Maxwell model references

In Eq. (11.1), P is permeability, < z is the volume fraction of the dispersed zeolite, the MMM subscript refers to the mixed-matrix membrane, the P subscript refers to the continuous polymer matrix and the Z subscript refers to the dispersed zeolite. The permeabiUty of the mixed-matrix membrane (Pmmm) can be estimated by this Maxwell model when the permeabilities of the pure polymer (Pp) and the pure zeoUte (Pz), as well as the volume fraction of the zeoUte (< ) are known. The selectivity of the mixed-matrix membrane for two molecules to be separated can be calculated from the Maxwell model predicted permeabiUties of the mixed-matrix membrane for both molecules. 

The rheological consequences of the Maxwell model are apparent in stress relaxation phenomena. In an ideal solid, the stress required to maintain a constant deformation is constant and does not alter as a function of time. However, in a Maxwellian body, the stress required to maintain a constant deformation decreases (relaxes) as a function of time. The relaxation process is due to the mobility of the dashpot, which in turn releases the stress on the spring. Using dynamic oscillatory methods, the rheological behavior of many pharmaceutical and biological systems may be conveniently described by the Maxwell model (for example, Reference 7, Reference 17, References 20 to 22). In practice, the rheological behavior of materials of pharmaceutical and biomedical significance is more appropriately described by not one, but a finite or infinite number of Maxwell elements. Therefore, associated with these are either discrete or continuous spectra of relaxation times, respectively (15,18). 

Another model referred to in the literature as a diffusion model [50] is similar in nature to the BFM, but is derived by assuming the membrane can be modelled as a dust component (at rest) present in the fluid mixture. The equations governing species transport are developed from the Stefan-Maxwell equations with the membrane as one of the mixture species. The resulting equation for species i is identical to Eq. (4.4) [50], thus the BFM and this diffusion model are equivalent. 

As has already been said and can be observed in Figures 12.5 and 12.7, results deviate from Maxwellian behavior at high frequencies. An upturn is observed in G", which has been ahributed in other systems to a transihon of the relaxation mode from reptation-scission (Cates model) to breathing or Rouse modes [28, 31]. In the systems presented in this chapter, if the relaxation mechanisms were just living reptation at long times -I- Rouse relaxations at short ones, results should be fitted by Equations 12.3 and 12.4, where a Rouse relaxation mode has been added to the Maxwell model, subscripts M and R referring to Maxwell and Rouse relaxations, respectively ... 

The only parameter in (11) having dimensions of time is C. Although this is not a relaxation time per se, it can be associated with a "Maxwell-type relaxation time, as follows. Although the linear Maxwell model predicts a constant (Newtonian) viscosity, it may be generalized by utilizing a co-rotational reference frame which follows the local rotation and translation of each fluid element [9]. When a term is added to account for the high shear limiting behavior, the result is the co-rotational form of the Jeffreys model ... 

The relaxation period defines the behavior of the system, in accordance with the Maxwell model with respect to the timescale of the applied stress. If the time t during which stress is applied is greater than the relaxation period, that is, t > t the system has properties similar to those of a viscous liquid, while at t t the system behaves like an elastic solid. The flow of glaciers and other processes of strain development in mountains and cliffs are representative examples of such behavior. In rheology, the ratio of a material s characteristic relaxation time to the characteristic flow time is referred to as the Deborah number. This parameter plays an important role in describing the response of various materials to different stresses. 

Fig. 8 Linear elastic and viscous modulus functions G co, T) and G"(a>, T) of gum EPDM2504, drawn using the G of a six elements generalized Maxwell model and the respective Cl, C2 parameters of a WLF type equation with 100 °C as reference temperature experimeutal data from a frequency-temperature sweep protocol at 1 deg. strain amplitude with a closed-cavity torsional harmonic rheometer are displayed for comparison with the calculated maps...

This model, often referred to as the upper convective Maxwell model, is weakly non-linear in that it predicts a first normal stress, but no shear thinning effects, i.e, the shear stress increases linearly with shear rate so that the viscosity is independent of shear rate. Combining Eqs. 2, 4, 5 and 6, we see that the tube model predicts the viscosity to be. 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l < n < 2) as y increases further above a certain critical value. 

Dividing the viscous parameter rj with a dimension unit of Pa-s by with a dimension unit of Pa provides the parameter X generally referred to as the relaxation time. The time-dependent retardation strain response of the Maxwell model can then be expressed as ... 

In this equation, P is permeabihty, is the volume fraction of the dispersed phase, the MM subscript refers to the mixed-matrix membrane, the M subscript refers to the continuous matrix, and the D subscript refers to the dispersed phase. Provided a researcher knows the volume fraction of the dispersed phase and the permeability through the two pure materials, the calculation is uncomphcated. A vast majority of researchers compared their mixed-matrix membrane results to either the Maxwell model or some extension of this equation. 

The Maxwell model and Hasselman-Johnson models were used to produce the curves provided in Figure 2. The equations are respectively provided below for reference ... 

In Chapter 2 when the Maxwell and Kelvin models were analysed, it was found that the time constant for the deformations was given by the ratio of viscosity to modulus. This ratio is sometimes referred to as the Relaxation or Natural time and is used to give an indication of whether the elastic or the viscous response dominates the flow of the melt. 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

We end this section with a comment referring to the Cauchy data for the scalars. In standard Maxwell theory, the Cauchy data are the eight functions A(i,6o<4M, and there is gauge invariance. In this topological model, they are the four complex functions (r, 0), 0 (r. 0), that is, eight real functions, constrained by the two conditions x V< >k) (V0 x V0 ) =0, k = 1,2, to ensure that the level curves of k will be orthogonal to those of 0. It is not necessary to prescribe the time derivatives 9o4>, 000 since they are determined by the duality conditions (138), as explained above. 

The local equivalence that we have just proved implies that the predictive contents of the Maxwell s theory and of this topological model are exactly the same when referred to local experiments, as most of them are. Accordingly, it is not possible to discern between the two by viewing locally. This is the operative meaning of local equivalence. 

The linear Maxwell equations appear in the model as the linearization by change of variables of nonlinear equations that refer to the scalars < ),0. This introduces a subtle form of nonlinearity that we call hidden nonlinearity. For this reason, the linearity of Maxwell s equations is compatible with the existence of topological constants of the motion. 

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

Reference [34] followed by several groups ([49, 51] to name a few) inspired from this statistical approach initiated in physics by Maxwell to develop a similar theory applied to the description of a neural population. Basically a neural population is described by its population density p(w,t) where w is the state of a neuron (scalar in the case of the LIF model, a vector containing the state variables in the case of... 

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