# Relation in Alginate-Based Magnetic Suspensions
## 3 Results  
As the first step in the dynamic mechanical analysis of alginate-based magneto-rheological suspensions, we tested water solutions of sodium alginate in concentrations 5, 10, and 15% (wt.). Frequency dependence experiments in the frequency range 0.1-100 Hz at 5 Pa stress showed that \(G^{\prime}\) and \(G^{\prime\prime}\) were linearly increasing in double log scale, with G\({}^{\prime\prime}\) \(>\) G\({}^{\prime}\) and their cross-over at approximately 80 Hz. Based on these results, the frequency of 1 Hz was chosen as the level for the study on stress-dependent rheological properties for MRSs. Typically, the process of gelation in polymer solutions is examined at a low frequency of the oscillatory stress and we took 1 Hz as a close value to a lower limit in frequency measurements (0.1 Hz). Figure 2 presents the shear stress dependence of storage modulus (\(G^{\prime}\)), loss modulus (\(G^{\prime\prime}\)), and shift angle (\(\delta\)) between the applied stress and the strain in DMA measurement at the frequency of oscillations 1 Hz for two concentrations of Na-alginate (5 and 10%). The dependences for the 15% solution were very much like that for the 10% solution but absolute values of moduli were higher by approximately an order of magnitude. In the range of shear stress, 1-50 Pa the values of dynamic moduli were constant for both concentrations, except the storage modulus for 5% solution. In that case, \(G^{\prime}\) kept constant in the range of shear stress 1-8 Pa and gradually diminished at higher values. According to common settlement [24] the plateau on the shear stress dependence of modulus is an intrinsic feature of the gel-like behavior of polymeric systems. In the case of Na-alginate it, in general, indicated the presence of physical gelation due to the formation of H-bonds. Meanwhile, for all concentrations of Na-alginate, the values of loss modulus (\(G^{\prime\prime}\)) were substantially higher than the values of storage modulus (\(G^{\prime}\)). It is clear from Figure 3, which presents the dependences of \(G^{\prime}\), \(G^{\prime\prime}\), \(\delta\) on Na-alginate concentration at 1 Hz frequency and selected shear stress \(\tau\) = 1 Pa. The values at 1 Pa were taken as characteristics for the modulus plateau. The prevalence of \(G^{\prime\prime}\) over \(G^{\prime}\) means that the balance in the viscoelastic behavior of the Na-alginate solution is biased to fluidic properties. It is clearly indicated by the value of the shift angle. For the idealized elastic material, the strain is in-phase with the stress and the shift angle is 0, while in the idealized fluid, it is 90 degrees. The shift angle was 85 degrees for the solution with 5% Na-alginate and it indicated that the viscous flow dominated the dynamic deformation. As the shift angle went down to 72 degrees for 10% solution and to 57 degrees for 15% solution, it meant that the elastic contribution to deformation progressively increased with the concentration of Na-alginate. It looks reasonable, as the upsurge in concentration had to result in the increase of the probability of inter-chain contacts, which were the origin of physical networking due to H-bonding. Figure 2: Shear stress dependence of the dynamic mechanical parameters for Na-alginate water solutions. Frequency of shear oscillations 1 Hz, with a temperature of 25 \({}^{\circ}\)C. Concentration 5% (wt. ): 1—storage modulus, \(G^{\prime}\); 2—loss modulus, \(G^{\prime\prime}\); 3—shift angle, \(\delta\). Concentration 10% (wt. ): 4—storage modulus, \(G^{\prime}\); 5—loss modulus, \(G^{\prime\prime}\); 6—shift angle, \(\delta\). The dependences for \(G^{\prime}\) and \(G^{\prime\prime}\) were found linear in the double-logarithmic scale. It means that \(G^{\prime}\) and \(G^{\prime\prime}\) had a power dependence on the concentration of Na-alginate in water (C). It led to the following scaling equations:  
\[ \begin{array}{l}G^{\prime}\propto C^{5.5}\\ G^{\prime\prime}\propto C^{3.9}\end{array}