The Continuous Collision of Air Molecules With a Surface Results in

Molecular Collision

Fisher Variational Principle and Thermodynamics

A. Plastino , ... M. Casas , in Variational and Extremum Principles in Macroscopic Systems, 2005

6.1 The relaxation approximation to the Boltzmann transport equation

We focus attention upon a gas in which the effect of molecular collisions is always to restore a local equilibrium situation described by the Maxwell–Boltzmann PD f ₀(r, v) [16]. In other words, we assume that (i) if the molecular distribution is disturbed from the local equilibrium so that the actual PD f is different from f ₀, then and (ii) the effect of collisions is simply to restore f to the local equilibrium value f ₀ exponentially with a relaxation time τ of the order of the average time between molecular collisions. In symbols, for fixed r, v, f changes as a result of collisions according to

(42) $f(t)\text{undefined}=\text{undefined}{f}_0+[f-f_0]\exp -[t\text{/}\tau ].$

In these conditions (Appendix A), the ensuing Boltzmann equation becomes [16]

(43) $\frac{\partial f}{\partial t}+{\displaystyle \sum _{i=1}^3\text{undefined}\left[v_i\frac{\partial f}{\partial x_i}+{\dot{v}}_i\frac{\partial f}{\partial v_i}\right]=-\frac{f-f_0}{\tau }},$

a linear differential equation for f. We consider now a situation slightly removed from equilibrium: f = f ₀ + f ₁ with f ₁ ≪ f ₀, so that Eq. (43) becomes

(44) $\frac{\partial f}{\partial t}+{\displaystyle \sum _{i=1}^3\text{undefined}\left[v_i\frac{\partial f}{\partial x_i}+{\dot{v}}_i\frac{\partial f}{\partial v_i}\right]=-f_1\text{/}\tau .}$

The left-hand side of Eq. (44) is small, since the right-hand side is, by definition, small. As a consequence, we can evaluate it by neglecting terms in f ₁ and write

(45) $\frac{\partial f_0}{\partial t}+{\displaystyle \sum _{i=1}^3\text{undefined}\left[v_i\frac{\partial f_0}{\partial x_i}+{\dot{v}}_i\frac{\partial f_0}{\partial v_i}\right]=-f_1\text{/}\tau .}$

Since f ₀ is the Maxwell–Boltzmann PD, independent of time ([∂f ₀/∂t] = 0), we finally get the so-called Boltzmann equation in the relaxation approximation [16]

(46) ${\displaystyle \sum _{i=1}^3\text{undefined}\left[v_i\frac{\partial f_0}{\partial x_i}+{\dot{v}}_i\frac{\partial f_0}{\partial v_i}\right]=-f_1\text{/}\tau .}$

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Cross Section Data

Jean W. Gallagher , in Advances In Atomic, Molecular, and Optical Physics, 1994

A CONTROLLED FUSION ATOMIC DATA CENTER AT OAK RIDGE NATIONAL LABORATORY (HEAD: D. R. SCHULTZ)

The ORNL Controlled Fusion Atomic Data Center (CFADC) was formally established in 1965 to identify, compile, evaluate, and recommend data on atomic and molecular collision processes which are important in fusion energy research.

A comprehensive bibliographical database of references to pertinent articles is maintained. Covering the period from 1978 to the present, the database is kept up to date by a group of expert consultants who scan 120 journals for appropriate articles. The database consists of 24,500 entries. Specific subject categories are given in Table I. Reactants emphasized by the program are given in Table II.

Table I. ORNL Atomic Collisions Bibliography Categories

Main Categories:

A. Heavy particle—heavy particle interactions

B. Interactions of atoms and molecules with fields

C. Particle penetration in macroscopic matter

D. Particle interactions with solid surfaces

E. Electron–particle interaction

H. Photon collisions with heavy particles

J. Data compilations

K. Reviews and Books

L. Bibliographies

Subcategories:

General

Elastic scattering

Excitation

Dissociation

Fluorescence

Electron capture

Ionization

Recombination

Energy transfer

Collisional de-excitation

Collisional line broadening

Involving H or He

Associative reactions

Detachment from negative ions

Interaction potentials

Angular scattering

Attenuation

Table II. ORNL CFADC Bibliographies—Most Common Reactants

The particles listed here may be either incident or target. All elements of the periodic table exist in the bibliography; only the most common are listed in this table.

Atoms and Ions:

H, deuterium, tritium, and their ions

He, Li, Be, C, O, and their ions

Al, Ar, Ti, Cr, Fe, Ni, and their ions

Cu, Zn, and their ions Kr, Mo, and their ions

Xe, Ta, W, and their ions

Electrons

Molecules:

H2, H3, HeH, N2, O2, CO, CO2, and their ions

OH, H2O, CH, CH2, CH3, CH4, and their ions

The database is maintained on a personal computer at the CFADC and can be searched interactively by specifying atomic process, reactant species, and years to be searched. Another type of search performed is an author search. The search is independent of position in the list of authors. All references with that author's name will be listed.

Search requests can be made by phone to the CFADC at (615) 574-4701. Future plans of the CFADC include development of on-line access to the bibliography, a UNIX-based workstation for user access via INTERNET.

The CFADC also publishes recommended atomic collision and spectroscopic numerical data in the series Atomic Data of Controlled Fusion Research (popularly called the Red Book). A list of recent volumes of the Red Book is given in Table III. In some cases, the data from these volumes are available in electronic format on request by external users; see Table IV.

Table III. Recent CFADC Redbook Numerical Compilations

(1) Thomas, E. W. (1985). "Particle Interactions with Surfaces. ORNL-6088.

(2) Weise, W. L. (1985). "Spectroscopic Data for Iron." ORNL-6089.

(3) Phaneuf, R. A., Janev, R. K., and Pindzola, M. S. (1987). "Collisions of Carbon and Oxygen Ions with Electrons, H, H2, and He." ORNL-6090.

(4) Wiese, W. L. (1990). "Spectroscopic Data for Titanium, Chromium, and Nickel."

Table IV. IAEA Atomic and Molecular Unit Databases

(1) Bell, K. L., Gilbody, H. B., Hughes, J. G., Kingston, A. E., and Smith, F. J., (1983). "Recommended Cross Sections and Rates for Electron Impact Ionisation of Light Atoms and Ions: Hydrogen to Oxygen." J. Phys. Chem. Ref. Data 12, 891.

(2) Itikawa, Y., Hara, S., Kato, T., Nakazaki, S., Pindzola, M. S., and Crandall, D. H. (1985). "Recommended Data on Excitation of Carbon and Oxygen Ions by Electron Collisions." At. Data Nucl. Data Tables (ADNDT) 33, 149.

(3) Bottcher, C., Griffin, D. C., Hunter, H. T., Janev, R. K., Kingston, A. E., Lennon, M. A., Phaneuf R. A., Pindzola, M. S., and Younger, S. M. (1987). "Recommended Data on Excitation of Carbon and Oxygen Ions by Electron Collisions." Nuclear Fusion, Special Supplement.

(4) Phaneuf, R. A., Janev, R. K., Pindzola, M. S. (1987). "Collisions of Carbon and Oxygen Ions with electrons, H, H2, and He." Atomic Data for Controlled Fusion Research, 5. Report ORNL-6090/V5, Oak Ridge National Laboratory, Tennessee.

(5) Lennon, M. A., Bell, K. L., Gilbody, H. B., and Hughes, J. G. (1988). "Atomic and Molecular Data for Fusion. Part II—Recommended Cross Sections and Rates for Electron Ionisation of Light Atoms and Ions; Fluorine to Nickel." J. Phys. Chem. Ref. Data 17, 1285.

(6) Kato, T., and Nakazaki, S. (1989). "Recommended Data for Excitation Rate Coefficients of Helium Atoms and Helium-like Ions by Electron Impact." At. Data Nucl. Data Tables (ADNDT) 42, 313.

(7) Janev, R. K., Langer, W. D., and Evans, Jr. K. (1987). Elementary Processes in Hydrogen–Helium Plasmas. Springer-Verlag, Berlin.

(8) Barnett, C. F. (1987). "Collisions of H, H2, He and Li Atoms and Ions with Atoms and Molecules," Vol. 1. Report ORNL-6086/VI, Oak Ridge National Laboratory, Tennessee.

(9) Higgens, M. J., Lennon, M. A., Hughes, J. G., Bell, K. L., Gilbody, H. B., Kingston, A. E., and Smith, F. J. (1989). "Atomic and Molecular Data for Fusion, Part III. Recommended Cross Sections and Rates for Electron Ionisation of Atoms and Ions; Copper to Uranium." Culham Report CLM-R294, Abingdon, Oxfordshire.

The CFADC also participates in an international network of atomic and molecular data centers, coordinated by the IAEA (see Section III.B). This effort facilitates the evaluation and exchange of existing data for fusion, and addresses needs of fusion energy research. The CFADC distributes the IAEA Aladdin Databases listed in Table IV.

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Theoretical, Experimental, and Numerical Techniques

GEORGE EMANUEL , in Handbook of Shock Waves, 2001

Vibrational Modes

At room temperature, the vibrational mode of most diatomic molecules is inactive. In other words, the molecule is in its vibrational ground state, and molecular collisions are energetically insufficient for populating the first excited vibrational state with more than a miniscule fraction of molecules. The key parameter is the characteristic vibrational temperature T_v of the mode. A large T_v value means a relatively stiff bond between the atoms, and a large energy spacing between adjacent vibrational levels of the mode. For instance, T_v is 2219 K for O₂ and 3352 K for N₂. The vibrational contribution starts to be significant when T/T_v is about 0.2, which is well above room temperature for air. (When modeling air, an average T_v value of 3056 K can be used.) One diatomic exception is I₂, where T_v is only 309 K.

For triatomics, and larger polyatomics, there are multiple vibrational modes whose characteristic temperatures may range from fairly low values, near 300 K, to large values that are in excess of 4000 K. A large polyatomic, especially one with heavy atoms, that is, a molecule with a relatively large molecular weight, will have a number of partially excited vibrational modes at room temperature. As a consequence, its constant volume specific heat c_v has a pronounced temperature variation, even at room temperature.

Triatomics and larger polyatomics are rarely represented near room temperature as a perfect gas with any degree of accuracy. This is because of a relatively large T_c value and one or more active vibrational modes. In this circumstance, Eqs. (3.1.3) are inadequate.

We thus replace these equations with the general relations

(3.1.21a, b) $\begin{array}{cc}p=p(T,\upsilon ),& c_{\upsilon }^0\end{array}=c_{\upsilon }^0(T)$

where $c_{\upsilon }^0$ is the part of c_v that accounts for the contribution of the translational and internal modes, and $c_{\upsilon }^0$ , only depends on the temperature. For example, for a molecule without electronic excitation, we can write

(3.1.22) $c_{\upsilon }^0=\frac{5+\delta }{2}R+R{\displaystyle \sum _{\upsilon }}{g}_{\upsilon }{\left(\frac{{\theta }_{\upsilon }}{sinh{\theta }_{\upsilon }}\right)}^2$

where g_v is the degeneracy of a distinct vibrational mode,

(3.1.23) ${\theta }_{\upsilon }=\frac{T_{\upsilon }}{2T}$

and

(3.1.24) $\begin{array}{cc}\delta & =-2,\text{monatomic}\text{gas};\hfill \\ & =0,\text{liner}\text{polyatomic};\text{and}\hfill \\ & =1,\text{nonlinear}\text{polyatomic}\hfill \end{array}$

The first term on the right accounts for translational and rotational excitation, while the summation term is the harmonic oscillator model for the vibrational modes.

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Generalized Analysis of Motion Using Magnetic Field Gradients

Paul T. Callaghan , Janez Stepišnik , in Advances in Magnetic and Optical Resonance, 1996

F SPECIAL CASES OF INTEREST

We now consider two special cases in which the diffusion spectra are nontrivial and Eq. (66) may be used to evaluate the result of a modulated spin-echo experiment. The first case concerns slow molecular collision rates. As pointed out by Einstein (1956), the result

(69) $〈{\left[x\left(t\right)-x\left(0\right)\right]}^2〉=2Dt$

holds only in the limit of large t. More generally, one may write [Uhlenbeck and Ornstein, 1930]

(70) $〈{\left[x\left(t\right)-x\left(0\right)\right]}^2〉=2D\left[t-{\tau }_c\left(1-\exp \left(-t/{\tau }_c\right)\right)\right]$

where the correlation time ${\tau }_c=m/f$ , f being the coefficient of friction and m the mass of the Brownian particle. The friction for small molecules in liquids results in correlation times on the order of 10⁻⁹ s or less. This is much less than the time of the shortest spin-echo sequence at around 10⁻⁴ s. However, for macromolecules, the correlation time may become large enough to be measured by the spin-echo method (Zupančič et al., 1985; Kveder et al., 1988; Fatkillin and Kimmich, 1994). Figure 2 shows that in some polymers (Grinberg et al., 1987) the self-diffusion constant measured by NMR does exhibit an unusual behavior at short times.

Fig. 2. Fit of the displacement for diffusion with memory to NMR spin-echo measurements in polymers. The data corresponds to the spin-echo signal from polysulfane-polybutadiene copolymer in different solvents comprising a mixture of trichloroethane (TCE) and octane. The fits were obtained using Eq. (83).

Using Eqs. (9) and (70) and writing $\zeta ={\tau }_c^{-1}$ , the diffusion spectrum is given by (Wang and Ornstein, 1945)

(71) $D\left(\omega \right)=\frac{D{\zeta }^2}{{\omega }^2+{\zeta }^2}$

Suppose that we evaluate the case of a usual PGSE sequence comprising finite pulses of width δ and gradient amplitude G. The phase spectrum follows from Eqs. (59) and (64) as

(72) $\mathbf{F}\left(\omega ,\text{Δ}\right)=\gamma \mathbf{G}\frac{\left(1-e^{i\omega \text{Δ}}\right)\left(1-e^{i\omega \delta }\right)}{{\omega }^2}$

with

(73) $|\mathbf{F}\left(\omega ,\text{Δ}\right){|}^2={\left[\gamma \delta \text{Δ}\left|\mathbf{G}\right|\frac{\sin \left(\omega \delta /2\right)\sin \left(\omega \text{Δ}/2\right)}{\left(\omega \text{Δ}/2\right)\left(\omega \delta /2\right)}\right]}^2$

Figure 3 shows both spectra. Hence it follows that

Fig. 3. The phase spectrum for PGSE sequence and the diffusion spectrum (dotted) for the Uhlenbeck time-dependent self-diffusion.

(74) $\beta \left(2\text{Δ}\right)=\frac{1}{\pi }{\displaystyle {\int }_0^{\infty }|\mathbf{F}\left(\omega ,\text{Δ}\right){|}^2\frac{D{\zeta }^2}{{\omega }^2+{\zeta }^2}d\omega }$

which can be evaluated to give

(75) $\beta \left(2\text{Δ}\right)={\gamma }^2{\mathbf{G}}^{2}D\left\{{\delta }^2\left(\text{Δ}-\frac{\delta }{3}\right)-\frac{2}{{\zeta }^2}\delta +\frac{1}{{\zeta }^3}\times \left[2+\exp \left(-\left(\text{Δ}-\delta \right)\zeta \right)+\exp \left(-\left(\text{Δ}+\delta \right)\zeta \right)-2\exp \left(-\text{Δ}\zeta \right)-2\exp \left(-\delta \zeta \right)\right]\right\}$

Note that the spin-echo attenuation depends in a characteristic way on the NMR parameters Δ and δ, as well as on the self-diffusion constant D and the frictional damping ζ. Equation (75) is quite general and applies to all time intervals.

Some special cases are of interest. When ${\tau }_c=1/\zeta \ll \text{Δ}$ , all terms but the first can be neglected. This is a well-known classic result (Torrey, 1956). In the limit of short intervals between the gradient pulses, when $\text{1/}\zeta \approx \text{Δ}$ , the spin-echo damping [Eq. (75)] follows the relationship

(76) $\beta \left(2\text{Δ}\right)=\frac{1}{2}{\gamma }^2{G}^2\zeta D{\text{Δ}}^2{\delta }^2$

When the pulses are very short, $\delta \ll \text{Δ}$ , Eq. (75) becomes

(77) $\beta \left(2\text{Δ}\right)={\gamma }^2{G}^2{\delta }^{2}D\left[\text{Δ}-\frac{1}{\zeta }\left(1-e^{-\text{Δ}\zeta }\right)\right]$

An inherent feature of Eq. (77) is its simple relationship [via Eq. (70)] to the mean-squared particle displacement, i.e.,

(78) $\beta \left(2\text{Δ}\right)=\frac{1}{2}{\gamma }^2{G}^2{\delta }^2〈{\left[x\left(\text{Δ}\right)-x\left(0\right)\right]}^2〉$

This relationship holds quite generally for the narrow pulse PGSE sequence provided that the phase distribution is Gaussian. This may be seen by considering the relationship between the particle displacement and the velocity correlation:

(79) $〈{\left[\mathbf{r}\left(t\right)-\mathbf{r}\left(0\right)\right]}^2〉={\displaystyle {\int }_0^{t}d{t}_1}{\displaystyle {\int }_0^{t}d{t}_2}〈\mathbf{v}\left(t_1\right)\mathbf{v}\left(t_2\right)〉$

By transforming to the frequency domain (Stepišnik, 1993) it may be shown that the displacement in one spatial dimension is linked to spectrum of velocity correlation through

(80) $〈{\left[x\left(t\right)-x\left(0\right)\right]}^2〉=\frac{2}{\pi }{\displaystyle {\int }_0^x{D}_{xx}\left(\omega \right){\left[\frac{\sin \left(\omega t/2\right)}{\left(\omega /2\right)}\right]}^{2}d\omega }$

Since it is also true that $\gamma \delta \text{Gsin}\left[\left(\omega \text{Δ/2}\right)/\left(\omega /2\right)\right]$ is the spectrum of the PGSE sequence [Eq. (73)] with short pulses separated by Δ, Eq. (78) follows directly. Note, however, that Eq. (78) gives the first term in the cumulant expansion and will only yield the correct echo attenuation if the phase distribution is Gaussian.

The next case concerns the trapping of molecules undergoing Brownian motion in the presence of some restoring force. While friction is the main parameter that determines Brownian motion in the short time limit (Wang and Omstein, 1945; Stepišnik, 1994), when dealing with random migration of molecules in a complex environment, other long range interactions may result in anomalous self-diffusion. This kind of deviation has been found in the modeling of Brownian movement in a periodic potential and in some specific cases of macromolecules in the random environment. The problem can be treated by using the Langevin equation along with a memory function, K(t),

(81) $\frac{dv\left(t\right)}{dt}+{\displaystyle {\int }_0^t\zeta \left(t-\tau \right)}v\left(\tau \right)d\tau =f\left(t\right)$

where the particle velocity v(t) is the dynamic variable and f(t) is a stochastic driving force defined by the coupling of the particle to the surroundings. The simplest form of memory function is the exponential function

(82) $\zeta \left(t\right)=\frac{\zeta }{T}{e}^{-t/T}$

where T is the relaxation time and is related to the degree of particle binding. With the assumption $f\left(t\right)=D\delta \left(t\right)$ , Eq. (82) gives the spectrum of velocity autocorrelations as

(83) $D\left(\omega \right)=D{\zeta }^2/{\left|i\omega +\frac{\zeta }{-i\omega T+1}\right|}^2$

Evaluation of Eq. (66) with the spectrum Eq. (83) is shown in detail in Stepišnik (1993, 1994). For the narrow pulse and the finite pulse PGSE experiment, two relatively simple closed form expressions are obtained. Figure 2 shows the fit of evaluated mean-squared displacement to the experimental data (Grinberg et al., 1987).

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Sound Absorption and Sound Absorbers

Frank Fahy , in Foundations of Engineering Acoustics, 2001

7.2.1 The origin of gas viscosity

As explained in Chapter 3, the translational motion of molecules underlies the phenomena of temperature and pressure of gases. The normal stress (pressure) acting on an imaginary plane surface in a gas is explained by molecular 'collision'. ^* In a gas of uniform temperature at rest in a continuum sense, in which molecules move randomly with equal probability in all directions, molecular collision produces zero average net flux of momentum across the surface, i.e., there is static equilibrium. Most of the molecules approaching the surface pass through it unscathed, and the momentum fluxes in opposite directions cancel. However, imagine what happens if the surface separates two regions of gas of equal temperature flowing at different speeds in a direction tangential to the surface, as shown in Fig. 7.1. This involves a discontinuity of tangential velocity at the interface. The molecules now carry the momentum associated with the mean flow plus that of random motion. Pressure equilibrium still exists, but molecules passing through the surface from the higher-speed flow, and ultimately 'colliding' with molecules of the slower-moving fluid, transfer to them greater flow-directed momentum than is passed in the opposite direction. In continuum terms, this effect can be attributed to the action of a stress component tangential to the surface: in other words, to a shear stress that acts so as to reduce the mean velocity difference. The postulated discontinuity of mean flow speeds clearly cannot be maintained. At very low relative speeds, the tangential velocity exhibits a continuous (smooth) variation with distance from the interface, the gradient of which is known as the 'rate of shear'. However, the shear layer so formed is inherently unstable and develops transverse waves. At a sufficiently high relative speed, this breaks up into turbulence. This is the origin of the jet noise of aircraft turbo-jet engines.

Fig. 7.1. Illustration of molecular transport across a discontinuity of mean flow speed.

The existence of shear stresses in fluids is attributed to the property termed 'viscosity'. Kinetic theory and experiment show that the viscous stress is linearly proportional to therate of shear, the factor of proportionality being termed the 'coefficient of dynamic viscosity'. Viscous stresses are therefore essentially non-conservative and dissipate fluid kinetic energy into heat. It is initially surprising to learn that shear, and hence viscous stresses, occur even in purely plane sound waves. Consideration of the diagonals of a fluid element under plane strain will show that shear distortion does occur (see Fig. 3.2). Not surprisingly, in view of its origin in molecular momentum transport, gas viscosity increases with temperature. On the other hand, liquid viscosity is caused largely by molecular attraction, which is weakened by temperature increase.

Another mechanism of conversion of sound energy into heat operates in gases that have more than one atom per molecule (diatomic or polyatomic). When a gas has work done on it by sudden compression, the kinetic energy of translational motion of the molecules increases virtually instantaneously, and the pressure, density and temperature rise. Some of this energy is subsequently fed into rotational and vibrational energy of the molecules, and the pressure falls: this is termed 'relaxation'. If the compression is reversed sufficiently quickly, negligible translational energy is lost and the work of compression can be fully recovered during expansion. If the compression–expansion cycle is sufficiently slow, thermodynamic equilibrium between the different energy 'modes' has time to be established, and again, the process is reversible. If the oscillation period lies somewhere in between these extremes, some sound energy will be irreversibly lost to the internal energy of the gas. Consequently the gas will not behave perfectly elastically, but will have a complex bulk modulus, and exhibit a form of hysteretic behaviour known as 'viscoelasticity' in which the pressure is a function of both volumetric strain and its time derivative. Pressures are then not fully in phase with the associated volumetric strains. This behaviour is attributed to the property of 'bulk viscosity', which is a rather misleading term since its origin is quite different from the momentum transport process that underlies dynamic viscosity.

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Theoretical, Experimental, and Numerical Techniques

LE ROY F. HENDERSON , in Handbook of Shock Waves, 2001

2.3 LENGTH AND TIME SCALES

One length scale that is always present is the thickness of the shock wave. The simplest example is that of a monatomic gas. Its shock wave thickness is about four mean free paths, that is, it takes about four molecular collisions to adjust the equilibrium state upstream of the shock to downstream of it. The molecular processes inside the shock wave are not in equilibrium. A shock wave is thicker in polyatomic gases because molecular rotation and vibration require more collisions for equilibrium to be attained. For weak shock waves in the atmosphere the thickness may be of the order of one 1 km because of the large number of collisions required to attain vibrational equilibrium in nitrogen, especially when moisture is present ( Johannesen and Hodgson, 1979). The shock wave thickness is also increased by chemical reactions as with detonations (Fickett and Davis, 1979), and by dissociation and ionization. More generally, the velocity and the thermal gradients inside the shock wave imply the importance of the material transport properties—particularly viscosity and heat conductivity (Zeldovich and Raizer; 1966; Thompson, 1972, p. 363). If the shock wave thickness length scale is too small to be of significance to the problem then it is sufficient to consider only the equilibrium states on both sides of the shock. One then has a shock Riemann problem.

Time scales are often important, but only two occurrences will be mentioned here. First, a time scale is present if the shock wave becomes unstable and splits into two waves moving in the same direction (Section 2.8.2). Suppose that an intense shock wave propagates into a metal, which is initially at atmospheric pressure and temperature, and suppose it also compresses the metal beyond its yield point. It is known that eventually the shock wave will split into two waves. The first is a precursor shock wave that compresses the metal to its yield point, and the second is a compressive plastic wave (Zeldovich and Raizer, 1966). Second, a shock wave may induce a change in phase of the material. A well-known example is the α → ɛ (body-centered-cubic to hexagonal-close-packed) phase transformation in iron that takes place at 12.8 Gpa, which can also cause splitting (Duvall and Graham, 1977). However, in many cases the time to attain equilibrium is orders of magnitude greater than the time for the shock wave to pass through the material. In this case there will be no phase change, and any equilibrium can only be metastable; there will then be no time scale. For example, if a shock wave compressed water at atmospheric pressure to a pressure P > 10⁴ atm (1000 MPa), and if thermodynamic equilibrium were attained, then ice (VII) would exist downstream of the shock wave (see Fig. 2.1). However, this does not usually happen because of the long time required for attaining equilibrium (Bethe, 1942). Instead, the water remains in the liquid phase but in metastable equilibrium. The time scale for thermodynamic equilibrium reduces rapidly, however, if the compressed state approaches a spinodal condition (Section 2.7).

FIGURE 2.1. Ice-water-phase diagram.

(adapted with changes from Eisenberg and Kauzman 1969). Copyright © 1969

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Theoretical, Experimental, and Numerical Techniques

DAVID F DAVIDSON , RONALD K. HANSON , in Handbook of Shock Waves, 2001

5.2.7.1 THEORY

Fluorescence is generally defined as radiation emitted by a molecule or atom when it decays by spontaneous emission of a photon in an optically allowed transition from a higher to a lower energy state. This de-excitation process occurs in parallel with other processes that de-excite the molecule, including collisional energy transfer to other molecular states. The population of the upper state can occur by chemical reaction, molecular collisions, and radiative interactions. In laser-induced fluorescence, the upper state is populated by absorption of laser radiation at an allowed resonant frequency between the discrete lower state and the excited upper state. After excitation the laser-populated upper state may undergo any of several different competing processes.

1)

The molecule could return to the ground state by laser-induced stimulated emission

2)

The molecule could absorb another photon and get still further excited.

3)

The rotational or vibrational internal energy distribution of the molecule could be rearranged by inelastic collisions with other molecules.

4)

Quenching or electronic energy transfer by inelastic collisions could occur.

5)

"Internal" collisions could cause dissociation of the molecule, which would include predissociation, the change from a stable to repulsive electronic arrangement.

6)

Or finally, The original state and its energetically nearby neighbors, populated by processes like that in 3), can emit spontaneously, or fluoresce.

The linear fluorescence equation, which describes the region where fluorescence is proportional to laser intensity, is used to model PLIF measurements in practice.

(5.2.6) $R_p=n_1{B}_{12}{I}_v\left[A_{21}/(A_{21}+Q_{21})\right]$

Where R_p (photons-cm⁻³s⁻¹) is the steady state fluorescence rate, n ₁ (cm⁻³) is the initial population of the lower absorbing state, B ₁₂ (s⁻¹ W⁻¹ cm² Hz) is Einstein B coefficient, I_v (Wcm⁻² Hz⁻¹) is the laser spectral intensity, A₂₁ (s⁻¹) is the Einstein A coefficient for spontaneous emission (i.e., fluorescence), and Q₂₁ (s⁻¹) is collisional transfer coefficient, which can be extended to include other loss rates. From kinetic theory, the collisional loss rate $Q_{21}=n{\sigma }_m〈\upsilon 〉$ , where n (cm⁻³) is the total number density, σ_m (cm²) is the mixture-averaged quenching crosssection, and $〈\upsilon 〉\left(\text{cm}{s}^{-1}\right)$ is the appropriate mean molecular speed.

From this equation we can see that the fluorescence signal is proportional to the lower state population and to a factor [(A₂₁/(A₂₁ + Q₂₁)] called the Stern-Vollmer factor or fluorescence quantum yield. The PLIF signal is then a measure of the light absorbed at each flow field point modified by the local fluorescence yield. This yield is affected by the collisional quenching rate. Current PLIF research is investigating strategies that avoid this problem. These include taking the ratio of two PLIF images to determine temperature, and monitoring velocity by Doppler-shifted absorption. By using signal differences and ratios, local collisional quenching effects can be effectively minimized.

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Motion of a single aerosol particle in a fluid

Warren H. Finlay , in The Mechanics of Inhaled Pharmaceutical Aerosols (Second Edition), 2019

3.4 Brownian diffusion

For very small particles, collisions with the randomly moving air molecules will cause the particle to undergo a nondeterministic random walk called Brownian motion. Consider, for example, the motion of a particle settling in air under the action of gravity, shown in Fig. 3.1.

Fig. 3.1. The trajectory of a spherical particle settling in air for (A) a particle of diameter d  ≫   mean free path of the air molecules and (B) a particle with diameter near that of the mean free path.

Very small particles (d  ≪   1 μm) diffuse readily due to molecular collisions with the gas. These molecular collisions are nondeterministic, and so, we cannot actually predict the motion of a given particle. However, if we only examine the particle motion over times that are much longer than the time between collisions with molecules, we can use a result developed by Einstein in 1905, which states that the root mean square displacement, x _d , of a particle in time t (where t  ≫   time between molecular collisions) due to Brownian motion is

(3.24) $x_d={\left(2D_{d}t\right)}^{1/2}$

where D is the particle diffusion coefficient and is given by

(3.25) $D_d={\mathit{kTC}}_c/\left(3\mathit{\pi \mu d}\right)$

Here, k  =   1.38   ×   10^{−   23}  J/K is Boltzmann's constant, T is the temperature in Kelvin, C _c is the Cunningham slip factor (Eq. 3.23), d is particle diameter, and μ is the viscosity of the surrounding fluid.

Because the diffusion coefficient D _d increases with decreasing particle size, diffusion becomes important for small particles. To decide at what particle size diffusion starts to become important, we can compare the distance x _s   = v _settling t that a particle will settle in time t with the distance x _d in Eq. (3.24) that the particle will diffuse in the same time t. The ratio x _d /x _s then is a measure of the importance of diffusion compared with sedimentation. Using Eq. (3.22) for v _settling , then we have

(3.26) $\frac{x_d}{x_s}=\frac{18\mu \sqrt{2D_{d}t}}{{\rho }_{\mathit{particle}}{\mathit{gd}}^2{C}_{c}t}$

which simplifies with the definition of D _d in Eq. (3.25) above to

(3.27) $\frac{x_d}{x_s}=\frac{1}{{\rho }_{\mathit{particle}}g}\sqrt{\frac{216\mathit{\mu kT}}{\pi {\mathit{td}}^5{C}_c}}$

Diffusion can be considered negligible if x _d /x _s   <   0.1 or so. Thus, substituting the value of x _d /x _s   =   0.1 into Eq. (3.27) allows us to solve for the time t above which diffusion will be negligible for a given particle diameter d. The result is shown in Fig. 3.2.

Fig. 3.2. The time t in Eq. (3.27) above which x _d   &lt;   0.1x _s , so that particle motion due to Brownian diffusion is estimated as being negligible compared with particle motion due to sedimentation.

The residence time t of a particle in a lung airway can be estimated for simplified models of the lung given in Chapter 5, and we find that for an inhalation flow rate of 18   L/min (typical of a tidal breathing delivery device, such as a nebulizer), the shortest residence time of a particle in any airway is approximately 0.03   s., so that from Fig. 3.2, we see diffusion can be considered to have negligible effect on a particle's motion in all lung air passages if the particle's diameter is larger than approximately 2.8   μm. For an inhalation flow rate of 60   L/min (typical of single-breath inhalers), residence times decrease to t  >   0.01   s, and Fig. 3.2 suggests that diffusion is negligible for particles with diameters larger than 3.5   μm. If we include a breath hold of 10   s duration (which can occur in the clinical use of single-breath inhalers), then Fig. 3.2 suggests that diffusion has negligible effect on the particle's motion compared with sedimentation for particles with diameters larger than 0.9   μm.

Thus, in deciding whether diffusion is an important mechanism of deposition for inhaled pharmaceutical aerosols, we must decide over what time interval we expect deposition to occur. If deposition occurs mainly during sedimentation with a breath hold, then diffusion is probably negligible for most inhaled pharmaceutical aerosols. However, if deposition occurs mainly during inhalation while the particle is in transit through the lung, then diffusion may need to be included for particles with diameter below a few microns. For larger particles, diffusion remains unimportant. Further discussion of this issue is given in Chapter 7.

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Aerothermodynamics of Vibrationally Nonequilibrium Gases

J. William Rich , Sergey O. Macheret , in Experimental Heat Transfer, Fluid Mechanics and Thermodynamics 1993, 1993

1 INTRODUCTION

Disequilibrium of the vibrational modes of motion of molecular gases is a pervasive phenomenon in many thermodynamic environments of engineering importance. Fig. 1, for example, shows a simple schematic indicating various modes of motion of diatomic gas molecules. The gas can store energy in each of the indicated modes, and each therefore can contribute to the specific heat of the gas. The total energy of each molecule or atom in the gas may be written [1]:

Fig. 1. Diatomic Molecular Energy Models.

(1) ${\text{E=E}}_{\text{TRANS}}{\text{+E}}_{\text{ROT}}{\text{+E}}_{\text{VIB}}{\text{+E}}_{\text{elec}}+{\text{E}}_{\text{INTERACTION}}$

where each of the energies shown correspond to the modes of motion shown in Fig. 1; nuclear energies are omitted as not participating in thermal gas processes of interest here. E_INTERACTION represents energies associated with the coupling of various modes (vibration with rotation, or vibration with electronic motion, etc.). The "internal" energy modes, rotation, vibration, electronic, are quantized; for engineering systems of macroscopic dimensions the translational energy modes are not quantized, and translational motion is described by classical mechanics.

It is convenient to designate the total energy of an atom or molecule corresponding to a particular array of specific internal quantum states as E_i, where the subscript i refers to the collection of quantum numbers designating the specific internal state

${\text{E}}_{\text{i}}{\text{=E}}_{{\text{i}}_{\text{INT}}}{\text{+E}}_{\text{TRANS}}\text{.}$

When the gas is in thermal equilibrium, the distribution of internal energy among the various modes is typically governed by Maxwell-Boltzmann statistics. It is well-known that for the equilibrium case the fractional number of molecules n_i in the i^th internal energy state is

(2) $\frac{{\text{n}}_{\text{i}}}{\text{N}}=\frac{{\text{g}}_{\text{i}}{\text{e}}^{-{\text{E}}_{{\text{i}}_{\text{,int}}\text{/kT}}}}{{\text{Q}}_{\text{int}},}$

where the internal partition function, Q_int, is given by

(3) ${\text{Q}}_{\text{int}}={\displaystyle \sum _{\text{i}}{\text{g}}_{\text{i}}{\text{e}}^{-{\text{E}}_{{\text{i}}_{\text{,INT}}\text{/kT}}}},$

where $\text{N}\equiv {\displaystyle \sum _{\text{i}}{\text{n}}_{\text{i}}}$ is the total number of atoms and molecules, and g_i is the statistical weight of the i^th internal energy state, k is Boltzmann's constant, and T is the temperature of the gas.

An isolated thermodynamic system will of course eventually go into the distribution of Eq. (2), as dictated by the Second Law. In the present paper, we wish to consider recent studies of some specific cases where the system is either initially prepared in a very nonequilibrium distribution, and the rate of relaxation to the equilibrium distribution Eq. (2) is slow, or, where the system is not isolated, and is maintained in a nonequilibrium steady-state. We will mostly confine ourselves to gaseous systems of diatomic molecules.

In what is to follow, it is important to recognize the greatly varying rates among different modes and gaseous species for the relaxation to equilibrium. Relaxation is principally controlled by collisions among the molecules, and the equilibration of the internal energy modes is controlled, by definition, by inelastic collisions transferring internal energy. Rotational relaxation rates are typically the fastest of these processes, requiring only a few molecular collisions to equilibrate the rotational and translational modes. At standard temperature and pressure, the mean time between molecular collisions in air is approximately 0.2 nsec. Rotational relaxation times are therefore of the order of a nsec.

The number of collisions required to equilibrate vibrational energy modes is commonly much longer than that required for rotation. For example, the N₂ relaxation time required to equilibrate the vibrational mode of pure nitrogen gas with the translational modes at standard conditions is approximately 1 sec. Such long relaxation times for vibrational modes are what make the lack of vibrational equilibrium common, as we mentioned at the outset. In such situations Eqs. (2)–(3) are inapplicable, equilibrium statistical mechanics can not be used to yield the usual thermodynamic functions, and the use of a unique temperature, T, to describe the thermodynamic state of the gas is incorrect. In the present paper, we will present some recent results in the study of such vibrationally nonequilibrium gaseous environments. Some examples that can be cited are:

• Low temperature glow plasmas in molecular gases. We refer here primarily to the positive column of molecular glow discharges, such as are used in electrically-excited gas lasers and in a variety of plasma-chemical reactors. Fig. 2, taken from Raizer [2] shows the situation for such an electric discharge in an N₂ plasma. The fraction of the total electrical energy input into the various modes of molecular motion, including ionization, is shown as a function of E/N, the ratio of the applied electrical field to the total molecular number density. The mean electron energy in the discharge increases with E/N. As it happens, at these E/N's, the cross-sections for the excitation of molecular vibrational states by electron impact are very large, and there is a strong coupling between the energy of the electron gas and the molecular vibrational mode. This is reflected in the figure; most of the input electrical energy goes into vibration, with smaller fractions into the other modes. Under such conditions, there is of course no unique temperature, and the discharge gases, although being maintained in steady state as long as the voltage remains on, are in extreme disequilibrium.

Fig. 2. N₂ Glow Discharge, fraction of energy transferred by electrons to (1) rotation, (2) vibration, (3) electronic excitation, and (4) ionization.

• Flow behind shock waves, such as occurs during planetary entry of space vehicles. Fig. 3, taken from Ref [3], is the result of a model analysis for a high altitude reentry shock wave. The model assumes that, while vibrational, rotational, and translational modes are not equilibrated with each other, they do each maintain an internal equilibration, such that a specific temperature, T_VIB, T_ROT, T_TRANS may be defined for each mode. Shown is the variation of each of these temperatures behind a strong shock in low density nitrogen. Temperatures are plotted vs. distance behind the shock in meters. The environments for which such internal mode temperatures may be safely assumed will be discussed later; for the result shown, one can see that all modes equilibrate by 20 cm after the shock. Clearly, prediction of radiation, heat transfer, and chemical process affecting the surface of the reentry vehicle requires detailed knowledge of the nonequilibrium energy transfer rates.

Fig. 3. Mode Disequilibrium Behind a Hypersonic Shock Wave. T_t=translational mode temperature; T_r=rotational mode temperature; T_v=vibrational mode temperature.

• Supersonic nozzle expansions, of high enthalpy gases, such as occur in rockets. Here, in a sense, is the opposite of the shock flow. In this case, upstream of the nozzle, the gas enthalpy is raised as the result of chemical reaction (conventional rocket) or as the result of input of electrical energy (arc rocket), and the gas is rapidly expanded to high Mach numbers. Typically, the gas begins at a high temperature equilibrium (stagnation conditions) and the translational mode temperature drops rapidly as the gas accelerates down the nozzle. The vibrational mode lags, retaining a great deal of the energy it had at stagnation. Energy flow in the nozzle is from the vibrational modes to the external ones, in contrast to the shock flow. In such a flow, extreme disequilibrium of energy storage among vibrational modes can be produced. Fig. 4, taken from recent work [4] of the Ohio State Group, shows the results of a model calculation for such a flow. In this example, a gas mixture of 60% Ar, 20% CO., and 20% N₂ by volume is expanded from equilibrium stagnation conditions of 100 atm. and 2000K in a 15° half-angle nozzle. An expansion up to M=10 is shown; the average energy in the vibrational mode of the CO and the N₂ in electron volts per molecule is given as a function of nozzle position downstream from the throat in centimeters. Under these conditions, while the two modes are essentially in equilibrium at the throat, there is a rapid transfer of energy from the N₂ to the CO. An assumption of equipartion would obviously be very erroneous. The reasons for this extreme disequilibrium require a look at the detailed vibrational energy transfer kinetics, which we shall proceed to review.

Fig. 4. Supersonic Nozzle Expansion. Average vibrational mode energy per molecule vs. nozzle position.

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Principles, Methods, Formation Mechanisms, and Structures of Nanomaterials Prepared via Gas-Phase Processes

Bangwei Zhang , in Physical Fundamentals of Nanomaterials, 2018

2.2.3.3 Laser Fabrication of Monocrystalline Wires and Films

Laser systems can be used to prepare nanowires and nanometer-thin films. Morales and Lieber [47] prepared Si nanowires via laser melting. They used an Si₉₀Fe₁₀ target, a furnace temperature of 1200°C, and an Ar gas flow of 50   cm³/min and produced uniform nanowires with diameters of 10   nm and lengths exceeding 1   μm (up to 30   μm), as shown in Fig. 2.15A and B. TEM analysis shows that the nanowire core is made up of Si, and its shell consists of SiO _x . HRTEM analysis indicates that Si(1   1   1) crystals with spacings of 0.31   nm are formed perpendicular to the direction indicated by the white arrow (growth) and that the SiO _x shell is amorphous.

Figure 2.15. TEM (A), TEM diffraction, (B) and HRTEM images (C) of nanoscale Si wires [47]. HRTEM, high-resolution transmission electron microscopy; TEM, transmission electronic microscopy.

The mechanism of nanowire growth can be divided into four steps as shown in Fig. 2.16. In step A, the Si_100−x Fe _x target absorbs energy from the laser bean, melts, and produces a dense Si/Fe particle vapor. Step B shows these particles and an inert gas quickly undergoing molecular collisions to form a Si-rich liquid nanoatomic cluster. Step C indicates deposition of Si from the saturated nanoatomic cluster to form a crystalline nanowire. The final step (D) shows that after arrival of a cold tip, the nanowire cures, and its head and shell became amorphous SiFe. This is referred to as the vapor–liquid–solid (VLS) growth mechanism and is different from the VS mechanism previously discussed. VLS starts with a vapor phase but forms a liquid before generating a solid.

Figure 2.16. Growth mechanism of Si nanowires prepared via laser melting [47].

Perrière et al. [48] prepared GaAs nanomembranes using the pulsed laser melting method. To avoid generation of distributions of large nanoparticles and possible oxidation of particles deposited on substrates, the researchers added a device to the manufacturing equipment that collected nanoparticles from the vapor phase. GaAs nanofilms produced in this manner do not require further processing.

In 2002, Yang et al. [49] prepared (FePt/Ag) _n multilayer films on MgO(1   0   0) substrates with a Nd:YAG 135   nm laser. The thicknesses of Ag and FePt ranged from 0.5 to 5   nm. The first layer always contained FePt. The layer deposition rate was approximately 0.02   nm/s for Ag and 0.05   nm/s for FePt.

Of the parameters that influence nanofabrication via laser preparation (the laser beam itself, the temperature in the reaction chamber, gas pressure and species, etc.), we discuss only the influence of the inert gas pressure and species on the laser deposition rate of an Ag film [50] (Fig. 2.17). Regardless of the atmosphere used, a maximum value exists. This maximum value increases as the mass of the inert gas decreases (Xe → Ar → Ne → He). With lighter inert gas atoms, it is easier for Ag atoms to reach the substrate and form a membrane after vaporization from the target, despite collisions with inert gas atoms. When a heavier inert gas is used, it is more difficult for Ag to reach the substrate and form a film. Why does the deposition rate change when the pressure increases? In ultrahigh vacuum, some of the particles in the plume generated by laser evaporation with particular energies can resputter onto the surface of the deposited film. As the pressure increases, the average kinetic energy of the evaporated atoms decreases, reducing the incidence of this sputtering effect and increasing the net deposition rate. However, at high pressures, evaporated particles due to scattering can cause some evaporated clusters to leave the target channel between the substrate and target, resulting in a lower deposition rate. Therefore, a deposition rate spike appears. When the inert gas consists of heavier atoms, the lower deposition rate that results is also easy to understand. Because their speeds are the same, larger atoms have more kinetic energy. Thus, evaporated atoms undergo more intense collisions and scattering, and fewer particles reach the substrate. Thus, He produces the largest film deposition rate of the four gases tested.

Figure 2.17. Laser deposition rates for Ag films vs reaction pressure in He, Ne, Ar, and Xe [50].

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