#### 3.1.2. Theoretical Approach to a Nanolipid Structure Coating with a Polycation: Concepts of Saturation

In general, stable colloidal dispersions are formed only in a polyelectrolyte concentration range between the saturation concentration,

C_{Sat}, i.e., the minimum concentration of a polymer required to cover the oppositely charged particles, and the depletion concentration,

C_{Dep}*, the ceiling for the depletion flocculation [

12]. These values are largely dependent on both the particle concentration and the polyelectrolyte properties, such as the radius of gyration, molecular weight, and concentration.

It is possible to calculate the saturation concentration,

C_{Sat}, from the Z-potential evolution, using an empirical model proposed by [

13] for liposomes covered by chitosan:

where

Z(c) is the zeta potential of liposomes at a chitosan concentration

c,

Z_{0} is the zeta potential at zero chitosan concentration (uncoated liposomes), and

Z_{Sat} is the zeta potential at the saturation concentration

C_{Sat}. In this work, the proposed model for chitosan coating of liposomes was extended to the coating of vegan liposomes by cationic guar gum. By fitting Equation (1) to the experimental data (R

^{2} ≥ 0.95), it was possible to obtain the values of

Z_{Sat} and

C_{Sat} for both standard liposomes and vegan liposomes (

Table 1). In particular, the saturation concentration was 0.00537%

w/

v of chitosan for liposomes and 0.0057%

w/

v of cationic guar gum for vegan liposomes.

As proposed in [

13], the so-called surface coating at saturation, Γ

_{Sat} (kg/m

^{2}), can be calculated using an approach similar to that used in [

11] for coating emulsion droplets with a biopolymer. In particular, for emulsions, Γ

_{Sat} can be calculated by the following equation:

where

$\varphi $ is the particle volume fraction and

r is the volume–surface mean radius of the particles, i.e., the Z-average value (all measure units must be in International System of Units, SI).

Unlike emulsion droplets, lipid vesicles are characterized by a core-shell structure, with only the shell made of lipids. Thus [

13], proposed to estimate the total surface area of liposomes as the area of a single liposome (a function of its average radius

r, multiplied for the numbers of liposomes,

n):

The numbers of liposomes,

n, can be calculated from the ratio between the total mass of phosphatidylcholine (PC)

M_{PC,total}, and the mass of PC needed to form a single vesicle,

M_{PC,liposome}, as follows:

where ρ

_{PC} is the PC density (1015 kg/m

^{3} at T = 298 K), and Δ

r is the thickness of the liposomal membrane, which is supposed to be around 4 nm [

13]. Therefore, the surface coating at saturation, Γ

_{Sat}, for vesicles can be written as the mass of the polycation adsorbed per unit of the surface:

By considering that the volume is the same for

M_{Sat} and

M_{PC,total}, their ratio can be re-written in terms of concentrations. Thus, Equation (5) becomes:

Taking into account that the Z-average radii were about 253 nm and 180 nm for liposomes and vegan liposomes, respectively, the surface coating at saturation was, therefore, 1.01 × 10

^{−7} kg/m

^{2} for liposomes and 1.06 × 10

^{−7} kg/m

^{2} for vegan liposomes (

Table 1).

A slightly larger amount of guar HC was required to achieve saturation. Thus, the higher surface coating for vegan liposomes is due to both the higher negative charge of the vegan liposomes and the Guar HC structure, having a single positive charge for an entire chain, different from chitosan with an amine group on each repeating unit, needing, therefore, a greater polymer amount to saturate the liposomes’ negative charge (

Figure 3). Another reason for the larger amount of Guar HC needed to saturate the liposomes’ surface could be the more flexible chain of Guar HC [

23], which accelerates flip-flop behaviour [

24]. In particular, the adsorption of this flexible chain on vegan liposomes induces the transfer of negatively charged molecules from the internal side to the external side of the membrane, leading to the formation of clusters composed of anionic lipid molecules. This phenomenon results from the formation of loops in the adsorbed polymer (due to its flexibility), where the repulsion between the positive charges located in the loops gives rise to the formation of packing defects in the lipid bilayer and facilitates the transfer of anionic lipids from the internal monolayer to the external monolayer. In contrast, the more rigid chain of chitosan (semi-flexible) [

25] forms rigid ordered layers, which do not contain extended loops witlyh uncompensated positive charges, thereby reducing the imperfection of the lipid bilayer. Therefore, chitosan has to saturate only external charges while Guar Gum has to saturate all negative charged lipid molecules.

Once the surfaces of both liposomes and vegan liposomes have become completely saturated, any additional polycation added will remain free in the continuous phase and will, consequently, generate a depletion attraction among the covered vesicles. When the depletion attraction is strong enough to overcome any repulsive interaction, large particles tend to flocculate, and it is possible to calculate the minimum amount of free polycation (

C_{Dep}) required to promote depletion flocculation, as suggested by [

12], for colloidal dispersions covered by oppositely charged polyelectrolytes:

where

M is the polycation molecular weight (reported in

Table 2);

N_{A} is Avogadro’s number;

v is the effective molar volume of the polyelectrolyte in the solution (in m

^{3}), calculated as

$v=4\pi {r}_{\mathrm{PE}}{}^{3}/3$, with

r_{PE} as the effective radius of the polyelectrolyte molecules in the solution (

Table 2); X is defined in the following equation according to [

12]:

where

${\theta}_{\mathrm{F}}$ and

${\theta}_{\mathrm{NF}}$ are the volumes available to the flocculated state and the non-flocculated state, respectively. In particular,

${\theta}_{F}$ was calculated by assuming that the volume fraction of particles when they are flocculated is approximately equal to the length of the depletion zone (2

r_{PE}) divided by the average distance between the surfaces of the particles, as shown in the following equation:

Moreover, the total amount of polycation required to promote depletion flocculation is given by Equation (11):

The values of

C_{Dep}* are reported in

Table 2.

To produce a stable dispersion against flocculation, it is necessary that enough polycation is used to completely saturate the surfaces of the vesicles, but not enough to promote depletion flocculation. Thus, the polycation concentration must be in the range C_{Sat} < C < C_{Dep}*, particularly for liposomes covered by chitosan 0.00537 < C [% w/v] < 0.04840 and for vegan liposomes covered by cationic guar gum 0.00570 < C [% w/v] < 0.2282.

Another important factor to assure stability against aggregation for the multilayered vesicles is the time required to be completely covered by the polycation, τ

_{Ads}, which must be less than the time of the particle–particle collisions, τ

_{Col}, i.e., τ

_{Ads}/τ

_{Col} < 1. The critical polycation concentration (

C_{Ads}) is obtained when the adsorption time τ

_{Ads} is just equal to the time between the particle–particle collisions, τ

_{Col} [

12]:

The values of

C_{Ads} for both liposomes and vegan liposomes are shown in

Table 2. If

C >

C_{Ads}, the adsorption time is faster than the collision time between the particles, and, therefore, little vesicle aggregation occurs. Therefore, it should be possible to make stable multilayer-coated vesicles without flocculation by using intermediate polycation concentrations, such as

C_{Ads} <

C <

C_{Dep}*. Lastly, for stable covered liposomes, the chitosan concentration should be between 0.0057 and 0.0484%

w/

v, and for stable covered vegan liposomes, the Guar HC concentration should be between 0.0086 and 0.2282%

w/

v.

#### 3.1.3. Combination of Experimental and Theoretical Approaches to Nanolipid Structure Coating with a Polycation

Combining the theoretical evaluation of

C_{Sat},

C_{Ads}, and

C_{Dep}* with the Z-potential evolution in

Figure 2 (where it is visible that a constant value of Z-potential was reached for liposomes after adding 0.00625%

w/

v of chitosan, and for vegan liposomes by adding 0.005%

w/

v of cationic guar gum), the range of useful polycation concentrations was restricted to 0.00625%–0.01%

w/

v of chitosan for liposomes and 0.0089%–0.01%

w/

v of Guar HC for vegan liposomes.

To further evaluate the best concentration for vesicle coating, the evolution of the size number distribution (the average value is the number mean size) and size intensity distribution (proportional to vesicle weight; its average value is the Z-average) was evaluated by increasing the polycation concentration, as shown in

Figure 4 and

Figure 5.

In particular, the liposomes’ size (both number (

Figure 4A) and Z-average (

Figure 4B) values) increased by adding chitosan up to the final investigated concentration of 0.01%

w/

v, without visible agglomeration among the vesicles, demonstrating that the added chitosan covered the vesicles. In particular, for Z-average values, liposomes changed their sizes from 253 nm without a coating to 323 nm at 0.00625%

w/

v, 420 nm at 0.0075%

w/

v, and finally to 505 nm at 0.01%

w/

v of chitosan. Vegan liposomes remained largely unchanged in their size (both number (

Figure 5A) and intensity (

Figure 5B) values) for all the investigated cationic guar gum concentrations. The Guar HC structure, which, unlike the chitosan structure with an amine group on each repeating unit, presents only one positive charge for an entire chain. Consequently, it is expected that the covering of the vegan liposomes with cationic guar gum is partial or non-homogeneous compared to the liposomes with chitosan.

From the theoretical analyses and experimental observations of the Z-potential and size evolution of liposomes by adding chitosan, 0.01% w/v was confirmed to be the best concentration to cover liposomes without aggregation. This concentration was also chosen for Guar HC to cover vegan liposomes because, even if their size was kept unchanged with all tested polymer concentrations, the Z-potential and theoretical evaluation produced this value.