge regulation model [24,25] allows chem-
end of this section it appears that two types of mechanism ically associated counterions and electrostatically con-
are responsible of the precipitation: one is purely electro- densed counterions to be distinguished. The computation
static and the other one is intimately related to a chemical of the fraction of counterions chemically associated with
association. In Section 5, entitled complexation between charged groups of the polyelectrolyte is performed using a
polyelectrolytes and multivalent counterions, we relate ex- mass action law that depends on the concentration [A−]
periments, which allow us to determine the quantities and of charged monomer and on the local counterion concen-
the types of complexes formed when chemical association tration [Mz+
] in the immediate vicinity of the charged
L
occurs before phase separation. The understanding of the cylinder representing the polyelectrolyte:
specific interactions between polyelectrolytes and coun-
[Mz+A−] = K1[A−][Mz+
]
L
terions allows us to give a criterion to distinguish when
electrostatic attraction or chemical association induces the
precipitation. In Section 6, the phase diagram of sodium
poly(acrylate) in presence of calcium chloride is described
using a mean field approach presented in Section 2.
and [Mz+]L = [Mz+
]bulk exp(−zeΨlocal/kT).
(2)
K1 is the first association constant (monocomplexation)
between monomers and counterions and Ψlocal is the local
electrostatic potential of the polyelectrolyte. The quantity
[Mz+
]
is the bulk concentration. The local concen-
bulk
2 Theoretical considerations
tration is determined by solving the Poisson Boltzmann
equations. This method was largely used for planar sur-
faces [24,25]. Now having introduced the different asso-
ciations between polyelectrolytes and counterions, we are
interested in the limit of stability of polyelectrolyte solu-
tions.
For polyelectrolyte with a sufficiently high charge density
along the backbone, some of counterions remain in the
vicinity of the polymeric chain. This phenomenon, known
as counterion condensation, was introduced for long
time [14–18]. For monovalent counterions and charged
monomers (−1), counterion condensation occurs when the
mean distance between two charged monomers, b, along
To describe solubility of the sodium polystyrene sul-
fonate in presence of z-valent counterions [8] the electro-
static model [9] was introduced. This model takes into
account a short-range electrostatic attraction between
negatively charged monomers (−1) and those carrying
condensed multivalent counterions which are positively
charged (z −1). The following spinodal equation has been
obtained:
the chain backbone is smaller than the Bjerrum length
2
˚
lB = e /4πεε0kT (= 7.2 A in water). This length defines
the distance at which electrostatic interaction between two
elementary charges is equal to the thermal energy kT. Let
us consider an anionic polyelectrolyte having counterions
of valency z. If the distance b is larger than lBz, the effec-
tive charge per monomer, zeff, remains constant (−1). If b
is lower than lBz, ion condensation occurs and the effec-
tive charge is reduced to (−1)b/zlB. When a salt with a
1
NC
σ3
ze2ff
+
+
− 2(1 − f − f0)f |V1Z| = 0.
(3)
1 − ϕ
2I
z-valent cation (concentration CS) is progressively added N is the degree of polymerization of the polymer, C is the
to a solution of anionic polyelectrolyte having monovalent number of monomers per unit volume, σ3 represents the
counterions (monomer concentration C), condensation of volume of the monomer and ϕ = Cσ3 is the volume frac-
z-valent counterions takes place and monovalent counteri- tion occupied by the polymer. The first term comes from
ons are released. Three different domains of condensation the entropy of polymer and the second one represents the
can qualitatively be defined:
steric excluded volume. The third term corresponds to the
electrostatic repulsion between monomers carrying an ef-
fective charge zeff = zf + f0 − 1, where f and f0 are
the fractions of monomer with a z-valent and monova-
lent condensed counterions, respectively. I represents the
ionic strength of free ions. The last term arises from an
attraction between the negatively charged monomers (−1)
and those possessing an effective positive charge of valency
(z − 1), due to the electrostatic condensation. The local
attraction V1Z depends on the valency of the counterion
and has the following form [9]:
CS < C(1 − b/lB)/z,
f0 = 1 − b/lB − zCS/C and f = CS/C;
C(1 − b/lB)/z ≤ CS ≤ C(1 − b/zlB)/z,
f0 = 0 and f = CS/C;
CS > C(1 − b/zlB)/z,
f0 = 0 and f = (1 − b/zlB)/z; (1)
π(z − 1)2l2
where f0 and f are the fractions of monomer carrying
monovalent and z-valent condensed counterion, respec-
V1Z = −
B B
(4)
√
8πlBI

I. Sabbagh and M. Delsanti: Solubility of highly charged polyelectrolytes with multivalent counterions
77
where B is a numerical factor of the order of 1. When the By taking ϕ = Cσ3, we recover the spinodal expression
ionic strength is high enough, the electrostatic attractions of the electrostatic model (Eq. (3)) with a supplementary
are completely screened. For a given monomer concentra- term due to the dicomplexation:
tion, the spinodal equation gives two solutions correspond-
1
NC
σ3
ze2ff
ing to the precipitation and the resolubilization limits as
the salt concentration increases. The salt concentration at
the resolubilization limit is independent of the polymer
concentration for a high degree of polymerization, and in-
creases with the valency z of counterion [9].
The spinodal equation (3) was derived from the di-
vergence of the monomer-monomer scattering function at
zero wave vector using a RPA or PRISM approach. As
spinodal equations obtained in the context of RPA can be
derived using Flory mean field approach [12], this more
simple approach is used here to describe the stability of
sodium −polyacrylate in presence of divalent chloride salt,
M++Cl2 , where chemical association between charged
monomers, A−, and divalent counterions, M++, are taken
into account. The free energy per site can be written as:
+
+
− 2(1 − f0 − f − fm − fd)
× (fm + f) |V12| − 2σ3χdfd = 0. (9)
1 − ϕ
2I
The term ze2ff/2I, with zeff = (f0+2f +2fm+fd−1), arises
from the entropy of the free ions [12] and represents the
screened repulsive electrostatic interactions between neg-
atively charged monomers, having an effective charge zeff
,
and it is basically responsible of polyelectrolyte stability.
It is important to point out that this model of mean field
Flory type is only well adapted to describe concentrated
polymer solutions [26].
3 Materials and methods
F/kT = (ϕ/N) log ϕ + (1 − ϕ) log(1 − ϕ) + ϕM log ϕM
+ ϕCl log ϕCl + ϕNa log ϕNa + χdfdϕ(1 − ϕ)
With the exception of one sample, the sodium dextran sul-
fate, all the polymers used had the same backbone but dif-
ferent ionic groups X: (-CH2-CHX-)N . Poly(acrylic acid)
samples (PAA: X = COO−, H+) were purchased from
Sigma Aldrich with nominal molecular weights: 2 × 103
(PAA1), 4.5 × 105 (PAA2) and 4 × 106 (PAA3) g/mole.
Sodium polyacrylate fully or partially charged were ob-
tained by adding sodium hydroxide to poly(acrylic acid)
stock solutions. Electrochemical techniques show that
the PAA1 and PAA2 have a carboxylation degree about
77% and 100%, respectively. Sodium poly(vinyl sulfonate)
(NaPVS: X = SO−3 , Na+) was purchased from Sigma
Aldrich. All these polymers were polydisperse (≈ 2,
see Tab. 1) and two of them were fractionated by ul-
trafiltration through a 5 KDalton membrane. PAA1F
and NaPVSF samples, with a polydispersity close to
1.3, were obtained from the fractionation of the sam-
ples PAA1 (acidic form) and NaPVS, respectively. Potas-
sium poly(vinyl sulfate) (KPVS: X = SO−4 , K+) was pur-
chased from Sigma Aldrich with a molecular weight close
to 3 × 105 g/mole. Sodium Dextran sulfate (NaDS) was
supplied by Pharmacia, with a nominal molecular weight
of 5 × 105 g/mole. This polyelectrolyte, with a polysac-
charidic backbone, contained 17% of sulfur corresponding
to 2.3 sulfate groups per glucosyle group.
+ χele(fm + f)(1 − f0 − f − fm − fd)ϕ2.
(5)
ϕNa, ϕCl, ϕM and ϕ are the volume fractions of free
ions and charged monomers, respectively. The parame-
ter fm represents the monomer fraction chemically as-
sociated with one divalent counterion (monocomplex-
ation A−M++), fd is the monomer fraction with a
dicomplex (AM1/2), f is the monomer fraction carrying
an electrostatically condensed counterion M++ and f0 is
the monomer fraction carrying an electrostatically con-
densed sodium ion Na+. The first five terms correspond
to polymer, solvent and free ion entropies, respectively.
The parameter χd is related to the interactions between
the dicomplexes (AM1/2) and water molecules. To obtain
expression (5) water is assumed to be a good solvent
(χ = 0) for charged entities: monomers and monocom-
plexes. The last term corresponds to the electrostatic at-
traction between bare monomers (A−) and those carry-
ing a divalent condensed counterion or a monocomplex
(A−M++); the parameter χele is equal to V12/σ3 < 0 (see
Eq. (4)).
The number of entities in this mixture is high but, the
electroneutrality condition
The polyelectrolytes used were characterized by aque-
ous gel permeation chromatography and light scattering
(see Tab. 1). The polyelectrolyte and salt solutions were
prepared using de-ionized water which had been passed
through a MilliQ Millipore system of pore size 0.22 µm (re-
sistivity of 18 MΩ cm). For the different mixtures with the
polyelectrolytes, the electrolytes used were: CaCl2, MgCl2,
BaCl2, CdCl2, MnCl2, CoCl2, NiCl2, CuCl2, LaCl3,
and trivalent spermidine (+H3N(CH2)4NH2+(CH2)3NH3+,
3Cl−) which is totally charged at pH 7 (pK ≈ 10). The
pH of the solutions was adjusted in such way that the
concentration of the hydrolyzed multivalent ions was neg-
ligible [27].
2ϕM + ϕNa = ϕCl + ϕ(1 − f0 − 2f − 2fm − fd),
and the knowledge of the fractions fi reduce the number
of independent variables needed to describe the system
to 3. The spinodal equation is found by considering [12]:
(6)
ꢀꢀ
ꢀꢀ
ꢀꢀ
ꢀꢀ
ꢀꢀ
ꢀꢀ
ꢀꢀ
ꢀꢀ
∂2F
det
= 0.
(7)
∂ϕi∂ϕj
The determinant (3 × 3 terms) can be written as:
1
1
(1 − f0 − 2f − 2fm − fd)2
4ϕM + ϕNa + ϕCl
+
+
+ 2χele
Nϕ 1 − ϕ
The cloud points, which correspond to the beginning
of the demixion, were determined by visual inspection
× (fm + f)(1 − f0 − f − fm − fd) − 2χdfd = 0. (8)

78
The European Physical Journal E
Table 1. Characteristics of the polymers.
Steric exclusion chromatography
Light scattering
Sample
PAA1
Exp. Cond.
0.1 M, PEG
0.1 M, PEG
0.5 M, SALS
–
Mw/Mn MW (g/mole)
Solvent
HCl 0.1 M
HCl 0.1 M
Dioxane
MW (g/mole)
2.8 × 103
5.1 × 103
6 × 105
≈ 2
1.4
2
–
PAA1F
PAA2
–
5.8 × 105
NaPVS
NaPVSF
KPVS
–
–
NaCl 1 M
NaCl 1 M
NaCl 1 M
NaCl 0.2 M
2.5 × 103
5.1 × 103
4.1 × 105
1.3 × 106
0.1 M, PEG
0.1 M, SALS
0.1 M, SALS
≈ 1.3
1.6
1.6
–
4.3 × 105
NaDS
1.0 × 106
In Steric Exclusion Chromatography measurements the eluent was aqueous solution of LiNO3 except for the NaDS where LiNO3
was replaced by NaNO3. In the experimental conditions (Exp. Cond.) is given the salinity of the eluent and the manner to
determine the characteristics of the samples: PEG the columns were calibrated with polyethylene glycol standards and SALS
a small angle light scattering detector was on line. Light scattering measurements where performed with a goniometer and the
solvent used was water in presence of an electrolyte or dioxane.
of the solutions (at room temperature). In the 4 Phase diagrams
NaPA2/CaCl2 mixtures, the cloud point was also deter-
mined from the abrupt increase of the absorption, at
400 nm, as a function of the concentration of calcium chlo-
ride added. Phase separations were always reversible and
the demixion limits were estimated to be determined with
a precision of the order of 10%.
4.1 Phase diagrams of polyelectrolytes in presence
of multivalent counterions without chemical
association: general behavior
To explain the phase diagrams of sodium poly(styrene
sulfonate) NaPSS in the presence of z-valent counteri-
ons [8] the electrostatic model, presented in the prece-
dent section (see Eq. (3)), was used [9]. With the elec-
trostatic model, the nature of polymer-backbone, of the
charged groups and of the counterions is not taken into
account. In order to find when this approach is valid dif-
ferent polyelectrolytes/counterions mixtures were investi-
gated and their phase diagrams are reported in this sec-
tion. At first, NaPSS was replaced by sodium poly(vinyl
sulfonate) (NaPVS), which possesses the same charged
group and practically the same charge density as NaPSS
(b/lB = 0.36). The NaPVS remains soluble in presence
of divalent counterions (Ca, Mg, Co, Cu, Ni, Mn, Cd) ex-
cept with barium. As in the case of NaPSS [4,6], only bar-
ium induces a precipitation. This point will be discussed
later in Section 4.2.2. Despite the fact that NaPVS and
NaPSS backbones are different, their behaviors in pres-
ence of LaCl3 (pH = 3) remain very similar (Fig. 1). The
demixion therefore seems to be mainly controlled by elec-
trostatic attractions and not by the nature of the back-
bone.
The compositions of the precipitate and of the super-
natant of few biphasique NaPA2/CaCl2 mixtures were de-
termined after centrifugation. The monomer concentra-
tions in the two phases were obtained by Total Organic
Carbon measurements (Dohrmann DC 80 or DC180).
Concentrations of calcium, lanthanum and sodium ions
were measured by atomic absorption spectroscopy.
The activity of the calcium ions in presence of sodium
poly(acrylate) was measured with an ion-sensitive elec-
trode (Orion 93-20). In first approximation, this gave a
measure of the concentration of the free calcium ions in so-
lution. Conductivity measurements were performed with
a conductometer Taccussel CD 810, that had a large sen-
sitivity ranging from 10−9 to 1 S/cm, and was used to
determine the association constants.
To determine the absorbance of metal ions, a spec-
trophotometer UV/VIS Perkin Elmer Lambda 900 was
used. The measurements, over a spectral domain ranging
from 200 to 800 nm, were performed using 1 cm quartz cell
and a reference cell containing pure water. This technique
was well suited to quantify the association of carboxylate
groups with cobalt ions but was inappropriate for calcium
ions.
In order to see if the nature of charged groups is a
pertinent parameter, the sulfonate groups (SO−3 ) of the
Extended X-ray Absorption Fine Structure spec- NaPVS were replaced by sulfate groups (OSO−3 ) by us-
troscopy (EXAFS) experiments were run on D21 spec- ing the KPVS. The KPVS was studied with the same
trometer at LURE (Orsay) with a fluorescence set-up. multivalent counterions and the same behavior as the
EXAFS provides information on the local environment two previous polyelectrolytes were found. The nature of
of a metal [28]. It consisted in scanning the incident charged group (sulfate and sulfonate) does not seem to
X-ray radiation energies in order to observe the oscillating have any incidence on the phase diagram. We also ex-
absorption beyond a photoionization edge of the metal. amined the solubility of sodium dextran sulfate (NaDS),
The present experimental set up was well adapted to per- which has practically the same charge density [29], but
form EXAFS measurements near the K edge of the cobalt a more hydrophilic and less flexible backbone than the
(7710 eV).
previous polyelectrolytes. The NaDS was also stable

I. Sabbagh and M. Delsanti: Solubility of highly charged polyelectrolytes with multivalent counterions
79
Fig. 2. Experimental NaPA2/CaCl2 phase diagram: cloud
points (full squares); composition of supernatant (full trian-
gles) and of precipitates (open circles). Only few initial com-
positions of solutions before the demixion (open lozenges) with
the corresponding tielines (dash lines) are reported. The full
line is only a guide for eyes to indicate the boundary of the
two-phases region “2ϕ”. No resolubilization was observed in
excess of calcium chloride.
Fig. 1. Experimental phase diagrams of different anionic poly-
electrolyte/trivalent cation systems: NaPSS/LaCl3 (full trian-
gles); NaPVSF/LaCl3 (full lozenges); KPVS/LaCl3 (full cir-
cles); NaDS/LaCl3 (full squares); NaPA2/spermidine (open
squares); KPVS/spermidine (open triangles). The finite two-
phases region “2ϕ” is surrounded by a one-phase region “1ϕ” at
high and low salt concentration. The full lines are only guides
for the eyes.
with all divalent counterions except with barium ion in
agreement with previous observations [3]. With LaCl3, at
pH 3, the solubility of NaDS is similar to all other poly-
electrolytes. However, the upper limit of the two-phases
region occurs at slightly higher salt concentration (Fig. 1).
Therefore, for sulfate or sulfonate groups, the nature of
backbone of the polymer has little effect on the phase
diagram. To see the generality of this behavior, the lan-
thanum in the KPVS/LaCl3 mixture was replaced by the
spermidine which is a linear organic cationic molecule fully
charged (+3) at pH 7. Although this molecule is very dif-
ferent from lanthanum, the result is largely unchanged
(Fig. 1). The lower concentration limit is nearly the same
for the two cations though the resolubilization occurs at
a higher spermidine concentration. Therefore, the electro-
static interactions can be considered, in first approxima-
tion, as the principal driving force for phase separation of
highly charged strong anionic polyelectrolytes in presence
of multivalent counterions, where no chemical associations
occurs.
A similar phase behavior was observed in the case
of a weak anionic polyelectrolyte, partially charged
(75%) NaPA2, in presence of spermidine (Fig. 1). In-
deed, conductivity measurements on mixtures of ac-
etate/methylamine and of vinylsulfonate/methylamine
show that there is no chemical association between the
primary ammonium (NH+3 ) and the carboxylate (or sul-
fonate) groups [30]. A general phase diagram seems to oc-
cur when only electrostatic forces are prevailing between
charged groups and counterions. At this level, it is inter-
esting to note that such phase diagrams were also observed
for DNA/polyamine mixtures [11,13].
ones, a lower phase separation line that remained at the
same position: [La+++]/[COO−] = 0.24 ± 0.1. The only
but major change was the absence of resolubilization at
high salt concentration. This difference comes from the
chemical association between the lanthanum and the car-
boxylate groups (see Tab. 2) which prevents the resolubi-
lization. Apart the resolubilization phenomenon in excess
of salt (case of trivalent counterions), the main difference
between NaPA and all strong polyelectrolytes, bearing
sulfonate or sulfate groups, appears with inorganic diva-
lent counterions. With all the divalent counterions studied
a phase separation without resolubilization was observed
(until the solubility limit of salt added). In order to un-
derstand the effect of chemical association on polyacry-
late solubility, the phase diagram of fully charged NaPA2
in presence of CaCl2 was carefully determined. For salt
concentrations above the concentration where the cloud
point appears, the solutions separate into a precipitate,
with a high polymer concentration, and a supernatant
with a very low polymer concentration (Fig. 2). When
the carboxylate concentration is higher than 0.003 M,
the salt concentration limit of the two-phases regime is
an increasing function of monomer concentration. More-
over this limit is nearly independent on the tempera-
ture (L-type). Above 0.005 M, the limit is quasi-linear:
[Ca++]/[COO−] = 0.36 ± 0.02.
We can notice that the demixion lines obtained in
the different systems, NaPVS/LaCl3, NaPA2/LaCl3 and
NaPA2/CaCl2, have a common law: [Counterions] ≈
[Charged Groups] × (0.7/z) where z is the counterion
valency. When [COO−] < 0.002 M, the salt concentra-
tion at demixion increases when the carboxylate con-
centration decreases and is more dependent on temper-
ature. Similar phase diagrams were observed in alginate
and acrylamide/acrylate copolymers [7]. Figure 2 shows
that the salt and monomer concentrations in the su-
pernatant are close to those determined from the ap-
pearance of cloud points. Within experimental error we
4.2 Phase diagrams of polyelectrolytes in presence
of multivalent counterions with chemical association
4.2.1 Sodium polyacrylate/multivalent counterions systems
The phase diagram of the NaPA2/LaCl3 (pH = 7) sys- found that the composition of the precipitate was nearly
tem was established and presented, similar to the previous independent of the initial polyacrylate concentration.

80
The European Physical Journal E
Table 2. Association constants with COO− groups.
˚
Ionic radius (A)
Ions
Mg++
Ca++
Ba++
Mn++
Co++
Ni++
Cu++
Cd++
La+++
log K1
0.55
0.57
0.45
0.70
0.70
0.84
1.85
1.35
1.82
log K2
− log k
11.4
–
≈ 0.30(*)
–
0.66
0.99
1.34
0.80
0.72
0.69
0.72
0.97
1.02
12.7
13.4
–
10.6
0.50
9.7
0.60
9.9
1.20
7.7
0.85
10.1
1.20
K3 = 100.7
8.5
(LaOH++
)
K1 and K2 are the formation constants of monocomplexes and dicomplexes, respectively. For divalent ions M++ the constants
are defined as follow: [M++COO−] = K1[M++][COO−] and [M(COO)2] = K2[M++COO−][COO−]. These constants are
estimated from a compilation of the stability constants of carboxylic acids with inorganic cations given in the literature [27,38]
at ionic strength of 0.1 M (or larger). (*) Constant determined from conductivity measurements [30]. k is the constant formation
of hydrolyzed ions MOH+ at zero ionic strength [27]: [MOH+][H+] = k[M++].
4.2.2 Phase diagram of NaPVSF/BaCl2 system
Now, let us consider the particular case of the barium
ion which is able to precipitate all polyelectrolytes (car-
rying carboxylate sulfate or sulfonate groups). As shown
in Figure 3, the phase diagram of the NaPVSF/BaCl2 is
similar to that of NaPA2/CaCl2. Furthermore no resolubi-
lization was observed in both systems. The precipitation of
poly(vinyl sulfonate) is probably due to a chemical associ-
ation between barium ions and sulfonate groups. It is well
known that the barium-vinyl sulfonate monomer ((C2H3–
Fig. 3. Experimental phase diagrams obtained for different
anionic polyelectrolyte/divalent cation systems: NaPA1/CaCl2
(open squares); NaPA1F/CaCl2 (open circles); NaPA2/CaCl2
(open lozenges); NaPA3/CaCl2 (open triangles); NaPA2 par-
tially charged (75%)/CoCl2 (full circles); NaPVSF/BaCl2 (full
squares). The lines joining cloud points are only guides for eyes.
SO3)2Ba) is poorly soluble (solubility product close to
5 × 10−4 M2) which means that the chemical binding of
barium with vinyl sulfonate monomers is strong. All di-
valent cations other than barium (M++ = Ba++) do not
precipitate poly(vinyl sulfonate), that is probably due to
the high solubility of their sulfonate monomers ((C2H3–
SO3)2M). In conclusion the demixion of polyelectrolytes
in presence of inorganic divalent counterions is due to
the chemical bonding between the counterions and the
In the precipitate, the concentration of polymer and cal-
charged groups of the polyelectrolyte.
cium are close to 4 M and 2 M, and a weak concentra-
tion of sodium ([Na+] ≈ [Ca++]/30) was detected. More
precisely we have [COO−]p/[Ca++]p = 2.1 ± 0.2, which is
5 Complexation between polyelectrolytes
and multivalent counterions
close to the stoichiometric composition (equivalent to nine
water molecules per carboxylate group).
In order to observe the influence of molecular weight Our aim was to determine the quantity and the type of
on phase separation boundaries, we determined the phase complexes formed before phase separation, especially for
diagrams of NaPA1, NaPA2, NaPA3 and NaPAF samples the system PANa/Ca. Unfortunately, the chemical asso-
in presence of CaCl2 (Fig. 3). At low concentration, pre- ciation of calcium with carboxylate groups is not well de-
cipitation appears first for the largest molecular weight. tected by spectroscopic methods [30,31]. Therefore we re-
At high polymer concentration, in the L-type regime, the placed the calcium ion by the cobalt ion for this study.
demixion lines become nearly independent of molecular Cobalt was chosen because calcium and cobalt have prac-
weight. In this domain, a study on NaPA2 with different tically the same ability to precipitate the polyacrylate
divalent counterions (Ba++, Ca++, Cd++, Co++, Cu++
,
(Fig. 3) and their complexation constants with COO−
Mg++, Mn++, Ni++) showed that their efficiencies to groups are close (see Tab. 2). Moreover, formation of com-
precipitate the sodium poly(acrylate) are practically the plexes between cobalt ion and COO− groups can be well
same.
characterized by UV/VIS spectroscopy.

I. Sabbagh and M. Delsanti: Solubility of highly charged polyelectrolytes with multivalent counterions
81
Am(λ) and Ad(λ) allows us to determine the cobalt frac-
tions in the monocomplex and dicomplex states. For in-
stance, for 0.0125 M of cobalt in presence of 0.04 M of
polyacrylate, the molar absorbance of the cobalt A(λ) is
very close to Ad(λ) (Fig. 4) and we find that the cobalt
fractions of the complexes are xd = 0.85 ± 0.05 and
xm = 0.15 ± 0.05. Hence, a weak polyacrylate concentra-
tion is equivalent to a high acetate concentration. This re-
sult shows that cobalt ions mostly form dicomplexes with
polyacrylate.
The wavelength shift, in the case of polyacrylate, can
be induced by a modification of the cobalt symmetry, due
to the high electrostatic potential induced by the poly-
electrolyte, or by the variation of cobalt coordination. We
checked these two possibilities by UV/VIS spectroscopy
on NaPVS/CoCl2 mixture and by EXAFS experiments on
NaPA1/CoCl2 mixture. In high concentration of NaPVS
and without NaPVS the cobalt absorbance amplitude is
the same, meaning that the electrostatic potential has no
influence on the electronic structure of the cobalt. This
result also proves the absence of complexation between
cobalt and sulfonate groups, which is consistent with the
absence of precipitation in the NaPVS/CoCl2 mixture.
EXAFS measurements were realized on a pure cobalt chlo-
ride solution (0.015 M) without added polymer and in
presence of 75% charged NaPA2 (0.08 M). Without poly-
mer, the distance between cobalt and oxygen was found
Fig. 4. Molar absorption spectra of cobalt ions and of
cobalt complexes (full lines). From the bottom to the top:
cobalt ions (A0(λ)); monocomplexes (Am(λ)); dicomplexes
(Ad(λ)). For comparison the experimental absorption spec-
trum of cobalt (0.0125 M) in presence of sodium polyacrylate
NaPA1 ([COO−] = 0.04 M) is reported (squares).
5.1 Monocomplexation and dicomplexation (UV/VIS
spectroscopy)
In order to determine the effect of complexation of the
cobalt ion with acrylate group on the absorption spec-
trum, we first determined the variation of the absorbance
of sodium acetate/cobalt chloride mixtures in the wave-
length range 400–650 nm. The cobalt concentration was
kept constant (0.002 M) while the acetate concentration
was varied between 0.01 and 3 M. The absorbance spec-
trum of cobalt, without sodium acetate shows a maximum
at a wavelength of 512 nm whereas the sodium acetate
(without cobalt) has no detectable absorbance. When the
acetate concentration increases the amplitude of the ab-
sorbance increases and the position of the maximum shifts
up to 520 nm.
˚
˚
to be 2.07 A, in agreement with the value 2.09 A reported
in reference [32]. The presence of NaPA does not change
the coordination number of cobalt, which remains hexa-
coordinated [33], but creates a very weak distortion: four
˚
˚
oxygen atoms at 1.96 A and two others at 2.10 A. The sec-
ond neighborhood of the cobalt should be water-oxygen
or carboxylate-carbon. But these elements have similar
electronic contrasts. Therefore, the second neighborhood
cannot be extracted quantitatively. The Fourier transform
of the signal, which expresses the electronic density as a
For a given wavelength λ the molar absorbance, A(λ),
can be written as follows:
˚
A(λ) = xmAm(λ) + xdAd(λ) + (1 − xm − xd)A0(λ)
function of the distance, presents a weak peak at 6 A from
(10)
the cobalt center, which is probably due to carbon. The
weak amplitude and the large thickness of the peak do
not allow quantitative analysis. Thus the wavelength shift
in UV/VIS is not due to the variation of cobalt coordina-
tion. As it appears for highly concentrated sodium acetate
solutions, the change of cobalt absorbance in presence of
polyacrylate is only due to chemical association between
cobalt ions and carboxylate groups.
where xm (= [COO−Co++]/[Co++
]initial) and xd (=
[Co(COO)2]/[Co++
]
initial) represent the fraction of mono-
complex and dicomplex, respectively. The quantities
Am(λ), Ad(λ) and A0(λ) are the molar absorbances of
monocomplex, dicomplex and non-associated cobalt, re-
spectively. As the fractions xm and xd can be calculated
using the complexation constants (K1 and K2) given in
Table 2, from the spectra obtained on seven mixtures it
was possible to determine the molar absorbance of the
cobalt in the monocomplex and dicomplex state. It is im-
portant to notice that for high acetate concentrations a
precise value of the second association constant is not
needed. Indeed, when log K2 = 0.5 is replaced by 0.3, the
dicomplex fraction only varies by 10%. We fitted the mea-
sured absorbances A(λ) to expression (10) where Am(λ)
and Ad(λ) were two adjustable parameters. The molar ab-
sorbances Am(λ) and Ad(λ) found are given in Figure 4.
5.2 Complexation and phase diagram
The fraction of dicomplexes near the phase separation
was determined in two different ways. First the ratio
[Co++]/[COO−] was fixed and secondly the [Co++] was
increased up to the phase separation at constant [COO−].
To begin the ratio [COO−]/[Co++] was fixed to
4 and the polyacrylate NaPA1 (charged at 95%) was
used. As the NaPA1 concentration increases the frac-
tions of cobalt ions in dicomplex and monocomplex forms
remain constant: [Co++
]
dicomplex/[Co++
]
≈ 80%
Now, from a given experimental spectrum, obtained
added
with a polyacrylate sample, the knowledge of A0(λ), and [Co++
]
monocomplex/[Co++
]
≈ 20%. This result
added

82
The European Physical Journal E
Fig. 5. Variation of fractions of acrylate groups in mono-
complex state (circles) and dicomplex state (triangles) at
the approach of the phase separation. Cloud point occurs
at [Co++]/[COO−] = 0.36. The carboxylate concentration of
NaPA2 sample (charged at 75%) was fixed to 0.0075 M. The
squares represent the total fraction of acrylate groups chemi-
cally associated with cobalt. It appears that all cobalt counte-
rions are associated with carboxylate groups.
Fig. 6. Variation of the viscosity of NaPA2 solutions in pres-
ence of CoCl2 and CaCl2 salts at the approach of the phase sep-
aration. The normalized viscosity of the mixture (polymer plus
salt) is equal to (ηpolymer+salt −ηwater)/(ηpolymer −ηwater ) where
ηpolymer is the viscosity of NaPA2 solution without salt. The
full squares and full triangles correspond to the NaPA2 par-
tially charged (75%), at a monomer concentration of 0.225 M,
in presence of CoCl2 and CaCl2, respectively. The open trian-
gles and open circles correspond to the NaPA2 fully charged
(100%) in presence of CaCl2 at a monomer concentrations of
0.28 and 0.028 M, respectively.
is in contradiction with a classical mass action law where
the cobalt concentration is calculated using the total vol-
ume of the solution. Even if the complexation constants
are large, we never expect that the fractions are constant.
The only way to explain this result is to suppose that
the relevant concentration is the local counterion concen-
tration in the immediate vicinity of the polyelectrolyte,
which should remain constant whatever is the polymer
concentration. Thus, the dicomplexation depends on lo-
cal charge density of polyelectrolyte and probably occurs
between two neighboring monomers.
To study the complexation when the demixion limit
(L-type) is approached, the concentration of NaPA2
(charged at 75%) was fixed to 0.01 M (0.0075 M of
COO−). The cobalt absorption spectrum was observed as
a function of the CoCl2 concentration up to phase sepa-
ration. We found that, all the cobalt ions were chemically
bound to carboxylate groups. In Figure 5 the fractions of
acrylate groups in monocomplex and dicomplex states are
reported. The fraction of cobalt in the dicomplex state is
close to 85% and increases by around 10% near the demix-
ion. This weak increase near phase separation should be
associated to the polymer shrinking, which increases the
number of contact between distant monomers [34] and the
counterion condensation [35].
In order to have information on the nature of di-
complexes, viscosity evolution studies as a function of
divalent counterion concentration, up to phase separa-
tion, were performed. Addition of divalent salt can in-
crease the viscosity if ionic cross-linking occurs (gel forma-
tion). With the NaPA2/CoCl2 system (charged at 75%)
at a polymer concentration of 2 wt% in the semi-dilute
regime ([COO−] = 0.21 M), we found that the viscosity
strongly decreases as the cobalt concentration increases
(Fig. 6). Cobalt ions provoke the collapse of the chains.
Dicomplexation essentially occurs between two neighbor-
ing monomers of the chain. Few bridges may be formed
but not enough to make a gel.
Fig. 7. Fractions of calcium associated (chemically or elec-
trostatically) with NaPA2, near the demixion, measured with
a specific calcium electrode for three concentrations of car-
boxylate: 0.0028 M (open squares); 0.0053 M (open triangles);
0.028 M (dots). All the calcium ions are associated with the
polyelectrolyte before phase separation.
(charged at 100%). The association constants of calcium
and cobalt with carboxylate groups are of the same or-
der of magnitude (see Tab. 2), so we can expect that all
calcium counterions are chemically associated with poly-
acrylate before phase separation. Moreover, specific elec-
trode measurements show that, before the demixion, all
the calcium ions are in the immediate vicinity of the poly-
electrolyte as cobalt counterions (Fig. 7). Indeed, Figure 6,
shows that calcium and cobalt ions reduce the viscosity in
the same way.
With highly charged polyelectrolytes, gel formation
can be induced with trivalent cations but not with diva-
lent cations. With trivalent cations, first the dicomplexa-
tion occurs between neighboring monomers and finally the
tricomplexation happens with a monomer which belongs
to another chain or a monomer of the same chain which is
The conclusions drawn for the system NaPA2/CoCl2 far from the two first complexed monomers. In semi-dilute
can reasonably be extended to NaPA2/CaCl2 system solution, bridges can be created if the third complexation

I. Sabbagh and M. Delsanti: Solubility of highly charged polyelectrolytes with multivalent counterions
83
Fig. 9. Concentration Cs∗ of the divalent salt M++Cl−2 at
the demixion limit for solutions of NaPA2 (partially charged
(75%), [COO−] = 0.01 M) in presence of different divalent
counterions without added NaCl (open bar chart) and with
0.2 M of NaCl (hatched bar chart). Without added NaCl, the
divalent salt concentrations Cs∗ are nearly the same (0.0035 M)
whatever is the nature of divalent cation.
Fig. 8. Variations of the fractions of cobalt in complex forms,
determined by UV/VIS spectroscopy, as a function of the
concentration of added NaCl: dicomplexes (triangles); mono-
complexes (squares). The thin solid line represents the total
fraction of cobalt chemically associated. The carboxylate con-
centration of NaPA2 (charged at 75%) was 0.028 M and the
cobalt concentration was 0.0065 M.
constant is high enough which is the case for Fe+++ (gel)
but not for La+++ (no gel).
precipitate the NaPA:
Mg++ < Ca++ < Ba++ < Mn++ < Co++ < Ni++
5.3 Affinity of divalent counterions to polyacrylate:
mixtures NaPA2/divalent counterions/NaCl
< Cd++ ꢀ Cu++
.
This classification is similar to the one established from
potentiometric measurements [36,37]. We can also classify
counterions as a function of their association constants
(see Tab. 2):
Let us first consider the effect of sodium chloride on the
NaPA2 (charged at 75%)/CaCl2 phase diagram. When the
concentration of NaCl increases, the cloud point appears
at higher CaCl2 concentrations. For instance, for a solu-
tion having a monomer concentration of 0.01 M, the ad-
dition of 0.01 M of NaCl changes the position of the cloud
point by only 5%, which is the order of magnitude of ex-
Ba++ < Mg++ < Ca++ < Mn++ = Co++ < Ni++
< Cd++ ꢀ Cu++
.
perimental uncertainty. But when NaCl concentration is Except for the barium, the two classifications are identi-
close to 0.2 M, which is much higher than divalent concen- cal. Thus, the demixion limit is directly correlated with
tration, the displacement of the cloud point is not negligi- the fraction of counterions associated with COO− groups,
ble (190%). The same behavior was observed with cobalt which depends on the association constant between diva-
ion, where it was possible to measure the variation of com- lent counterions and COO− groups. The barium counte-
plexation fraction (UV/VIS spectroscopy) as a function of rion has the larger ionic radius (see Tab. 2) which could fa-
added NaCl concentration. Figure 8 shows that the cobalt vor the dicomplexation between two neighboring charged
dicomplex fraction decreases when the NaCl concentration monomers. In contrast with the other mixtures, the solu-
increases. Thus we conclude that the solubility of sodium bility of NaPA2/CuCl2 decreases when sodium chloride is
polyacrylate is intimately related to the dicomplexation. added. Because of its high association constant, the frac-
The cobalt dicomplex fraction depends on the competi- tion of copper associated with COO− groups is insensitive
tion between Na+ and Co++. So we can expect that for to sodium counterions, but NaCl salt screens the repulsion
divalent counterions having a larger affinity to carboxylate between the negatively charged monomers which favors
groups, the ability of sodium counterion to displace them the precipitation of the polyelectrolyte.
should be reduced. Let us compare the effect of sodium
chloride on a 0.01 M polyacrylate solution in presence of
different divalent counterions. Without monovalent added 5.4 Criterion for the complexation of divalent
salt, the divalent counterion concentration at the demix- counterions with anionic polyelectrolytes
ion is nearly the same (0.0035 ± 0.0003 M) whatever is
the nature of divalent cation (Fig. 9). Except for copper, The stability of polyelectrolyte solutions in presence of
the concentration of divalent counterion needed to precip- multivalent counterions depends on the affinity between
itate the polyacrylate in presence of 0.2 M NaCl increases their functional groups and counterions. Two extreme
and the amplitude of the variation strongly depends on cases can be encountered: in the first one the electro-
the nature of divalent counterion. Then it is possible to static attraction is the principal driving force for phase
classify divalent counterions, according to their power to separation whereas in the second one the precipitation

84
The European Physical Journal E
plexes are computed using mass action laws
[MA+]L = K1([A−]L − [MA+]L − 2[M(A)2]L)
× ([M++]L − [MA+]L − [M(A)2]L),
[M(A)2]L = K2[MA+]L([A−]L − [MA+]L − 2[M(A)2]L).
(12)
Using the complexation constants, K1 and K2, of the
cobalt with carboxylate groups, given in Table 2, we ad-
just the parameter R to fit the complex fractions of cobalt
in dicomplex state (85%) and monocomplex state (15%)
Fig. 10. Fractions of divalent counterions M++ in com-
plex forms as a function of the first association constant K1.
Assuming that the first and second constants are related,
log K1/ log K1K2 = 0.6 (see text and Tab. 2), fractions of deduced from UV/VIS spectroscopy measurements. We
˚
monocomplexes (short dash) and of dicomplexes (long dash)
find that the best value is R ≈ 7 A when b is chosen to
are computed using the tube model with [M++]/[A−] = 0.36.
When no dicomplexation occurs (K2 = 0), the fraction of
monocomplexes are computed in two manners: with the charge
regulation model (full line) and the tube model (dash dot line).
˚
2.5 A.
Now, whatever the values of the complex constants,
the fractions of divalent counterions in complex states can
˚
be computed using expressions (11, 12) with R ≈ 7 A.
For instance, in Figure 10 the fraction of divalent counte-
rions in monocomplex state is reported when the second
constant K2 is fixed to zero. We can see that the fraction
determined with the tube method has the same behavior
as the one obtained using charge regulation model.
is induced by complexation of the polyelectrolyte with the
multivalent ions. Now, let us try to find a criterion which
allows us to distinguish between these two limiting cases
from the simple knowledge of association constants be-
tween the divalent counterions and the functional groups
of the polymer.
Empirically, we noticed that for a given functional
group, the ratio log K1/ log K1K2 is approximately a con-
stant [27,38], this constant is of the order of 0.6 for acry-
late groups. Thus fractions of counterions in monocomplex
and dicomplex states as function of K1 can be computed
(Fig. 10). When the first association constant is small
(K1 < 10−2), divalent counterions are not chemically asso-
ciated and are basically electrostatically condensed (which
is certainly the case for most inorganic counterions with
sulfonate groups). In those conditions, no phase separa-
tion is expected for divalent counterions. When K1 > 1, it
is more than 65% of counterions which form dicomplexes.
This condition is fulfilled for all divalent inorganic coun-
terions with carboxylate groups (see Tab. 2).
Assuming that there is no dicomplexation the fraction
of divalent counterions in monocomplex state can be cal-
culated using charge regulation model. The polyelectrolyte
˚
is assumed to be a cylinder of diameter 5 A with a mean
˚
distance between charges b = 2.5 A. In Figure 10 the frac-
tion of counterions in monocomplex form as a function
of the value of monocomplexation constant is reported.
Figure 10 shows that the complexation fraction sharply
increases above K1 = 10−2
.
Unfortunately, the introduction of two complexation
constants is not possible in the charge regulation model
presented above. In order to take into account the dicom-
plexation, which is the driving force for phase separation,
we use another procedure. The added divalent counterions
are assumed to be confined in a cylindrical tube of radius
R around the polyelectrolyte having a thread shape. Of
course this hypothesis is only meaningful when no free di-
valent counterions are present in the solution (which is
experimentally the case). The local monomer and counte-
rion concentrations depend on both the distance between
charged monomers and the tube radius: one monomer is
confined in a volume of πbR2. The local charged monomer,
[A−]L, and divalent counterion, [M++]L, concentrations
(in moles per unit volume) are
6 Phase diagrams of NaPA/CaCl2 systems:
mean field model
First, let us try to describe the NaPA2/CaCl2 phase di-
agram by using expressions (4, 10). It is necessary to
know the parameters: N, σ3, f0, f, fm, fd, B, χd. As the
NaPA2/CaCl2 and NaPA2/CoCl2 systems are very simi-
lar, when the concentration of divalent salt (CS) is lower
than 0.4 times the monomer concentration (C), we assume
that all added calcium ions are chemically bound to the
polymer with a fraction of calcium in the dicomplex state
of 90% (Figs. 10 and 12). In this domain of concentration,
the three monomer fractions f, fm and fd are estimated as
follow: f = 0, fm = 0.1CS/C, and fd = 2(0.9CS/C). The
fraction of monomer f0 carrying a condensed sodium is
computed following expression (1): f0 = f00 −2CS/C when
CS < f00 C/2 and f0 = 0 when CS > f00 C/2, where the ini-
tial sodium fraction, f00 = 1−b/lB is fixed to 0.65 (sodium
[A−]L = 1/(6 × 1023πbR2) and [M++
= [M++]([A−]L/[A−])
]
L
(11)
where [M++] and [A−] are the macroscopic concentra- ions are stoichiometrically replaced by calcium ions). The
tions. The concentration of monocomplexes and dicom- values of B and σ are chosen to be equal to 0.2 and to 5 A
˚

I. Sabbagh and M. Delsanti: Solubility of highly charged polyelectrolytes with multivalent counterions
85
Fig. 12. Theoretical phase diagrams of mixtures sodium
polyacrylate/CaCl2. Lines are the spinodal curves computed
for different degrees of polymerization N: from the top to the
bottom N is equal to 38, 8 × 103, and 4 × 105.
Fig. 11. Variation of interaction terms of the spinodal equa-
tion near the phase separation as a function of the added cal-
cium (the polymer concentration is fixed to 0.01 M): repulsive
electrostatic term (thin full line); attractive term due to the
dicomplexation in absolute value (thick full line); attractive
electrostatic term in absolute value (dash line).
regime. When CS ≥ 0.4C, condensation is stopped (see
Eq. (1)) and the number of monomer that carry cobalt
in complex state increases slightly. In this complicated re-
gion, we have assumed that fm and fd remain constant:
fm = 0.1 × 0.41 = 0.041 and fd = 2 × 0.9 × 0.41 = 0.74.
The position of the minimum of the demixion line depends
on the degree of polymerization and on χd value. The the-
oretical phase diagrams describe the experimental results
semi-quantitatively.
It is important to notice that the association constants
decrease with salt concentration at low ionic strength in
the Debye-Hu¨ckel approximation [43] but at high ionic
strength they tend to a constant [27]. At the same time
the complex fraction always increases with salt concen-
tration. The absence of resolubilization is the signature
of a dominant dicomplexation. In contrast, if the mono-
complexation is dominant precipitation is expected to be
followed by a resolubilization due to charge inversion [10].
(as in Ref. [9]). Using expressions (4, 9), the adjustable pa-
rameter χd is determined from the experimental demixion
limit measured in the system NaPA2/CaCl2 in the con-
centration range 0.005 M < C < 0.06 M, where most of
experimental determinations of demixion were done. At
this monomer concentration range the entropic term is
small and the value of χd deduced is weakly dependent
on the exact value of the effective degree of polymeriza-
tion taken. We find that the results are correctly described
∼
with χd 20 (N = 8000). This large value corresponds to
=
a high attraction between monomers that carry calcium
in dicomplex state. In other words the dicomplexes are
hydrophobic, which indicates that the chemical associa-
tion is followed by a partial dehydration of Ca++ ion and
COO− group [39–42]. It is important to notice that the
value of χd is strongly dependent on the value of σ. To give
an idea about the relative importance of the different in-
teractions controlling the precipitation, we have reported
in Figure 11 the different interaction terms of the spin-
odal equation calculated for C = 0.01 M. By increasing
the salt concentration the hydrophobic term (2σ3χdfd),
due to the dicomplexation, increases and becomes nearly
a constant above CS/C ≈ 3 whereas the repulsive electro-
static term, which stabilizes the polyelectrolyte, decreases
sharply. In first approximation, precipitation occurs when
the hydrophobic term is equal to the repulsive term in
absolute value, which corresponds to he crossing point of
the two upper curves in Figure 11. The crossing point
is weakly dependent on the variation of χd. Therefore, for
two different divalent counterions which are completely as-
sociated with carboxylate groups, similar demixion curves
will be observed if their χd values are of the same order
of magnitude. When complexation is absent, the electro-
static attractions are not strong enough to induce phase
separation.
7 Conclusion
In the case of strong polyelectrolytes (sulfonates, sulfates)
no chemical associations occur and their solubility de-
pends on electrostatic interactions. Other factors such as
the nature of polymer backbone (hydrophobicity, rigid-
ity), the nature of charged groups and the nature of coun-
terions have little effect. The important parameter is the
counterion valency. A finite diphasic zone exists only for
counterions with a valency higher or equal to 3. For weak
polyelectrolytes, like poly(acrylate), complexes are formed
with inorganic counterions. The polyacrylate chemically
reacts with all inorganic multivalent counterions and a
L-type demixion takes place. Before phase separation, the
high electrostatic polyelectrolyte potential leads nearly all
counterions to form complexes with charged monomers.
Chemical association partially dehydrates counterions and
charged groups of the polymers therefore the complexes
In Figure 12 the spinodal curves calculated for the formed are hydrophobic. The demixion limit is related to
three degrees of polymerization of the samples PAA1, the fraction of monomer in dicomplex state, which de-
PAA2, PAA3 are reported. The increasing of the demix- pends on the polyelectrolyte charge density. The com-
ion limit for weak polymer concentrations (Fig. 12) is due plexation increases as the counterion concentration in-
to the entropic term, which becomes dominant in this creases and therefore inhibits polymer resolubilization

86
The European Physical Journal E
at high counterion concentration. For highly charged poly- 18. F. Oosawa, Polyelectrolytes (Marcel Dekker, New York,
1971).
19. A.D. MacGillivray, J.J. Winkleman, J. Chem. Phys. 45,
2184 (1966).
20. A. Katchalsky, Z. Alexandrowicz, O. Keden, in Chemical
Physics of Ionic Solutions, edited by B.E. Conway, R.G.
Barradas (Wiley, New York, 1966), p. 295.
21. L. Belloni, Ph.D. thesis, Paris, France, 1982.
22. L. Belloni, M. Drifford, P. Turq, Chem. Phys. 83, 147
electrolytes in presence of divalent counterions the major-
ity of counterions form dicomplexes between two neigh-
boring monomers on the same chain. No gel formation
is observed before the demixion. The understanding of
these local interactions allows us to describe the phase
diagrams of sodium poly(acrylate)/CaCl2 mixtures semi-
quantitatively taking into account the chemical bonding
between COO− and Ca++ counterions.
(1984).
23. L. Belloni, Coll. Surf. A: Physicochem. Eng. Aspects 140,
We thank L. Belloni, M. Olvera de la Cruz, B. Cabane, F.
Lafuma, and J.P. Jolivet for many valuable discussions. We 24. J.A. Davis, R.O. James, J.O. Leckie, Trans. Faraday Soc.
thank again L. Belloni for providing computer programs to
63, 480 (1978).
calculate the fractions of counterions chemically associated or 25. J.N. Israelachvili, Intermolecular and Surface Forces (Aca-
electrostatically condensed. We acknowledge P. Lixon for the
demic Press, London, 1992).
help in the characterization of the different polyelectrolytes. 26. P.-G. De Gennes, Scaling Concepts in Polymer Physics
We are also grateful to R. Cortes for helping us in EXAFS
(Cornell University, Ithaca, NY, 1979).
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