Stress tensor molecular expressions

This hydrostatic approach also yields a formal closed formula for y in terms of the components of the stress tensor. When the stress tensor is expressed in terms of molecular variables, the resulting statistical mechanical formula for y provides a direct means for the calculation of surface tension. For example, it may be used directly to compute the surface tension of dilute ionic solutions (6). It also illustrates in molecular detail the iterative subtractive procedures that lead to the excess functions of the familiar phenomenological approach. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

The macroscopic relationship between the total molecular stress tensor, which includes both the hydrostatic- or thermodynamic pressure and the viscous stresses is expressed as ... 

To derive a molecular expression for the stress tensor component which is the basic quantity if one wishes to compute pseudo-experimental data F (/i) /R, we start from the thermodynamic expression for the exact differential given in Eq. (1.59), where the strain tensor [Pg.202]

By analogy with Appendix E.3 we derive molecular expressions for various (diagonal) components of the stress tensor (7 — x, y, or z) by realizing that w e may write... 

This completes the discussion of the various contnbutions to the stress tensor. Table 1 gives a summary of the expressions for the flux expressions that are obtained by taking the first term in the Taylor-senes expansions of the fluxes, and Table 2 summarizes the complete expressions (except for the inter-molecular contributions). 

Microscopic expression for die stress tensor Let us now study the viscoelastic properties using molecular models. As was discussed in Chapter 3, the macroscofnc stress of the polymer solutions is written as (see eqn (3.133))... 

Following the scheme of MNET, a Fokker-Planck equation was obtained from which a coarse-grained description in terms of the hydrodynamic equations was derived in turn. Molecular deformation and diffusion effects become coupled and a class of non-linear constitutive relations for the kinetic and elastic parts of the stress tensor are obtained. The expression for the stress tensor can be written in terms of dimensionless quantities like... 

In a non-ideal (viscous) fluid, momentum may also be transferred across a surface through the molecular motions and interactions within the fluid. These are expressed through the total stress tensor, n, where Ttij is the flux of positive j momentum in the ith direction. 2 is a second-order symmetric tensor, and the rate of flow of momentum as a result of molecular motions is given by (see Figure 3.2) ... 

Although the experiments described in the foregoing section are very helpful for developing our intuition about the behavior of viscoelastic fluids such as polymers, they are not suitable for obtaining and cataloging information about specific polymeric materials. For the characterization of polymers it is necessary to make careful measurements of stresses in systems where the velocity or displacement field is known within rather strict limits. These rheometric experiments provide information about one or more of the stress components as functions of shear rate, frequency, or of other controllable variables these functions are generally referred to as material functions , since they are different for each material. Once these material functions have been measured, they can be used to test various empirical or molecular expressions for the stress tensor (that is, the constitutive equation), or they can be used to establish the values of the parameters that appear in these stress-tensor expressions. 

Now, the material element a is subjeaed to a small, virtual, isochoric displacement E. The unit cell exhibits an isochoric deformation due to this displacement (see Figure Al). Both the microscopic and maaoscopic energies change on displacement, and comparison of these changes allows us to derive a microscopic (molecular) expression of the deviatoric part of the stress tensor, oj, as explained below. 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

In general the net macroscopic pressure tensor is determined by two different molecular effects One pressure tensor component associated with the pressure and a second one associated with the viscous stresses. For a fluid at rest, the system is in an equilibrium static state containing no velocity or pressure gradients so the average pressure equals the static pressure everywhere in the system. The static pressure is thus always acting normal to any control volume surface area in the fluid independent of its orientation. For a compressible fluid at rest, the static pressure may be identified with the pressure of classical thermodynamics as may be derived from the diagonal elements of the pressure tensor expression (2.189) when the equilibrium distribution function is known. On the assumption that there is local thermodynamic equilibrium even when the fluid is in motion this concept of stress is retained at the macroscopic level. For an incompressible fluid the thermodynamic, or more correctly thermostatic, pressure cannot be deflned except as the limit of pressure in a sequence of compressible fluids. In this case the pressure has to be taken as an independent dynamical variable [2] (Sects. 5.13-5.24). 

For energy minimizations at least the first derivatives of to the atomic coordinates are necessary. In molecular dynamics the pressure tensor is needed as well. Complete expressions for energies, forces, and stresses have been given by Karasawa and Goddard for n = I and n =6 note that their h corresponds to our 2nH. 

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