Continuity equation polymer

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The third necessary ingredient for the model is a continuity equation for the non-volatile polymer melt... 

The expressions for the stress tensor together with the equations for the moments considered as additional variables, the continuity equation, and the equation of motion constitute the basis of the dynamics of dilute polymer solutions. This system of equations may be used to investigate the flow of dilute solutions in various experimental situations. Certain simple cases were examined in order to demonstrate applicability of the expressions obtained to dilute solutions, to indicate the range of their applicability, and to specify the expressions for quantities which were introduced previously as phenomenological constants. 

Of course, membrane thickness can be calculated supposing that its volume is given by the arithmetic addition of solvent and polymer volume. Assuming that the most important effect of matrix swelling/shrinking is to modify the solute diffusion coefficient Dg (and the Peppas-Reinhart equation [Equation 15.14] can be considered to account for this), permeation can be described by the continuity equation (Equation 15.1) where the generative term Rj is set to zero and the following initial and boundary conditions hold ... 

A modified version of this model was developed for a nonuniform polymer film (2). With these assumptions, the continuity equations for the gas and polymer phase may be written as... 

A mass balance enforced by the continuity equation governs the equation for the air-polymer interface profile ... 

Chiu FC, Ung MH (2007) Thermal properties rmd phase morphology of melt-mixed poly (trimethylene terephthalate)/polycarbonate blends-mixing time effect. Polym Test 26 338-350 Chuah HH (2004) Effect of process variables on bulk development of air-textured poly (trimethylene terephthalate) bulk continuous filament. J Appl Polym Sci 92 1011-1017 Chuah HH, Lin VD, Soni U (2001) PTT molecular weight and Mark-Houwink equation. Polymer 42 7137-7139... 

Transient molecular deformation and orientation in the systems subjected to flow deformation results in transient and orientation dependent crystal nucleation. Quasi steady-state kinetic theory of crystal nucleation is proposed for the polymer systems exhibiting transient molecular deformation controlled by the chain relaxation time. Access time of individual kinetic elements taking part in the nucleation process is much shorter than the chain relaxation time, and a quasi steady-state distribution of clusters is considered. TVansient term of the continuity equations for the distribution of the clusters scales with much shorter characteristic time of an individual segment motion, and the distribution approaches quasi steady state at any moment of the time scaled with the chain relaxation time. Quasi steady-state kinetic theory of nucleation in transient polymer systems can be used for elongation rates in a wide range 0 < esT C N. ... 

It should be noted that for termination by recombination (Fig. 10.5A), inherently an error is introduced regarding its contribution in the classical moment equations. For the individual living polymer continuity equations, it can be derived that the contribution of termination by recombination is... 

Equation (8.1) is the continuity equation for the polymer Eq. (8.2) describes the evolution of v due to the BZ reaction and the transport of the catalyst with the movement of the polymer network Recall that the catalyst is tethered to the chains and, thus, does not diffuse through the solution. Equation (8.3) characterizes the changes in u due to the BZ reaction, the transport of this activator with the solvent, and the diffusion of the activator within the solvent We assumed for simplicity that it is solely the polymer-solvent interdiffusion that contributes to the gel dynamics hence, in Eq. (8.3), we took into account that < vfPi + (1 — )vf i = 0, where vf i is the solvent velocity [2]. 

Throughout this text we will assume that polymer melts are incompressible liquids, by which we mean that the density never changes with position or time. This is clearly an approximation that must be relaxed in some applications - injection molding, for example, where the compressibility of the melt becomes important because of the extremely high pressures - but the incompressibility assumption will suffice for our purpose here. If the density never changes in time or space, rates of change with respect to these variables (i.e., derivatives) must be zero dp/dt = 0, dp/dx = 0, etc.), and the continuity equation simplifies to... 

Equation 3.1 is the continuity equation for the polymer Equation 3.2 describes the evolution of v due to the BZ reaction and the transport of the catalyst with the movement of the polymer network. It may be recalled that the catalyst is... 

At time Z > 0, a polymer bank of radius rp t) displaces a brine bank of annular radius r (Z) — r (Z) within the reservoir. Substituting the aforementioned viscosity model into Darcy s law, invoking the continuity equation, and integrating over the velocity through the polymer bank, we obtain... 

Consider the time evolution of the number concentration of the polyelectrolyte chains c(jc, t) at the location x along the jc-direction and at time t. Here we have taken the jc-value for the polymer to be its center-of-mass value. According to the continuity equation (Section 6.4, Equations 8.1 and 8.2),... 

Consider the simplest situation of Figure 9.4a, where an absorbing spherical sink of radius R is at the center of the coordinate system and polymer chains are present in the solution around the sink. Following the classical theory of Smoluchowski (Chandrasekhar 1943), we assume that the sink absorbs the polymer chains as soon as the center-of-mass of a polymer chain approaches the surface of the sink. We identify the capture rate of polymer chains as the steady-state net flux of polymer chains into the absorbing sink. Let the initial number concentration of the polymer chains be cq. The polymer concenuation is continuously maintained as co at distances far from the sink. The polymer chains undergo only diffusion and there are no other convective contributions. The continuity equation for the number concentration of polymer chains in three... 

Since the total concentration of each polymer is conserved, then the continuity equation must be obeyed,... 

Assume that the structure of the porous material could be completely described by the subset of space in a bulk material where transport can occur. In principle, the governing equations of fluid motion could be solved, with providing the relevant boundary conditions for the flow problem. For example, for fluid flow in a porous polymer, the continuity equation and the equations of motion may be written in general form ... 

The inclusion of the complexation reaction in facilitated transport greatly increases the mathematically complexity of the flux equations at steady state. The concentration through the polymer of the penetrant, ca, the carrier, Cc, and the penetrant-carrier complex, ca c, must satisfy the following continuity equations ... 

The continuity equation relates the mass polymer flow rate, m, to Vi as follows ... 

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