Macroscopic forces

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n -t 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-Iimited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

It is difficult to define turbulence. Intuitively, we associate it with the fine-structure of the fluid motion, as opposed to the flow pattern of the large-scale currents. Although it is not possible to describe exactly the distribution in space and time of this small-scale motion, we can characterize it in terms of certain statistical parameters such as the variance of the current velocity at some fixed location. A similar approach has been adopted to describe the motion at the molecular level. It is not possible to describe the movement of some individual molecule, but groups of molecules obey certain characteristic laws. In this way the individual behavior of many molecules sums to yield the average motion in response to macroscopic forces. 

This section breaks down as follows first, the macroscopic forces acting on the droplet growing at the pore opening will be discussed, then the balance equations where these forces are involved will be dealt with. However, the accurate derivation of these forces is not reported here because it is beyond the aim of this contribution and therefore only the expressions of the forces used in balance equations will be presented. 

The theory of Kaluza and Klein [89, 90] is based on an observation that of two macroscopic forces of Nature only gravitation can be ascribed to geometric features of four-dimensional space-time. In order to incorporate another interaction the logical development would be to consider an additional dimension and to examine if extra degrees of freedom provided by 15 covariant components of the five-dimensional symmetric tensor needed to specify the line element... 

Hitherto, the drift velocity has been related to macroscopic forces (e.g., the Stokes viscous force F = 6nrrf or the electric force F = zc X) through the relation... 

The adhesive force between a neutral particulate contaminant and the wafer is expected to be due to the attractive Van der Waal s interaction between molecules.This is a macroscopic force found by averaging over the force between all the molecules of a particle and the neighboring surface. For a spherical particle sitting on a flat wafer, it is known that surface roughness will cause the mean distance of separation between the particle and the wafer to be nonzero. The attractive force between these two entities acts along the normal between the sphere and the wafer, and is given by ... 

A mechanophore (blue in Fig. 2a) is a strategically designed chemical entity which responds to mechanical force in a predictable and useful manner (Fig. 2d-f). The polymer strand here acts as an actuator to transmit macroscopic force to the target. For a fully extended polymer chain, the maximum tension force is at the middle point of the chain contour. So the mechanophore should be incorporated into the middle of the chain with its active bond along the chain contotu (Fig. 2a) [15, 29, 32]. Examples of mechanochemical reactions include homolytic scission of weak bonds (diazo [33]), electrocyclic ring-opening (benzocyclobutenes [29], spiropyrans [32, 34 5], gem-dichlorocyclopropanes [46-49], ge/n-difluorocyclo-propanes [30, 50], and epoxide [51]), cycloreversion reactions (cyclobutane derivatives [52-56], Diels-Alder adducts [57, 58], 1,3-dipolar adducts [59, 60], and 1,2-dioxetanes [61]), dative bond scission [62-64], and flex-activated reactions [34, 65, 66], as recently reviewed by Bielawski [67]. 

In a later study, bis-NHC ruthenium-alkylidene complex was activated under compressive strain [87] (Fig. 16). In order to initiate Ru-mediated polymerisation of norbomene in solid state, polymer catalyst (34 kg mol ) and a norbomene monomer were incorporated in a high molecular weight poly(tetrahydrofuran) (pTHF) matrix (Mn=170 kDa, PDI=1.3) which provided the physical cross-linking through the crystalline domains and allowed macroscopic forces to be transferred to the metal-ligand bonds. Consecutive compressions showed that up to 25% of norbomene monomer was polymerised after five loading cycles. 

The pressure is the normal force per unit area due to molecular collisions. If there is a difference in pressure across a volmne of air in some direction, a macroscopic force, called the pressure gradient force, will act on the volmne accelerating the air toward the lower pressure. The pressure gradient force per unit mass is written... 

Traditionally, the collision is treated at the microscopic level, generating a molecular force that is multiplied by the number of molecules for producing the macroscopic force. The Formal Graph theory allows such modeling with Formal Objects working at the level of few entities that are the singletons (cf. Chapter 4, Section 4.4.4). However, as this concept has not been much developed in this book, this system is modeled in terms of collection, using only poles as Formal Objects. 

How can the spreading law V dd) be explained The macroscopic force pulling on the droplet is the unbalanced Young s force... 

If instead of imposing on the line a certain displacement velocity, we exert on it a macroscopic force per unit length given by... 

Here t) is the dynamic viscosity and V is the potential of external body forces. The latter may include macroscopic forces, such as gravity, as well as intermolecular forces exerted on the fluid by a bounding solid wall or support. The classic boundary condition on the solid wall is no slip ti = 0. The conditions on the free boundary are given by the tangential and normal stress balances and the kinematic condition which states that the interface moves with the local normal velocity of the flow. 

The force given by Eq. (10), when summed over all the filaments that are present at a particular instant, is equal to the macroscopic force of adhesion. But this will be true only if the force, Eq. (10), is less than the force required for interfacial separation, which we will now examine in detail. 

The net effect of this distortion of the substrate is that the direction of the force at the perimeter is no longer parallel to the plane of the polymer/substrate interface. Hence the requirement of normal force on the filament, to cause a particular rate of base shrinkage, will be larger, and the macroscopic force of adhesion will be larger. 

We can now examine the predictions made by this model in regard to the force of adhesion. The macroscopic force will be small, if one of two microscopic conditions hold ... 

If inequality (27c) holds during a very early stage of the separation process (regardless of the speed of separation) the macroscopic force will be low. The increment of work needed to start the separation of a filament base from the solid is small, and after a small force has been applied (sufficient to furnish that work) an even smaller force will be sufficient to complete the base separation. 

The macroscopic force will be large if ft > 1, if the product r] eA is large for individual filaments, and if the number of filaments that are present in unit area of the separation zone (Figure 4) is large. 

The predictions that we have just made, in regard to macroscopic force of adhesion, are in agreement with common observations, e.g., those reported by Kaelble,(34) by Gardon, (35,36) and by Aubrey.(37,38) See also, below, the discussion of Table 1. 

At the macroscale, capillary forces seldom have a noticeable effect on physical phenomena. The reason is that their magnitude is much smaller than that of the usual macroscopic forces, such as gravity. However, in two cases capillary forces may become important in space where gravity is... 

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