Collision frequency function motion

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

Aerosols are unstable with respect to coagulation. The reduction in surface area that accompanie.s coalescence corresponds to a reduction in the Gibbs free energy under conditions of constant temperature and pressure. The prediction of aerosol coagulation rates is a two-step process. The first is the derivation of a mathematical expression that keeps count of particle collisions as a function of particle size it incorporates a general expression for tlie collision frequency function. An expression for the collision frequency based on a physical model is then introduced into the equation Chat keep.s count of collisions. The collision mechanisms include Brownian motion, laminar shear, and turbulence. There may be interacting force fields between the particles. The processes are basically nonlinear, and this lead.s to formidable difficulties in the mathematical theory. 

The quantity XjXj = 0 because the motion of the two particles is independent. The collision frequency function first derived by Smoluchowski is then obtained by substitution in (7.13) ... 

The various collision mechanisms are compared in Fig. 7.7 which shows the collision frequency function for l- m particles interacting with particles of other sizes. Under conditions corresponding to turbulence in the open atmosphere ( j ss 5cm /sec- ), either Brownian motion or differential sedimentation plays a dominant role. Brownian motion controls for particles smaller than 1 jam. At lower altitudes in the atmosphere and in turbulent pipe flows, shear becomes important. 

Consider the flow of an aerosol through a 4-in. duct at a velocity of 50 ft/sec. Compare the coagulation rate by Brownian motion and laminar shear in the viscous sublayer, near the wall. Present your results by plotting the collision frequency function for particles with dp = 1 /im colliding with particle.s of other sizes. Assume a temperature of 20 C. Hint In the viscous. sublayer, the velocity distribution is given by the relation... 

In many, if not most, cases of practical interest, the fluid in which the particles are suspended is in turbulent motion. In Chapter 7. the effects of turbulence on the collision frequency function for coagulation were di.scussed. In the last chapter, nucleation in turbulent How was analy ,ed through certain scaling relations based on the form of the concentration and velocity fluctuations in the shear layer of a turbulent jet. In this section the GDE for turbulent flow is derived by making the Reynolds assumption that the fluid velocity and size distribution function cun be written as the sum of mean and fluctuating components ... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

Expressions of power absorption are also obtained for systems in which ion-molecule reactions occur. Comisarow s theory involves two types of colhsion frequencies the reduced nonreactive collision frequency c, and the chemically reactive collision frequency k, the first-order reaction rate of an IMR. c is introduced into the equation of motion (5) from which A is derived, k is introduced into a kinetic equation which gives the ion currents for primary, secondary and tertiary ions as a function of time. The calculation of the total power absorption in the case of IMR is a counting procedure according to Eq. (9 a) which sums all the ions produced in all cell regions and all the power absorptions. 

Nuclear motion becomes important, then, when neutron energies are in the order oi kT kT = 0.025 ev for T = 20°C). The effects of the nuclear motion are significant in connection with two different aspects of the interaction phenomenon, namely, the frequency function for scattering collision and the specification of neutron cross sections for neutron energies around kT, It will be recalled that the frequency functions for... 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

Luminescence lifetime spectroscopy. In addition to the nanosecond lifetime measurements that are now rather routine, lifetime measurements on a femtosecond time scale are being attained with the intensity correlation method (124), which is an indirect technique for investigating the dynamics of excited states in the time frame of the laser pulse itself. The sample is excited with two laser pulse trains of equal amplitude and frequencies nl and n2 and the time-integrated luminescence at the difference frequency (nl - n2 ) is measured as a function of the relative pulse delay. Hochstrasser (125) has measured inertial motions of rotating molecules in condensed phases on time scales shorter than the collision time, allowing insight into relaxation processes following molecular collisions. 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

Photon Correlation. Particles suspended in a fluid undergo Brownian motion due to collisions with the liquid molecules. This random motion results in scattering and Doppler broadening of the frequency of the scattered light. Experimentally, it is more accurate to measure the autocorrelation function in the time domain than measuring the power spectrum in the frequency domain. The normalized electric field autocorrelation function g(t) for a suspension of monodisperse particles or droplets is given by ... 

The wavenumber displacement of a solute vibration is a complex function of both solute and solvent properties and can be explained in terms of weak nonspecific electrostatic interactions (dipole-dipole, dipole-induced dipole, etc.) and of strong specific association of solute with solvent molecules, usually of the hydrogen-bond type [140], It should be realized that the duration of vibrational transitions is very short with respect to motion of the solvent molecules e.g. for an O—H stretching vibration, the frequency is ca. 10 s ). Thus, it is possible to observe such transitions even for short-lived entities such as may arise after a collision in the liquid phase (collision complexes) [140],... 

Block equations can be constructed that describe the macroscopic magnetization of the spin system including the relaxation mechanism involving spin exchange caused by molecular collisions. These equations were solved as a function of the exchange frequency, wex, for the nitroxide free radical case. Spectra were computed by assuming the degrees of motional freedom observed with the labeled androstane or cholestane molecules localized at the air-water interface. 

We next evaluate the lineshape function (8.16) for two concrete situations in gases, (The complexity of molecular motions in liquids precludes computation of their dipole correlation functions in a text of this scope.) In the first situation, we imagine that we are examining lineshapes in the far-infrared spectrum of a collision-free, rotating polar molecule. Its dipole moment /Iq is assumed to rotate classically without interruption with angular frequency cwq about an axis normal to /Iq. In a dilute gas, we would then have... 

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