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Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2012 | OriginalPaper | Buchkapitel

2. Basic Phenomena and Definitions

verfasst von : Rainer Kimmich

Erschienen in: Principles of Soft-Matter Dynamics

Verlag: Springer Netherlands

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Abstract

This chapter provides a compilation of the basic concepts on which soft-matter dynamics is founded. This implies interaction forces, equations of motions, and translational and rotational categories of molecular dynamics. The objective is to have a closed representation in a nomenclature consistent with the subsequent applications. Moreover, a number of less common topics such as Casimir forces and other electrochemical effects will be addressed as well in view of the attention these subjects are attracting in the context of colloidal suspensions and micro-electromechanical systems (MEMS).

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Vorheriges Kapitel Introduction
Nächstes Kapitel Noninvasive Methods
Fußnoten
1
We restrict ourselves to the treatment of two-particle interactions. Three- or more-particle contributions can be finite but are practically (and didactically) of minor importance. References [1‐3] are recommended for further reading in this matter. For crude estimations, it suffices to sum up all possible pair interactions and assume that these binary contributions are independent of each other. This is called the pairwise additivity approximation providing the total potential energy in the system with an accuracy of about 10%.
 
2
The term “dispersion” alludes to the fact that this interaction is mainly due to the weakly bound electrons that are responsible for dispersion of light.
 
3
In this context, it should be mentioned that there is a certain analogy to forces based on fluctuations of a non-quantum-electrodynamic nature: The critical Casimir force between colloid particles and surfaces is the result of concentration fluctuations at the critical point of fluid mixtures [9]. At the critical point, the correlation length of such fluctuations diverges. Depending on the chemical nature of the confining surfaces, the forces can be attractive or repulsive.
 
4
An analogous and illustrative wave phenomenon is the following example familiar to everybody: A piece of timber floating in harbor water apparently tends to “stick” to the quay wall. It is literally “attracted” because water waves steadily existing between wall and timber are restricted with respect to the wavelength. The nature and origin of the waves are totally different in this case, of course.
 
5
Apart from this, the tiniest fluctuation of the initial conditions of the involved particles tends to cause deterministic chaos making the resulting trajectories unpredictable. Expressed in a more flowery language, this is also known as the “effect of the stroke of a butterfly” on a macroscopic system (see, e.g., Ref. [19]).
 
6
Applying a second-rank (Cartesian) tensor \( \mathop {A {\scriptstyle \sim}} = \left({\begin{array}{llll} {{{a}_{{11}}}} & {{{a}_{{12}}}} &{{{a}_{{13}}}}\\{{{a}_{{21}}}} & {{{a}_{{22}}}} & {{{a}_{{23}}}}\\{{{a}_{{31}}}} & {{{a}_{{32}}}} & {{{a}_{{33}}}}\\\end{array} } \right) \) to a (column) vector \( {B} = \left( {\begin{array}{llll} {{{b}_1}} \\{{{b}_2}} \\{{{b}_3}} \\\end{array} } \right) \) gives a new (column) vector \( {C} = \mathop {A {\scriptstyle \sim}}\cdot {B} \) with elements \( {{c}_i} = \sum\nolimits_{{j = 1}}^3 {{{a}_{{ij}}}} {{b}_j} \), \( i = 1,2,3 \) (rule of thumb: “row elements times column elements”).
 
7
The Kronecker symbol is defined by \( {{\delta}_{{ij}}} = 1 \) for \( i = j \) and \( {{\delta}_{{ij}}} = 0 \) otherwise.
 
8
Provided that the time scale is long enough, any material subjected to thermal molecular motions will flow indeed. However, it is a widespread misbelief that medieval glass windows in churches are thicker at the bottom due to viscous flow over the centuries. It is true that window glass flows under gravitational stress in principle. But this can perceptibly occur only on time scales much longer than the age of the oldest churches.
 
9
For a discussion of the thermodynamically somewhat delicate term “pressure” under flow conditions, see Ref. [20], for example.
 
10
To make clear that viscoelasticity is not under consideration here, one better speaks of the zero-shear viscosity, where “zero shear” means that the strain rate is slow relative to any molecular dynamic processes. Sometimes, the viscosity per se introduced above is called the dynamic or absolute viscosity. It is distinguished from the kinematic viscosity defined as the dynamic viscosity divided by the density. Further terms in use are the volume viscosity (or bulk viscosity) which however matters only with sound phenomena in compressible fluids.
 
11
In the case of compressible fluids, the so-called second viscosity coefficient \( \lambda \) must be taken into account in addition. The diagonal elements are then extended to \( {{\tau}_{{ii}}} = \lambda {\nabla} \cdot {v} + 2{{\eta}_0}\partial {{v}_i}/\partial i \), where \( i = x,y,z \) [21].
 
12
For an explanation of exponential correlation functions, see Sect. 1.1.3. A well-known example for this sort of statistics is radioactive decay with the mean lifetime as the time constant. Note that Eq. (2.142) follows from the Langevin equation as well if supplemented by an inertial term as will be shown below in the context of Eq. (2.161) and the explicit expression for the time constant \( {{\tau}_v} \) given in Eq. (2.162).
 
13
The central-limit theorem [22] states that sums of the form \( {{\sigma}_N} = \sum\nolimits_{{j = 1}}^N {{{\xi}_j}} \) which are repeatedly formed of \( N \) random and statistically independent variables \( {{\xi}_j} \) are distributed according to a Gaussian function in the limit \( N\to \infty \). An important characteristic of this theorem is that the random distribution, with which the variables vary, may be arbitrary provided that it is the same for each variable and that its second moment is finite.
 
14
The \( \nu \)-th moment of a distribution function \( g\left( \Omega \right) \) of a variable \( \Omega \) is defined by \( {{M}_{\nu }} = \int_{{ - \infty }}^{\infty } {g\left( \Omega \right)} {{\Omega}^{\nu }}\rm{d}\Omega \). If the distribution is even, all odd moments vanish and vice versa. The second moment is the mean-square fluctuation of the variable. In the case of a Gaussian distribution (which is an even function), the second moment determines all higher moments according to \( {{M}_4} = 3M_2^2 \), \( {{M}_6} = 15M_2^3 \), … or generally \( {{M}_{{2j}}} = 1\cdot 3\cdot 5\cdots \left( {2j - 1} \right)M_2^j \) where \( j \) is a natural number.
 
15
Synonymous expressions in use are probability distribution and probability density function.
 
16
In macroscopically isotropic media, the correlation length is defined analogous to the correlation time (see Eq. (1.7), for instance): If \( G(r) = \left\langle {{A}(r)\cdot {{{B}}^{*}}(0)} \right\rangle \) is the spatial correlation function linking two spatial functions \( {A}\left( {{{{R}}_1}} \right),{B}\left( {{{{R}}_2}} \right) \) in a distance \( {r} = {{{R}}_2} - {{{R}}_1} \), the correlation length is given by \( \xi = {{\left[ {G(0) - G(\infty )} \right]}^{{ - 1}}}\int_0^{\infty } {\left[ {G(x) - G(\infty )} \right]{\rm d}{\it x}} \), where \( x \) is an arbitrary coordinate of \( {r} \).
 
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Metadaten
Titel
Basic Phenomena and Definitions
verfasst von
Rainer Kimmich
Copyright-Jahr
2012
Verlag
Springer Netherlands
Buch
Principles of Soft-Matter Dynamics

Print ISBN: 978-94-007-5535-2

Electronic ISBN: 978-94-007-5536-9

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-94-007-5536-9_2