Open Cell Metal Foams for Beam Liners ?

Full electromagnetic modeling of reticulated metal foams is still to come. A numerical approach based on Weiland’s finite integration technique (FIT, [7]) has been proposed by Zhang et al. [8] to compute the (frequency, thickness and angle of incidence dependent) reflection coefficient of SiC reticulated foam, and optimize its design. A simplified model, consisting of stacked square-mesh grids has been used by Losito et al. [9],[10] to investigate the RF shielding properties of metallic foams.
The main limitation of Zhang’s analysis is in the use of a simple body-centered-cubic unit-cell foam model, for easiest numerical implementation. The FIT scheme, however may accommodate in principle more complicated and realistic foam-cell geometries, including in principle the WP one.
In the limit where bubbles and metal struts are much smaller than the smallest wavelength of interest, the DC conductivity of a metal foam can be computed using effective medium theory (EMT), for which several formulations exist (see, e.g., [11]-[13] for a review). These include: i) the “infinite dilution” approximation, where inclusions do not interact, and are subject to the field which would exist in the homogeneous host; ii) the self-consistent approach [14], credited to Bruggemann, where inclusions are thought of as being embedded in the (yet to be modeled) effective medium; iii) the differential scheme, whereby inhomogeneities are incrementally added to the composite¹¹1 In this approach, the total concentration of inhomogeneities does not coincide with the volume fraction $p$, because at each step new inclusions may be placed where old inclusions have already been set., until the final concentration is reached, so that at each step the inclusions do not interact, and do not modify the field computed at the previous step [15]; iv) the effective-field methods, whereby interaction among the inclusions is described in terms of an effective field acting on each particle, accounting for the presence of the others. Two main versions of this method exist, credited to Mori-Tanaka [16] and Levin-Kanaun [17], differing in the way the effective field is computed (average over the matrix only, or average over the matrix and the inclusions, respectively).
Both the infinite-dilution and the self-consistent approaches yield

$$\sigma_{\mbox{\it eff}}=\sigma_{0}(1-p\nu),$$

(1)

where $\sigma_{0}$ is the bulk metal conductivity, $p$ is the porosity (volume fraction of the vacuum bubbles). and $\nu$ is a morphology-dependent factor. The differential approach yields

$$\sigma_{\mbox{\it eff}}=\sigma_{0}(1-p)^{\nu},$$

(2)

while the Mori-Tanaka/Levin-Kanaun approaches yield

$$\sigma_{\mbox{\it eff}}=\sigma_{0}/(1+\frac{\nu p}{1-p}).$$

(3)

All equations (1)-(3) merge, as expected, in the $p\rightarrow 0$ limit. The various models are synthetically compared in Figure 5.

All these models predict larger conductivity then observed in measurements on Al foams²²2 It should be noted that open and closed cell metal foams behave similarly in terms of electrical conductivity, while being markedly different as regards thermal conductivity, due to the different role of convective flow.. This has been attributed to significant oxide formation on the Al conducting web [18]. Equation (2) agrees in form with predictions based on percolation theory [19] - although strictly speaking there’s no threshold here beyond which the conducting component disconnects.

The vacuum dynamics for each molecular species which may be desorbed from the wall by synchrotron radiation can be described by the following set of (coupled) rate equations [20]

$$\left\{\begin{array}[]{l}\displaystyle{V\frac{dn}{dt}=q-an+b\Theta}\\ \\ \displaystyle{F\frac{d\Theta}{dt}=cn-b\Theta}\end{array}\right.$$

(4)

Here $n\mbox{ }[m^{-3}]$ and $\Theta\mbox{ }[m^{-2}]$ are the volume and surface densities of desorbed particles, respectively, and $V$ and $F$ represent the volume and wall-area of the liner per unit length, respectively.
The first term on the r.h.s. of the first rate equation represents the number of molecules desorbed by synchrotron radiation per unit length and time, and is given by

$$q=\eta\dot{\Gamma}$$

(5)

where $\eta$ is the desorption yield (number of desorbed molecules per incident photon) and $\dot{\Gamma}$ is the specific photon flux (number of photons hitting the wall per unit length and time). The second term represents the number of molecules which are removed per unit time and unit length by either sticking to the wall, or escaping through the holes. The $a$ coefficient in (4) can be accordingly written

$$a=\frac{\langle v\rangle}{4}(s+f)F$$

(6)

where $\langle v\rangle\approx(8kT/\pi m)^{1/2}$ is the average molecular speed, $m$ being the molecular mass, $k$ the Boltzmann constant ant $T$ the absolute temperature, $\langle v\rangle/4$ is the average numer of collisions of a single molecule per unit time and unit wall surface, $s$ is the sticking probability, and $f_{h}$ is the escape probability. The third term accounts for thermal or radiation induced re-cycling of molecules sticking at the walls. The $b$ coefficient in (4) can be accordingly written

$$b=\kappa\dot{\Gamma}+F\nu_{o}\exp(-E/kT)$$

(7)

Here the first term accounts for radiation induced recycling, described by the coefficient $\kappa\mbox{ }[m^{2}]$, while the second term describes thermally-activated recycling, $\nu_{0}$ being a typical molecular vibrational frequency, and $E$ a typical activation energy.
The $b\Theta$ term appears with reversed sign on the r.h.s. of the second rate equation, where it represents the the number of molecules de-sticking from the wall surface per unit time and unit length. The first term on the r.h.s. of this equation represents the number of molecules sticking to the wall, per unit time and unit length, whence (compare with eq. (6))

$$c=\frac{\langle v\rangle}{4}sF$$

(8)

At equilibrium, $\dot{n}=\dot{\Theta}=0$, and the rate equations yield:

$$\left\{\begin{array}[]{l}\displaystyle{n_{eq}=\frac{4\eta\dot{\Gamma}}{\langle v\rangle fF}}\\ \\ \displaystyle{\Theta_{eq}=\frac{s}{f}\frac{\eta\dot{\Gamma}}{\kappa\dot{\Gamma}+F\nu_{o}\exp(-E/kT)}}\end{array}\right.$$

(9)

Typical values (from LHC) of the parameters in (9) are collected in Table II below [20].

The equilibrium molecular densities in (9) should not exceed some critical values for safe operation [20].

Beam coupling impedances provide a synthetic description of the beam-pipe interaction, for investigating beam dynamics and stability [2]. For the simplest case of a circular pipe of radius $b$ with on-axis beam, the longitudinal beam-coupling impedance per unit length is given by

$$Z_{\parallel}(\omega)=\frac{Z_{wall}}{2\pi b}$$

(14)

where $Z_{wall}$ is the wall impedance, and a Leontóvich assumption is implied⁴⁴4 Equation (14) is a special case of a general formula which allows to compute the longitudinal and transverse beam-coupling impedances of a pipe with complicated geometric and constitutive properties [25].. Similarly, the nonzero components of the diagonal transverse beam coupling impedance dyadic are

$$Z_{\perp}(\omega)=\frac{cZ_{wall}}{\omega\pi b^{3}}$$

(15)

where $c$ is the speed of light in vacuum. The parasitic loss (energy lost by the beam per unit pipe length) is directly related to the longitudinal impedance via

$$\Delta{\cal E}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|I(\omega)|^{2}\Re e~Z_{\parallel}(\omega)d\omega.$$

(16)

where $I(\omega)$ is the beam-current frequency spectrum [2]. The beam coupling impedances (and parasitic losses) should not exceed some critical values for safe operation [1].