External force definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

By definition, the equilibrium distribution /o is one which does not depend on time. For simplicity assume further that the system is uniform in space so that /o is not a function of position, and set all external forces F = 0. The LHS of equation 9.32... 

The vertical spring and mass is an example of a stable system and by definition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a finite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail. 

When describing the effect of an external force, we must first define the force itself. A lay person s definition of a force is the amount of effort to get the desired effect. As scientists, we need a more precise definition of force. With a precise definition we can understand and quantify the effect of an applied force on a polymeric material. The mathematical definition of force is the work (which is a form of energy) required to move an object over some distance. Another way to define a force is in terms of the acceleration it creates when applied to some object of a mass m. In our everyday experiences, the first explanation is a simple idea to relate to. When we push a stalled car we exert a force on it. We could easily quantify the force from the weight of the car, the slope of the hill it is sitting on, and how far we must push it. Once we begin to talk about forces in polymer systems, the ideas become a bit more complicated. For example, the force required to open a bag of candy is defined by the work required to deform the bag until it ruptures by overcoming the intermolecular forces which hold the plastic together. 

On the contrary, the definition of the collision process, Eq. (429), is such that through a sequence of such events, the perturbation caused by the external force may be propagated at long distances. For instance, in the diagram of Fig. 21, corresponding to a typical term of the iterative solution of Eq. (428), the T operators are not localized around the B-particle. This allows long-range hydrodynamical effects. 

The sign convention for work W follows namrally from the definition in Equation (3.3). When the external force vector is in the same direction as the displacement vector, the surroundings do work on the system, 0 is equal to 0, cos 0 is equal to 1, and W is positive. When the external force vector is in the opposite direction from that of the displacement vector, the system does work on the surroundings, 9 is equal to tt, cos 0 is equal to —1, and W is negative. Thus, the convention that W represents the work done on the system by its surroundings follows namrally from the definition of work. ... 

There are three definitions needed to accurately describe this process. (1) Compaction is the compression and consolidation of a two-phase (particulate solid/gas) system by the application of an external force (2) compression causes an increase in the apparent density (or a reduction in volume) by the displacement of air and (3) consolidation is defined as an increase in mechanical strength due to particle-particle interaction [1,2]. 

Figure 21.1 (a) An environmental system is a subunit ( IN ) of the world separated from the rest of the world ( OUT ) by a boundary (bold line). The dynamics of the system is determined by internal processes and by external forces (see Box 21.1 for definitions). There is an output of the system to the environment, whose effect on the external forces is neglected in the model (no feedback from IN to OUT . 

Note that the employed definition of a system contains an important asymmetry between the system and the external world. The system s description includes the influence of the external forces on the system, but not the reverse, the influence of the system on the outside world. From a mathematical viewpoint the equations that describe the system s behavior contain the external forces, but the latter have to be taken from information outside the model. As will been shown, the specific hierarchy between system and outside world can often (but not always) be justified based on the respective strength of the interactions. Take the system of the earth. Solar radiation is a very strong driving force for the earth, but the back-radiation from the earth to the sun is so tiny that nobody would want to include it as a feedback mechanism in a radiation model of the sun. 

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

For the more complicated molecular models such as, for example, those that assume central forces, we replace the above set of parameters by a new set involved in defining the force field. If we add to this the problem of complex molecules (i.c., those with internal structure), then there is the additional set of parameters needed to define the interactions between the internal molecular motions and the external force fields. From the point of view of the hard sphere model this would involve the definition of still more accommodation coefficients to describe the efficiency of transfer of internal energy between colliding molecules. 

The first, the quantity of compensated heat, has a definite value when we know the initial state 0 and the final state 1 between which the modification is taken. This is not so for the second like the quantity P, it cannot be determined if the initial and final states alone are known, but only when all the intermediate states through which the stem has passed and the external forces which have obliged the gfystem to pass from each to the following are known. 

The electromotive force of a galvanic cell is a measure of the electrical work which can be obtained from the reaction in the cell. The total or maximum work which can be obtained from the cell reaction includes also the work which is done against the external forces owing to the changes in volume (formation of gas, etc.) of the reacting substances. From the definition of affinity (p. 318) it follows, therefore, that the electromotive... 

Diffusion, by definition, is a spontaneous tendency to eliminate a concentration gradient. Because it involves no external forces, it is a constant but slow process. On the whole, however, diffusion is a fringe force in terms of the speed and scale of gas migration, and other more rapid processes are superimposed upon it. 

The mechanical definition of work is the product of the external force on a body times the distance through which the force acts. If a body moves in a straight line from point f to with a constant force, F, applied along the direction of the path, the work done on the body is... 

A number of papers appeared in the 1980s by Nielsen and Martin [47-49] and one in 2002 by Pendas [50] that employ the classical approach in the definition of pressure as explored by Slater [14] and others and embodied in Equation (17). This approach identifies the pv product with the virial of the external forces acting on the nuclei relating the pressure, in "analogy to classical thinking" [50], to the trace of a stress tensor, Equation (29)... 

The definition of the atomic contribution to the virial of the external forces of constraint is nontrivial. Keith has developed the procedure for the atomic partitioning of null properties properties such as the sum of the Feynman forces on the nuclei, that sum to zero for the entire molecule [51]. Thus in analogy with the expression for a total system, the energy of atom A is given by... 

We may wish, for example, to know the probability of finding a gas molecule at a definite spot in the box within which we suppose the gas to have been enclosed. If no external forces act on the molecules, we shall be unable to give any reason why a particle of gas should be at one place in the box rather than at another. Similarly, in this case there is no assignable reason why a particle of the gas should move in one direction rather than in another. We therefore introduce the following hypothesis, the principle of molecular chaos For the molecules of gas in a closed box, in the absence of external forces, all positions in the box and all directions of velocity are equally probable. 

The mobility can also be viewed as the drift velocity that would be attained by the particles under unit external force. Recall (9.50), which is the mobility in the special case of an electrical force. By definition, the electrical mobility is related to the particle mobility by Be = qB, where q is the particle charge. A particle with zero charge, has a mobility given by (9.78) and zero electrical mobility. 

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