# Salt-induced reentrant stability of polyion-decorated particles with tunable
surface charge density

Simona Sennato  Laura Carlini  Domenico Truzzolillo  Federico Bordi  CNR-ISC
UOS Roma- c/o Dipartimento di Fisica - Sapienza Università di Roma - P.zzle A.
Moro, 2 - 00185 Roma - Italy  Dipartimento di Fisica - Sapienza Università di
Roma - P.zzle A. Moro, 2 - 00185 Roma - Italy  Laboratoire Charles Coulomb
(L2C), UMR 5221 CNRS-Université de Montpellier, Montpellier, France

###### Abstract

The electrostatic complexation between DOTAP-DOPC unilamellar liposomes and an
oppositely charged polyelectrolyte (NaPA) has been investigated in a wide
range of the liposome surface charge density. We systematically characterized
the ”reentrant condensation” and the charge inversion of polyelectrolyte-
decorated liposomes by means of dynamic light scattering and electrophoresis.
We explored the stability of this model polyelectrolyte/colloid system by
fixing each time the charge of the bare liposomes and by changing two
independent control parameters of the suspensions: the polyelectrolyte/colloid
charge ratio and the ionic strength of the aqueous suspending medium. The
progressive addition of neutral DOPC lipid within the liposome membrane gave
rise to a new intriguing phenomenon: the stability diagram of the suspensions
showed a _novel reentrance_ due to the crossing of the desorption threshold of
the polyelectrolyte. Indeed, at fixed charge density of the bare DOTAP/DOPC
liposomes and for a wide range of polyion concentrations, we showed that the
simple electrolyte addition first (low salt regime) destabilizes the
suspensions because of the enhanced screening of the residual repulsion
between the complexes, and then (high salt regime) determines the onset of a
new stable phase, originated by the absence of polyelectrolyte adsorption on
the particle surfaces. We show that the observed phenomenology can be
rationalized within the Velegol-Thwar model for heterogeneously charged
particles and that the polyelectrolyte desorption fits well the predictions of
the adsorption theory of Winkler and Cherstvy [1]. Our findings unambiguously
support the picture of the reentrant condensation as driven by the correlated
adsorption of the polyelectrolyte chains on the particle surface, providing
interesting insights into possible mechanisms for tailoring complex colloids
via salt-induced effects.

###### keywords:

liposomes , aggregation of polyion-decorated particles , charge patch
attraction , polyion desorption

††journal: ArXiv

##  1 Introduction

The phenomenon of complexation between oppositely charged colloids or
macromolecules in aqueous solution is a fundamental process, widely exploited
by nature, as for example in DNA packaging within biological cells [2], but
also by technology, as in industrial colloidal stabilization, water treatment
and paper making [3].
Being the result of a delicate balance of forces of different nature, small
variations in the physico-chemical parameters may induce large changes in the
resulting complexes [4]. Electrostatics is clearly the relevant interaction in
driving the aggregation, although non-electrostatic interactions can be
important in modulating the process and can deeply affect the characteristics
of the resulting self-assembled structures. However, due to the great
complexity of these systems, also the details of purely electrostatic
interactions remain to be completely clarified, despite the intense
experimental, theoretical and computational research aimed to understand the
mechanisms driving the formation and the stabilization of the complexes [5, 6]
in the physical context of macroion-multivalent counterion interactions [7].
The possibility to get a fine control of the stability of the complexes by
tuning the competition between attractive and repulsive electrostatic
interactions justifies the actual enduring attention on polyion-colloid
complexation, being key to the technological development of nano-structured
materials or nano-devices to be used for example in drug delivery or gene-
therapy [8, 9].
In particular, a rich literature exists on the variety of structures formed by
charged liposomes interacting via electrostatic forces with oppositely charged
linear polyions, whether they are synthetic polymers, polypeptides or DNA [10,
11, 12]. A feature characterizing the phase behavior of these systems is the
presence of a ”reentrant condensation” accompanied by a marked ”overcharging”,
or charge inversion [13, 14]. Indeed, when a given volume of a liposome
suspension is mixed with the same volume of a solution containing oppositely
charged polyelectrolytes, the rapid formation of clusters is systematically
observed. For low enough polyelectrolyte concentration the clusters are small
(they are formed by a few liposomes), and after their rapid formation they are
very stable. The measured $\zeta$-potential of these clusters is only slightly
lower, in absolute value, than that measured for the liposomes in absence of
adsorbed polyelectrolytes. As the polyion concentration (and hence the ratio,
$\xi$, between the number of stoichiometric charges on the polymer and on the
particles) is increased, larger but still stable clusters are formed, with a
$\zeta$-potential which is further reduced in absolute value. At a
sufficiently high polyelectrolyte concentration, the liposome suspension is
completely destabilized, the large clusters that form are not stable, their
$\zeta$-potential is close to zero and they rapidly coalesce in macroscopic
”flocs”. Going beyond the _isoelectric point_ , hence further increasing the
polyelectrolyte concentration, stable clusters form again, but now their size
_decreases_ upon increasing the polyelectrolyte concentration in the mixture:
the polyelectrolyte-induced condensation of the liposomes is hence
”reentrant”. At the same time, the $\zeta$-potential after having passed
through the zero, increases again in absolute value, and the sign of the
excess charge of the clusters is that of the polyelectrolyte, i.e. opposite to
that of the original liposomes (charge inversion).
As the polyelectrolyte concentration is even further increased some ”free”
polyelectrolyte (non-complexed) chains begin to coexist with the overcharged
small clusters or single liposomes. Eventually, at even higher polyelectrolyte
content and due to the strong electrostatic screening, the colloidal
suspension is completely destabilized, as expected from the Derjaguin-Landau-
Verwey-Overbeek (DLVO) theory [15].
It can be shown [16] that both the overcharging and the reentrant condensation
are consequences of the _correlated_ adsorption of polyelectrolyte chains on
the particle surface. Indeed, the electrostatic attraction of the
polyelectrolytes to the oppositely charged particle surface, the repulsion
between the like-charged adsorbed chains and their confinement entropy [17,
16, 10, 18] compete to give rise to adsorption patterns that are characterized
by long range correlations. This correlated adsorption is also a consequence
of the large difference between the diffusivities of the colloidal particles
and the polyelectrolyte chains [19] that reflects in different timescales of
adsorption and aggregation. When the polyelectrolyte and the oppositely
charged colloidal particles are mixed together, the adsorption of the
polyelectrolyte on the particle surface is a much faster process than the
aggregation. The huge surface/volume ratio which characterizes the colloid,
and the distribution of this surface within the whole volume of the host
phase, speed up dramatically the adsorption process, decreasing the time
needed by for the polyelectrolyte chains to reach the adsorbing surface by
diffusion. Conversely, the aggregation, being controlled by the diffusivity of
the bulkier colloidal particles, occurs on longer timescales [20]. As a
result, when the concentration of the polyelectrolyte is not enough to
”saturate” the available surface of the particles, the latter appears
”decorated” by statistically regular patterns rather than uniformly coated by
the polyelectrolytes [14]. This alternation of domains of ”polyelectrolyte-
covered” and ”bare” particle surface results in a non-uniform surface charge
distribution [6]. Such a ”patchy” distribution of charge, together with the
screening effect given by the small counterions, generates a short range
attraction between these ”polyelectrolyte-decorated particles” (hereafter pd-
particles), that has been widely studied in the last two decades [21, 22, 23,
24]. The superposition of this attraction, the ubiquitous van der Waals
interaction and the electrostatic repulsion due to the residual net charge of
the complexes, results in a inter-particle potential characterized by a
barrier that, in proper conditions, yields the formation of stable, finite
size clusters [25], even if all the primary particles bear the same net charge
[18], the adsorbed polyelectrolyte layer representing a sort of electrostatic
glue that sticks together the particles [26]. Noteworthy, this mechanism works
in principle not only for liposomes, but for whatever colloid and
polyelectrolyte, as long as the proper ”patchy” charge distribution can be
established. In fact in the presence of an oppositely charged polyelectrolyte
many colloids, characterized by different structures and chemical
compositions, like bare [27] and lipid-coated latex particles [28], ferric
oxide particles [29], dendrimers [30] and micelles [31], show the ”reentrant
condensation” phenomenon.

In a recent paper [32], we have systematically explored the phase diagram and
the reentrant condensation of cationic DOTAP liposomes in the presence of the
anionic polyelectrolyte sodium polyacrylate (NaPA) and at varying the ionic
strength of the aqueous suspending medium. In this work we focus instead on
the effect on the phase diagram of the liposome surface charge density. To
this aim, to vary the surface charge, we employed liposomes prepared from
mixtures of a cationic lipid, Di-oleoyl-trimethyl-ammonium-propane [DOTAP],
and a zwitterionic phospholipid, Di-oleoyl-phosphatidyl-coline [DOPC], at
different molar fraction of the two components. The reentrant condensation of
these mixed DOTAP-DOPC pd-liposomes was explored by varying the ionic strength
of the solution with the addition of proper amounts of NaCl.
DOTAP-DOPC-NaPA is a good model system to study the complex but general
phenomenology that appears by mixing oppositely charged colloidal particles
and polyelectrolytes. Moreover, due to the good biocompatibility of NaPA [33,
34], the DOTAP-DOPC-NaPA complexes have also a specific interest for their
potential use as multi-compartment vectors for multi-drug delivery [8]. A
deeper understanding of the relation between polyion adsorption, pd-liposome
interactions and stability of the resulting aggregates is key for further
developments of functional colloidal particles with application in
biotechnology and material science.

##  2 Experimental

###  2.1 Materials

The cationic lipid 1,2-dioleoyl-3-trimethylammonium-propane (chloride salt)
[DOTAP] and the zwitterionic phospholipid 1,2-dioleoyl-sn-
glycero-3-phosphocholine [DOPC] were purchased from Avanti Polar Lipids
(Alabaster, AL) and used without further purification. The negatively charged
polyelectrolyte sodium poly-acrylate, $[-CH_{2}CH(CO_{2}Na)-]_{n}$ [NaPA],
with nominal molecular weight 60 kD, was purchased from Polysciences Inc.
(Warrington, PA) as 25 $\%$ aqueous solution. All liposome samples and
polyelectrolyte solutions were prepared in Milli-Q grade water, with
electrical conductivity less than 1$\cdot$10-6 S/cm. Sodium Chloride
(analytical grade) was from Merck (Germany).

###  2.2 Preparation of cationic lipid-polyion complexes

DOTAP/DOPC mixed liposomes were prepared by dissolving appropriate amounts of
the two lipids in methanol-chloroform solution (1:1 vol/vol). In all the
experiments the total lipid concentration was kept constant at approximately
1.6 mg/ml, while the molar fraction of the cationic DOTAP,
$X=n_{DOTAP}/(n_{DOTAP}+n_{DOPC})$ (where $n_{i}$ is the number of moles of
the specie $i$) was varied from 0.2 to 0.75. The solvent was evaporated by
overnight roto-evaporation in vacuum. The dried lipid film was then re-
hydrated with Milli-Q quality water, for 1 hour at a temperature of
$40^{\circ}$C, well above the main phase transition temperature, $T_{m}$, of
both DOPC and DOTAP ($\approx-17^{\circ}$C and $\approx 0^{\circ}$C,
respectively). In order to form small uni-lamellar vesicles, the lipid aqueous
suspension was sonicated for 1 hour at a pulsed power mode until the solution
appeared optically transparent in white light. Finally, the vesicle dispersion
was filtered using a Millipore 0.4 $\mu m$ polycarbonate filter. The liposome
dispersions were stored at $4^{\circ}$C.
For all the liposome preparations, we checked the size distribution by Dynamic
Light Scattering (DLS). The distribution was log-normal with a mean
hydrodynamic radius of $40\pm 5$ nm and a polydispersity of the order of
$0.2\div 0.3$.

Each liposome-polyelectrolyte mixture was prepared according to the following
standard procedure [13]: a volume of 0.4 ml of the polyelectrolyte solution,
at the required concentration and ionic strength, was added to an equal volume
of the liposome dispersion in a single mixing step and gently agitated by
hand. Before mixing, the liposome suspension and polyelectrolyte solution were
kept in a thermostatted bath at the temperature of the experiment, to avoid
interference of thermal gradients during the following measurement. After
mixing the two components, the sample was immediately placed in the
thermostatted cell holder of the instrument for the measurement of the
electrophoretic mobility, the size and the size distribution of the resulting
aggregates. Electrophoretic mobility measurements were systematically
performed 5 minutes after the mixing, immediately followed by the size
determination. During the experiments the temperature was controlled within
$\pm 0.1^{\circ}$C.
The ionic strength due to the 1-valent counterions Na+ and Cl- in the pd-
liposome suspensions was varied from 0.005 M to 0.8 M by adding appropriate
amounts of NaCl to the polyelectrolyte solution to be mixed with the liposome
suspensions.

###  2.3 Dynamic light scattering measurements

Size and size distribution of liposomes and polyions-liposome aggregates were
characterized by means of dynamic light scattering (DLS) measurements. For
light-scattering measurements, a MALVERN NanoZetasizer apparatus equipped with
a 5 mW HeNe laser was employed (Malvern Instruments LTD, UK). This system uses
backscatter detection, i.e. the scattered light is collected at an angle of
173o. The main advantage of this detection geometry, when compared to the more
conventional 90o, is its inherent larger insensitiveness to multiple
scattering effects [35]. Intuitively, since nor the illuminating laser beam,
nor the detected scattered light need to travel through the entire sample, the
chance that incident and scattered photons will encounter more than one
particle is reduced. Moreover, as large particles scatter mainly in the
forward direction, the effects of dust or, as is our case, of large irregular
aggregates (lumps or clots) on the size distribution are greatly reduced.
To obtain the size distribution, the measured autocorrelation functions were
analyzed by means of the CONTIN algorithm [36]. Decay times are used to
determine the distribution of the diffusion coefficients $D$ of the particles,
which in turn can be converted in a distribution of apparent hydrodynamic
radii, $R_{H}$, using the Stokes$-$Einstein relationship
$R_{H}=K_{B}T/6\pi\eta D$, where $K_{B}T$ is the thermal energy and $\eta$ the
solvent viscosity. The values of the radii shown here correspond to the
average values on several measurement and are obtained from intensity weighted
distributions [36, 37].

###  2.4 Electrophoretic mobility measurements

The electrophoretic mobility of the suspended particles was measured by means
of the same NanoZetaSizer apparatus employed for DLS measurements. This
instrument is integrated with a laser Doppler electrophoresis technique, and
the particle size and electrophoretic mobility can be measured almost
simultaneously and in the same cuvette. In this way, possible experimental
uncertainties due to different sample preparations, thermal gradients and
convection are significantly reduced.
Electrophoretic mobility is determined using the Phase Analysis Light
Scattering (PALS) technique [38], a method which is especially useful at high
ionic strengths, where mobilities are usually low. In these cases the PALS
configuration has been shown to be able to measure mobilities two orders of
magnitudes lower than traditional light scattering methods based on the
shifted frequency spectrum (spectral analysis).
In general, the conversion of the electrophoretic mobility into the
$\zeta$-potential for a colloidal particle is not a trivial matter, because it
involves a rather complex hydrodynamical problem [39]. However, if the reduced
electrokinetic radius $\kappa R$ (with $\kappa$ the inverse Debye’s screening
length and $R$ the particle s radius), is large enough, the $\zeta$-potential
can be straightforwardly obtained from the Helmholtz/Smoluchowski (HS)
approximation:

$\mu=\varepsilon_{0}\varepsilon_{r}\zeta/\eta$ (1)

where $\epsilon_{0}$ and $\epsilon$ are the vacuum and solvent permittivity,
respectively.

##  3 Theoretical and phenomenological background

Figure 1: (Color online) The potential of mean force
$\langle\Phi_{AB}(r)\rangle$ between the pd-liposomes (equation (2)) in units
of the thermal energy $k_{B}T$ as a function of the distance $r$ between the
particle’s surfaces, calculated for typical values of the surface potential
average value, $\Psi_{A}=\Psi_{B}=25\,mV$, and standard deviation,
$\sigma_{A}=\sigma_{B}=15\,mV$, at $T=298.16\,K^{\circ}$, with the Hamaker
constant $\mathcal{H}=10^{-20}\,J$ and with a thickness of the liposome
bilayer $h_{A}=h_{B}=5\,nm$. In panel A the curves are calculated at two
different values of the ionic strength and hence of the electrostatic
screening, i.e. at $\kappa=1\,nm^{-1}$ (full lines) and $\kappa=1.1\,nm^{-1}$
(dotted lines) and for several different values of the radius $R$ of the
particles, to show that at increasing the screening, a same height of the
potential barrier is reached for a larger radius of the aggregates. As the
salt content is increased, panel B ($\kappa=1.25\,nm^{-1}$, full lines), the
secondary minimum in front of the potential barrier may become deep enough to
destabilize the suspension. As expected, at much higher ionic strengths
($\kappa=2.0\,nm^{-1}$, dotted lines) the repulsive barrier disappears, there
is only one deep minimum and the colloid dispersion is completely
destabilized.

Monte Carlo simulations have recently shown that the reentrant condensation of
pd-particles can be rather satisfactorily described by assuming that two
decorated particles ($A$ and $B$) interact via the potential of mean force
[40, 18, 32]

$\begin{split}\langle\Phi_{AB}(r)\rangle=\frac{R_{A}R_{B}}{R_{A}+R_{B}}\Biggl{\\{}\varepsilon_{0}\varepsilon_{r}\Biggl{[}(\Psi_{A}^{2}+\Psi_{B}^{2}+\sigma_{A}^{2}+\sigma_{B}^{2})\ln(1-e^{-2\kappa
r})+\\\ 2\Psi_{A}\Psi_{B}\ln\left(\coth\frac{\kappa
r}{2}\right)\Biggr{]}-\frac{\mathcal{H}}{6}\Biggl{(}\frac{1}{r+h_{A}+h_{B}}-\frac{1}{r+h_{A}}-\frac{1}{r+h_{B}}+\frac{1}{r}\Biggr{)}\Biggr{\\}}\\\
-\frac{\mathcal{H}}{6}\ln\left[\frac{r(r+h_{A}+h_{B})}{(r+h_{A})(r+h_{B})}\right].\end{split}$
(2)

Here $R_{i}$ is the radius of the particle $i$ (with $i\equiv A,B$), $r$ is
the minimal distance between the two interacting particle surfaces, and
$\Psi_{i}$ and $\sigma_{i}$ are the values of the mean and standard deviation
of the surface potential of the particles (assumed spherical).
The last two terms, proportional to the Hamaker constant $\mathcal{H}$, take
into account the van der Waals forces (between shelled spheres with radii
$R_{i}$ and shell thickness $h_{i}$ [41]), while the terms within the square
brackets represent the screened electrostatic interaction between two
inhomogeneously charged spheres in an electrolyte solution, taking into
account the presence of the ionic double layer at their surface [23]. Such
potential corresponds to an extension of the classical Hogg-Healy-Fuerstenau
(HHF) theory for homogeneously charged spheres [42] to which reduces for
$\sigma_{A}=\sigma_{B}=0$.
Both the electrostatic and the van der Waals contributions in eq. (2) are
calculated assuming the validity of the Derjaguin approximation [43, 44].
Within this approximation, the force between the surfaces of two interacting
particles is proportional to an effective radius, whose reciprocal is the
arithmetic mean of the inverse curvature radii of the surfaces involved,
$R_{A}$ and $R_{B}$ (the prefactor in front of the curly brackets in eq. (2)).
The Derjaguin approximation holds for inter-particle distances $d\ll R_{A}$,
$R_{B}$ and for $\kappa^{-1}\ll R_{A}$, $R_{A}$. In these conditions, the
approximation has been recently investigated experimentally and found
”surprisingly robust” [45].
It is easy to see that the first electrostatic term in eq. (2) is always
attractive (negative) also for like-charged particles. The presence of this
attractive term is consistent with the results of Poisson-Boltzmann equation
(PBE) approaches and simulations [46, 47, 18], pointing out the existence of a
short range attraction between polyelectrolyte-decorated macroions. The term
”short range” is well justified since the decaying length of the attractive
term $(2\kappa)^{-1}$ is one half of the Debye screening length, that sets the
scale for the screened electrostatic interactions.
Figure (1) shows the typical behavior of the potential of mean force
$\langle\Phi_{AB}(r)\rangle$ as a function of the distance $r$ between the
particle surfaces. The values of the potential, in units of the thermal energy
$k_{B}T$, are calculated for typical values of the average surface potential,
$\Psi_{A}=\Psi_{B}=25\,mV$, and standard deviation,
$\sigma_{A}=\sigma_{B}=15\,mV$, at temperature $T=298.16\,K^{\circ}$, with the
Hamaker constant $\mathcal{H}=10^{-20}\,J$ and for a thickness of the liposome
bilayer $h_{A}=h_{B}=5\,nm$. The curves are calculated at three different
ionic strengths and for several different values of the particle’s radius $R$.
At a given value of the ionic strength and hence of the electrostatic
screening length, $\kappa^{-1}$, the height of the potential barrier increases
with the Derjaguin effective radius $R$ of the aggregates [40].
Within this framework, the aggregation of the pd-particles can hence be
described as a thermally activated process [25] where, since the potential
barrier height increases with their own size, a finite size of the clusters
turns out to be eventually stable [40, 48].
Panel A of figure (1) shows how the ionic strength tunes the size of the
stable clusters: as the salt concentration is increased a larger size of the
aggregates is needed in order to get the same barrier profile.
At much higher ionic strengths (Panel B of figure (1), dotted lines) the
repulsive barrier disappears, there is only one deep global minimum, and the
colloid dispersion is completely destabilized. However, already at
intermediate values of the salt content (full lines in panel B) the secondary
minimum in front of the barrier begins to deepen enough to destabilize the
dispersion.

Figure 2: (Color online) The average hydrodynamic radius (panel A) and the
corresponding $\zeta$-potential (panel B) of polyion-liposome clusters as a
function of the polyion/lipid charge ratio $\xi$, in the presence of different
amounts of a simple electrolyte, NaCl, added to a final concentration of: 0.4
M ($\lhd$); 0.2 M ($\triangledown$); 0.005 M ($\circ$). Temperature was fixed
at 25 ∘C. As an example, the arrow spans the instability range of the 0.4 M
NaCl sample. Adapted with changes from [32].

This theoretical picture is in excellent agreement with the observed
phenomenology, as we will further point out later. Figure (2) shows the
typical reentrant condensation behavior of pure DOTAP/NaPA decorated liposomes
in the presence of NaCl. Consistently with the prediction of the theory, for a
given value of the polyelectrolyte/lipid charge ratio $\xi$ the size of the
aggregates increases with the salt concentration. The symbols in the figure
corresponds to stable aggregates, i.e. aggregates whose size, once formed (on
a much shorter timescale than the measurement time, which is typically a few
minutes), remains constant for several weeks. At the lowest concentration of
the electrolyte the suspension is unstable only in a very narrow zone close to
the isoelectric condition and the reentrant condensation curve appears almost
continuous. Indeed, it may be even difficult to locate experimentally the
precise polyelectrolyte/lipid charge ratio that results in instability and
phase separation. However, as the electrolyte concentration is further
increased, an ample gap of instability appears around the isoelectric point,
where stable aggregates does not form and whose width increases with the salt
content. Within this range of polyelectrolyte/lipid charge ratios, as the
polyelectrolyte solution and the liposome suspension are mixed together, large
mesoscopic ”flocs” immediately begins to form, that rapidly coalesce and
eventually separate from the aqueous solution, floating on its surface
(flocculation).
These features appear to be rather general. A similar behavior was observed,
for example, also in the aggregation of oppositely charged polyelectrolytes
and micelles [49]. In that system, consistently with the reentrant
condensation scenario, an increasing salt concentration causes the onset and
further expansion of coacervation phenomena close to the isoelectric point.

Figure 3: (Color online) Phase diagram of the pd-liposome complexes formed by
pure DOTAP liposomes in the presence of added salt. At each given
polyion/lipid molar ratio $\xi$, and for a defined concentration of NaCl
added, full symbols represent stable samples, open symbols represent unstable
samples (data partially published in [32]).

The behavior of the $\zeta$-potential of the pd-liposome aggregates is shown
in Panel B of fig. 2. As expected from the Gouy-Chapman-Stern theory [50, 44],
the absolute value of the $\zeta$-potential decreases as the salt content
increases both below and above the isoelectric point and for a given
polyelectrolyte-lipid charge ratio. Beyond the isoelectric point the
$\zeta$-potential decreases even more markedly (in absolute value) with the
increasing salt content. This effect can be interpreted within the framework
of the ”screening-reduced” adsorption regime described by Muthukumar [51].
This behavior has to be expected when in the adsorption of a highly charged
polyelectrolyte on an oppositely charged surface the driving attraction is
mainly electrostatic, and the added salt screens equally well the chain chain
repulsion and the chain surface attraction. Evidence of this regime can be
also found in the MC simulations of Truzzolillo et al. [18], where the
dependence of the thickness of the polyelectrolyte layer adsorbed on single
pd-particles is investigated as a function of the ionic strength of the
solution.
Fig. 3 shows the NaCl-$\xi$ phase diagram of pd-liposomes complexes formed by
pure DOTAP liposomes. Stable mesoscopic aggregates or unstable large clusters,
which eventually flocculate, are formed depending on $\xi$ and on the amount
of the added salt (plotted in full and open symbols, respectively). The
position of the isoelectric point is marked by a vertical dashed line. Below
the isoelectric point, the boundary line between the stable and the unstable
regions is approximately straight, on the contrary once the isoelectric point
is crossed, there is a clear change of the slope of the boundary line at
$\xi\approx 1.3$. This inflection point corresponds to the maximum
overcharging of the liposomes: after this point the particles surface is
completely saturated by the polyelectrolyte and the ”stable aggregates”
correspond to single liposomes coated by an almost homogeneous layer of
polyelectrolyte. Beyond the inflection point ($\xi\approx 1.3$) a further
increase of the polyelectrolyte concentration turn out simply into an increase
of the ”free polyelectrolyte” amount in solution, and does not affect the net
charge of the complexes. The cause of this decrease of the boundary slope is
that in this regime, also the free polyelectrolyte (and its counterions)
contribute to the electrostatic screening, so that a comparatively lower NaCl
concentration produces a similar destabilizing effect on the suspension [15].
The complexation experiments have been performed increasing the NaCl
concentration up to 1 M. Above 0.6 M NaCl the colloidal suspension is
completely destabilized also at the lowest polyelectrolyte/lipid ratio
investigated, and above 0.8 M NaCl even in the absence of the polyelectrolyte
(see also [52]).

##  4 Results

We focused here on the effect of the surface charge density of the liposomes
on the stability/instability diagram of pd-liposomes as a function of the
polyelectrolyte/lipid charge ratio $\xi$ and of the ionic strength of the
solution. To this aim we prepared DOTAP/DOPC liposomes at different molar
fraction $X$ of the charged lipid and measured the hydrodynamic radius and the
electrophoretic mobility of the resulting pd-liposome complexes, varying the
polyelectrolyte/lipid charge ratio $\xi$ by changing the polyelectrolyte
content at a fixed NaCl concentration, or vice versa. In what follows the
polyelectrolyte/lipid charge ratio $\xi$ is defined in terms of moles of the
_charged_ lipid only, i.e. as the ratio $\xi=N_{P}/N_{L^{+}}$ between the
nominal (i.e. assuming complete dissociation of the ionizable groups) number
of moles of charges on the polyelectrolyte, $N_{P}$, and the number of moles
of the charged lipid (DOTAP), $N_{L^{+}}$, in the liposomes.
The ionic strength of the aqueous medium was varied from 0.005 to 0.8 M by
adding appropriate amounts of NaCl to the polyelectrolyte solution to be mixed
with the liposome suspension. Since the contribution of the liposome and
polyelectrolyte counterions to the ionic strength is of the order of 1$-$2 mM,
the largest contribution to the screening, except for the very low salt
concentrations, comes from the added electrolyte.
A preliminary study was performed in order to check the stability of the
different DOTAP/DOPC liposome suspensions in the presence of increasing
amounts of added salt (but in absence of polyelectrolyte). Within the
investigated NaCl concentration range (i.e. up to 0.8 M) the DOTAP/DOPC
liposome suspensions are stable, the vesicles showing only a progressive
reduction of their size (about 20$\%$), probably due to osmotic shrinkage
[53], as it has been already observed for pure DOTAP liposomes [15]. The
measured $\zeta$ potential decreases at increasing the ionic strength, in
agreement with the Gouy Chapman theory [50], and consistently with the results
of Barenholz et al. [54] for similar DOTAP/DOPC liposome suspensions.

Figure 4: (Color online) The average hydrodynamic radius (panel A) and the
corresponding $\zeta$-potential (panel B) of pd-liposome clusters formed by
DOTAP/DOPC 0.5/0.5 liposomes as a function of the polyion/lipid molar ratio
$\xi$, without added salt ($\blacksquare$) and in the presence of different
concentrations of NaCl: 0.05 M ($\bigcirc$); 0.1 M ($\bigtriangleup$); 0.5 M
($\bigtriangledown$). The temperature was fixed at 25 ∘C. Panel A, inset: the
potential of mean force $\langle\Phi_{AB}(r)\rangle$ of equation (2)
calculated for several values of the aggregate radii and at two different
values of the surface electrostatic potential $\Psi$ : 25 mV (full lines) and
22 mV ; as in figure (1) $\sigma_{A}=\sigma_{B}=15\,mV$,
$T=298.16\,K^{\circ}$, and $\mathcal{H}=10^{-20}\,J$. The inverse Debye
constant is $\kappa=1.1\,nm^{-1}$. At this ionic strength a small difference
in the surface potential moves the system from the ”barrier-limited” to the
”secondary-minimum” aggregation regime.

In Fig. 4 we show an example of the behavior of the mixed DOTAP/DOPC pd-
liposomes in the $R-\xi$ and $\zeta-\xi$ planes. The curves are obtained for
liposomes prepared at equimolar concentrations of DOTAP and DOPC at different
salt content and are compared with the curves obtained for the pd-liposomes
without any added salt.
As expected on the basis of the universality of this kind of aggregation, that
does not depend on the details of the chemical structure of the
polyelectrolyte and the colloid, in the absence of salt we found the typical
reentrant condensation curve: stable pd-liposomes aggregates form in the whole
range of $\xi$, except close to the isoelectric point. The addition of 0.05 M
NaCl already causes a significant widening of the instability gap (marked by
the dashed arrows in panel A of figure 2, and the addition of increasing
amounts of salt makes the gap progressively larger. Notably, after 0.1 M NaCl,
considering the same amount of added salt, the instability region widens at a
lower charge ratio if compared with the one observed with pure DOTAP
liposomes. That is, at the same polyion/lipid ratio $\xi$, pd-liposome
aggregates formed by DOTAP-DOPC liposomes are destabilized by a lower amount
of salt compared to DOTAP liposomes. This finding is consistent with the
expectations based on the potential described by equation (2). Indeed, it is
easy to calculate that the amount of added salt needed to move from the
”barrier-limited” aggregation regime, where the size of the stable aggregates
is determined by the increase of the potential barrier with the size of the
aggregates, to the ”secondary-minimum” regime, where the deepening of the
secondary minimum causes the instability, decreases with the value of the
surface potential (see the inset of figure (4 \- panel A). In other words, the
secondary minimum deepens comparatively faster with the increasing salt
content for lower values of the surface electrostatic potential $\Psi$.

Figure 5: Panel A: the average hydrodynamic radius ($\square$) and the
corresponding $\zeta$-potential ($\bullet$) of pd-liposome clusters formed by
mixed DOTAP/DOPC 0.5/0.5 liposomes as a function of the added NaCl at a
polyion/lipid molar ratio $\xi$= 0.4. Panel B: phase diagram of the DOTAP/DOPC
0.5/0.5 pd-liposome complexes formed in the presence of added salt. At each
given polyion/lipid molar ratio $\xi$ and NaCl concentration, full symbols
represent stable samples, open symbols represent the unstable ones. The dashed
line marks the position of the isoelectric point, where the inversion of
$\zeta$-potential occurs. As in all the previous experiments the temperature
was fixed at T= 25 ∘C.

The addition of 0.15 M NaCl it is sufficient to destabilize the pd-liposomes
aggregates at all the charge ratios larger than $\xi>0.4$. Conversely, at
$\xi<0.4$ small stable complexes are observed, with a radius of $100\div 200$
nm and a $\zeta$-potential of $\backsim 20mV$. These values are significantly
different from those characterizing the bare liposomes (R $\backsim$ 40 nm,
$\zeta$ pot $\backsim 40$ mV), thus indicating that these particles are small
aggregates of pd-liposomes. For a larger amount of salt, up to 0.4 M NaCl, a
complete salt-induced destabilization of the pd-liposome is observed, at all
the charge ratio $\xi$. Unexpectedly, a further salt addition promotes a
return to small stable complexes. These complexes are observed only at low
$\xi$ values in the presence of 0.45 M NaCl and over the whole $\xi$ range
explored at 0.5 M, as it is shown in fig. 2 ($\bigtriangledown$ symbols). The
fact that the radius of these objects is around 50 nm, close the one measured
for pure liposomes with same NaCl concentration added in solution, suggest
that they are bare liposomes with scarce or null polyion adsorption.
Panel B of fig. (4) shows the corresponding behavior of the $\zeta$-potential
of the samples whose data are shown in panel A. Again, the curve obtained in
absence of added salt is reported (black circles) for comparison. As observed
for pure DOTAP liposomes, shown in fig. (2), the addition of salt up to 0.15 M
promotes an evident decrease of the $\zeta$-potential at $\xi<1$, due to a
progressive screening effect. In the overcharged region the same amount of
added salt does not cause significant differences in the $\zeta$-potential,
which remains approximately constant.
Above 0.4 M NaCl, a different behavior is found, which is not observed in pure
DOTAP liposomes. What is new here is that the charge inversion is not observed
anymore, but the $\zeta$-potential stays positive over the whole $\xi$ range
investigated, with only a weak decrease above $\xi$=1. Notably, as for the
hydrodynamic radius (panel A), the value of the $\zeta$-potential of
DOTAP/DOPC 0.5/0.5 liposomes in the presence of 0.5 M NaCl does not change
appreciably with or without the polyelectrolyte. Actually, above 0.4 M NaCl,
both the size and $\zeta$-potential of the pd-liposomes correspond to those of
the bare liposomes at the same concentration of salt, a clear indication that
in this condition, due to the screening effect of the electrolyte the
polyelectrolyte adsorption on the particle surface is very much reduced or
absent.
In order to gain further insight on the effect on the aggregation induced by
the added salt, it is instructive to considered the behavior of size and
$\zeta$-potential of the aggregates as a function of the salt concentration
and at a fixed polyion/lipid charge ratio. An example is shown in panel A of
Fig. (5) for the complexes of DOTAP/DOPC 0.5/0.5 liposomes at $\xi$= 0.4. A
three phase behavior appears, ruled by salt concentration. At low salt, an
increase of the NaCl content promotes the formation of stable pd-liposomes
aggregates with increasing size and decreasing $\zeta$-potential, until the
”secondary-minimum driven” instability region is reached. Beyond a critical
concentration $C^{*}$, approximately between 0.45 and 0.5 M, a new region of
coexisting ”free” liposomes and polyelectrolytes is found. The polyelectrolyte
does not adsorb anymore on the liposome.
Panel B of Fig. (5) shows this behavior in terms of a phase diagram, analogous
to that shown in figure (3) for pure DOTAP pd-liposomes. In this diagram the
onset of a stable region of isolated bare liposomes at high salt content is
evident, in contrast with the case of pure DOTAP pd-liposomes (fig. (3)).
Clearly, the aggregation of pd-liposomes can be enhanced or suppressed in a
non trivial way by changing the ionic strength of the solution.
The behavior described for mixed DOTAP/DOPC 0.5/0.5 pd-liposomes has been
observed with similar features in all the other mixed liposome preparations.

Figure 6: (Color online) Phase diagrams of the pd-liposome complexes formed by
mixed DOTAP/DOPC liposomes in the presence of added salt, at the different
DOTAP molar fractions (from top to bottom) X=0.2 (A), X = 0.35 (B) X = 0.75
(C). At each given polyion/lipid molar ratio $\xi$ and NaCl concentration,
full symbols represent stable samples, open symbols represent the unstable
ones. Dashed lines mark the position of the isoelectric point, where the
inversion of $\zeta$-potential occurs. As in all the previous experiments the
temperature was fixed at T= 25 ∘C.

Fig. (6) shows the phase diagrams of pd-liposomes built up at different DOTAP
molar fractions X (with X increasing from top to bottom: panel A, X=0.2; B , X
= 0.35; C, X = 0.75). For all the liposome preparations, a complete series of
experiments was performed, up to 0.5 M NaCl at least, and in many cases up to
1 M NaCl. Differently from the case of pure DOTAP pd-liposomes (fig. (3)), all
the phase diagrams of mixed DOTAP-DOPC pd-liposomes are characterized by the
presence of a stable region at high salt content. Within this region,
independently of the charge ratio, stable complexes are observed with a size
which remains close to that of the primary liposomes. The onset of this region
for the different DOTAP/DOPC mixed liposomes occurs at different salt
concentration. Such critical salt content increases when the molar fraction X
of the charged lipid DOTAP composing the liposomes is increased.

Notably, for all the mixed DOTAP-DOPC liposomes with a DOTAP molar fraction
larger than X=0.3 (including the pure DOTAP liposomes, fig (3)), the
isolectric point occurs at $\xi>1$. Only for the liposome suspension with the
lowest DOTAP concentration (X = 0.2) the charge inversion is observed at a
charge ratio $<1$, i.e. at $\xi\approx 0.8$. With the given definition of the
stoichiometric polyelectrolyte/lipid charge ratio $\xi=N_{P}/N_{L+}$, in which
the presence of the neutral DOPC lipid is not considered, a fixed position of
the isoelectric point would imply a regular ”compensation” between the
decreased liposome charge and the amount of polyelectrolyte which is necessary
to neutralize it.
In several studies on complexes formed by DNA and equimolar DOTAP-DOPC or
DOTAP-DOPE mixed liposomes, it was observed that the charge neutralization
point occurs at the same charge ratio of pure DOTAP liposomes [55, 56]. Among
these investigations, Koltover et al. [56] varied the fraction of DOPC in
mixed liposomes from 0 to 0.7, studying the structure of the resulting
aggregates (lipoplexes) in pure water and in the presence of 0.15 M of NaCl.
For DOTAP fractions smaller than 0.25, in excess of the neutral DOPC lipid,
they observed a lipid demixing when DNA adsorbs to the liposome surface.
Interestingly, DOTAP-DOPC liposomes also demix in the absence of DNA at the
same DOPC fraction, forming stable lipid bilayers with the DOPC excess
segregating in one-phase domains. A similar effect has been also observed in
mixed DOTAP/DPPC monolayers [57] and liposomes [58], where in absence of
polyelectrolytes at room temperature and at low molar fractions of DOTAP
($X\approx 0.2$) the mixtures show a tendency to separate in DOTAP-rich and
DOTAP-poor domains.
As consequence of this demixing, in 0.2-0.8 DOTAP-DOPC liposomes an
heterogeneous charge distribution may occur due to the formation of one-phase
uncharged DOPC regions and mixed charged DOTAP-DOPC region. Spontaneous
”charge patches” can form at the bare liposome surface, thus giving rise to an
attractive contribution which adds to the ”charge patch” attraction due to
correlated adsorption. So in the 0.2-0.8 DOTAP-DOPC mixed liposomes the
(total) attraction is probably larger compared the other mixed liposomes, at
the same polyion coverage, i.e, at the same $\xi$, and a smaller amount of
polyelectrolyte is sufficient to reduce the repulsions via charge
neutralization and to promote the aggregation of like-charged pd-liposomes.
Such demixing could justify the occurrence of the isoelectric point and of the
aggregation maximum at a lower value of $\xi$ for the X=0.2 mixture (fig. (6)
panel A). Indeed, a smaller number of adsorbed chains with loops and dangling
ends can efficiently neutralize the charged lipids when they are concentrated
in a few domains (more chains would be needed where the charged lipids
distributed more uniformly).

The possibility to displace the adsorbed polyions from the surface of
liposomes upon increasing the concentration of salt in the solution has been
observed also in other liposome systems. Yaroslavov and coworkers [59]
observed the desorption of the positively charged polycation (poly-N-
ethyl-4-vinylpyridinium bromide) from the surface of anionic liposomes
(composed of phosphatidylserine (PS) and phosphatidylcholine (PC)) when a
critical concentration of salt is reached. Since the salt concentration needed
to completely displace the polyelectrolyte from the surface is roughly
proportional to the amount of the charged lipid (PS) in the liposome
composition, the authors have hypothesized that the desorption is basically
controlled by the surface charge density. In that work the liposomes are
formed by two naturally occurring mixtures of PS and PC lipids. In this
system, since the exact composition of the hydrophobic tails of the lipid is
unknown, the determination of the area per molecule occupied at the bilayer
surface, and hence the surface charge density of the liposomes, is uncertain.
So that it is comparatively more difficult to correlate the effect of the salt
and of the surface charge on the adsorption/desorption of the polyelectrolyte.

These findings can be rationalized within the theory first developed by
Muthukumar [51] for pure electrostatic adsorption, proving that polyion
adsorption can be finely controlled by changing the surface charge of the
liposomes on the molecular scale and, more generally, by tuning the
electrostatic interaction, which, in turn, rule the stability of the pd-
liposome complexes.
The effect on the adsorption/desorption of a polyelectrolyte onto an
oppositely charged surface of the presence of added salt in the aqueous
solution bathing the surface has been investigated theoretically by several
authors [60, 51, 61, 62, 63, 1]. In particular, based on a generalized mean-
field approach, Winkler and Cherstvy [1] proposed a model for the adsorption
of a flexible polyelectrolyte on a spherical surface. According to this model,
the non-adsorption regime occurs when the inverse of the Debye screening
length $\kappa$ exceeds a critical value:

$\kappa_{c}^{3}=\frac{24|\sigma_{c}||\rho|}{j^{2}_{0}\varepsilon_{0}\varepsilon_{r}k_{B}Tl_{K}}$
(3)

where $j_{0}=2.405$ is the first positive root of the Bessel functions of the
first kind $J_{0}$, $l_{K}$ is the Kuhn length of the polyelectrolyte,
$\sigma_{c}$ and $\rho$ the surface and linear charge density of spherical
surface and polyelectrolyte, respectively. Equation (3) holds in the limit
$\kappa R\rightarrow\infty$, where $R$ is the particle radius. In our
experiments $\kappa R$ is typically $\approx 100$.
From the experimental NaCl vs $\xi$ phase diagram of the mixed DOTAP-DOPC pd-
liposomes (figure (5 panel B and (6)) the critical concentration of NaCl
needed to induce the polyion desorption can be estimated as midway between the
last line of ”unstable” samples and the the first of stable ones. From these
values, assuming that the contribution of the counterions of the liposomes and
the polyelectrolyte is negligible, a critical screening length
$\kappa_{c}^{-1}$ can be determined. In this way, since the structural
parameters that appear in equation (3), the Kuhn length etc., of our system
are known from the literature, this equation can be used to analyze the
observed dependence of the critical salt concentration for desorption on the
liposome surface charge.
The persistence length (which is half the Kuhn length) of NaPA is $\approx
12\,$Å [64], so that $l_{K}\approx 24\,$Å. The distance between the
(monovalent) ionizable groups of NaPA is $b=2.52$Å [65], so that the nominal
(i.e. without taking into account the counterion condensation) linear charge
density on the chain can be readily determined.
The stoichiometric surface charge density of mixed DOTAP-DOPC liposomes can be
estimated considering the values of the area per molecule occupied in a
bilayer by pure DOPC and DOTAP, 72.5 and 70 Å2 respectively [66]. Then,
assuming the cationic DOTAP headgroup completely ionized and the DOPC (which
is zwitterionic) completely neutral, the surface charge density $\sigma_{c}$
can be calculated as a function of the composition as

$\sigma_{c}=\frac{e}{[A_{DOTAP}+A_{DOPC}(\frac{1}{X}-1)]}$ (4)

where $e$ is the unit charge, $X$ is the DOTAP molar fraction in the mixture,
and $A_{DOTAP}$ and $A_{DOPC}$ are the values of the area per molecule of
DOTAP and DOPC, respectively. Actually, however, the nominal or ”bare” charge,
$Q_{bare}$, of these liposomes must be replaced by a renormalized one [67],
since, due to the rather high density of ionizable groups on the surface, part
of the counterions remain ”entrapped” in its immediate vicinity. Indeed,
experimentally, the effective charge $Q_{eff}$ of many colloids is
considerably smaller than the ”structural” surface charge estimated, for
example, by titration [68]. The relation between bare charge and effective
charge of mixtures of charged and zwitteionic lipids has been thoroughly
investigated for the DOTAP-DOPE system [69], and both in the case of spherical
vesicles (liposomes), and of planar monolayers (Langmuir monolayers) it has
been shown that the effective charge can be rather satisfactorily evaluated
applying the model of Aubouy et al. [70].
Based on an approximate nonlinear Poisson-Boltzmann equation, in the limit of
large $k_{D}R$ the model gives an equation relating the effective charge
$Q_{eff}$ and the bare charge $Q_{bare}=eZ_{bare}$ of spherical colloids

$Q_{eff}\frac{l_{B}}{R}=4k_{D}RT(x)+2\left[5-\frac{T(x)^{4}+3}{T(x)^{2}+1}\right]T(x)$
(5)

where the function $T(x)$ is defined as

$T(x)=\frac{1}{x}(\sqrt{1+x^{2}}-1)$ (6)

and $x$ as

$x=\frac{Z_{bare}l_{B}}{R(2k_{D}R+2)}$ (7)

Equation (5) can be employed to calculate the effective surface charge density
of mixed DOTAP-DOPC liposomes at the NaCl concentration corresponding to the
desorption condition that must be used in eq. (3) to fit the experimental
data. Panel A of fig. (7) shows the behavior of the nominal and the effective
liposome charge (full and open symbols, respectively) calculated for the
liposomes prepared at the different molar fractions of the charged lipid
(DOTAP) in the mixture. As expected, at increasing liposome surface charge the
renormalization becomes more relevant.

Also in the case of the polyelectrolyte employed, NaPA, the linear charge
density should be reduced by a factor $f$, due to the counterion condensation
phenomena. For polyelectrolytes in the infinite dilution limit, Manning’s rule
allows to calculate $f$ as $f=b/l_{B}$, where $l_{B}$ is the Bjerrum length
$e^{2}/(4\pi\varepsilon_{0}\varepsilon_{r}k_{B}T)$, about 7.14 Å in water at
T=25 ∘C, which gives for NaPA $f=0.35$ [71]. However, experimentally both the
persistence length and the fraction of uncondensed counterions changes with
the concentration and with the ionic strength of the solution (for $f$, see
for example [72]). Moreover, the determination of the effective polyion charge
in adsorption condition is a controversial issue. Different theories show that
in the process of adsorption of polyelectrolytes onto an oppositely charged
colloidal particle the counterions can be completely released [73], mostly
released [74], partially released [7], not always released [75], or not
released at all (in the case of flexible polyelectrolytes) [76].
As a check of the validity of the model represented by equation (3) to
describe the polyelectrolyte desorption in our system, we plotted the surface
charge densities of the DOTAP-DOPC liposomes prepared at the different
concentrations of the charged lipid versus the cube of the ”critical” inverse
Debye length, $\kappa_{c}$, obtained experimentally from the phase diagrams of
figs. (5)- Panel B and (6). If equation (3) holds a linear trend passing
through the origin of the $\sigma_{C}-\kappa_{C}^{3}$ plane is expected.
The result is shown in panel B of fig. (7). The open symbols represent the
nominal values of the surface charge densities calculated as a function of the
charged lipid molar fraction from equation (4). In this case the relation (3)
is not satisfied. However, if the renormalized values calculated from the
Aubouy’s model (5) are used (full symbols) the agreement is excellent
(R-squared value 0.949, P $<$ 0.026). From the value of the slope obtained
from the linear fit, being all the other quantities known, the factor $f$,
introduced in order to renormalize the polyion charge density $\rho$ can be
calculated. It is worth to remind here that this is the effective charge of
the polyelectrolyte when it is _adsorbed_ on the oppositely charged particle
surface. In practice $f$ can be interpreted as the fraction of the counterions
that are not condensed, i.e. do not keep staying in the proximity of the
chain, but are ”free” in the solution, when the polyion is adsorbed. The value
of $f$ obtained from the fit is $f=0.55\pm 0.1$, thus indicating that the
counterions are only _partially released_ in the process of adsorption. This
value for the fraction of free counterions can be compared with the analogous
value for the polyelectrolyte when it is not adsorbed. As we stated above, the
Manning’s rule gives for NaPA a values $f=0.35$. Experimentally, for this
highly charged polyelectrolyte a value of $f=0.25\div 0.30$ has been estimated
[72] from low frequency conductivity measurements, in aqueous solution at T=25
∘C and in absence of added salt. This value increases up to $f=0.4\div 0.6$ in
the presence of a large excess of added salt [77].
In addition to that, a partial release of the condensed counterions when the
polyelectrolyte adsorbs on an oppositely charged particle has been also
observed in the case of NaPA adsorbed on pure DOTAP liposomes, with a
completely different experimental approach based on electrical conductivity
measurements [78].

Figure 7: Panel A, the nominal ($\circ$) and effective ($\bullet$) charge,
calculated according to the Aubouy’s model, as a function of the molar
fraction of the cationic lipid in the mixed liposomes, lines guides the eye
only. Panel B, when plotted against the cube of the experimental inverse Debye
length of desorption, $\kappa_{c}$, the nominal surface charge of the
DOTAP/DOPC mixed liposomes ($\circ$) does not show the linear trend through
the origin expected from equation (3) (the line is a guide for the eye only).
On the contrary, when the renormalized charge calculated from Aubouy’s model
(5) is employed the agreement is excellent (($\bullet$), coefficient of
determination, R-squared= 0.949, P$<0.026$).

##  5 Conclusions

We have investigated the ”reentrant condensation” and the charge inversion of
polyelectrolyte-decorated liposomes in a wide range of surface charge
densities of liposomes, from $\approx 0.25$ C/m2 of pure DOTAP liposomes [32]
to $\approx 0.05$ C/m2, correspondent to the DOTAP/DOPC mixture with only
$20\%$ DOTAP.
In all cases the Velegol-Twar potential [23], extended by the inclusion of the
van der Waals interactions [40, 18], well captures the observed phenomenology.
When the surface charge density of the mixed liposomes is lowered via
progressively adding neutral lipid within the liposome membrane, a new
intriguing phenomenon shows up when the concentration of a simple electrolyte
in the dispersing medium increases above a critical value: the stability
diagram of the suspensions shows a novel reentrance due to the crossing of the
desorption threshold of the polyelectrolyte. Indeed at fixed charge density of
the bare DOTAP/DOPC liposomes and for a wide range of polyion concentrations,
we showed that the simple electrolyte addition first (low salt regime)
destabilizes the suspensions because of the enhanced screening of the residual
repulsion between the complexes, and then (high salt regime) determines the
onset of a new stable phase originated by the absence of polyelectrolyte
adsorption on the particle surfaces. At a fixed linear charge density of the
polyelectrolyte, the onset of this desorption regime depends on both the ionic
strength of the bathing solution and on the liposome surface charge density.
Once an effective surface charge density, taking into account the effect of
the counterions, is calculated using the Aubuoy’s model [70] for each
DOTAP/DOPC mixture, the dependence of this effective charge from the cube of
the critical inverse Debye length $\kappa_{c}$ shows an excellent agreement
with the linear trend predicted by the adsorption theory of Winkler and
Cherstvy [1]. Via a linear fit we estimated a value of the renormalized linear
charge density on the polyelectrolyte which is consistent with previous
estimates based on independent measurements [78].
These findings give strong evidence that, in presence of adsorption from a
bathing aqueous solution of highly charged polyelectrolytes onto an oppositely
charged colloids, both the adsorbing chains and the particles maintain part of
their ”condensed” counterions, and give further support to the overall picture
of the reentrant condensation of pd-colloids as driven by the correlated
adsorption of the polyelectrolyte on the particle surface.
In the wider framework of bionanotechnology, this investigation find its
natural applications in the engineering of nanoparticle systems, having
implications in the definition of its colloidal stability and also of its
biological fate, which it has been shown to be strongly influenced by the
formation of the so called ”protein corona” [79]. Actually, these two points
are closely related each other, since the protein corona itself contributes to
shape the surface properties, the charge and the resistance to aggregation of
the nanoparticles. It was also shown that the adsorption of proteins onto the
nanoparticle surfaces can strongly affect the protein conformation, which may
ultimately culminate in the loss of its biological activity [80]. In this
view, these results give further support to the possibility to trigger the
protein adsorption at the surface of charged nanoparticles in a physiological
environment by tuning the surface charge. This last aspect is also envisaged
as relevant for the perspective to include strategically a protein corona into
designed composite nanoparticles for the load and the controlled release of
different drugs [81].

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