It was possible to determine such kinetics by following the change in solution elastic modulus as a function of time.
Moreover, df can be scaled with entanglement density and solution plateau modulus (GN) through df ∼ (ϕ/ϕe)2.6 ∼ GN.
Sodium silicate solutions of modulus ranging from 1.6 to 2.2 were used and the samples were cured at 25, 40 and 60 °C.
To obtain the analytical solutions, elasticity modulus and mass density are assumed to vary exponentially through the plate thickness, while Poisson's ratio and length scale parameter are set to constant values.
NaOH, KOH solutions and silicate solutions with low modulus were tested.
Jacobi elliptic solutions are generalized to the hyperbolic exact solutions and trigonometric exact solutions when the modulus m → 1 and m → 0, respectively.
The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus m → 0 and m → 1, respectively.
The RF domain precoder optimization is commonly required to provide a precoder solution with unit modulus matrix entries, to enable implementation with simple phase-shifter elements, and to be constant over the entire system bandwidth, to facilitate application in the analog domain.
However, the corresponding optimization problem (P3) or (P4) will in general not output a unit-modulus solution.
Analytical solutions for the modulus and Poisson's ratios of the cellular structure in all principal directions were deduced.
