# Relation in Alginate-Based Magnetic Suspensions
## 3 Results
As a result, there was a cross-over in the \(G^{\prime}\) vs. \(G^{\prime\prime}\) relation at high content of Ni@C, Ni, Fe MPs in the suspension. While at low concentrations of these MPs \(G^{\prime\prime}\) was larger than \(G^{\prime}\), at high concentration this relation had reversed to \(G^{\prime\prime}<G^{\prime}\). It meant that at high MPs content (except Fe\({}_{3}\)O\({}_{4}\) MPs) elastic deformation prevailed over the viscous dissipation of mechanical energy. Figure 4: Shear stress dependence of the storage modulus (\(G^{\prime}\)), the loss modulus (\(G^{\prime\prime}\)), and the shift angle (\(\delta\)) for the magneto-rheological suspension of 2.8% Fe\({}_{3}\)O\({}_{4}\) MPs dispersed in 5% Na-alginate solution. Frequency of shear oscillations 1 Hz, temperature 25 \({}^{\circ}\)C.  
It can be illustrated by Figure 6 which gives the dependence of the shift angle on MPs concentration in the suspensions. Figure 5: Concentration dependences of storage modulus \(G^{\prime}\) and loss modulus \(G^{\prime\prime}\) for the magneto-rheological suspensions with different magnetic particles. (**a**)—Fe and Fe\({}_{3}\)O\({}_{4}\) MPs, (**b**)—Ni and Ni@C MPs. \(G^{\prime}\) and \(G^{\prime\prime}\) are averaged over the plateau in 0.1–1 Pa range of shear stress. Frequency 1 Hz. Temperature 25 \({}^{\circ}\)C.  
Figure 6: Concentration dependences of shift angle \(\delta\) for magneto-rheological suspensions with different magnetic particles. \(\delta\) is averaged over the plateau at 0.1–1 Pa range of shear stress. Frequency 1 Hz. Temperature 25 \({}^{\circ}\)C.  
It is clear from Figure 6 that at low concentrations of MPs the dynamic deformation of suspensions was predominantly the viscous flow, as the shift angle was above 80 degrees close to the value of 90 degrees for the ideal fluid. In the case of suspensions with Fe\({}_{3}\)O\({}_{4}\) the fluidic nature of deformation preserved up to the highest content of MPs (30% wt., 8.3% vol.). Meanwhile, for the suspensions with other MPs, the turn-over occurred for the high concentration of MPs. The shift angle for the suspensions with Ni@C, Ni, and Fe MPs dropped down to values below 10 degrees which were close to the characteristic shift angle \(\delta=0\) for the ideal solid. It meant that magneto-rheological suspensions with Ni@C, Ni, and Fe MPs in a concentration above 3-4% (vol.) became elastic gels rather than viscous fluids. Note that even in the fluidic state the effect of the particle concentration on the moduli is much higher than that predicted by the classical [PERSON] theory of effective viscosity of suspensions. According to the degree of the descent of the shift angle, the suspensions form a sequence Fe\({}_{3}\)O\({}_{4}\)-Ni@C-Ni-Fe, which is identical to that marked above concerning \(G^{\prime}\) data. What is the underlying reason for the transition of fluid to elastic gel to occur in the case of suspensions with Ni@C, Ni, and Fe MPs, and not to happen in the case of suspension with Fe\({}_{3}\)O\({}_{4}\)? Our hypothesis was that its background is in different interfacial interactions in these suspensions. To check this supposition, we studied the interfacial adhesion of Na-alginate polymer to the surface of MPs using the thermodynamic approach, which was successfully elaborated for this purpose in earlier studies [26; 27; 28]. From the viewpoint of basic thermodynamics, the intensity of the adhesion of polymer at the interface with solid might be characterized by the value of the enthalpy of adhesion (\(\Delta H_{adh}\)), which is the gain in enthalpy if the polymer is put in contact with a solid surface. The thermodynamic equation for this process is:  
\[\text{Na-alginate polymer + solid MNP}=\text{Composite}+\Delta H_{adh} \tag{2}\]  
The heat effect of this process cannot, certainly, be measured directly as both interacting substances are solids. In this case the appropriate thermodynamic cycle is elaborated, which includes the sequence of measurable steps from the initial state to the final state.