Macroscopic particle statics and dynamics

Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10 m. The retarded dispersion energy varies as R at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties... 

We first examine the statics and dynamics of a one-dimensional macroscopic particle of mass M. Specifically, we study a relaxation process xp xp=o that takes the particle from an initial point of static equilibrium xp to a final point xp = o (Fig- 3.1). 

Thus, microscopic particle statics is very similar to and hardly more complex than macroscopic particle statics. We will see, however, that the similarity of microscopic and macroscopic particle behavior will not persist when we move from statics to dynamics. 

In the absence of external fields the suspension under consideration is macroscopically isotropic (W = const). The applied field h (we denote it in the same way as above but imply the electric field and dipoles as well as the magnetic ones), orienting, statically or dynamically, the particles, thus induces a uniaxial anisotropy, which is conventionally characterized by the orientational order parameter tensor (Piin h)) defined by Eq. (4.358). (We remind the reader that for rigid dipolar particles there is no difference between the unit vectors e and .) As in the case of the internal order parameter S2, [see Eq. (4.81)], one may define the set of quantities (Pi(n h)) for an arbitrary l. Of those, the first statistical moment (Pi) is proportional to the polarization (magnetization) of the medium, and the moments with / > 2, although not having meanings of directly observable quantities, determine those via the chain-linked set [see Eq. (4.369)]. 

In Section II we compare particle mechanics in the slow and fast variable timescale regimes. We start the discussion by showing the following. For damped macroscopic particles, the potential energy function whose minima locate the particle s points of static equilibrium also produces the forces which drive its dynamics. For damped microscopic particles, in contrast, the potential that determines the particle s statics may or may not produce the forces that drive its dynamics. 

Figure 3.1. The statics and relaxation dynamics of a one-dimensional macroscopic particle. A particle of mass M bound to the origin by a potential U(x) and also subject to a constraint force F, is initially in static equilibrium at the minimum xp of Uf(x) = U x) —xF.F is then removed so that Up(x) V x), and the particle relaxes to a new point of static equilibrium = 0, the minimum of U x). As discussed in the Appendix, Gibbs has noted that particle relaxation processes xp o xp are analogous to thermodynamic relaxation processes F Fp.

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

As a result of magnetic inter-particle interactions, each nanoparticle will feel a net local interaction field that can be modelled by a time-dependent local applied field, Hint(t). If the time variation of the local interaction field is either very fast or very slow compared to the relevant characteristic times (e.g., i) of the particle, then Hi t(t) can in turn be modelled as a static local field, that will, of course, depend on temperature and macroscopic applied field. The distributions of particle positions, orientations, and supermoments will determine the distribution of local interaction fields. These interaction fields are present in zero applied field and dramatically affect the behaviors of the individual nanoparticles and, consequently, of the sample as a whole. They achieve this in two important ways. First, they change the equilibrium magnetic properties of the sample, giving rise, for example, to superferromagnetic ordering or interaction Curie-Weiss behaviors (see below). Second, and possibly more importantly, they affect dynamic response, via their influence on SP dwell times. 

However, before turning to dynamics, we note that microscopic and macroscopic statics actually do difler in one significant way. This occurs because only a particle of molecular size is sensibly affected by microscopic fluid fluctuations like those noted after Eq. (3.7). 

Within the context of cross-scaling systems, they are homogeneous with both microscopic molecular dynamics and macroscopic smoothed particle hydrodynamics techniques. From a numerical standpoint, we do not need to switch over from particles to a static grid with control parameters, such as the Knudsen number. 

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