# Relation in Alginate-Based Magnetic Suspensions
## 3 Results
 \tag{1}\]  
The power exponents in Equation (1) are substantially larger than the classical exponent 1/3 in Flory-Rehner theory for the elasticity of polymeric networks [25]. Most likely it is the consequence of the dominance of the fluidity of the solutions. Although the elastic contribution to the deformation of Na-alginate solutions is clearly present, it is still less pronounced than the contribution from the viscous flow. In other words, the solutions of Na-alginate are more \"liquids\" than \"elastomers\". At the molecular level, it probably means that the H-bonds, which presumably act as the cross-links in the physical networking of the solution, are unstable and they constantly appear and dissolve in a stochastic fashion. It is also likely that the polyelectrolyte nature of Na-alginate contributes considerably to the fluidity of the solution. Basically, interaction among negatively charged ionized base-units of the Na-alginate chain is repulsive and it hampers the formation of stable cross-links in the physical network of inter-chain H-bonds. Let us now turn to the dynamics of magneto-rheological suspensions (MRS) based on Na-alginate solutions with dispersed magnetic particles (MPs). For technical reasons, MRS were prepared using a 5% solution of Na-alginate only. The viscosity of 10% and 15% solutions was too high to provide a uniform mixing with MNPs. Figure 4 presents the shear stress dependences of storage modulus (\(G^{\prime}\)), loss modulus (\(G^{\prime\prime}\)), and shift angle (\(\delta\)) at 1 Hz for MRS based on 5% Na-alginate solution with 2.8% (wt.) of Fe\({}_{3}\)O\({}_{4}\) MPs. The dependence of all these dynamic characteristics was essentially the same as that presented in Figure 2 for the 5% Na-alginate solution without MPs. The storage modulus had a plateau at a low level of shear stress and moderately diminished with its increase. The loss modulus preserved an almost constant value and was substantially higher than the storage modulus. The shift angle was almost constant, and it was close to 90 degrees. It all meant that MRS with 2.8% (wt.) of Fe\({}_{3}\)O\({}_{4}\) MPs demonstrated the same liquid-type behavior as the carrier solution of Na-alginate. Figure 3: Concentration dependences of the dynamic moduli and shift angle at 1 Hz frequency and shear stress 1 Pa. Similar features were observed for suspensions with all other MPs, which contained 2.8% (wt.) of MPs. The dependences of \(G^{\prime}\), \(G^{\prime\prime}\), \(\delta\) were qualitatively very much like that for the initial 5% Na-alginate solution. However, the situation changed drastically if the content of MPs in suspensions was enlarged. Figure 5 presents the dependence of the dynamic moduli (\(G^{\prime}\) and \(G^{\prime\prime}\)) on the volume fraction of MPs in suspension. The values of \(G^{\prime}\) and \(G^{\prime\prime}\) were averaged over the range of their constancy, which was 0.1-1 Pa. In the case of Fe\({}_{3}\)O\({}_{4}\) MPs, both dynamic moduli moderately increased with particle volume fraction. At any concentration, the loss modulus was higher than the storage modulus. In the case of Fe, Ni, and Ni@C MPs there was a steep increase in both moduli if the content of particles exceeded 3-4% (vol.) Note that the ordinate in Figure 5 is logarithmic and the growth in the moduli was several orders of magnitude. For instance, the storage modulus of 12% (1.7% vol.) suspension of Fe MPs was \(G^{\prime}\) = 3.1 Pa and for 30% (5.2% vol.) it had lifted to 42.3 kPa, which was exactly 4 orders of magnitude boost. According to the degree of elevation of \(G^{\prime}\) with concentration the MPs fell in a sequence: Fe\({}_{3}\)O\({}_{4}\)-Ni@C-Ni-Fe. The loss modulus of suspensions also enlarged with MPs concentration. The influence of the nature of MPs on the degree of elevation was the same as for \(G^{\prime}\). At the same time the increase in \(G^{\prime\prime}\) was not as drastic as the increase in \(G^{\prime}\).