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Regulation of the structure of fibres from polymer solutions

“…The method of mathematical modeling of gelation in spinning fibres can be used for basic, qualitative evaluations [37][38][39][40][41]. There are phase diagrams and diffusion coefficients of the solvents and precipitant water into the polymer diffusion and gel for the PANsolvent system (DMF and aqueous solution of NaSCN).…”

Gelation in Spinning Chemical Fibres from Solutions of Polymers

“…Mathematical models of this phenomenon both with consideration of all mechanisms simultaneously [7] and based on each mechanism separately [8][9][10][11] have been proposed based on the above approach to gelation. The kinetics of isothermal and thermotropic gelation was modeled according to [6] for the PANDMFwater system [8][9][10][11].…”

“…The kinetics of isothermal and thermotropic gelation was modeled according to [6] for the PANDMFwater system [8][9][10][11]. In analyzing the results of modeling, the qualitative difference in the gelation kinetics caused by diffusion alone (Fig.…”

“…The study of gelation in spinning chemical fibres [8][9][10][11] from polymer solutions allowed determining the basic quantitative characteristics of gelation: total gelation time t g ; gelation time stages and their duration t g = t 0 + t cr , where t 0 is the time required for the concentration of precipitator or the temperature in the center of the jet (r = 0) change by approximately 1% relative to the initial value; t cr is the duration of the change in these quantities from the initial value to the value necessary for ; the rate of the change in the gel thickness in time dR g (t)/dt. During gelation, there are two qualitative time stages in the polymer solution: the time required for attaining a metastable state in the system, i.e., attaining the spinodal in the phase diagram, t 0 , and the time required for gelation of the system in the complete thickness of the fibre t g .…”

“…Table 1 reports the calculated ratios for estimations of the gelation characteristics obtained from approximate analytical mathematical models for each mechanism separately [8][9][10]. The following notation is used: α is the heat transfer coefficient; λ is the thermal conductivity; β is the coefficient in the expression ν(x) = ν 0 exp(βx) from [11]; ρ is the density of the polymer solution; η is the dynamic longitudinal viscosity; θ c = (c g -c cr )/(C g -C 0 ) is the reduced concentration of precipitator in the fibre; θ cr is the dimensionless temperature of the solution or concentration of precipitator in the fibre corresponding to the values of T cr or C cr ; θ T = [T(r, t) -T g ]/(T 0 -T g ) is the reduced dimensionless temperature; a is the thermal diffusivity; A = (1 + n/Bi) -1 ; B = 2(1 + m/Bi)(1/(m + 2) + 1/Bi) -1 ; n and m are quantities determined from the corresponding equations in [9]; Bi = αR/λ is the Biot number; C 0 is the initial concentration of precipitator in the polymer solution; C cr is the concentration of precipitator at the time of the phase transition, whose values are determined from the phase diagram of the polymersolventprecipitator system at a certain fibre temperature; C g is the concentration of precipitator in the spinning bath; D is the diffusion coefficient of the precipitator; erf -1 is the inverse function of erf the Gaussian error function; Fo C = Dt/R 2 is the Fourier number for diffusion; t is the current time; Fo 1a is the Fourier number for heat transfer determined from the corresponding equations in [9]; Fo 1d = 0.081 is the Fourier number for diffusion, defined in [8]; R is the fibre radius; S = πR 2 is the fibre radius; S = πR 2 is the area of the section of the polymer jet; T 0 is the initial temperature of the solution; T cr is the temperature of the polymer solution at the time of the phase transition, determined from the phase diagram of the polymersolventprecipitator system at certain precipitator and polymer concentrations; T g is the ambient temperature (in the spinning bath); t ∞ is the total diffusion time (when concentration equilibrium appears); ν, ν′ are the longitudinal velocity of the fibre and its gradient; ν 0 is the initial longitudinal velocity; x is the coordinate in the direction of movement; x 0 is the value of coordinate x corresponding to the value of ν′ cr at which the phase transition is possible with all other parameters fixed.…”