The Krieger–Dougherty model was applied to describe the relation between relative high shear viscosity of the solution and volume fraction of the protein at both pD in order to elucidate the charge effect on the interaction potential.

The overall data reduction was carried out according to the standard NIST procedures.¹⁹

The so-called “primary electroviscous effect” is due to the interactions of the diffuse double layer around each particle; the “secondary electroviscous effect” can be explained in terms of balance of electrostatic repulsive force and hydrodynamic compressive force on each particle; the “tertiary electroviscous effect” is influenced by the particle shape.

A solution of strongly interacting colloidal particles at high volume fractions and low electrolyte concentrations orders into crystalline lattices at rest.

If shear is applied, the flow concentrates stress above all at lattice dislocations where particles are loosely trapped.

Under flow, the solution's microstructure can be modeled as a ‘blend’ made up of a solid ordered phase coexisting with a fluid disordered phase and when shear rate increases the disordered phase rises above the ordered one.

The rheological behavior of the solutions we investigated is consistent with this model: the shear leads to a destruction of the ordered structure and the so-called shear-thinning behavior is observed (i.e. the shear viscosity decreases as shear rate increases).

In order to quantitatively describe the flow curves the Sisko approach has been used (eqn. (1)).

This relatively simple model^25,26 is useful to describe a shear-thinning behavior in presence of the high shear plateau only: η = η_∞ + Kⁿ⁻¹where K is the consistency index, n is the flow behavior index and η_∞ is the limit viscosity at infinite shear rate.

The parameter K gives an indication of the non-Newtonian nature of the sample and can be assimilated to the yield stress in a Bingham-type fluid.

When K = 0 or n = 1, the model describes a simple Newtonian fluid.

This last sample has been fitted using the Cross model^23,25 (Fig. 3b): where η₀ and η_∞ are the zero shear and the infinite shear viscosities, 4.56 and 1.78 Pa.s, respectively.

The viscosity trend of samples at high volume fraction can be rationalized in terms of electrostatic charge on the molecules and their aggregation behavior.

Electrostatic interactions influence the rheological behavior of solutions: in high concentrated solutions electrostatic interactions overcome Brownian interactions and order occurs, and obviously the electrostatic interactions experienced by the particles depend on their superficial charge.

This means that another effect contributes to the rheological behavior of the investigated systems, i.e. the aggregation phenomenon is surely favored at pD 11.0 due to the attractive surface of the low charged protein; moreover, this effect enhances with concentration.

One of the well-known correlations is the Krieger–Dougherty model: where the parameters to be fitted [η] and ϕ_max are the intrinsic viscosity and the volume fraction corresponding to the maximum packing, respectively, while η₀ is the solvent viscosity.

If we assume a face center cubic (fcc) like structure we can calculate the mean intermolecular distance, d, from the protein concentration using the formula: where N_A is Avogadro's number and [c] is the protein molar concentration.

This spatial disposition allows all the charged macro-ions to be at the same distance to their first neighbors, while the simple cubic ordering forces some macro-ion to stay closer than others.

In order to fit the experimental data quantitatively we used the following equation: I(Q) = A_pP(Q)S(Q)where A_p is the amplitude factor.

In the case of a globular protein cytochrome C, it is given by: 10⁻³[p]N_A[b_pa + N_ex(b_D − b_H)χ + mb_solv − V_Hb_solv/v_ω]²where N_A is Avogadro's number, [p] the protein concentration in mM, b_pa = ∑_p b_i = 258.5164 × 10⁻¹² cm the total scattering length of the protein, N_ex the number of labile protons that the protein can exchange with the solvent, b_H and b_D are respectively the scattering length of the hydrogen and deuterium, m is the number of solvent molecules in the hydration shell, V_H is the hydrated protein volume, v_ω = 30 ų is the volume of a water molecule, b_solv is the solvent scattering length calculated each time according to the sample composition using the formula b_solv = χb_D2O + (1 − χ)b_H2O where χ is the volume fraction of D₂O in the solvent and b_D2O and b_H2O are respectively the D₂O and H₂O scattering length.

The protein structure factor, S(Q), has been determined according to the generalized one-component macro-ion model (GOCM) already tested.³⁰

As said above the GOCM extends the OCM based on DLVO interaction, which is valid only in dilute solutions, to finite macro-ion concentrations.

A protein solution is described as made up of charged macro-ions experiencing screened Coulomb interaction.

The protein charge reported in the tables is an interaction charge and not the actual charge on the protein surface.

The interaction charge is the charge really experienced by protein and it is the surface charge due to the ionization of external residues reduced by the counter ions that adsorb at interface and are part of the protein itself.

The stronger NaSCN effect can be related to the well-known Hofmeister effect.⁶